NBER WORKING PAPER SERIES DOWNSIDE RISK AND THE … · Downside Risk and the Momentum Effect Andrew Ang, Joseph Chen and Yuhang Xing NBER Working Paper No. 8643 December 2001 JEL
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NBER WORKING PAPER SERIES
DOWNSIDE RISK AND THE MOMENTUM EFFECT
Andrew AngJoseph ChenYuhang Xing
Working Paper 8643http://www.nber.org/papers/w8643
NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue
Cambridge, MA 02138December 2001
The authors would like to thank Brad Barber, Alon Brav, Geert Bekaert, John Cochrane, Randy Cohen, KentDaniel, Bob Dittmar, Cam Harvey, David Hirschleiffer, Qing Li, Terence Lim, Bob Stambaugh, AkhtarSiddique and Zhenyu Wang for insightful discussions. We especially thank Bob Hodrick for detailedcomments. We thank seminar participants at Columbia University and USC for helpful comments. This paperis funded by a Q-Group research grant. The views expressed herein are those of the authors and notnecessarily those of the National Bureau of Economic Research.
exposureson F , ratherthan for greaterexposuresto downsiderisk. To investigatethis, we
performa sort on > ? , after controlling for F . Eachmonth,we placehalf of the stocksbased
on their F ’s into a low F groupandthe otherhalf into a high F group. Then,within eachFgroup,werankstocksbasedon their > ? into threegroups:a low > ? group,amedium> ? group
anda high > ? 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 F group,theaverage
returnsincreasefromthelow > ? portfolio to thehigh > ? portfolio,with anannualizeddifference
of 2.40%(0.20%permonth).Moving acrossthelow F group,meanreturnsof the > ? portfolios
increase,while thebetaremainsflat ataroundF = 0.66.In thehigh F group,weobservethatthe
within the high F group, is 3.24%per annum(0.27%per month), with a t-statisticof 1.98.
However, the F decreaseswith increasing> ? . Therefore,the higher returnsassociatedwith
higherdownsiderisk arenot rewardsfor bearinghighermarket risk, but arerewardsfor bearing
higherdownsiderisk.
In PanelB of Table(5), for each>@? group,we take the simpleaverageacrossthe two Fgroupsandcreatethreeportfolios,whichwecall the F -balanced> ? portfolios.Moving fromthe
F -balancedlow > ? portfolio to the F -balancedhigh > ? portfolio, meanreturnsmonotonically
increasewith > ? . This increaseis accompaniedby a monotonicdecreasein F from F = 0.94
to F = 0.87.HenceF is not contributing to thedownsiderisk effect, sincewithin eachF 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 o and prq , andtake the intersectionsto
bethe6 o /prq portfolios. This procedureproducesa similar result,but givesanaveragemonthlyreturnof 0.22%
(t-stat= 1.88),which is significantat the 10% level. This procedureproducespoor dispersionon pjq becauseoand prq arehighly correlated,sothe independentsortplacesmorefirms in the low o /low prq andthehigh o /highpjq portfolios,thanin thelow o /high prq andthehigh o /low pjq portfolios.Oursortingprocedurefirst controlsforo andthensortson prq , creatingmuchmorebalancedportfolioswith greaterdispersionover pjq .
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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 > ? have negative loadingson CMC. That is, thelow > ?portfoliosarenegatively correlatedwith theCMC factor. SincetheCMC factorshortslow > ?stocks,many of the stocksin the low > ? portfolios have shortpositionsin the CMC factor.
Whenweaugmenttheregressionin equation(9) with theFama-Frenchfactors,theintercept
coefficientsan�
aresmaller. However, thefit of thedatais notmuchbetter, with theadjustedw 6’s
thatarealmostidenticalto theoriginal modelof around90%. While theloadingsof SMB and
HML arestatisticallysignificant,theseloadingsgothewrongway. Low > ? portfolioshavehigh3 SMB is long morefirms thanit is shortsincethebreakpointsaredeterminedusingmarket capitalizationsof
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-freeratefromCRSPandtakeCRSP’s value-weightedreturnsof all stocksasthemarketportfolio.
Higher Moment PortfoliosWe constructportfoliosbasedon correlationsbetweenassetÆ ’s excessreturnÇfÈ andthemarket’sexcessreturn Ç�Éconditionalon downsidemovesof the market (p q ) andon upsidemovesof the market (p�Ê ). We alsoconstuctportfoliosbasedon coskewness,cokurtosis,o , o conditionalon downsidemarketmovements(oËq ), o conditionalon upsidemarketmovements(o Ê ). At thebeginningof eachmonth,we calculateeachstock’s momentmeasuresusingthepastyear’sdaily log returnsfrom theCRSPdaily file. For themomentswhich conditionon downsideorupsidemovements,we definean observationat time Ì to bea downside(upside)market movementif the excessmarket returnat Ì is lessthanor equalto (greaterthanor equalto) the averageexcessmarket returnduring thepastoneyearperiodin consideration.We requirea stockto have at least220observationsto be includedin thecalculation.Thesemomentmeasuresarethenusedto sort the stocksinto decilesanda value-weightedreturniscalculatedfor 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 Í;Î]Ï of thereturnsof theworstperformersonemonthago,plus ÍBÎ�Ï of thereturnsof theworstperformerstwo monthsago,etc.
is thenumberof availablestocksat monthÌ , ê Ó is thetotaldollar valueof theincludedstocksat theendof month Ì æ Í , andê Ö is thetotal dollar valuefo thestocksat theendof July 1962.Theinnovationsin liquidityarecomputedastheresidualsof thefollowing regression:è=éÑ Ó ×¸ð Ù£ñ èòéÑ Ó q Ö Ù£ó î ê Ó Î ê Ö ï éÑ Ó q Ö Ùõô�Ó ä (A-3)
Finally, theaggregateliquidity measure,ÐçÓ , is takento bethefitted residuals,ÐöÓ × éô Ó .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,oö÷È , by runningthefollowing regressionusingthemostrecentfive yearsof monthlydata:
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 Ì and Ì æ Í� ,whereÌ 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 andsecondordercanscale.Thatis, anannualestimateof themean andvolatility � canbe estimatedfrom meansandvolatilities estimatedfrom daily data Õ and � Õ , by the time aggregatedrelations × ��� � Õ and� × � ��� �� Õ . Hence,daily measurefor secondordermoments,suchasprq , p Ê , o , oËq ando Ê areequivalentto their correspondingmonthlymeasures.We now prove thatdaily coskewnessandcokurtosisdefinedin equations(5) and(6) areequivalentto monthlycoskewnessandcokurtosis.
With theassumptionof infinitely divisibledistributions,cumulantsscalebut notcentralmoments(“cumulantscumulate”).Thecentralmoment,�� , of � is definedas:
�� × �q �
î � æ Ö ï � â�� for Ç × � á��;á��;á¡ä`ä ä`á (B-1)
28
integratingover thedistribution of returns� , and Ö ×�� î � ï . Theproductcumulants,� � , arethecoefficientsintheexpansion:
C Computing Hansen-Jagannathan (1997) Distances and P-values
JagannathanandWang(1996)derive theasymptoticdistribution of theHJ distance(equation(24)), showing thatthedistributionof þIH î �KJ ï . involvesaweightedsumof
î ë æ ü æ Í ï�L . Ö statistics.Theweightsaretheë æ ü æ Í
non-zeroeigenvaluesof:
M × � 76NPO 76�QN R æ O 76N å N î åTSN O N å N ï q Ö åTSN O76)QN q Ö O 76N �
76)QN áwhere�
76N and O76N aretheupper-triangularCholesky decompositionsof � N and O N respectively, and å N ×�UWV�XUZY .
JagannathanandWang(1996)show thatM
hasexactlyë æ ü æ Í positive eigenvaluesØ Ö á¡ä¡ä�ä á Ø ì q\[yq Ö . The
asymptoticdistributionof theHJdistancemetricis:
þ]H î �^J ï .`_ì qa[yq Öb Ø b L . Ö
asþ _dc. 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:
whereÇ ÈHÒ Ó is thereturnonthe Æ -th portfolio, Ç eÓ is therisk-freerate,o È is an úfH Í vectorof factorloadings,and �'Óis the úgH Í vectorof factors.We assumethattherisk-freerateandthefactorsfollow a first-orderVAR process.Let h Ó × î ÇeÓ á,� Ó ï S , andh Ó follows:
h Ó × Ù M h Ó q Ö ÙõôjÓ(á (C-2)
whereôjÓ�i ë î á�j ï . WeestimatethisVAR systemanduseé ,
éMand
éj astheparametersfor our factorgeneratingprocess.In eachsimulation,wegenerate432observationsof factorsandtherisk-freeratefrom theVAR systeminequation(C-2). For theportfolio returns,we usethesampleregressioncoefficient of eachportfolio returnon thefactors,
éj � æ éo S éj�k éo , whereéj � is thecovariancematrix of theassets
andéj k 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
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Panel B: Portfolios Sorted on Past p ÊPortfolio Mean Std Auto o Size B/M p Ê o Ê High–Low t-stat1 Low p Ê 1.13 4.56 0.17 0.82 2.88 0.60 0.50 0.63 –0.06 –0.382 1.05 4.63 0.19 0.90 3.08 0.59 0.63 0.783 1.09 4.61 0.16 0.92 3.24 0.58 0.68 0.824 1.06 4.67 0.15 0.94 3.44 0.56 0.70 0.855 0.99 4.62 0.14 0.95 3.66 0.54 0.76 0.916 1.03 4.62 0.12 0.97 3.91 0.54 0.78 0.907 1.00 4.70 0.09 1.01 4.23 0.52 0.84 0.978 1.11 4.67 0.08 1.01 4.65 0.52 0.85 0.969 1.12 4.63 0.07 1.02 5.27 0.48 0.92 1.0210 High p Ê 1.07 4.52 0.00 1.00 6.65 0.36 0.95 1.06
The table lists the summarystatisticsof the value-weightedprq and p Ê portfolios at a monthly frequency,where prq and p Ê aredefinedin equation(3). For eachmonth,we calculateprq (p Ê ) 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 o is thepost-formationbetaof theportfolio with respectto themarketportfolio. At thebeginningof eachmonthÌ , wecomputeeachportfolio’ssimpleaveragelog market capitalizationin millions (size)andvalue-weightedbook-to-market ratio (B/M).The columnslabeled p q (pjÊ ) and o q (ovÊ ) 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 o 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.
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Table3: PortfoliosSortedon PastF , F ? andF �
Panel A: Portfolios Sorted on Past oPortfolio Mean Std Auto o High–Low t-stat1 Low o 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 o 1.13 7.63 0.05 1.57
Panel C: Portfolios Sorted on Past o ÊPortfolio Mean Std Auto o o Ê p Ê m Ê High–Low t-stat1 Low o Ê 1.05 5.46 0.16 0.93 0.77 0.46 1.67 -0.05 -0.212 1.06 4.33 0.19 0.83 0.67 0.59 1.143 1.05 4.06 0.16 0.80 0.69 0.67 1.044 1.01 4.10 0.11 0.83 0.82 0.75 1.095 0.98 4.03 0.13 0.84 0.79 0.75 1.056 1.05 4.07 0.06 0.87 0.86 0.84 1.027 1.07 4.35 0.06 0.94 0.90 0.86 1.058 1.02 4.65 0.04 1.01 0.98 0.88 1.119 1.12 5.25 0.05 1.12 1.13 0.86 1.3110 High o Ê 1.00 6.77 0.06 1.41 1.45 0.80 1.81
Thetablelistssummarystatisticsfor value-weightedo , oËq ando Ê portfoliosatamonthlyfrequency, whereoËq ando Ê aredefinedin equation(4). For eachmonth,wecalculateo (o q , o Ê ) of all stocksbasedondailycontinuouslycompoundedreturnsoverthepastyear. Werankthestocksinto deciles(1–10),andcalculatethevalue-weightedsimplepercentagereturnover thenext month. We rebalancetheportfoliosmonthly. Meansandstandarddeviationsarein percentagetermsper month. Std denotesthe standarddeviation (volatility),Auto denotesthefirst autocorrelation,ando is post-formationthebetaof theportfolio. ThecolumnslabeledoËq (o Ê ) and pjq (p Ê ) show thepost-formationdownside(upside)betasanddownside(upside)correlationsof theportfolios. Thecolumnlabeledm Ê ( m�q ) lists theratio of thevolatility of theportfolio to thevolatilityof themarket,bothconditioningon thedownside(upside).High–Low is themeanreturndifferencebetweenportfolio 10 andportfolio 1 andt-statgivesthet-statisticfor this difference.T-statisticsarecomputedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags. Thesampleperiodis from January1964to December1999.
PanelA of this tableshowsthetime-seriesregressionof excessreturnÇ È onfactorsúýü�þ , �sú�� and � ú Ð .Theten prq portfoliosof Table(1) areusedin theregression.Ì î ï is thet-statisticof theregressioncoefficientcomputedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags. The regressiono .is adjustedfor the numberof degreesof freedom. p o � is the � -statisticof Gibbons,RossandShanken(1989),testingthehypothesisthat theregressioninterceptarejointly zero. q î p o � ï is the q -valueof p o � .The sampleperiodis from January1964to December1999. PanelB reportsthe ð ’s andt-statisticsin thetime seriesregressionin two subsamples.Column“10–1” is thedifferenceof the ð ’s for the10thdecileandthefirst decile.
Std= 5.82 Std= 5.28 Std= 4.89 t-stat=1.98o = 1.23 o = 1.16 o = 1.09p q = 0.89 p q = 0.94 p q = 0.96
Panel B: o -Balanced prq Portfolios
Low pjq Medium prq High pjq High prq - Low pjqo -balanced Mean=0.86 Mean=0.98 Mean=1.10 Mean= 0.23Std=4.60 Std=4.29 Std=3.88 t-stat= 2.35o =0.94 o =0.93 o =0.87pjq =0.88 pjq =0.92 pjq =0.96
Summarystatisticsfor the portfoliosusedto constructdownsiderisk factor r ú r at a monthly frequency.Eachmonth,we rankstocksbasedon their o , calculatedfrom thepreviousyearusingdaily data,into a lowo group anda high o group,eachgroupconsistingof onehalf of all firms. Then, within eacho group,we rank stocksbasedon their pjq , which is alsocalculatedusingdaily dataover the pastyear, into threegroups:a low prq group,a mediumpjq groupanda high pjq group,with cutoff pointsat 33.3%and66.7%.We computethe monthly value-weightedsimplereturnsfor eachportfolio. The o -balancedgroupsaretheequal-weightedaverageof the portfolios acrossthe two o 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.
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Table7: Regressionof WML ontoVariousFactors
Constant MKT SMB HML CMC SKS Adj o .Model A: coef 0.75 0.66 0.12
This tableshows theresultsfrom theFama-MacBeth(1973)regressiontestson the40momentumportfoliossortedby past6 monthsreturns.MKT, SMB andHML areFamaandFrench(1993)’sthreefactorsandCMCis the downsiderisk factor. WML is returnon the zero-coststrategy going long pastwinnersandshortingpastlosers(constructedfollowing Carhart(1997)). In the first stagewe estimatethe factor loadingsoverthe whole sample.The factorpremia, s , areestimatedin the second-stagecross-sectionalregressions.Wecomputetwo t-statisticsfor eachestimate.Thefirst oneis calculatedusingtheuncorrectedFama-MacBethstandarderrors. The secondoneis calculatedusingShanken’s (1992)adjustedstandarderrors. The o . isadjustedfor thenumberof degreesof freedom. Thelastcolumnof thetablereportsp-valuesfrom L . testsonthejoint significanceof thebetasof eachmodel.Thefirst p-valueis computedusingtheuncorrectedvariance-covariancematrix,while thesecondoneusesShanken’s (1992)correction.T-statisticsthataresignificantatthe5%(1%) level aredenotedwith * (**). Thesampleperiodis from January1964to December1999.
40
Table9: GMM Testsof theMomentumPortfolios
Constant MKT SMB HML CMC WML J-Test HJTest
Model A: CAPM
Coefficient(t ) 1.01 –4.68 57.81 0.59t-stat 70.87l�l –4.14l�l [0.03]l [0.00]l�lPremium( s ) 0.92t-stat 4.14l�lModel B: Fama-French Model
This tablelists the optimalGMM estimationresultsof the modelsusing40 momentumportfolios with therisk-freerate.Coefficient ( t ) refersto thefactorcoefficientsin thepricingkernelandPremia( s ) refersto thefactorpremia( s ) in monthlypercentageterms.P-valuesof JandHJ testsareprovidedin [], with p-valuesoflessthan5% (1%) denotedby * (**). TheJ-testis Hansen’s (1982)L . 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.
TERM coef –0.02 0.02 –0.01 0.03lt-stat –2.21l 1.42 –0.82
This tableshows the resultsof the regressionsbetweenCMC andthe macroeconomicvariables. PanelAlists the resultsfrom the regressionsof r ú r on lagged r ú r 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 Ì andtime Ì æ Í� , whereÌ 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,F , > ? 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
1
1.5
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,o and prq of the JegadeeshandTitman (1993)momentumportfolios. J refersto formationperiodand ü refersto holding periods. For eachmonth,wesortall NYSE andAMEX stocksinto decileportfoliosbasedon their returnsover thepast J =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)). y refersto formationperiodand z refersto holdingperiods.For eachmonth,we sortallNYSEandAMEX stocksinto decileportfoliosbasedontheir returnsoverthepasty =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 y]{}| or the risk-freeasset.The first ten portfolioscorrespondto the z~{�� monthholdingperiod,thesecondtento the z~{�| monthholdingperiod,thethirdten to the z�{�� monthholdingperiod,andfinally the fourth ten to the z�{���� holdingperiod. The41stassetis therisk-freeasset.Thegraphsshow theaveragepricingerrorswith asterixes,with two standarderrorbandsin solid lines. Theunitson the � -axisarein percentageterms.Pricingerrorsareestimatedfollowingcomputationof theHansen-Jagannathan(1997)distance.