Downside Correlation and Expected Stock Returns Andrew Ang Columbia University and NBER Joseph Chen University of Southern California Yuhang Xing Columbia University This Version: 12 Mar, 2002 JEL Classification: C12, C15, C32, G12 Keywords: asymmetric risk, cross-sectional asset pricing, downside correlation, downside risk, momentum effect This paper was previously circulated under the title “Downside Risk and the Momentum Effect.” The authors thank Brad Barber, Geert Bekaert, Alon Brav, John Cochrane, Randy Cohen, Kent Daniel, Bob Dittmar, Rob Engle, Cam Harvey, David Hirschleifer, Qing Li, Terence Lim, Toby Moskowitz, Akhtar Siddique, Bob Stambaugh and Zhenyu Wang. We especially thank Bob Hodrick for detailed comments. We thank seminar participants at Columbia University, NYU, USC, the Five Star Conference, and the NBER Asset Pricing Meeting for helpful comments. The authors acknowledge funding from a Q-Group research grant. Andrew Ang: [email protected]; Joe Chen: [email protected]; Yuhang Xing: [email protected].
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PanelB of Table(1) reportsthesummarystatisticsof stockssortedby downsidebeta, E�F .The columnslabeled‘ E ’ and ‘ E F ’ list the post-formationunconditionalbetasanddownside
In this section,we investigatewhy the conditionalbetameasuresfail to producea significant
relationbetweendownsidebetasandexpectedreturns.Onereasonwhy theeffect of downside
betais weakis thatthereis little differenceacrossdownsideandupsidebetasin thedata,sothe
downsidebetapicksup very little asymmetryin risk.
PanelA of Figure(1) shows theaveragedownsideandupsidebetafor variousconditioning
levels on the R -axis acrossthe 48 Fama-French(1997) industry portfolios at the monthly
frequency. On the LHS of the R -axis for RTS � , thefiguredisplaystheaverageE�F #&GM( across
all 48 industryportfolios, whereG
is R standarddeviationsbelow the unconditionalmeanof
the market. For example,at R � U�/ , the figure plots E�F #&GV� 47698 U5WQX"Y[Z]\ ( , where47698 is theunconditionalmeanof theexcessmarket returnandWQXBY[Z]\
is theunconditional
volatility of theexcessmarket return. At R �^� , the figureplots E�F #&G_� 47698 ( . Similarly,
on the RHS of the R -axis for RV` � , the figure displays E #&GM( , forG
representingR standard
deviationsabove the meanof the market. Therearetwo pointsplottedat R �^� representingE F #$GP� 47698 (.a E F and E #&GP� 47698 (Qa E . To constructtheaverageindustry E F #$GH( , we
first selectthesampleof observationswhich satisfiestheconditioningrequirementbasedonG.
Then,the individual E F #$GH( for eachindustryis computedfor eachsample.Thefiguregraphs
theaverageE�F #&GH( acrossthe industriesfor eachG. The procedureis repeatedfor theaverage
6
E #$GH( acrossthe48 industries.
In PanelA of Figure (1), the averageE�F acrossthe 48 industriesis only slightly higher
(4.8%) than the average E atGb� 47698 ( R � � ). As we condition on more extreme
market moves(asG
becomeslarger in absolutevalue),theplot shows little differencebetweenE F #$GH( and E #$GH( . Whenwe examinethe differencesbetweenE F and E for the industries
individually, we seethereasonwhy. PanelB shows theratio of E�F to E acrosseachof the48
industriesatG� 4�698 . Thedownsidebetais greaterthantheupsidebetafor only 25 out of
the48 industries.In summary, thereis little asymmetryin conditionalbetasacrossupsideand
downsidemovementsof themarket.
To further investigatethefailureof theconditionalbetas,we decomposethedownsideand
upsidebetasinto a conditionalcorrelationterm and a ratio of conditional total volatility to
conditionalmarketvolatility:
E F #&GH(.� cov#$����� �c� 47698 �dI 47698 ��KeGH(
var# 47698 �JI 47698 ��KTGH( �?f F #$GH(hgji F #&GH(
and E #&GH(.� cov#$����� �c� 47698 �dI 47698 ��OeGH(
As with thenotationfor downsideandupsidebeta,whenGp� 47698 , we abbreviateto
f F #&G��47698 (.a7f F , f #$Gq� 47698 (Qa7f , i F #&Gq� 47698 (Qa<i F , andi #$GP� 47698 (.a<i .
The asymmetryin conditionalcorrelationsis muchstrongerthan the asymmetryin betas
acrossmarket downsideandupsidemovements.PanelC of Figure(1) looks at the effectsof
downsideandupsidecorrelationsacrossvariousG. Thereis a marked asymmetryacrossthe
averagedownsideandupsidecorrelationsfor the48 industryportfolios,with a sharpbreakatGr� 47698 ( R �s� ). Ang andChen(2001) show that if returnsare drawn from a normal
distribution, asG
becomeslarger in absolutemagnitude,thedownsideandupsidecorrelations
must be symmetricand tend to zero. While upsideconditional correlationsdecreaseasG
7
increases,downsidecorrelationsdo not decreaseasG
decreases.PanelD of Figure(1) shows
that atGt� 4�698 , the point estimatesof the downsidecorrelations,
f F , arehigher thanthe
upsidecorrelations,f
, for every industryportfolio.
The secondterm in equation(4) is the reasonwhy thereis an asymmetryin conditional
correlationsbut not in conditionalbetas. The terms,i F #&GM( and
i #$GH(, arethe ratiosof total
assetvolatility to market volatility, conditionalon downsideand upsidemarket moves. We
plot thesevolatility ratiosin PanelE of Figure(1). Downsidevolatility ratiosaremuchlower
thanvolatility ratioson theupside.ThecorrespondingPanelF shows theratioi F *Hi for each
industry. In all but oneof 48 industries,thedownsidevolatility ratio is higherthantheupside
volatility ratio.
Therearetwo effectsthatexplainwhy onaveragei F #&GM(uKTi #&GM( . First,thedenominatorofi F #&GH( and
i #$GH(in equation(6) is marketvolatility. Marketvolatility is asymmetricandhigher
for momentumin the time-seriesregressionsreducesthe alphasof the downsidecorrelation
portfolios. We show in this sectionthatdownsidecorrelationsarenot mechanicallylinked to
pastreturns,hencethemomentumeffect.
To disentangletheeffectsof pastreturnsanddownsidecorrelations,we performa double� g �sortacrosspast6 monthsreturnsanddownsidecorrelations.At eachmonth,wefirst sort
all stocksinto quintilesbasedon their past6 monthreturns.Thento control for pastreturns,
we sort stockswithin eachpastreturn quintiles into additionalquintiles basedonf F . This
procedurecreates25portfolios,andwetake theaveragesof thef F portfoliosacrosspastreturn
quintiles.
We report the alphasfrom the Fama-Frenchthree-factormodelof thesefive portfolios in
Panel A of Table (5). Controlling for past returns,theseaveragesof downsidecorrelation
portfolios show cross-sectionaldispersioninf F . Their alphasare statistically significant,
and the differencebetweenthe first and fifth portfolio alphasis 0.33%per month, which is
alsosignificantwith a p-value= 0.00. In Table(4), controlling for momentumover the first
subsampleperiod(Jan1964- Dec1981)yieldedonly aborderlinesignificantresult.Now, when
we control for themomentumeffect by thedoubleportfolio sort,we find thatthedifferencein
WML still doesnot affect the significanceof CMC. The GRS test suggeststhat this model,2 SMB is long morefirms thanit is shortsincethebreakpointsaredeterminedusingmarket capitalizationsof
very low levelsof thet-statistics.Nevertheless,thepoint estimatesdo show thatlosersperform
betterthanwinnersduringmarket downturns.3
We observe thesameeffect for thedownsidecorrelationportfoliosfrom Table(2) in Panel
B. High downside correlationstockshave higher unconditionalreturnsthan low downside3 For periodswhenthe market return is lessthantwo standarddeviationsbelow the mean,althoughwinners
under-performlosers,the patternmoving acrossthe decileportfolios from losersto winnersis not monotonic.
Theloadingsareestimatedfrom thetime-seriesregressionsof themomentumportfolioson the
factorsfrom the first stepof the Fama-MacBeth(1973)procedure.We seethat for eachset
of portfolios, as we go from the past loser portfolio (decile 1) to the pastwinner portfolio
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(decile10), the loadingson themarket portfolio remainflat, so thatbetahaslittle explanatory
power. The loadingson SMB decreasefrom the losersto thewinners,exceptfor the last two
deciles.Similarly, the loadingson theHML factoralsogo in thewrongdirection,decreasing
monotonicallyfrom thelosersto thewinners.
In contrastto the decreasingloadingson the SMB andHML factors,the loadingson the
CMC factor in Figure (4) almostmonotonicallyincreasefrom stronglynegative for the past
loserportfolios to slightly positive for the pastwinner portfolios. The increasingloadingson
CMC acrossthedecileportfoliosfor eachholdingperiod 6 areconsistentwith the increasingf F statisticsacrossthedecilesin Figure(3). Winnerportfolioshavehigherf F , higherloadings
on CMC, andhigherexpectedreturns.Thenegative loadingsfor loserstocksimply that losers
have higherdownsidecorrelationexposurethanwinners. This reflectsthe evidencein Table
(9), which shows that pastwinner stocksdo poorly whenthe market haslarge moveson the
Using GMM cross-sectionalestimations(describedin the Appendix),we canconductsome
additionalhypothesistestsfor thegoodnessof fit for thevariousmodelsin Table(10) to price
the momentumeffect. Taking Model C as an unconstrainedmodel and using its weighting
matrix to re-estimateModel A, we canconducta ª ; over-identificationtest. This testsrejects
thenull hypothesisof theFama-Frenchmodelwith a p-value=0.02.Hence,CMC doesprovide
additionalexplanatorypower for the cross-sectionof momentumportfolios which the Fama-
Frenchmodeldoesnotprovide. ModelD of Table(10)nestsModelC, whichusesMKT, SMB,
HML andCMC factors.We run a ª ; over-identificationtestwith thenull of Model C against
the alternative of Model D, which rejectswith a p-value of 0.01. Hence,we concludethat
WML still hasfurtherexplanatorypower, in thepresenceof CMC, to pricethecross-sectionof
momentumportfolios.
We graphthe averagepricing errorsfor the modelsin Figure(5), following Hodrick and
Zhang(2001).Thepricing errorsarecomputedusingtheweightingmatrix,%-\9�«�"� £ � £ !� � F � ,
where£ �
is a vector of grossreturnsof the baseassets. Sincethe sameweighting matrix
is usedacrossall of the models,we can comparethe differencesin the pricing errors for
differentmodels.Figure(5) displayseachmomentumportfolio on the R -axis, wherethe first
ten portfolios correspondto the 6 �¬ monthholding period, the secondten to the 6 �¯®monthholdingperiod,thethird tento the 6 �b° monthholdingperiod,andfinally thefourth
ten to the 6 �/ = holding period. The 41stassetis the risk-freeasset.The figure plots two
23
standarderrorboundsin solid lines,andthepricingerrorsfor eachassetin *’ s.
Figure (5) shows that the CAPM hasmost of its pricing errorsoutsidethe two standard
error bandsandshows that the loserportfolios arethe mostdifficult for the CAPM to price.
The Fama-Frenchmodelhasmostdifficulty pricing pastwinners;the pricing errorsof every
highestwinnerportfolio lies outsidethetwo standarderrorbands.ThemodelusingMKT and
CMC factorsis theonly modelthathasall thepricing errorswithin two standarderrorbands.
However, addingCMC to the FamaFrenchmodelor the Carhartmodeldoesnot changethe
pricingerrorsof theassetsverymuch.
While Figure(5) cangive usa visual representationof thepricing errors,we canformally
test if all the pricing errorsarezeroby usinga Hansen-Jagannathan(1997)(HJ) test(seethe
wouldalsoassigna largerrole to conditionalbetasthanto conditionalcorrelations.
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Appendix
A Data and Portfolio Construction
Data SourcesWe usedatafrom the Centerfor Researchin SecurityPrices(CRSP)to constructportfolios of stockssortedbyvariouscharacteristicsof returns.We confineour attentionto ordinarycommonstockslistsedon NYSE, AMEXandNASDAQ, omitting ADRs, REITs,closed-endfunds,foreignfirms andothersecuritieswhich do not have aCRSPsharetypecodeof 10 or 11. We usedaily returnsfrom CRSPfor theperiodcoveringJanuary1st,1964toDecember31st,1999,includingNASDAQ datawhich is only availablepost-1972.Weusetheone-monthrisk-freeratefrom CRSPandtake CRSP’s value-weightedreturnsof all stocksasthemarket portfolio. All our returnsareexpressedascontinuouslycompoundedreturns.
At thebeginningof eachmonth,we calculateeachstock’s momentmeasuresusingthe pastyear’s daily logreturnsfrom theCRSPdaily file. For themomentswhich conditionon downsideor upsidemovements,we defineanobservationat time Ð to beadownside(upside)marketmovementif theexcessmarket returnat Ð is lessthanorequalto (greaterthanor equalto) theaverageexcessmarketreturnduringthepastoneyearperiodin consideration.Werequireastockto haveat least220observationsto beincludedin thecalculation.Thesemomentmeasuresarethenusedto sortthestocksinto decilesanda value-weightedreturnis calculatedfor all stocksin eachdecile.Theportfoliosarerebalancedmonthly.
SKS and WML Factor ConstructionHarvey and Siddique(2000) use60 monthsof datato computethe coskewnessdefinedin equation(A-1) forall stocksin NYSE, AMEX andNASDAQ. Stocksaresortedin orderof increasingnegative coskewness. ThecoskewnessfactorSKSis thevalue-weightedaveragereturnsof firms in thetop 3 deciles(with themostnegativecoskewness)minusthevalue-weightedaveragereturnof firmsin thebottom3 deciles(stockswith themostpositivecoskewness)in the61stmonth.
Following Carhart(1997),we constructWML asthe equally-weightedaverageof firms with the highest30percenteleven-monthreturnslaggedonemonthminusthe equally-weightedaverageof firms with the lowest30percenteleven-monthreturnslaggedonemonth.In constructingWML, all stocksin NYSE,AMEX andNASDAQareusedandportfoliosarerebalancedmonthly.
26
Liquidity Factor and Liquidity BetasWe follow PastorandStambaugh(2001)to constructanaggregateliquidity measure,Ñ . Stockreturnandvolumedataareobtainedfrom CRSP. NASDAQ stocksareexcludedin theconstructionof theaggregateliquidity measure.Theliquidity estimate,Ò ³�¿ ´ , for anindividualstock ± in month Ð is theordinaryleastsquares(OLS) estimateof Ò ³�¿ ´in thefollowing regression:²dÓ³�¿ Ô ¹�Õ ¿ ´ »×Ö ³¥¿ ´+ØÚÙM³�¿ ´)Û² ³�¿ Ô�¿ ´HØ Ò ³¥¿ ´nÜ ±�Ý�Þ Ê ²dÓ³¥¿ Ô�¿ ´ Ë+ß ³¥¿ Ô�¿ ´�Ø ¾ ³�¿ Ô ¹�Õ ¿ ´ Ãtà »'á à Ï�Ï�Ï Ã,â Ï (A-3)
In equation(A-3), Û² ³¥¿ Ô�¿ ´ is theraw returnon stock ± on day à of month Ð , ² Ó³�¿ Ô�¿ ´ »×² ³�¿ Ô�¿ ´�Ä ²�¶ ¿ Ô�¿ ´ is thestockreturnin excessof themarket return,and ß ³�¿ Ô�¿ ´ is thedollar volumefor stock ± on day à of month Ð . Themarket returnon dayon day à of month Ð , ²�¶ ¿ Ô�¿ ´ , is takenasthereturnon theCRSPvalue-weightedmarketportfolio. A stock’sliquidity estimate,Ò ³�¿ ´ , is computedin a givenmonthonly if thereareat least15 consecutiveobservations,andifthestockhasa month-endsharepricesof greaterthan$5 andlessthan$1000.
Theaggregateliquidity measure,Ñ , is computedbasedontheliquidity estimates,Òã³¥¿ ´ , of individualfirmslistedon NYSE andAMEX from August1962to December1992.Only theindividual liquidity estimatesthatmeettheabove criteria is used.To constructthe innovationsin aggregateliquidity, we follow PastorandStambaughandfirst form thescaledmonthlydifference:äPåÒ�´}»Læwç ´ç Õãè áé êë ³íì Õ�î Òã³¥¿ ´ Ä Òã³�¿ ´ ¸�Õ�ï à (A-4)
whereé
is thenumberof availablestocksat month Ð , ç ´ is thetotaldollar valueof theincludedstocksat theendof month Ð Ä á , and ç Õ is thetotal dollar valueof thestocksat theendof July 1962.Theinnovationsin liquidityarecomputedastheresidualsin thefollowing regression:äPåÒ ´ »ñð Øóò ä�åÒ ´ ¸�Õ ØÚô î ç ´$õ ç Õ ï åÒ ´ ¸�Õ Ø÷öÍ´ Ï (A-5)
Finally, theaggregateliquidity measure,Ñ}´ , is takento bethefitted residuals,Ñ�´y» åö ´ .To calculatethe liquidity betasfor individual stocks,at the endof eachmonthbetween1968and1999,we
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 á õ�� of thereturnsof theworstperformersonemonthago,plus á õ�� of thereturnsof theworstperformerstwo monthsago,etc.
Macroeconomic VariablesWe usethe following macroeconomicvariablesfrom the FederalReserve Bank of St. Louis: the growth ratein the index of leadingeconomicindicators(LEI), the growth rate in the index of Help WantedAdvertising inNewspapers(HELP), the growth rateof total industrialproduction(IP), the ConsumerPriceIndex inflation rate(CPI), the level of theFedfundsrate(FED), andthetermspreadbetweenthe10-yearT-bondsandthe3-monthsT-bills (TERM). All growth rates(includinginflation) arecomputedasthedifferencein logsof theindex at timesÐ and Ð Ä á�� , whereÐ is monthly.
in which � ù is a scalar, � is a Ç á vectorof factorpremia,and º ³ is an Ç á vectorof factorloadingsforportfolio ± . TheFama-MacBeth(1973)is a two-stepcross-sectionalestimationprocedure.
where � ´ is a Ç� á vectorof factors,� ù is a scalar, and � Õ is a Ç� á vectorof coefficients.Therepresentationin equation(B-7) is equivalentto a linearbetapricingmodel:¼ î � ³í´ ï »�� ù Ø ����º ³ à (B-8)
which is analogousto equation(B-1) for excessreturns.Theconstant� ù is givenby:
To testwhethera factor� is priced,we testthenull hypothesisÿ ù�� ���]»! .Letting �:´ denoteané á vectorof grossreturns�]´y» î � Õ ´ à Ï�Ï�Ï Ã � ê ´ ï � , anddenotingtheparametersof the
2 » ü43Î� + 3Î65�87:9 Ä + 3Î� â � î â ��;+ � â � ï ¸�Õ â ��,+ 3Î<5�!= ¸�Õ + 3Î� ü43Î<5� Ãwhereü�3Î� and + 3Î� aretheupper-triangularCholesky decompositionsof ü � and + � respectively, and â � »?><@BA> ( .Thematrix ü � is theoptimalweightingmatrix,where +DC� » ü ¸�Õ� » ½ ÈFE cov î Ý � Ã Ý �� ï Á ¸�Õ . JagannathanandWangshow that
To calculatea small samplep-valuefor the HJ distance,we assumethat the linear factormodelholdsandsimulatea datageneratingprocess(DGP) with 432 observations,the samelengthasin our samples.The DGPtakestheform: ² ³¥¿ ´ »Æ² �´ ¸�Õ Ø ºN�³ ��´+Ø ¾ ³í´ à (B-12)
where² ³�¿ ´ is thereturnonthe ± -th portfolio, ² �´ is therisk-freerate,º ³ is an ÇO á vectorof factorloadings,and ��´is the Ç� á vectorof factors.We assumethattherisk-freerateandthefactorsfollow a first-orderVAR process.Let Ph´y» î ² �´ à � ´ ï � , and Ph´ follows:
P ´ »�Q Ø 2 P ´ ¸�Õ Ø÷öM´ à (B-13)
where ö ´SR é î Ã � ï . We estimatethis VAR systemandusethe estimatesåQ ,å2
andå� as the parametersfor
our factorgeneratingprocess.In eachsimulation,we generate432observationsof factorsandthe risk-freeratefrom the VAR systemin equation(B-13). For the portfolio returns,we usethe sampleregressioncoefficient ofeachportfolio returnon the factors,
å� V 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.
29
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31
Table1: PortfoliosSortedon Past E , EQF and E Panel A: Portfolios Sorted on Past º
The table shows the results from Fama-MacBeth(1973) regressiontests on 20 downside correlationportfolios. Theseportfolios are formed in the following fashion. Stocksare first sortedinto two groupsaccordingto their pastbetaover the pastyear, usingdaily returns(high betaversuslow beta). Eachgroupconsistsof one half of all firms. Then, within eachbetagroup, we rank stocksbasedon their ·Í¸ , alsocomputedusingdaily dataover thepastyearinto decileportfolios.This givesus2 ( º ) 10 ( ·Í¸ ) portfolios,makinga total of twenty downsidecorrelationportfolios. 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)). T-statisticsarecomputedusingShanken (1992)adjustedstandarderrors. GRSdenotestheF-testof Gibbons,RossandShanken(1989)testingthehypothesisthatthe Å ’sof all 20portfoliosare jointly zero. T-statisticsthat aresignificantat the 5% (1%) level aredenotedwith * (**). The sampleperiodis from January1964to December1999.
Table4: SubsampleAnalysisoff F PortfoliosHigh - Low � ’s
PanelA shows thesummarystatisticsof thefactors.MKT is theCRSPvalue-weightedreturnsof all stocks.SMB andHML arethesizeandthebook-to-marketfactors(constructedbyFamaandFrench(1993)),WML isthereturnonthezero-coststrategy of goinglongpastwinnersandshortingpastlosers(constructedfollowingCarhart(1997)),andSKS is the returnon going long stockswith the mostnegative pastcoskewnessandshortingstockswith themostpositivepastcoskewness(constructedfollowing Harvey andSiddique(2000)).CMC is the returnon a portfolio going long stockswith thehighestpastdownsidecorrelationandshortingstockswith the lowestpastdownsidecorrelation. The first two columnsshow the meansandthe standarddeviationsof thefactors,expressedasmonthlypercentages.Skew andKurt aretheskewnessandkurtosisoftheportfolio returns.Auto refersto first-orderautocorrelation.Factorswith statisticallysignificantmeansatthe5% (1%) level aredenotedwith * (**), usingheteroskedastic-robustNewey-West(1987)standarderrorswith 3 lags.Thecorrelationmatrix betweenthefactorsis reportedin PanelB. PanelC reportstheregressionof CMC ontoMKT, SMB, HML andWML factors,with t-statisticscomputedusing3 Newey-Westlags.T-statisticsthataresignificantat the5%(1%) level aredenotedwith * (**). Thesampleperiodis from January1964to December1999.
The table shows the results from Fama-MacBeth(1973) regressiontests on 20 downside correlationportfolios. Theseportfolios are formed in the following fashion. Stocksare first sortedinto two groupsaccordingto their pastbetaover the pastyear, usingdaily returns(high betaversuslow beta). Eachgroupconsistsof one half of all firms. Then, within eachbetagroup, we rank stocksbasedon their ·Í¸ , alsocomputedusingdaily dataover thepastyearinto decileportfolios.This givesus2 ( º ) 10 ( · ¸ ) portfolios,makinga total of twenty downsidecorrelationportfolios. 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)).CMC is thereturnonaportfolio goinglongstockswith thehighestpastdownsidecorrelationsandshortingstockswith the lowestpastdownsidecorrelations.T-statisticsarecomputedusingShanken (1992) adjustedstandarderrors. GRS denotesthe F-testof Gibbons,RossandShanken (1989)testingthehypothesisthat the Å ’s of all 20 portfoliosarejointly zero. T-statisticsthataresignificantat the5% (1%) level aredenotedwith * (**). Thesampleperiodis from January1964to December1999.
TERM coef –0.02 0.02 –0.01 0.03Ct-stat –2.21C 1.42 –0.82
This tableshows the resultsof the regressionsbetweenCMC andthe macroeconomicvariables. PanelAlists the resultsfrom the regressionsof ] Ç ] on lagged ] Ç ] 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 rates(including inflation) arecomputedasthe differencesinlogsof the index at time Ð andtime Ð Ä á6� , where Ð 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.
Thetoppanelshowsportfolio alphasfrom the20 ·�¸ portfolios.Theseportfoliosareformedin thefollowingfashion.First, stocksaresortedinto two groupsaccordingto their pastbetaover the pastyear, usingdailyreturns(highbetaversuslow beta).Eachgroupconsistsof onehalf of all firms. Then,within eachbetagroup,we rankstocksbasedon their ·�¸ , alsocomputedusingdaily dataover the pastyearinto decileportfolios.This givesus 2 ( º ) 10 ( ·Í¸ ) portfolios, makinga total of twenty downsidecorrelationportfolios. Theportfolios 1-10 (11-20)arefrom the low (high) betagroup. The Å ’s arefrom a modelof the Fama-French(1993)factors,augmentedwith Carhart(1997)’sWML momentumfactorandareshown over threeperiods:over thefull sample,from Jan1964- Dec1981,andfrom Jan1982- Dec1999.Thebottompanelshows theportfolio factorloadingson theMKT, SMB, HML andWML factorsover thefull sample.
42
Figure3: AverageReturn,E , f F of MomentumPortfolios
2 4 6 8 100
0.5
1
1.5
2J=6 K=3
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Mean β ρ−
2 4 6 8 100
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Mean β ρ−
2 4 6 8 100
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Mean β ρ−
2 4 6 8 100
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Mean β ρ−
Theseplots show the averagemonthly percentagereturns, m and n�o of the JegadeeshandTitman (1993)momentumportfolios. p refersto formationperiodand q refersto holding periods. For eachmonth,wesortall NYSE andAMEX stocksinto decileportfoliosbasedon their returnsover thepast p =6 months.Weconsiderholding periodsover the next 3, 6, 9 and12 months. This procedureyields 4 strategiesand40portfoliosin total. Thesampleperiodis from January1964to December1999.
43
Figure4: Loadingsof MomentumPortfoliosonFactors
2 4 6 8 10−1
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0
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1.5J=6 K=3
Load
ings
on
Fact
ors
Decile
MKTSMBHMLCMC
2 4 6 8 10−1
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1.5J=6 K=6
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ings
on
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ors
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MKTSMBHMLCMC
2 4 6 8 10−1
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ings
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ors
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MKTSMBHMLCMC
2 4 6 8 10−1
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ings
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ors
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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(B-2)). p refersto formationperiodand q refersto holdingperiods.For eachmonth,we sortallNYSEandAMEX stocksinto decileportfoliosbasedontheir returnsoverthepast p =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 Section5.2. Eachstar in the graphrepresentsoneof the 40 momentumportfolios with psrit or the risk-freeasset.The first ten portfolioscorrespondto the qur�v monthholdingperiod,thesecondtento the qur�t monthholdingperiod,thethirdten to the qwryx monthholdingperiod,andfinally the fourth ten to the qOr?z�{ holdingperiod. The41stassetis therisk-freeasset.Thegraphsshow theaveragepricingerrorswith asterixes,with two standarderrorbandsin solid lines. Theunitson the | -axisarein percentageterms.Pricingerrorsareestimatedfollowingcomputationof the Hansen-Jagannathan(1997)distance.TheCarhart(1997)four-factormodelconsistsofMKT, SMB, HML andWML factors.