downside correlation and expected stock returns · 2014-07-30 · downside correlation and expected...
TRANSCRIPT
DownsideCorrelationandExpectedStockReturns�
Andrew Ang
ColumbiaUniversityandNBER
JosephChen
Universityof SouthernCalifornia
YuhangXing
ColumbiaUniversity
ThisVersion:12Mar, 2002
JELClassification:C12,C15,C32,G12
Keywords:asymmetricrisk, cross-sectionalassetpricing,downside
correlation,downsiderisk, momentumeffect
�Thispaperwaspreviouslycirculatedunderthetitle “DownsideRiskandtheMomentumEffect.” The
authorsthankBradBarber, GeertBekaert,Alon Brav, JohnCochrane,RandyCohen,KentDaniel,Bob
Dittmar, RobEngle,CamHarvey, David Hirschleifer, Qing Li, TerenceLim, Toby Moskowitz, Akhtar
Siddique,Bob StambaughandZhenyu Wang.We especiallythankBobHodrick for detailedcomments.
We thankseminarparticipantsat ColumbiaUniversity, NYU, USC, the Five StarConference,andthe
NBERAssetPricingMeetingfor helpful comments.Theauthorsacknowledgefundingfrom aQ-Group
researchgrant. Andrew Ang: [email protected];JoeChen: [email protected]; Yuhang
Xing: [email protected].
Abstract
If investorsaremoreaverseto therisk of lossesonthedownsidethanof gainsontheupside,
investorsoughtto demandgreatercompensationfor holdingstockswith greaterdownsiderisk.
Downsidecorrelationsbettercapturetheasymmetricnatureof risk thandownsidebetas,since
conditionalbetasexhibit little asymmetryacrossfalling andrisingmarkets.Wefind thatstocks
with high downsidecorrelationswith the market, which are correlationsover periodswhen
excessmarket returnsare below the mean,have high expectedreturns. Controlling for the
market beta,the sizeeffect, andthe book-to-market effect, the expectedreturnon a portfolio
of stockswith the greatestdownsidecorrelationsexceedsthe expectedreturnon a portfolio
of stockswith the leastdownsidecorrelationsby 6.55%per annum. We find that part of the
profitability of investingin momentumstrategiescanbeexplainedascompensationfor bearing
highexposureto downsiderisk.
1 Introduction
According to the Capital AssetPricing Model (CAPM), a stock’s expectedexcessreturn is
proportionalto its market beta,which is constantacrossdown-marketsandup-markets.While
this modelhasbeenrejected,modernfactormodels,like the FamaandFrench(1993) three-
factormodel,continueto maintainthesymmetricnatureof factorloadingsacrossdown-markets
andup-markets.However, asearlyasMarkowitz (1959),economistshaverealizedthatinvestors
caredifferentlyaboutdownsiderisk, thanthey careabouttotal market risk. Markowitz advises
constructingportfoliosbasedon semi-variances,ratherthanonvariances,sincesemi-variances
weightupsiderisk (gains)differentlyfrom downsiderisk (losses).More recently, in Kahneman
andTversky (1979)’s lossaversionutility andin Gul (1991)’s first-orderrisk aversionutility,
lossesareweightedmoreheavily thangainsin aninvestor’sutility function. If investorsdislike
downsiderisk, they oughtto demandhighercompensations– in the form of higherexpected
returns– for holdingassetswith greaterdownsiderisk.
Onenaturalextensionof the CAPM, that takesinto accountthis asymmetrictreatmentof
risk, is the useof downsideandupsidebetas(Bawa andLindenberg, 1977). Thesedownside
betasarethemarketbetascomputedoverperiodsfor which themarket returnis below its mean
(downsideperiods).However, downsidebetasproducelittle variationsin thecross-sectionof
expectedreturnsin thedata,sincethey areaffectedby thechangesin idiosyncraticvolatility and
in market volatility acrossdownsideandupsideperiods.In particular, many authors,including
Campbellet al. (2001),find that market volatility increasesin down-marketsandrecessions.
Moreover, Duffee(1995)findsthatidiosyncraticvolatility decreasesin down-markets.Both of
theseeffectscauseconditionalbetato havelittle asymmetryacrossthedownsideandtheupside.
In contrast,conditional correlationsare immunefrom different volatility effects acrossup-
marketsanddown-markets,andexhibit significantasymmetriesacrossdownsideversusupside
movesby themarket (Ang andChen,2001).Thissuggeststhatconditionalcorrelationsmaybe
betterableto capturetheasymmetricnatureof risk thanconditionalbetas.
Wefind thatstockswith highdownsidecorrelations,whichwemeasureashighly correlated
movementswith the aggregatemarket in periodswhen markets fall, provide high expected
returns. The portfolio of greatestdownsidecorrelationstocksoutperformsthe portfolio of
lowestdownsidecorrelationstocksby 4.91%perannum.We show thatdownsidecorrelations
arenot linkedto low liquidity in down marketsnor mechanicallylinkedto pastreturns.After
controlling for the market beta, the size effect, and the book-to-market effect, the greatest
downsidecorrelationportfolio outperformsthelowestdownsidecorrelationportfolio by 6.55%
perannum.
1
Our researchdesign follows the customof constructingand adding factors to explain
deviationsfrom theCapitalAssetPricingModel(CAPM).1 While thisapproachdoesnotspeak
to the natureof the risk premia,our goal is not to presenta theoreticalmodel that explains
how downsiderisk is priced in equilibrium. Our goal is to demonstratethat a part of the
factor structurein stock returnsreflectsvariationsin downsiderisk, measuredby downside
correlations.Not surprisingly, we find that while the Fama-French(1993)three-factormodel
cannotexplain the variationsin expectedreturnsof stockssortedby downsidecorrelations,
a factor reflectingthe spreadin expectedreturnsinducedby downsidecorrelationsexplains
thesevariations. We term this factor ‘CMC’, andfind that it alsohelpsto forecasteconomic
downturns.
As anapplicationof theCMC factor, we link theprofitability of theJegadeeshandTitman
(1993)momentumstrategiesto downsiderisk. Existingexplanationsof themomentumeffect
are largely behavioral in natureand use modelswith imperfect formation and updatingof
investors’expectationsin responseto new information(Barberis,ShleiferandVishny, 1998;
Daniel,HirshleiferandSubrahmanyam,1998;HongandStein,1999).Theseexplanationsrely
on the assumptionthat arbitrageis limited, so that arbitrageurscannoteliminatethe apparent
profitability of momentumstrategies.Mispricingmaypersistbecausearbitrageursneedto bear
someundiversifiablerisk,andrisk-aversearbitrageursdemandcompensationfor acceptingsuch
risk (Hirshleifer, 2001). We argue that thesemomentumstrategies have high exposuresto
a systematicdownsidecorrelationfactor. The intuition behindthis story is that pastwinner
stockshavehighreturns,in part,becauseduringperiodswhenthemarketexperiencesdownside
moves,winnerstocksmovedown morewith themarket thanpastloserstocks.
The momentumportfolios load positively and significantly on the downsidecorrelation
factor. In particular, a linear two-factormodelwith the market andthe CMC factorexplains
someof the cross-sectionalvariationsamongmomentumportfolio returns. The downside
correlationfactorcommandsa significantlypositive risk premiumin cross-sectionaltests,and
retainsits statisticalsignificancein the presenceof the Fama-Frenchandmomentumfactors.
However, the downsidecorrelationfactor only modestlyreducesthe Carhart(1997) WML
momentumfactorpremiumby 2.18%per annum,andhypothesistestsreject that this factor
canfully accountfor themomentumeffect.1 Otherauthorsusefactorswhich reflectthesizeandthebook-to-market effects(FamaandFrench,1993and
1996),macroeconomicfactors(Chen,Roll andRoss,1986),productionfactors(Cochrane,1996),labor income
(JagannathanandWang,1996),marketmicrostructurefactorslikevolume(Gervais,KanielandMingelgrin,2001)
or liquidity (PastorandStambaugh,2001),andfactorsmotivatedfrom corporatefinancetheory(Lamont,Polkand
Saa-Requejo,2001).
2
Our findings are closely relatedto other studieswhich usefactor modelsto accountfor
the high momentumreturns. Harvey and Siddique (2000) demonstratethat skewnessis
priced,andshow that momentumstrategiesarenegatively skewed. Unlike skewnessor other
centeredmoments,our conditional correlationmeasureemphasizesthe asymmetryof risk
acrossdownsideandupsidemarketmoves.Ourfindingsarealsorelatedto DeBondtandThaler
(1987)whofind thatpastwinnerstockshavegreaterdownsidebetasthanupsidebetas.Wefind
that thespreadsin expectedreturnsfrom downsidebetaarevery weaksinceconditionalbetas
are roughly constantacrossupsideanddownsideperiods. In contrast,downsidecorrelation
portfoliosproducelargecross-sectionalvariationsin expectedreturns.
The restof this paperis organizedasfollows. Section2 investigatesthe relationbetween
higher-order momentsand expectedreturns. We show that portfolios sortedby increasing
downsidecorrelationshave increasingexpectedreturns.Ontheotherhand,portfoliossortedby
otherhighermomentsdonotproduceany discernablepatternin theirexpectedreturns.Section
3 exploresif thepatternsacrossportfoliosof downsidecorrelationsarerobustaftercontrolling
for someknown effects. Section4 detailstheconstructionof our downsidecorrelationfactor
andshowsthatit commandsaneconomicallysignificantrisk premium.Weapplythedownside
correlationfactorto pricethemomentumportfoliosin Section5. Section6 concludes.
2 Higher-Order Moments and Expected Returns
We start with the relationsbetweencenteredmomentsand expectedreturnsin Section2.1.
Since this framework fails to producesignificant spreadsin expectedreturns,we turn our
attentionto downsideandupsidebetasadvocatedby Bawa andLindenberg (1977)in Section
2.2. High downsidebetastockshave only slightly higherexpectedreturnsthanlow downside
betastocks.Section2.3examinesthecauseof this failure,andshowsthattheeffectof changing
idiosyncraticandmarketvolatilitiesacrossdownsideandupsideperiods,maskstheasymmetry
of conditional betasacrossthe downside and the upside. On the other hand, downside
correlationis notaffectedbychangingvolatility effects,andexhibitshighlyasymmetricpatterns
acrossthedownsideandupsideperiods.Portfoliosortsbasedondownsidecorrelationsproduce
largespreadsin expectedreturns,whichwedemonstratein Section2.4.
2.1 Centered versus Conditional Moments
Economictheorypredictsthattheexpectedreturnof anassetis linkedto higher-ordermoments
of theasset’sreturnthroughthepreferencesof amarginal investor. ThestandardEulerequation
3
in anarbitrage-freeeconomyis: ��������� ������� �� ��� �����(1)
where��� �
is thepricing kernelor thestochasticdiscountfactorand����� �� �
is theexcessreturn
on asset� . If we assumethat consumptionandwealthareequivalent,thenthe pricing kernel
is the marginal rateof substitutionfor the marginal investor:��� ��� �"!$#&%'�� �)(�*+�"!$#,%-�$(
. By
takingaTaylorexpansionof themarginal investor’sutility function,�
, wecanwrite:��� �.�0/21 %-�,� !3!� !547698 �� �:1 %<;� � !3! != � !747698 ;�� � 1?>�>@>�� (2)
where 47698 �� � is therateof returnon themarketportfolio, in excessof therisk-freerate.
The coefficient on 4�698 �� � in equation(2),%-�A�B!3!C*+�"!
, correspondsto the relative risk
aversion of the marginal investor. The coefficient on 47698 ;�� � is studiedby Kraus and
Litzenberger (1976)andmotivatesHarvey andSiddique(2000)’s coskewnessmeasure,where
risk-averseinvestorspreferpositivelyskewedassetsto negativelyskewedassets.Dittmar(2001)
examinesthe cokurtosiscoefficient on 4�698BD�� � and arguesthat investorswith decreasing
absoluteprudencedislikecokurtosis.
If the systematiccomponentof skewnessor kurtosis are priced, then stockssortedby
coskewnessor cokurtosisshouldexhibit cross-sectionalspreadsin expectedreturns. When
stocksaresortedinto decileportfolios by increasingpastcoskewness,we do find that stocks
with more negative coskewnesshave higher returns. However, the differencebetweenthe
portfolio of stockswith themostnegativecoskewnessandtheportfolio of stockswith themost
positive coskewnessis only 1.79%per annum,which is not statisticallysignificantat the 5%
level (t-stat= 1.17). Whenwe sort on cokurtosis,high cokurtosisstockshave slightly lower
expectedreturnsthanlow cokurtosisstocks,which is oppositeof thatpredictedby theory.
Why might centeredmomentslike coskewnessandcokurtosisfail to pick up muchpattern
in thecross-sectionof stockreturns?If themarginal investor’s utility is kinked,skewnessand
othercenteredmomentsmay not effectively capturethe asymmetryin aversionto risk across
upsideanddownsidemoves.Empiricaltestsrejectstandardspecificationsfor�
, suchaspower
utility, and leave unansweredwhat the most appropriaterepresentationfor�
is. However,
economictheorydoesnot restricttheutility function�
to besmooth.For instance,Kahneman
andTversky (1979)’s lossaversionutility function andGul (1991)’s first-orderrisk aversion
utility function have a kink at the referencepoint to which an investorcomparesgainsand
losses.Theseasymmetric,kinked utility functionssuggestthat polynomialexpansionsof�
,
suchas the expansionusedby Bansal,Hsieh and Viswanathan(1993), may not be a good
global approximationsof�
. In particular, standardpolynomialexpansionsmay not capture
4
risk which differsacrossdown or up markets. In aneffort to capturetheasymmetricnatureof
risk, we turn to momentsconditionedon downsideandupsidemovesof themarket.
2.2 Downside and Upside Betas
A naturalstartingpoint for examiningasymmetriesin risk is to considerdownsideandupside
betas.Following Bawa andLindenberg (1977),we definedownsidebeta E�F andupsidebetaE as:
E F #$GH(.� cov#$����� �A� 47698 �JI 47698 ��KLGM(
var# 47698 �NI 47698 ��KLGM(
and E #$GH(.� cov#$����� �A� 47698 �JI 47698 ��OLGM(
var# 47698 �NI 47698 ��OLGM( � (3)
where����� �
is theexcessstockreturnand 47698 � is theexcessmarket return.TheparameterG
is
a conditioninglevel. Hence,E F #&GH( is thebetaamongobservationswherethemarket returnis
lessthanG, and E #&GH( is thebetaamongobservationswherethemarket returnis greaterthanG
. In thecasewhereGP� 47698 , where 47698 is themeanexcessmarket return,weabbreviate
thenotationto E�F #$G�� 47698 (Q� E�F and E #$G�� 47698 (Q� E .Downside and upsidebetascapturethe notion of asymmetricexposuresto risk across
periodswhenthemarket falls andperiodswhenthemarket rises.Thesemomentsaredifferent
from centeredmomentsbecausethey emphasizethe asymmetryacrossupsidemarket moves
anddownsidemarketmovesexplicitly by theconditioninglevelG. ComputingE�F #&GM( and E #$GH(
is simple: we take thoseobservationswhich satisfythe conditioningrequirementbasedonG,
andthencomputethebetason thissubsampleof observations.If theagentsin theeconomyare
moresensitive to lossesthanthey areto gains,thenstockswith greaterdownsiderisk should
providegreatercompensation.
To examineif downsideor upsidebetasarerelatedto expectedreturns,wesortstocksbased
on theirdownsidebetasandon theirupsidebetasto form portfoliosbasedon thesesorts.Since
we have fewer observationsavailableto usasG
movesfurtheraway from themean,we focus
on the conditioninglevel ofG?� 47698 , so that roughly half of the observationsare used
to computeE�F and E for eachstock. We rank stocksinto deciles,andcalculatethe value-
weightedholding period return of the portfolio of stocksin eachdecile over the following
month. We rebalancetheseportfolios eachmonth. Appendix A provides further detail on
portfolio construction.
Table (1) presentsthe characteristicsof portfolios formedby the sortsof stockson their
downsidebetasandon their upsidebetas,aswell ason their unconditionalbetas.PanelA of
5
Table(1) showsthesummarystatisticsof stockssortedby theunconditionalbetas.Thecolumn
labeled‘ E ’ showstheunconditionalbetasof eachportfolioscalculatedatthemonthlyfrequency
over the whole sample.This columnshows that the portfolios constructedby rankingstocks
on pastunconditionalbetasretaintheir beta-rankingsin the post-formationperiod. However,
confirmingmany previousstudies,PanelA showsthatthereis nopatternin theexpectedreturns
of thebeta-sortedportfolios.
PanelB of Table(1) reportsthesummarystatisticsof stockssortedby downsidebeta, E�F .The columnslabeled‘ E ’ and ‘ E F ’ list the post-formationunconditionalbetasanddownside
betas,respectively. Portfolioswith higherpastdownsidebetahave higherunconditionalbetas.
Sorting on past EQF also produceslarge ex-post formationspreadsin downsidebeta,that is,
downsidebeta is also persistent. There is a weakly increasing,but mostly humped-shaped
patternin themeanreturnsof the E�F portfolios. However, thedifferencein the returnsis not
statisticallysignificant.In PanelC,stockssortedonpastE exhibit nospreadin averagereturns.
Hence,while thereappearsto bea weakspreadin theexpectedreturnsacrossdownsidebetas,
upsidebetadoesnotseemto bepriced.
2.3 The Failure of Conditional Beta Measures
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#$����� �c� 47698 �dI 47698 ��OeGH(
var# 47698 �JI 47698 ��OTGH( �?f #$GH(hgji #&GH(d� (4)
wheredownsideandupsidecorrelation(f F #$GH( and
f #&GM(, respectively) aregivenby:f F #$GH( � corr
#$����� �c� 47698 �dI 47698 ��KTGH(and
f #$GH(k�corr#$����� �A� 47698 �JI 47698 ��OLGM(l� (5)
andi F #&GM( and
i #&GM(areratiosof conditionalvolatilities:i F #$GH(.� m #n����� �JI 4�698 �oKTGH(m # 47698 �dI 47698 ��KLGM(
andi #$GH(.� m #n����� �JI 4�698 �oOTGH(m # 47698 �dI 47698 ��OLGM( > (6)
As with thenotationfor downsideandupsidebeta,whenGp� 47698 , we abbreviateto
f F #&G��47698 (.a7f F , f #$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 #$GH(in equation(6) is marketvolatility. Marketvolatility is asymmetricandhigher
afternegativeshocksto expectedreturns.Hence,conditionalonthedownside,thedenominator
of equation(6) is larger fori F thanfor
i . Second,Duffee (1995)finds that cross-sectional
dispersionis asymmetricandidiosyncraticvolatility decreaseswhenthestockmarketfalls. This
makesthe ratio of total volatility to market volatility larger fori
thanfori F . Both of these
effectscontributeto downsidei F #&GM( beinglower than
i #$GH(on average,asPanelsE andF of
Figure(1) demonstrate.
The decreaseofi F #&GM( on the downsiderelative to
i #&GM(on the upsidecounter-actsthe
increaseoff F #$GH( onthedownsiderelative to
f #&GH(ontheupside.Multiplying thepointsin the
conditionalcorrelationline in PanelC togetherwith thecorrespondingpoint in theconditional
volatility ratio line in Panel E producesa relatively flat effect for the conditional betasin
PanelA. Hence,theconditionalbetais relatively flat acrossdownsideandupsidemovements
becausethe increasein correlationson the downsideis mutedby the decreasein conditional
idiosyncraticvolatility andby theincreasein marketvolatility.
2.4 Downside Correlations and Expected Returns
While thecross-sectionalspreadsof expectedreturnsof stockssortedon downsideandupside
betasare small, we now demonstratethat sorting on the correlationcomponentof the beta
producesa large spreadin expectedreturns, in particular for downside correlations. The
conditionalcorrelationsareasymmetricover downsideandupsidemovements,asopposedto
therelatively low asymmetryin conditionalbetasacrossthedownsideandupside.Conditional
correlationsareunaffectedby different idiosyncraticandmarket volatility acrossupsideand
downsidemoves,whereastheseeffectscauseconditionalbetato have little downsideversus
8
upsideasymmetry. Sortingontheothercomponentof beta,theratioof total to marketvolatility,
producesno patternin expectedreturns.Hence,conditionalcorrelationsmaybebetterableto
captureasymmetriesin risk thanconditionalbetas.
Table(2) listsmonthlysummarystatisticsof theportfoliossortedbyf F and
f . Wechoose
thesameconditioninglevel,GP� 47698 , asthesortsfor theconditionalbetasin Table(1). Panel
A of Table(2) containstheresultsfor stockssortedon pastf F . Thefirst columnlists themean
monthlyholdingperiodreturnsof eachdecileportfolio. Stockswith thehighestpastdownside
correlationshave thehighestreturns.Goingfrom theportfolio of lowestdownsidecorrelations
(portfolio 1), to theportfolio of highestdownsidecorrelations(portfolio 10), theaveragereturn
almostmonotonicallyincreases.The returndifferentialbetweenthe portfolios of the highest
decilef F stocksandthelowestdecile
f F stocksis 4.91%perannum(0.40%permonth).This
differenceis statisticallysignificantat the 5% level (t-stat= 2.26),usingNewey-West(1987)
standarderrorswith 3 lags.
Theportfolio of stockswith thehighestpastdownsidecorrelationshave thehighestbetas.
Sincethe CAPM predictsthat high betaassetshave high expectedreturns,we investigatein
Section3 if the high returnsof theseportfolios areexplainedby the market betas.However,
the high returnson theseportfolios do not appearto be attributableto the sizeeffect or the
book-to-market effect. Thecolumnslabeled“Size” and“B/M” show thathighf F stockstend
to be large stocksandgrowth stocks. Sizeandbook-to-market effectswould predicthighf F
stocksto have low returnsratherthanhigh returns.Thef F portfoliosarealsoflat in leverage,
soleveragealsocannotbedriving thepatternin expectedreturns.
Wealsocontrolfor thesizeandbook-to-marketeffectusingtheFama-French(1993)three-
factormodel.We take thetime-seriesalphasfrom aregressionof af F decile’sexcessportfolio
returnsontoMKT, SMB andHML factors:���v�w�7xH�y1{z|� 47698 �}1r~@��W 47� �}1r����� 47� �y1r������> (7)
Thesealphasarereportedin the columnlabeled‘FF � ’ of Table(2), andthey maintaintheir
nearlymonotonicrankings. The differencein the Fama-Frenchalphasbetweenthe decile10
portfolio andthe decile1 portfolio is 0.53%per month,or 6.55%per annumwith a p-value
0.00. Hence,thevariationin downsiderisk in thef F portfolios is not explainedby theFama-
Frenchmodel. In fact,controllingfor themarket, thesizefactorandthebook-to-market factor
increasesthedifferencesin thereturnsfrom 4.91%to 6.55%perannum.
Thesecondto the lastcolumncalculatesthepost-formationconditionaldownsidecorrela-
tion of eachdecileportfolio. Thesepost-formationperiodf F aremonotonicallyincreasing,
which indicates that the top decile portfolio, formed by taking stocks with the highest
9
conditionaldownsidecorrelationover thepastyearat thedaily frequency, is theportfolio with
thehighestdownsidecorrelationover thewholesampleat themonthlyfrequency. This implies
thatusingpastf F is agoodpredictorof future
f F andthatdownsidecorrelationsarepersistent.
The last column lists the downsidebetas, E�F , of eachdecile portfolio. The E�F column
showsthatthef F portfolioshavea fairly flat E F pattern.Hence,thespreadin expectedreturns
of thef F portfolios is not dueto E�F . We contrastthis with the ex-post formation
f F of theE�F portfolios in PanelB of Table(1). The E�F portfolioshave a slightly humpedshapepattern
(increasingandthendecreasing)of expectedreturns.Thef F statisticsof the E F portfolioshave
thesamehumpedshapepattern.
PanelB of Table(2) shows the summarystatisticsof stockssortedbyf
. In contrastto
the stockssortedbyf F , thereis no discernablepatternbetweenthe meanreturnsandupside
correlations.However, thepatternsin the E ’s,marketcapitalizations,andbook-to-marketratios
of stockssortedbyf
are similar to the patternsfound inf F sorts. In particular, high
f stocksalsotendto havehigherbetas,tendto belargestocks,andtendto begrowth stocks.The
last two columnslist thepost-formationf
and E statistics.Here,bothf
and E increase
monotonicallyfrom decile1 to 10,but portfolio sortsbyf
do notproduceany patternin their
expectedreturns.
In summary, Table (2) shows that assetswith higher downsidecorrelationshave higher
returns.Thedifferencebetweenthefirst andtenthdecileof raw returnsis 4.91%perannum,but
this increasesto 6.55%perannumcontrolling for theFama-French(1993)factors.Downside
betadoesnotpick up thisspreadin expectedreturnsbecauseit exhibits little asymmetryacross
downside and upsidemovements. In contrastto the strong relationshipbetweenexpected
returnsanddownsidemoments,expectedreturnsdonotseemto berelatedto upsideconditional
moments( E orf
).
3 Pricing the Downside Correlation Portfolios
In this sectionwe conducta batteryof teststo try to price the downsidecorrelationeffect.
Section3.1examinesif theexpectedreturnsondownsidecorrelationportfolioscanbeexplained
by themarket beta.Section3.2examinesif theseportfolio returnscanbeexplainedby thesize
effect,thebook-to-marketeffect,or themomentumeffect. Section3.3asksif thehighdownside
correlationexpectedreturnsare robust acrossvarioussubsamples.We show that downside
correlationis not mechanicallylinkedto pastreturnsnor relatedto periodsof low liquidity in
Sections3.4and3.5,respectively. Section3.6 interpretsour findings.
10
3.1 Can Market Risk Price Downside Correlation?
In Table(2), the downsidecorrelationportfolios display increasingbetaswith increasingf F .
This raisestheconcernthat theexpectedreturnson thedownsidecorrelationportfolioscanbe
explainedby themarket beta.To allay suchconcerns,we work with twentyportfoliosformed
in the following manner. Stocksarefirst sortedinto two groups(high betaversuslow beta)
accordingto their pastbetasover the pastyear at the daily frequency. Eachgroup consists
of onehalf of all firms. Then,within eachbetagroup,we rankstocksbasedon theirf F , also
computedusingdaily dataoverthepastyearinto decileportfolios.Thisgivesus2 ( E ) g 10(f F )
portfolios. Whenwe averageacrossthebetaportfolios for eachf F decile,we find thespread
inf F controlling for the betaeffect in ten decileportfolios. Thesedecilebeta-controlled
f Fportfolioshave near-flat uniform ex-postformationbetas,which indicatesthat thedouble-sort
is successfulat controllingfor beta.
Thedifferencebetweenthetenthandthefirst decilebeta-controlledf F portfoliosis 0.56%
permonth,or 6.95%perannum,with a p-valueof 0.00.A Gibbons,RossandShanken(1989)
(GRS)F-testto jointly testif the portfolio alphasarezerorejectswith a p-valueof 0.00. We
alsoconsiderCarhart(1997)four-factormodel,whichconsistsof theFama-Frenchfactorsplus
amomentumfactor, WML:���v�w�7xH��1rz|� 4�698 ��1e~��nW 47� �y1e���n� 47� ��1��h�n% 47� �}1{���v��> (8)
Similar to the Fama-Frenchtime-seriesregressions,the alphasare still significant, with a
differenceof 0.44%per month, or 5.40%per annum,betweenthe tenth and the first decile
portfolios. A GRS test also rejectswith a p-value of 0.00. Since the effect of downside
correlationremainsaftercontrollingfor themarketbeta,weconcludethatdownsidecorrelation
cannotbeexplainedby market risk.
3.2 Can Other Risk Factors Price Downside Correlation?
While wecancontrolfor theeffectof marketbetaby formingadditionaldouble-sortportfolios,
this strategy quickly becomesdifficult to implementif we try to control for the effects of
multiple factors.In this section,we run Fama-MacBeth(1973)cross-sectionaltests(described
in theAppendix)to testif multiplerisk factorscanpricethespreadin expectedreturnsproduced
by downsidecorrelation. In particular, we focuson the standardFama-French(1993)model,
andaugmentthis with theCarhart(1997)WML momentumfactor.
To run thesetests,we work with the 20f F portfoliosdescribedin theprevious sectionto
controlfor thebetaeffect. Table(3) reportstheFama-MacBethcross-sectionalestimatesof the
11
Fama-French(1993)factorpremiums:�"#$�����&(����}�o1e�}Y[Z]\�z|��1e�}��Y[��~@��1r���]Y��y���A�(9)
where���
is thepremiumof factor � . WealsoreporttheCarhart(1997)modelfactorpremiums:�"#$���v�&(.�7����1r��Y[Z]\�z|��1e�}��Y[��~@��1r���]Y��y����1r���PY����h�,>(10)
In both cases,the factor premiumsare all statistically insignificantand the size premiumis
estimatedto benegative. Theseresultsaredrivenby the inability of thesestandardfactorsto
pricethef F portfolios.Werejectthattheportfolio alphasarejointly zerousingaGRStest.
In Figure(2), we plot the alphasandfactor loadingsfrom the Carhartfour-factormodel.
Figure(2) ordersthe20portfoliossothatportfolios1-10correspondto thelow betagroup,and
portfolios11-20correspondto thehighbetagroup.Theportfolio alphasin thetoppanelreflect
thespreadin expectedreturnsmoving from lowf F to high
f F , but aremuchmorepronounced
in thehigh betagroup. The factorloadingsfor MKT in thebottompanelreflectthe low-high
betasorting,andarelargely flat within eachbetagroup. The factorloadingsfor SMB go the
wrongway, sothathighf F firmswith high returnshavesmallSMB loadings.TheHML factor
alsodoesnotaccountfor thef F effect,sincetheHML factorloadingsincreasein thehighbeta
groupandareflat acrossthelow betagroup.Finally, theWML factorloadingsarelargely flat
andverysmall.
3.3 Is the Downside Correlation Effect Stable Across Subsamples?
A simplerobustnessexerciseis to checkif thespreadin thef F correlationportfolios remains
statisticallysignificantin varioussubsamples.Thetop panelof Figure(2) alsoplotsthealphas
from the four-factor model from the full sample(Jan1964 to Dec 1999), from Jan1964 to
Dec1981andfrom Jan1982to Dec1999.Thealphashavealmostthesameincreasingpattern
in eachsubsample.Hence,thesamequalitative relationshipbetweenf F andexpectedreturns
holdsin differentsubperiods.
To conducta more formal test for stability over different subsamples,we computethe
differencebetweenthe alphasof the tenth and the first decile beta-controlledf F portfolios,
which arereportedin Table(4). The differencein the alphasbetweenportfolio 10 and1 are
statisticallysignificantat the1%level for boththethree-factorandthefour-factormodelsacross
thewholesampleperiod.For theFama-Frenchmodelalphasin thefirst column,thealphasare
still largeandstatisticallysignificantwhenthesampleis split into two separatecalendarperiods,
andremainsignificantwhenthesampleis split into NBER expansionsandrecessions.
12
In thelasttwo columnsof Table(4) we reportthealphasfrom thefour-factormodel.These
alphasareslightly smallerthan the alphasfrom the Fama-Frenchmodelandarealsohighly
significantacrossNBER expansionsandrecessions.However, whenthe sampleis split into
two calendarperiods,thedifferencein thealphasis near-significant(p-value= 0.06)over Jan
1964- Dec1981,but is highly statisticallysignificantover thesecondperiod(Jan1982- Dec
1999).Nevertheless,thealphasarestill of alargemagnitudeacrossvarioussubsamples.A more
seriousconcernis that sincethealphasincluding theWML factoraresmallerthanthealphas
from the Fama-Frenchmodel, this raisesthe questionthat a large part of the high expected
returnsinducedby highdownsidecorrelationsmaybedueto momentum.
3.4 Is Downside Correlation Capturing Past Returns?
Thereare several similiarities betweenthe Jegadeesh-Titman (1993) momentumeffect and
downsiderisk, which raisesthe concernthat downsidecorrelationis merelya noisy measure
of pastreturns.First, like momentum,thedownsidecorrelationalphasareexacerbatedby size
andvalueeffects (FamaandFrench,1996; Grundy andMartin, 2001). Second,controlling
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
thealphasbecomeshighly significant(t-stat= 2.50)overthisfirst subsampleperiod(Jan1964-
Dec1981).Hence,aftercontrollingfor momentum,highdownsidecorrelationstocksstill have
high returns.
13
3.5 Is Downside Correlation Liquidity?
A numberof studiesfind that liquidity of the market driesup during down markets. Pastor
andStambaugh(2001)constructanaggregateliquidity measurewhich usessignedorderflow,
andfind that their liquidity measurespikesdownwardsduring periodsof extremedownward
moves, such as during the October 1987 crash, and during the OPEC oil crisis. Jones
(2001)alsofind that thebid-askspreadsincreasewith market downturns,while Chordia,Roll
andSubrahmanyam (2000) find a positive associationat a daily frequency betweenmarket-
wide liquidity and market returns. Thesedown markets, which seemto be correlatedwith
systematicallylow liquidity, are preciselythe periodswhich downsiderisk-averseinvestors
dislike.
To studytherelationbetweendownsiderisk andliquidity, we follow PastorandStambaugh
(2001) and reconstructtheir aggregate liquidity measure,� , detailedin Appendix A. After
constructingtheliquidity measure,we assigna historicalliquidity beta,E � , at eachmonth,for
eachstocklisted on NYSE, AMEX andNASDAQ. This is doneusingmonthly dataover the
previous5 yearsfrom thefollowing regression:��������xH��1 E � � ��1{zJ� 47698 �}1e~���W 4�� �}1r�y��� 47� �}1{���v��� (11)
where� � is theaggregateliquidity measure.
Sinceeachstock � in our samplehasa downsidecorrelation(f F��� � ) anda liquidity beta( E ���� � )
for eachmonth,we canexaminetheunconditionalrelationsbetweenthe two measures.First,
wecomputethecross-sectionalcorrelationbetweenf F��� � and E ���� � ateachtime � , andthenaverage
over time to obtaintheaveragecross-sectionalcorrelationbetweendownsiderisk andliquidity.
Theaveragecross-sectionalcorrelationis –0.0108,whichis closeto zero.Weobtaintheaverage
time-seriescorrelationbetweenf F��� � and E ���� � by computingthe correlationbetweenthesetwo
variablesfor eachfirm acrosstime, andthenaveragingacrossfirms. The averagetime-series
correlationis -0.0029,whichis alsoalmostzero.Hence,ourmeasureof downsiderisk is almost
orthogonalto PastorandStambaugh’smeasureof aggregateliquidity risk.
To further investigatethe relation betweendownside risk and aggregate liquidity, we
perform a� g �
doublesort basedon liquidity andf F . At eachmonth, we independently
sortstocksinto two quintile groupsbasedon E � andf F . Theintersectionof thesetwo quintile
groupsforms25portfoliossortedby E � andf F . Wetaketheaverageof the
f F portfoliosacrossE � quintiles,andreporttheinterceptcoefficientsfrom aFama-French(1993)factortime-series
regressionof theseaverageportfoliosin PanelB of Table(5).
Weobserveasimilarpatternin theaveragereturnsmoving from lowf F tohigh
f F portfolios
asin Table(2). Thereis negative mispricingin the lowf F portfoliosandpositive mispricing
14
in the highf F portfolio. The differencebetween
x��andx��
is 0.26% per month, which is
statisticallysignificantat the 1% level. Hence,even after controlling for liquidity risk using
thePastor-Stambaughliquidity measure,thereremainssignificantmispricingof downsiderisk
relative to theFama-Frenchthreefactormodel.
3.6 Is Downside Correlation Risk?
Sincethemean-varianceframework is rejectedby variousassetpricingtests,it is notsurprising
thathigher-ordermomentsplayarole in explainingcross-sectionalvariationsin expectedstock
returns.However, which higher-ordermomentsareimportantfor cross-sectionalpricing is still
a subjectof debate.We have shown that portfolios sortedby downsidecorrelationsproduce
large spreadsin expectedreturnswhich cannotbe explainedby the market beta,the Fama-
French(1993)SMB andHML factors,by momentum,or by liquidity. Thespreadin returnsis
alsorobustacrosssubsamples.
One puzzling result is why high downside correlationstocksexhibit significantly high
variationsin expectedreturns,while thedifferencein expectedreturnsbetweenstockswith high
downsidebetaversuslow downsidebetais weak.Downsidecorrelationis scaledto emphasize
comovementsin only direction,while downsidebetameasuresboth magnitudeanddirection.
Weacknowledgethatit is hardto think of amodelwherethemagnitudedoesnotmatterbut only
thedirectiondoes.Evenin modelswith one-sidedconstraints,for examplebindingshort-sales
constraints(Chen,HongandStein,2001)or wealthconstraints(Kyle andXiong, 2001),there
shouldbebothdirectionandmagnitudeeffects.
However, downsidecorrelationis the componentof downsidebetawhich is unaffected
by changesin idiosyncraticand market volatilities acrossdownsideand upsidemovements.
Conditional correlationsstrongly differ acrossup and down markets. However, in down
markets, idiosyncraticstock volatility decreaseswhile market volatility increases(Duffee,
1995). This meansthat the ratio of total volatility to market volatility decreasesin down
markets, which causesthe betasto have little variationsacrossdownsideor upsidemoves
of the market. Conditionalcorrelationsareunaffectedby changingidiosyncraticandmarket
volatilities andthey aremuchmoreasymmetricacrossmarket downsideandupsideperiods.
Hence,conditionalcorrelationsareagoodstatisticto measureasymmetriesin risk.
Lacking an economicmodel, we are reluctant to say that the high expectedreturns
commandedby high downsidecorrelationportfolios are due to risk. However, the standard
risk factorscannotprice,or evenexacerbate,theexpectedreturnsof thedownsidecorrelation
portfolios. In order to summarizethis effect in a model, we capturethe spreadin returns
15
producedby downside correlationsby constructinga factor that mimicks this downside
correlationeffect. This factorshouldbe ableto price the downsidecorrelationportfolios (by
construction)andmayalsohelpexplain othervariationsin thecross-sectionof expectedstock
returns.
4 A Downside Correlation Factor
In this section,we build a factorthat reflectsthe high expectedreturnsearnedby stockswith
highdownsidecorrelations.Wedescribetheconstructionof this factorin Section4.1,andshow
that it pricesthe downsidecorrelationportfolios in Section4.2. Section4.3 shows that the
downsidecorrelationfactorsignificantlypredictseconomicrecessions.
4.1 Constructing the Downside Correlation Factor
We constructa downsidecorrelationfactor that capturesthe returnpremiumbetweenstocks
with high downsidecorrelationsandstockswith low downsidecorrelations,which we call the
CMC factorfor “high CorrelationsMinus low Correlations”.TheCMC factorgoeslongstocks
with high downsidecorrelations,which have high expectedreturns,andshortsstockswith low
downsidecorrelations,whichhave low expectedreturns.
In constructingtheCMC factor, we arecarefulto control for thepositive relationshown in
Table(2) betweenthebetaandthedownsidecorrelation.TheCMC factorextractsthespread
in expectedreturnsdueto downsidecorrelation,controllingfor thebeta.Eachmonth,weplace
half of thestocksbasedon their E ’s into a low E groupandtheotherhalf into a high E group.
Then,within eachE group,werankstocksbasedontheirf F into threegroups:a low, amedium
anda highf F groupswith thecutoffs at 33.3%and66.7%.This sortingprocedurecreatessix
portfoliosin total.
We calculatemonthly value-weightedreturnsfor eachof these6 portfolios. Within the
low E group,theportfolio returnsincreasefrom the lowf F portfolio to thehigh
f F portfolio,
with an annualizeddifferenceof 2.40%(0.20%per month). Moving acrossthe low E group,
portfolio returnsof thef F portfolios increase,while thebetaremainsflat at aroundE = 0.66.
Thereturnalsoincreaseswith increasingf F within thehigh E group.Within thehigh E group,
thedifferencein returnsof thehighf F andlow
f F portfolios is 3.24%perannum(0.27%per
month),with a t-statisticof 1.98,but the E decreaseswith increasingf F . Therefore,thehigher
returnsassociatedwith portfolioswith highf F arenot rewardsfor bearinghighermarket risk,
but arerewardsfor bearinghigherdownsidecorrelation.
16
For eachf F group,we take the simpleaverageacrossthe two E groupsandcreatethree
portfolios, which we call the E -balancedf F portfolios. Moving acrossthe E -balanced
f Fportfolios, meanreturnsmonotonicallyincreasewith
f F . This increaseis accompaniedby a
monotonicdecrease,ratherthananincrease,in beta.Wedefineourdownsiderisk factor, CMC,
asthereturnonazero-coststrategy of goinglongthe E -balancedhighf F portfolio andshorting
the E -balancedlowf F portfolio. Thisstrategy is rebalancedmonthly. Thereturnonthisstrategy
is 2.80%perannum(0.23%permonth)with a t-statisticof 2.35andap-valueof 0.02.
Sincewe includeeveryfirm listedon NYSE/AMEX andNASDAQ, andusedaily data,the
impactof small illiquid firms might be a concern.We addressthis issuein two ways. First,
all of our portfoliosarevalue-weighted,which reducestheinfluenceof smallerfirms. Second,
we performthe samesortingprocedureasabove, but excludefirms that aresmallerthanthe
tenthNYSE percentile.With this alternative procedure,we find thatCMC is still statistically
significantwith anaveragemonthlyreturnof 0.23%andat-statisticof 2.04.Thesechecksshow
thatour resultsarenotbiasedby smallfirms.
Table (6) lists the summarystatisticsfor the CMC factor in comparisonto the market,
SMB and HML factorsof Famaand French(1993), the SKS coskewnessfactor of Harvey
andSiddique(2000),andtheWML momentumfactorfrom Carhart(1997). TheCMC factor
hasamonthlymeanreturnof 0.23%,which is higherthanthemeanreturnof SMB (0.19%per
month)andapproximatelytwo-thirdsof the meanreturnof HML (0.32%per month). While
thereturnsonCMC andHML arestatisticallysignificantat the5% confidencelevel, thereturn
on SMB is not statisticallysignificant.CMC hasa monthlyvolatility of 2.06%,which is lower
thanthevolatilitiesof SMB (2.93%)andHML (2.65%).CMC alsohascloseto zeroskewness,
andit is lessautocorrelated(10%) thanthe Fama-Frenchfactors(17% for SMB and20% for
HML). The Harvey-SiddiqueSKS factorhasa small averagereturnper month (0.10%)and
is not statisticallysignificant.In contrast,theWML factorhasthehighestaveragereturn,over
0.90%permonth.However, unliketheotherfactors,WML is constructedusingequal-weighted
portfolios,ratherthanvalue-weightedportfolios.
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%).Thecorrelation
matrix shows that SKS and CMC have a correlationof –3%, suggestingthat asymmetric
downsidecorrelationrisk hasa differenteffect thanskewnessrisk.
Table(6) shows thatCMC is quitenegatively correlatedwith SMB (–64%). To allay fears
that CMC is not merely reflectingthe inverseof the size effect, we examinethe individual
17
firm compositionof CMC andSMB. On average,3660firms areusedto constructSMB each
month,of which SMB is long 2755firms andshort905firms.2 We find thattheoverlapof the
firms,thatSMB is goinglongandCMC is goingshort,constitutes,onaverage,only 27%of the
total compositionof SMB. Thus,theindividual firm compositionsof SMB andCMC arequite
different.Wefind thatthehighnegativecorrelationbetweenthetwo factorsstemsfrom thefact
thatSMB performspoorly in thelate80’sandthe90’s,while CMC performsstronglyover this
period.
To besurethatCMC is notmerelyreflectingtheinformationalreadycapturedby themarket,
SMB,HML andWML, PanelC of Table(6) regressesCMC onthesefour factorsandaconstant.
TheCMC factorloadsnegatively on SMB andHML, which is consistentwith thecorrelation
patternsin PanelB andthefactthathighdownsidecorrelationstockstendto belargestocksand
growth stocks.Theloadingof CMC on WML is very small (0.07)comparedto themagnitude
of theSMB (-0.44)andHML (-0.21)loadings.Netof theseloadings,theintercepttermremains
positiveandsignificant,which indicatesthatCMC is notexplainedby theotherfactors.In fact,
the meanreturn left unexplainedincreasesto 0.33%per month,comparedto the unadjusted
meanreturnof CMC of 0.23%permonth.
4.2 Pricing the Downside Correlation Portfolios
The 20 downside correlationportfolios examinedin Section2 cannotbe explained by the
market,SMB,HML andWML factors.OurCMC factoroughtto explainthevariationsof these
downsidecorrelationportfolios,sinceby constructionit reflectsthespreadin expectedreturns
dueto cross-sectionalvariationinf F . Table(7) revisits theFama-MacBeth(1973)regression
testsin Section2 to seeif CMC is successfulin pricing thedownsidecorrelationportfolios.
Model A of Table(7) is the traditionalCAPM augmentedwith theCMC. Theestimateof
thepremiumon CMC is positiveandstatisticallysignificant.Moreover, theGRSF-testcannot
rejectthehypothesisthatthemarket factoraugmentedwith CMC canpricethevariationsin the
downsidecorrelationportfolio returns.ThepremiumonCMC continuesto remainpositiveand
significantafteraddingthe two Fama-Frenchfactorsin Model B. It is alsoapproximatelythe
sameasthemeanCMC premiumin Table(6). TheGRStestsuggeststhatsomeof theportfolio
returnsarenot explainedby this model,but this result is only weakly significant(p-value=
0.04). Furthermore,in thepresenceof themarket, theFama-Frenchfactors,andCMC factor,
WML still doesnot affect the significanceof CMC. The GRS test suggeststhat this model,2 SMB is long morefirms thanit is shortsincethebreakpointsaredeterminedusingmarket capitalizationsof
NYSEfirms,eventhoughtheportfolio formationusesall NYSE,AMEX andNASDAQ firms.
18
which incorporatesCMC, explainsthevariationsin returnsof downsidecorrelationportfolios.
In short,CMC successfullypricesthedownsidecorrelationeffect.
4.3 Forecasting Macroeconomic Variables
We briefly explore the relation betweendownside correlation and the businesscycle by
investigatinghow thedownsidecorrelationfactorcovarieswith andmacroeconomicvariables.
The investigationin this sectionshouldbe regardedasan exploratoryexercise,ratherthanas
a formal test of the underlyingeconomicdeterminantsof downsiderisk. Our analysishere
is motivatedby studiessuchasLiew andVassalou(2000),who show that otherreturn-based
factors,suchastheFama-French(1993)SMB andHML factors,canpredictGDPgrowth and
hencemayreflectsystematicrisk.
We considersix macroeconomicvariableswhich reflectunderlyingeconomicactivity and
businessconditions. Our first two variablesare leadingindicatorsof economicactivity: the
growth ratein theindex of leadingeconomicindicators(LEI) andthegrowth ratein theindex of
HelpWantedAdvertisingin Newspapers(HELP).Wealsousethegrowth rateof total industrial
production(IP). Thenext threevariablesmeasurepriceandtermstructureconditions:theCPI
inflation rate,the level of the Fedfundsrate(FED) andthe term spreadbetweenthe 10-year
T-bondsandthe3-monthsT-bills (TERM). All growth rates(includinginflation) arecomputed
asthedifferencein logsof theindex at times � and � UL/ = , where� is monthly.
To examinethe connectionbetweendownsiderisk andmacroeconomicvariables,we run
two setsof regressions. The first set regressesCMC on laggedmacrovariables,while the
secondsetregressesmacroeconomicvariableson laggedCMC. Thefirst setof regressionsare
of theform: � 4 � ���?x[1 D �C¡ � z|� 4�¢ ��£[¤ � F ��1 D �¥¡ �o¦ � � 4 � � F ��1r��� (12)
whereweusevariousmacroeconomicvariablesfor 4�¢ �P£[¤ � .PanelA of Table(8) lists theregressionresultsfrom equation(12). Thereis no significant
relationbetweenlaggedmacroeconomicvariablesandtheCMC factor, exceptfor thefirst lag
of LEI, which is significantlynegatively relatedwith CMC. A 1% increasein thegrowth rate
of LEI predictsa 27 basispoint decreasein the premiumof the downsidecorrelationfactor.
However, thep-valuefor the joint test(in the lastcolumnof Table(8)) thatall laggedLEI are
equalto zerofails to rejectthenull with p-value=0.09.Overall,with theexceptionof LEI, there
is little evidenceof predictivepowerby macroeconomicvariablesto forecastCMC returns.
19
To exploreif thedownsidecorrelationfactorpredictsfuturemovementsof macroeconomic
variableswe run regressionsof theform:
4�¢ ��£[¤ �§��x[1 D �¥¡ � zJ� � 4 � � F ��1 D �¥¡ �o¦ � 4?¢ �P£[¤ � F �y1{����> (13)
We alsoincludelaggedmacroeconomicvariablesin theright handsideof theregressionsince
mostof themacroeconomicvariablesarehighly autocorrelated.PanelB of Table(8) lists the
regressionresultsof equation(13). We reportonly thecoefficientson laggedCMC. While the
macroeconomicvariablesprovide little forecastingpower for CMC, theCMC factorhassome
forecastingability for futuremacroeconomicvariables.In particular, highCMC forecastslower
futureeconomicactivity (HELP, p-value= 0.00;IP, p-value= 0.03),lower futureinterestrates
(FED, p-value= 0.01) and lower future term spreads(TERM, p-value= 0.03), wherethe p-
valuesrefer to a joint testthat thethreecoefficientson laggedCMC in equation(13) areequal
to zero.
In general, these results show that high CMC forecastseconomic downturns. The
predictionsof high CMC and future low economicactivity is seendirectly in the negative
coefficients for HELP and IP. Term spreadsalso tend to be lower in economicrecessions.
Estimatesof Taylor (1993)-typepolicy ruleson the FED over long samples,wherethe FED
rateis a linear functionof inflation andrealactivity, show shortratesto belower whenoutput
is low. Hence,thepositivecorrelationof high CMC with futurelow HELP, low IP, low TERM
andlow FED shows thathigh CMC forecastseconomicdownturns.In otherwords,thereward
for holdingstockswith highdownsiderisk is greaterwhenfutureeconomicprospectsturnsour,
perhapsbecausetheincidenceof extrememarket downsidemovesincreasesduringrecessions.
5 An Application to Pricing the Momentum Effect
That a CMC factor, constructedfrom thef F portfolios, explainsthe cross-sectionalvariation
acrossf F portfoliosis nosurprise.Indeed,wewouldbeconcernedif theCMC factorcouldnot
price thef F portfolios. As an applicationof the CMC factor, we demonstratethat CMC has
explanatorypower to accountfor someof themomentumeffect.
5.1 Data and Motivation
Why might we expectCMC to help explain someof the momentumeffect? We first present
evidencethat momentumstrategies tend to performpoorly when the market makesextreme
20
downward moves. To show this, we work with Jegadeeshand Titman (1993)’s momentum
portfolios,correspondingto the ¨ =6 monthformationperiod,which is standardin theliterature
(Chordiaand Shivakumar, 2001). After stocksare sortedinto decilesbasedon their past6
month returns,they are held for the next 6 monthsholding periods,where 6 = 3, 6, 9 or
12. We form anequal-weightedportfolio within eachdecileandcalculateoverlappingholding
periodreturnsfor thenext 6 months.
Figure(3) plotstheaveragereturnsof the40portfoliossortedonpast6 monthsreturns.The
averagereturnsareshown with *’ s. Thereare10 portfolioscorrespondingto eachof the 6 =3,
6, 9 and12 monthsholdingperiods.Figure(3) shows averagereturnsto be increasingacross
thedeciles(from losersto winners)andareroughly thesamefor eachholdingperiod 6 . The
differencesin returnsbetweenthewinnerportfolio (decile10) andtheloserportfolio (decile1)
are0.54,0.77,0.86and0.68percentpermonth,with correspondingt-statisticsof 1.88,3.00,
3.87and3.22,for 6 =3,6, 9 and12respectively. Hence,thereturndifferencesbetweenwinners
andlosersaresignificantat the1%level exceptthemomentumstrategy correspondingto 6 =3.
Figure(3) alsoshowsthef F of themomentumportfolios,which increasegoingfrom thelosers
to thewinners,exceptat thehighestwinnerdecile.Hence,themomentumstrategiesgenerally
haveapositiverelationwith downsidecorrelationexposure.
PanelA of Table(9) shows the exposureof momentumstrategiesto downsiderisk. This
tableshows theFama-Frenchalphasfor the losers(first decile)andwinners(tenthdecile)for
eachholdingperiod 6 . Theexposureof themomentumstrategiesto downsiderisk is revealed
by comparingthealphasfrom thefull sampleto thealphasfrom asubsamplewherethemarket
experiencesextremedownwardmoves. An extremedownwardmove is definedto be a move
thatis morenegativethantwo standarddeviationsbelow theunconditionalmean.Thereare41
suchobservationsoutof 432totalmonths.
Duringthefull sample,theFama-Frenchalphasfor winnersarehigherthanlosers.However,
duringperiodsof marketdistress,thispatternis reversedsowinnersperformworsethanlosers.
The very few observationsduring theseextremedown periodsmeansthat we shouldbe very
reluctantto concludethatwinnershave higherdownsiderisk thanlosers,especiallygiventhe
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.
However, the10thdecile(winners)alwaysunderperformsthebottomdecile(losers)acrossall © holdingperiods.
21
correlationstocksto compensatefor their much lower returnswhen the market crashes. If
arbitraguersare averseto downside risk in the momentumstrategies, they would demand
compensationinto order to bearsuchrisk. We hopethat the CMC factormay pick up some
of the downsideexposureof the momentumportfolios. Indeed,Table (9) suggeststhat the
momentumportfoliosandthedownsidecorrelationportfoliosbothreflectdownsiderisk.
PanelC of Table(9) examinestheeconomicreductionin themomentumpremiumby adding
a CMC factor. Model A of Table(9) regressesWML ontoa constant,theMKT andtheCMC
factor. This regressionhasan£ ;
of 12%,andasignificantlypositive loading.TheCMC factor
reducesthemomentumraw return(0.90%permonth)to 0.72%permonth,areductionof 2.18%
perannum.In Model B, theFama-French(1993)WML alphais 1.05%permonth,andadding
theCMC factorreducesthis, in Model C, to 0.86%permonth,which is a reductionof 2.30%
per annum.Hence,while themomentumeffect cannotbe completelyexplainedby downside
correlation,PanelC of Table(9) showsthatCMC hassomeexplanatorypowerfor WML which
theotherfactors(MKT, SMB, HML) donothave.
5.2 Fama-MacBeth (1973) Cross-Sectional Test
We now conductformal cross-sectionalestimationsof therelationbetweendownsiderisk and
expectedreturnsof momentumreturnsin Table(10) with Fama-MacBethtests.Usingdataon
the40 momentumportfolioscorrespondingto the ¨ =6 formationperiod,we first examinethe
Fama-French(1993)modelin Model A. Theestimatesof the risk premiafor SMB andHML
arenegative, which reflectthe fact that the loadingson SMB andHML go thewrongway for
themomentumportfolios.
In comparison,Model B addsCMC asa factor togetherwith the market. The estimated
premiumon CMC is 8.76%perannum(0.73permonth)andstatisticallysignificantat the5%
level. TheseresultsdonotchangewhenSMB andHML areaddedin ModelC. In ModelD, we
augmenttheCarhart(1997)four-factormodel(MKT, SMB, HML andWML) with CMC. The
factorpremiaon WML andCMC arebothsignificant.The fact thatCMC remainssignificant
at the5% level (t-stat=2.01)in thepresenceof WML shows thatCMC is picking up someof
the momentumeffect reflectingdownsiderisk. Moreover, the magnitudeof the CMC factor
loadingremainsrelatively unchangedafteraddingWML.
Figure(4) graphstheloadingsof eachmomentumportfolioonMKT, SMB,HML andCMC.
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
22
(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
downside,while pastloserstocksperformbetterin theseextremeperiods.
5.3 GMM Hypothesis Tests
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
Appendixfor furtherdetails).This testsoverwhelminglyrejects,bothasymptoticallyandwith
small-samplesimulations,the null that the pricing errosare jointly zero for all the models.
Althoughall pricing errorsfor themodelof MKT andCMC fall within thetwo standarderror
bands,theHJ testsrejectthis modelbecausebecausetheHJ distancedoesnot assignanequal
weight to all the portfolios in the test. Hence,while momentumportfolios do seemto have
exposureto downsiderisk, theCMC factoronly modestlyreducesthemomentumpremiumby
2.18%,andalthoughit comesout significantin cross-sectionaltestswith momentumassets,
CMC cannotcompletelyaccountfor all themomentumeffect.
6 Conclusion
We find that stockswith high downsidecorrelationshave higherexpectedreturnsthanstocks
with low downsidecorrelations.Theportfolio of stockswith thegreatestdownsidecorrelations
outperformstheportfolio of stockswith thelowestdownsidecorrelationsby 4.91%perannum.
Downsidecorrelationis distinct from market risk andliquidity risk, andit is not mechanically
linkedto pastreturns.Moreover, controllingfor themarketbeta,thesizeeffectandthebook-to-
marketeffectincreasesthedifferencein thereturnsbetweenthehighestandthelowestdownside
correlationportfoliosto 6.55%perannum.To capturethisasymmetry, weconstructadownside
correlationfactor(CMC) thatgoeslong stockswith high downsidecorrelationsandgoesshort
stockswith low downsidecorrelations.TheCMC factoris pricedby portfoliosof stockssorted
by downsidecorrelations,andexposureto this downsidecorrelationfactorhelpsexplain some
of theprofitability of momentumstrategies.
While mosteconomicmodelswould suggestthatboth the magnitudeandthe directionof
risk oughtto matter, our downsidecorrelationmeasureonly capturesthedirectionof downside
24
comovements,but not the magnitudeof the comovements. The decompositionof the betas
show thattheactionof total andmarket volatilities confoundthemagnitudesof joint downside
movements. In particular, the ratio of total to market volatility decreaseson the downside,
which causesconditionalbetasto exhibit little asymmetryacrossthedownsideandtheupside.
In contrast,conditionalcorrelationsareimmuneto theseeffectsandexhibit highly asymmetric
patternsacrossdownsideandupsidemarketmovements.Hence,conditionalcorrelationsappear
to beacleanermeasureof asymmetricexposureto risk thanconditionalbetas.
While we show thathigh downsidecorrelationstockscommandhigh expectedreturnsthat
cannotbe accountedfor by the standardrisk factors,our empiricalwork leavesunexplained
what underlyingeconomicmechanismscausesomestocksto exhibit greaterdownsiderisk,
and why investorsdemandcompensationfor exposuresto suchrisk. A more difficult task
is capturingthe interactionbetweenidiosyncraticand market volatility acrossup and down
marketswhich causesconditionalbetasto have little asymmetryacrossdownsideandupside
markets, while conditionalcorrelationsexhibit pronouncedasymmetryacrossdownsideand
upsidemarkets. Economieswith frictions andhiddeninformation(Hong andStein,2001)or
with agentsfacing binding wealth constraints(Kyle and Xiong, 2001) have both significant
directionandmagnitudeeffects.Althoughrepresentativeagentmodelswith asymmetricutility
functions,like first-orderrisk aversion(Bekaert,Hodrick andMarshall,1997)or lossaversion
(Barberis,HuangandSantos,2001),havenotbeencalibratedin thecross-section,thesemodels
wouldalsoassigna largerrole to conditionalbetasthanto conditionalcorrelations.
25
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.
The48 industryportfoliosarefrom FamaandFrench(1997)andareobtainedfrom KennethFrench’swebsiteat http://web.mit.edu/kfrench/www/datalibrary.html. The FamaandFrench(1993)factors,SMB andHML, arealsofrom thedatalibrary atKennethFrench’swebsite.
Higher Moment PortfoliosWe constructportfolios basedon correlationsbetweenasset± ’s excessreturn ²)³µ´ andthe market’s excessreturn²�¶ ´ conditional on downsidemoves of the market ( ·�¸ ) and on upsidemoves of the market ( ·@¹ ). We alsoconstuctportfoliosbasedon coskewness,cokurtosis,º , º conditionalon downsidemarket movements( º�¸ ), andº conditionalon upsidemarketmovements( º�¹ ).
Coskewnessis defined,following Harvey andSiddique(2000),as:
coskew » ¼:½ ¾ ³�¿ ´ ¾AÀ¶ ¿ ´�Á ¼:½ ¾ À³�¿ ´ Á ¼]½ ¾ À¶ ¿ ´ Á)à (A-1)
where ¾ ³�¿ ´ »V² ³�¿ ´§ÄÆÅ�³ Ä º ³�Ç ©qÈ ´ , is the residualfrom the regressionof ² ³�¿ ´ on the contemporaneousexcessmarketreturn,and ¾ ¶ ¿ ´ is theresidualfrom theregressionof themarketexcessreturnonaconstant.Similar to thedefinitionof coskewnessin equation(A-1), we definecokurtosisas:
cokurt » ¼:½ ¾ ³�¿ ´ ¾Aɶ ¿ ´�Á ¼]½ ¾ À³�¿ ´ ÁMÊ ¼:½ ¾ À¶ ¿ ´�ÁCËÍÌÎ}Ï (A-2)
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
identify stockslistedon NYSE,AMEX andNASDAQ with at leastfive yearsof monthlyreturns.For eachstock,we estimatea liquidity beta, º�ø³ , by runningthefollowing regressionusingthemostrecentfive yearsof monthlydata: ²�³�¿ ´}»º+ù³ Ø º ø³ Ñ�´ Ø º�ú³ Ç ©qÈH´ Ø º�û³�ü Ç-ý ´ Ø º�þ³2ÿ Ç Ñ}´ Ø ¾ ³¥¿ ´ à (A-6)
where² ³¥¿ ´ denotesasset± ’s excessreturnand Ñ ´ is theinnovationin aggregateliquidity.
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.
27
B Fama-MacBeth and GMM Cross-Sectional Tests
Fama-MacBeth (1973) Cross-Sectional TestsWe considerlinearcross-sectionalregressionalmodelsof theform:¼ î ²)³µ´ ï »�� ù Ø ����ºÍ³ à (B-1)
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.
In thefirst step,we usetheentiresampleto estimatethefactorloadings,º ³ :² ³í´ » Å�³MØ�� �´ º ³ÍØ� d³í´ à Ð�»já à � à Ï�Ï�Ï È Ã (B-2)
whereÅ�³ is ascalarand ��´ is a Ç� á vectorof factors.In thesecondstep,we run a cross-sectionalregressionateachtime Ð over
éportfolios,holdingthe º ³ ’s fixedat their estimatedvalues,
åº ³ , in equation(B-2):²)³µ´}»�� ù Ø ��� åºÍ³ Ø ö ³µ´ à ±y»'á à � à ÏíÏ�Ï é Ï (B-3)
Thefactorpremia,� , areestimatedastheaveragesof thecross-sectionalregressionestimates:å�"» áÈ �ë ´¥ì Õ å� ´ Ï (B-4)
Thecovariancematrix of � , ��� , is estimatedby:å����» áÈ À�ë ´¥ì Õ î å� ´�Ä Û� ï î å� ´�Ä Û� ï � à (B-5)
where Û� is themeanof � .Sincethefactorloadingsareestimatedin thefirst stageandtheseloadingsareusedasindependentvariables
in thesecondstage,thereis anerrors-in-variablesproblem.To remedythis, we useShanken’s (1992)methodtoadjustthestandarderrorsby multiplying
å��� with theadjustmentfactor î á Ø å� � å� ¸�Õ� å� ï ¸�Õ , whereå� � is theestimated
covariancematrixof thefactors� ´ .GMM Cross-Sectional TestsThestandardEulerequationfor a grossreturn, �]³í´ , is givenby:¼ î ç ´ � ³í´ ï »'á Ï (B-6)
Linearfactormodelsassumethatthepricingkernelcanbewrittenasa linearcombinationof factors:ç ´ »�� ù Ø ���Õ ��´ à (B-7)
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:
� ù » á¼ î ç ´ ï » á� ù Ø � �Õ � î � ´ ï Ãthefactorloadings,ºM³ , aregivenby: º ³ » cov î ��´ à � �´ ï ¸+Õ cov î ��´ à � ³µ´ ï Ãandthefactorpremia,� , aregivenby:
�B» Ä á� ù cov î � ´ à � �´ ï � Õ Ï28
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
pricingkernelas �Q» î � ù Ã � �Õ ï � , thesamplepricingerroris:
Ý � î � ï » áÈ�ë ´¥ì Õ î ç ´"�]´ Ä á ï Ï (B-9)
TheGMM estimateof � is thesolutionto:
#%$'&(*) »È Ý ��,+ � Ý � à (B-10)
where+ � is aweightingmatrix.
Hansen-Jagannathan (1997) TestJagannathanandWang(1996)derive theasymptoticdistributionof theHJdistancemetric:ÿ ) ».- Ý � î � ï � ¼:½ �]´/� �´ Á ¸+Õ Ý � î � ï Ã (B-11)
whichcanbeinterpretedastheleast-squaredistancebetweenagivenpricingkernelandtheclosestpoint in thesetof thepricing kernelsthatcanpricethebaseassetscorrectly. Theasymptoticdistribution of È î ÿ ) ï À involvesaweightedsumof î é Ä © Ä á ï10 À Õ statistics.Theweightsarethe
é Ä © Ä á non-zeroeigenvaluesof:
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
2hasexactly
é Ä © Ä á positiveeigenvaluesÖ Õ Ã Ï�Ï�Ï Ã Ö ê ¸HGo¸+Õ . Theasymptoticdistributionof theHJdistancemetricis: È î ÿ ) ï ÀJI ê ¸KGo¸�Õë � Ö � 0 À Õas È IML . 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: 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,
åºM³ , asour factorloadings.We assumethe error termsof the baseassets,¾ ´ ,follow IID multivariatenormaldistributionswith meanzeroandcovariancematrix,å�UT Ä åº � å�UV åº , where
å�UT is thecovariancematrixof theassetsand
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
Portfolio 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 B: Portfolios Sorted on Past º�¸Portfolio Mean Std Auto º º�¸ ·Í¸ WM¸ High–Low t-stat1 Low º�¸ 0.78 4.21 0.16 0.67 0.89 0.71 1.26 0.31 1.042 0.93 3.74 0.14 0.68 0.74 0.73 1.023 0.99 3.71 0.09 0.73 0.82 0.83 0.984 1.09 3.92 0.05 0.80 0.88 0.89 0.995 1.05 4.00 0.06 0.85 0.89 0.91 0.986 1.06 4.52 0.07 0.98 0.98 0.93 1.067 1.11 4.82 0.04 1.04 1.02 0.92 1.118 1.24 5.39 0.05 1.17 1.12 0.92 1.219 1.22 6.26 0.04 1.32 1.30 0.89 1.4610 High º�¸ 1.09 7.81 0.08 1.57 1.52 0.84 1.82
Panel C: Portfolios Sorted on Past º�¹Portfolio Mean Std Auto º º�¹ ·@¹ Wã¹ High–Low t-stat1 Low º�¹ 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�¹ 1.00 6.77 0.06 1.41 1.45 0.80 1.81
Thetablelistssummarystatisticsfor value-weightedº , º�¸ and º�¹ portfoliosatamonthlyfrequency, whereº�¸ and º�¹ aredefinedin equation(3), setting Ö » Ç ©qÈ . For eachmonth,we calculateº ( º�¸ , º�¹ )of all stocksbasedon daily continuouslycompoundedreturnsover the pastyear. We rank the stocksintodeciles(1–10),andcalculatethevalue-weightedsimplepercentagereturnoverthenext month.Werebalancethe portfoliosmonthly. Meansandstandarddeviationsarein percentagetermspermonth. Stddenotesthestandarddeviation (volatility), Auto denotesthefirst autocorrelation,and º is post-formationthebetaof theportfolio. Thecolumnslabeledº�¸ ( º�¹ ) and ·Í¸ ( ·@¹ ) show thepost-formationdownside(upside)betasanddownside(upside)correlationsof theportfolios. ThecolumnlabeledWã¹ ( WM¸ ) lists theratio of thevolatilityof theportfolio to thevolatility of 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.
32
Table2: PortfoliosSortedon ConditionalCorrelations
Panel A: Portfolios Sorted on Past ·Í¸Portfolio Mean Std Auto º FF Å Size B/M Lev ·�¸ º�¸1 Low ·Í¸ 0.77 4.18 0.15 0.69 –0.37 2.61 0.63 4.96 0.74 0.942 0.88 4.34 0.17 0.81 –0.30 2.92 0.62 4.71 0.80 0.973 0.87 4.32 0.15 0.83 –0.31 3.19 0.60 4.88 0.82 0.954 0.94 4.39 0.15 0.87 –0.24 3.46 0.58 4.38 0.83 0.975 0.97 4.39 0.10 0.90 –0.19 3.74 0.56 5.15 0.85 0.956 1.00 4.45 0.09 0.94 –0.16 4.04 0.53 4.49 0.90 1.017 1.00 4.64 0.09 1.00 –0.15 4.39 0.50 7.06 0.92 1.028 1.03 4.58 0.08 1.00 –0.10 4.82 0.48 4.04 0.94 1.059 1.12 4.77 0.02 1.05 0.02 5.36 0.46 5.42 0.96 1.0810 High ·Í¸ 1.17 4.76 0.01 1.04 0.16 6.38 0.39 4.40 0.94 0.97
High - Low 0.40 2.26
Panel B: Portfolios Sorted on Past ·@¹Portfolio Mean Std Auto º FF Å Size B/M Lev ·�¸ º�¸1 Low ·@¹ 1.13 4.56 0.17 0.82 0.02 2.88 0.60 7.52 0.50 0.632 1.05 4.63 0.19 0.90 –0.13 3.08 0.59 6.18 0.63 0.783 1.09 4.61 0.16 0.92 –0.10 3.24 0.58 4.51 0.68 0.824 1.06 4.67 0.15 0.94 –0.13 3.44 0.56 4.57 0.70 0.855 0.99 4.62 0.14 0.95 –0.18 3.66 0.54 4.47 0.76 0.916 1.03 4.62 0.12 0.97 –0.17 3.91 0.54 4.42 0.78 0.907 1.00 4.70 0.09 1.01 –0.18 4.23 0.52 4.84 0.84 0.978 1.11 4.67 0.08 1.01 –0.06 4.65 0.52 4.25 0.85 0.969 1.12 4.63 0.07 1.02 –0.01 5.27 0.48 4.71 0.92 1.0210 High ·@¹ 1.07 4.52 0.00 1.00 0.07 6.65 0.36 4.35 0.95 1.06
High - Low –0.06 –0.38
Thetablelists summarystatisticsof thevalue-weighted·Í¸ and ·�¹ portfoliosat a monthlyfrequency, where· ¸ and · ¹ aredefinedin equation(5), settingÖ2» Ç ©qÈ . For eachmonth,wecalculate· ¸ ( · ¹ ) of all stocksbasedon daily continuouslycompoundedreturnsover thepastyear. We rankthestocksinto deciles(1–10),andcalculatethevalue-weightedsimplepercentagereturnover thenext month.We rebalancetheportfoliosat a monthly frequency. Meansandstandarddeviationsarein percentagetermspermonth. Stddenotesthestandarddeviation (volatility), Auto denotesthefirst autocorrelation,and º is thepost-formationbetaof theportfolio with respectto themarket portfolio. ThecolumnlabeledFF Å lists the Å from a regressionof theexcess· ¸ portfolio returnon the Fama-French(1993)modelat a monthly frequency. At the beginningofeachmonth Ð , we computeeachportfolio’s simpleaveragelog market capitalizationin millions (size)andvalue-weightedbook-to-market ratio (B/M). The columnLev is the simpleaverageof firms leverageratiowhich is definedasthe ratio of bookvalueof assetto book valueof equity. The columnslabeled·Í¸ ( ·�¹ )and º�¸ ( º�¹ ) show the post-formationdownside(upside)correlationsanddownside(upside)betasof theportfolios. High–Low is the meanreturndifferencebetweenportfolio 10 andportfolio 1 and t-statgivesthet-statisticfor this difference.T-statisticsarecomputedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags.Thesampleperiodis from January1964to December1999.
33
Table3: PricingtheDownsideCorrelationPortfolios(1)
FactorPremiums�� ù MKT SMB HML WML GRSTest
Model A: Fama-French Model
Premium( � ) 0.54 0.65 -0.46 0.06 3.24t-stat 2.41C 0.20 -1.78 0.19 p-val=0.00CXCModel B: Fama-French Factors and WML
Premium( � ) 0.49 0.12 -0.43 0.13 0.69 2.31t-stat 2.03C 0.36 -1.60 0.39 0.99 p-val=0.00CXC
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
FF FF + WML10-1 t-stat 10-1 t-stat
Full Sample 0.56 4.73 0.44 3.45
Two SubsamplesJan1964- Dec1981 0.44 2.66 0.32 1.84Jan1982- Dec1999 0.66 3.97 0.52 2.90
NBERBusinessCyclesExpansions 0.44 3.59 0.28 2.10Recessions 1.13 3.55 1.14 3.65
Wereportthedifferencein monthly Å ’s for thetenthandthefirst decilebeta-controlled·�¸ portfolios,with t-statisticscomputedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags.FF denotestheFamaandFrench(1993)model,andFF+WML denotestheFama-Frenchmodelaugmentedwith Carhart(1997)’sWML momentumfactor.
34
Table5: DownsideCorrelationControllingfor PastReturnsandLiquidity
Panel A: Downside Correlation Portfolios Controlling for Past Returnsð ò Ü Y Ð î ð ï Ð î ò ï Ð î Ü ï Ð î Y ï � À1 Low ·�¸ -0.40 0.80 0.69 0.41 -5.00 35.02 18.30 11.58 0.882 -0.40 0.90 0.60 0.34 -5.19 36.09 15.72 7.22 0.913 -0.27 0.99 0.46 0.23 -3.79 42.29 12.37 4.67 0.934 -0.17 1.07 0.32 0.12 -2.49 50.12 8.80 2.60 0.945 High ·�¸ -0.07 1.10 0.11 -0.10 -1.09 59.96 3.77 -2.93 0.95ð[Z - ð Õ = 0.33t-stat= 3.03
Panel B: Downside Correlation Portfolios Controlling for Liquidityð ò Ü Y Ð î ð ï Ð î ò ï Ð î Ü ï Ð î Y ï � À1 Low ·�¸ -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 ·�¸ 0.08 1.07 -0.13 -0.13 1.80 90.83 -7.70 -4.70 0.98ð Z - ð Õ = 0.26t-stat= 2.62
This table shows the time-seriesregressionsof excessreturnson portfolios formed by a doublesort thatcontrols for past returnsand a double sort that controls for liquidity. In Panel A, we first sort stocksinto quintiles by their past6 monthsreturns(momentum). Then, we sort stockswithin eachpastreturnquintilesinto additionalquintilesaccordingto ·�¸ . The intersectionof thesequintilesforms 25 portfolios.The portfolios in Panel A are the averagesacrosspast return quintiles of the portfolios sortedby · ¸ .In Panel B, we sort stocks into quintiles by º�ø and independentlysort stocks into separatequintilesaccordingto ·Í¸ . The intersectionof thesequintiles forms 25 portfolios. The portfolios in Panel B arethe averagesacrossº ø quintilesof the portfolios sortedby · ¸ . In both PanelA andPanelB, we reportthe coefficientsfrom a time-seriesregressionof the portfolio returnsonto the Fama-French(1993)factors:² ³í´ »×ð ³JØPòc³�Ç ©PÈ ´dØPÜ)³ ü Ç-ý]´�Ø\Yͳ ÿ Ç Ñ ´�Ø ¾ ³µ´ . ColumnslabeledÐ î ï show thet-statisticsof theregressioncoefficients computedusing Newey-West (1987) heteroskedastic-robust standarderrorswith 3 lags. Theregression� À is adjustedfor thenumberof degreesof freedom.January1968to December1999. ð Z - ð Õ isthedifferencein thealphasð betweenthe5th quintile andthefirst quintile.
35
Table6: SummaryStatisticsof theFactors
Panel A: Summary Statistics
Factor Mean Std Skew Kurt AutoMKT 0.55C 4.40 –0.51 5.50 0.06SMB 0.19 2.93 0.17 3.84 0.17HML 0.32C 2.65 –0.12 3.93 0.20WML 0.90CBC 3.88 –1.05 7.08 0.00SKS 0.10 2.26 0.69 7.45 0.08CMC 0.23C 2.06 0.04 5.41 0.10
Panel B: Correlation Matrix
MKT SMB HML WML SKS CMCMKT 1.00SMB 0.32 1.00HML –0.40 –0.16 1.00WML 0.00 –0.27 –0.14 1.00SKS 0.13 0.08 0.03 –0.01 1.00CMC –0.16 –0.64 –0.17 0.35 –0.03 1.00
Panel C: Regression of CMC onto Various Factors
Constant MKT SMB HML WMLcoef 0.33 -0.03 -0.44 -0.21 0.07t-stat 4.02CXC -1.47 -11.02CBC -5.84CBC 2.65CXC
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.
36
Table7: PricingtheDownsideCorrelationPortfolios(2)
FactorPremiums�� ù MKT SMB HML WML CMC GRSTest
Model A: MKT and CMC
Premium( � ) 0.62 -0.04 0.23 1.28t-stat 4.08CXC -0.15 2.18C p-val=0.18
Model B: Fama-French Factors and CMC
Premium( � ) 0.57 0.03 -0.36 0.02 0.22 1.65t-stat 2.55C 0.09 -1.08 0.07 2.07C p-val=0.04CModel C: Fama-French Factors, WML, and CMC
Premium( � ) 0.50 0.10 -0.39 0.11 0.64 0.21 1.34t-stat 2.11C 0.29 -1.15 0.33 0.88 2.01C p-val=0.15
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.
37
Table8: MacroeconomicVariablesand� 4 �
Panel A: ] Ç ] ´y»ñð ØF^ ɳ�ì Õ ò ³ Ç 2 ]_�a`:´ ¸ ³ ØF^ ɳ�ì Õ ô ³b] Ç ] ´ ¸ ³ Ø ¾ ´Ç 2 ]_�a`�´ ¸+Õ Ç 2 ]_�_`:´ ¸ À Ç 2 ]_�a`:´ ¸ É JointSigLEI coef –0.27 0.22 0.06 0.09
t-stat –2.27C 1.24 0.60HELP coef –0.00 –0.04 0.05 0.24
t-stat –0.12 –1.26 1.97IP coef –0.13 0.18 –0.02 0.16
t-stat –1.39 1.43 –0.22CPI coef 0.17 –0.03 –0.18 0.64
t-stat 0.43 –0.05 –0.46FED coef 0.22 –0.19 –0.02 0.48
t-stat 1.47 –0.83 –0.12TERM coef 0.10 –0.39 0.26 0.64
t-stat 0.46 –1.11 1.10
Panel B:Ç 2 ]_�a` ´ »×ð Ø ^ ɳ�ì Õ òc³ ] Ç ] ´ ¸ ³MØ ^ ɳíì Õ ô�³�Ç 2 ]_�a` ´ ¸ ³ÍØ ¾ ´] Ç ] ´ ¸�Õ ] Ç ] ´ ¸ À ] Ç ] ´ ¸ É JointSig
LEI coef –0.02 0.02 0.01 0.62t-stat –1.04 0.73 0.31
HELP coef –0.48 0.03 0.18 0.00CXCt-stat –5.34CBC 0.42 1.77
CPI coef –0.01 0.00 –0.01 0.51t-stat –0.84 –0.17 –1.14
IP coef –0.04 –0.06 0.00 0.03Ct-stat –1.77 –2.04C 0.03
FED coef 0.00 –0.03 –0.03 0.01CXCt-stat 0.30 –1.37 –2.42C
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.
38
Table9: Momentum,DownsideCorrelationandWML Returns
Panel A: Momentum Portfolio Å ’s from the Fama-French Model
K=3 K=6 K=9 K=121 L 10 W 1 L 10 W 1 L 10W 1 L 10 W
Full SampleÅ -0.54 0.44 -0.64 0.46 -0.67 0.47 -0.62 0.41t-stat -2.74 2.61 -3.65 2.88 -4.47 3.12 -4.53 2.86
ConditioningonMKT c mean- 2SEÅ -0.69 -1.36 -0.76 -1.32 -0.73 -1.16 -0.60 -1.35t-stat -0.84 -1.52 -1.06 -1.66 -1.23 -1.64 -1.16 -2.41
Panel B: Decile ·Í¸ Portfolio Å ’s from the Fama-French Model
Decile1 2 3 4 5 6 7 8 9 10
Full SampleÅ -0.37 -0.30 -0.31 -0.24 -0.19 -0.16 -0.15 -0.10 0.02 0.16t-stat -3.35 -3.13 -3.56 -2.56 -2.35 -2.07 -2.12 -1.47 0.33 2.80
Conditioningon MKT c mean- 2SEÅ 2.33 1.79 1.19 1.02 0.15 -0.03 0.89 0.63 0.12 -0.39t-stat 2.10 1.84 0.98 0.74 0.14 -0.04 1.34 1.22 0.36 -0.73
Panel C: Regression of WML onto Various Factors
Constant MKT SMB HML CMC Adj � ÀModelA: coef 0.72 0.05 0.68 0.12t-stat 4.19CXC 0.73 4.82CBC
ModelB: coef 1.05 0.02 –0.41 –0.26 0.10t-stat 6.25CXC 0.33 –3.20CXC –2.19C
ModelC: coef 0.86 0.04 –0.19 –0.15 0.47 0.13t-stat 4.87CXC 0.55 –1.18 –1.27 2.81CBC
In PanelsA andB, thetablereportsÅ ’s from a Fama-French(1993)modelfor the40 momentumportfoliosandthedecile ·Í¸ portfolios. The40 momentumportfoliosareformedusinga J=6monthformationperiod,with holdingperiodsof © = 3, 6, 9 or 12months,with 10decileswithin in each© . Thedecile ·Í¸ portfoliosarethesameportfolios in Table(2). Alpha’s from two samplesarereported:over thefull sample,andovera sampleconditionedon themarket return(MKT) beingbelow two standarddeviationsfrom its mean.Thefull sampleperiodis from January1964to December1999. Thereare41 observationswherethemarket islessthantwo standarddeviationsfrom its unconditionalmean,whereboththemeanandstandarddeviationarecomputedusingthefull sample.In PanelC, thetablereportsthetime-seriesregressionof themomentumfactor, WML, onto variousother factors. MKT is the market, SMB and HML are Fama-French(1993)factors,CMC is thedownsiderisk factor, andSKSis theHarvey-Siddique(2000)skewnessfactor. Thet-statis computedusingNewey-West (1987) heteroskedastic-robust standarderrorswith 3 lags. In Panel C, t-statisticsthataresignificantat the5%(1%) level aredenotedwith * (**). Thesampleperiodis from January1964to December1999.
39
Table10: Fama-MacBethRegressionTestsof theMomentumPortfolios
FactorPremiums�� ù MKT SMB HML CMC WML JointSig
Model A: Fama-French Model
Premium( � ) –0.49 2.04 –0.50 –0.98 p-val=0.07t-stat –0.45 1.66 –1.65 –2.17CModel B: Using MKT and CMC
Premium( � ) –0.66 1.98 0.73 p-val=0.03Ct-stat –0.92 2.32C 2.38CModel C: Fama-French Factors and CMC
Premium( � ) –0.65 1.73 –0.11 –0.52 1.02 p-val=0.00CXCt-stat –0.57 1.22 –0.25 –0.81 2.43CModel D: Fama-French Factors, CMC and WML
Premium( � ) –0.24 0.45 0.50 0.64 0.98 0.84 p-val=0.01CXCt-stat –0.19 0.29 1.04 1.08 2.01C 2.74CBC
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 thefirst stagewe estimatethefactorloadingsover thewholesample.Thefactorpremia, � , areestimatedin thesecond-stagecross-sectionalregressions.The lastcolumnof the tablereportsp-valuesfrom 0 À testson the joint significanceof thebetasof eachmodel. AllstatisticsarecomputedusingShanken(1992)standarderrors.T-statisticsthataresignificantat the5% (1%)level aredenotedwith * (**). Thesampleperiodis from January1964to December1999.
40
Figure1: DownsideandUpsideMomentsof IndustryPortfolios
−1.5 −1 −0.5 0 0.5 1 1.5
0.8
1
1.2
1.4
Panel A: Average conditional betas
−1.5 −1 −0.5 0 0.5 1 1.5
0.4
0.5
0.6
0.7
0.8
Panel C: Average conditional correlations
−1.5 −1 −0.5 0 0.5 1 1.51
1.5
2
2.5
3Panel E: Average conditional k’s
0 10 20 30 40 500.5
1
1.5
2Panel B: Ratio of β− to β+
0 10 20 30 40 501
1.5
2
2.5Panel D: Ratio of ρ− to ρ+
0 10 20 30 40 500.6
0.8
1
1.2
1.4Panel F: Ratio of k− to k+
Theleft columnof thefigureshowsupsideanddownsidebetasin PanelA, upsideanddownsidecorrelationsin Panel C and the ratio of upsideanddownsidetotal portfolio volatility to market volatility in Panel E.All of theseareaveragedacrossthe 48 FamaandFrench(1997)industryportfolios. Thegraphin PanelAis constructedas follows (PanelsC andE aresimilar). The figure displaysthe averageº}¸ î Ö ï acrosstheportfolios, where Ö is d standarddeviationsbelow the unconditionalmeanof the market. For example,atd » Ä á , thefigureplotstheaverageindustry º ¸ î Ö�» Ç ©qÈ Ä ü � ú G � ï , whereÇ ©PÈ is theunconditionalmeanof theexcessmarket returnand ü � ú G � is theunconditionalvolatility of theexcessmarket return.AtdÆ»e , the figure plots the averageº�¸ î Ö » Ç ©qÈ ï . Similarly, on the RHS of the d -axis for dgfh , thefigure displaysthe averageº ¹ î Ö ï , for Ö representingd standarddeviationsabove the meanof the market.Thereare two pointsplottedat d-»i representingthe averageº}¸ î Öó» Ç ©PÈ ïkj º�¸ and the averageº�¹ î Ö » Ç ©PÈ ïlj º�¹ . The 95% standarderror boundsshown in dottedlines in PanelsA, C, andE areproducedby bootstrapwith 10,000simulations.PanelsB, D andF, in theright columnof thefigure,reporttheratioof º�¸ õ º�¹ , (PanelB), theratioof ·Í¸ õ ·@¹ (PanelD) andtheratioof WM¸ õ W ¹ (PanelE) for eachof the48 industryportfolios (numbered1-48 on the d -axis). All of theseratiosarecomputedat the conditioninglevel of Ö�» Ç ©qÈ . Datais sampledmonthlyfrom Jan1964- Dec1999.
41
Figure2: AlphasandFactorLoadingsof theDownsideCorrelationPortfolios
0 2 4 6 8 10 12 14 16 18 20−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6α in FF +WML model
α
Decile
Full 64−8182−99
0 2 4 6 8 10 12 14 16 18 20−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4Loading in FF+WML model
Fact
or L
aodi
ng
Decile
MKTSMBHMLWML
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
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, 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
−0.5
0
0.5
1
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
Load
ings
on
Fact
ors
Decile
MKTSMBHMLCMC
2 4 6 8 10−1
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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(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.
44
Figure5: PricingErrorsof GMM Estimation(HJ method)
10 20 30 40
−0.8
−0.6
−0.4
−0.2
0
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0.4
CAPM
Prici
ng E
rror
Portfolio10 20 30 40
−0.8
−0.6
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Fama−French Model
Prici
ng E
rror
Portfolio
10 20 30 40
−0.8
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Market and CMC
Prici
ng E
rror
Portfolio10 20 30 40
−0.8
−0.6
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Fama−French Model and CMC
Prici
ng E
rror
Portfolio
10 20 30 40
−0.8
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Carhart Four Factor Model
Prici
ng E
rror
Portfolio10 20 30 40
−0.8
−0.6
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0
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0.4
Carhart Model and CMC
Prici
ng E
rror
Portfolio
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.
45