downside correlation and expected stock returns · 2014-07-30 · downside correlation and expected...

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¶ ¿ ´�ÁÂ ¼:½ ¾ À³�¿ ´ Á ¼]½ ¾ À¶ ¿ ´ Á)Ã (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�¿ ´ ¾AÉ¶ ¿ ´�ÁÂ 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�¿ ín thefollowing regression:²dÓ³�¿ Ô ¹�Õ ¿ ´ »×Ö ³¥¿ ´+ØÚÙM³�¿ ´)Û² ³�¿ Ô�¿ ´HØ Ò ³¥¿ ńÜ ±�Ý�Þ Ê ²dÓ³¥¿ Ô�¿ ´ Ë+ß ³¥¿ Ô�¿ ´�Ø ¾ ³�¿ Ô ¹�Õ ¿ ´ Ãtà »'á Ã 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 Ã (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�ý» 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¥¿ ´ Ã (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 Ã (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´ Ã Ð�»já Ã � Ã 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´ Ã ±y»'á Ã � Ã 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 � Ã (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 ��´ Ã (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 Ã (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 ��´ Ã � �´ ï ¸+Õ cov î ��´ Ã � ³µ´ ï Ãandthefactorpremia,� , aregivenby:

�B» Ä á� ù cov î � ´ Ã � �´ ï � Õ Ï28

To testwhethera factor� is priced,we testthenull hypothesisÿ ù�� ]»! .Letting �:´ denoteané á vectorof grossreturns�]ý» î � Õ ´ Ã 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 � Ã (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´ Ã (B-12)

where² ³�¿ ´ is thereturnonthe ± -th portfolio, ² �´ is therisk-freerate,º ³ is an ÇO á vectorof factorloadings,and ��ís the Ç� á vectorof factors.We assumethattherisk-freerateandthefactorsfollow a first-orderVAR process.Let Phý» î ² �´ Ã � ´ ï � , and Ph´ follows:

P ´ »�Q Ø 2 P ´ ¸�Õ Ø÷öM´ Ã (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\YÍ³ 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^ É³�ì Õ ò ³ Ç 2 ]_�a`:´ ¸ ³ ØF^ É³�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 ^ É³�ì Õ òc³ ] Ç ] ´ ¸ ³MØ ^ É³íì 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.

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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

1

1.5

2J=6 K=9

Decile

Mean β ρ−

2 4 6 8 100

0.5

1

1.5

2J=6 K=12

Decile

Mean β ρ−

Theseplots show the averagemonthly percentagereturns, 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

−0.5

0

0.5

1

1.5J=6 K=6

Load

ings

on

Fact

ors

Decile

MKTSMBHMLCMC

2 4 6 8 10−1

−0.5

0

0.5

1

1.5J=6 K=9

Load

ings

on

Fact

ors

Decile

MKTSMBHMLCMC

2 4 6 8 10−1

−0.5

0

0.5

1

1.5J=6 K=12

Load

ings

on

Fact

ors

Decile

MKTSMBHMLCMC

Theseplots show the loadingsof the JegadeeshandTitman (1993)momentumportfolios on MKT, SMB,HML and CMC. Factor loadingsare estimatedin the first stepof the Fama-MacBeth(1973) procedure(equation(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

0.2

0.4

CAPM

Prici

ng E

rror

Portfolio10 20 30 40

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Fama−French Model

Prici

ng E

rror

Portfolio

10 20 30 40

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Market and CMC

Prici

ng E

rror


−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Fama−French Model and CMC

Prici

ng E

rror

Portfolio

10 20 30 40

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Carhart Four Factor Model

Prici

ng E

rror


−0.8

−0.6

−0.4

−0.2

0

0.2

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.

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downside correlation and expected stock returns · 2014-07-30 · downside correlation and expected...

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