Aggregate volatility expectations and threshold CAPM

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ContentslistsavailableatScienceDirect

North

American

Journal

of

Economics

and

Finance

Aggregate

volatility

expectations

and

threshold

CAPM

Yakup

Eser

Arısoy

a,∗

_,

_Aslıhan

_Altay-Salih

b,1

_,

Levent

Akdeniz

b,2

a_Université_{Paris-Dauphine,}_DRM_Finance,₇₅₇₇₅_Paris_Cedex_16,_France b_Bilkent_University_Faculty_of_Business_{Administration,}₀₆₅₃₃_Ankara,_Turkey

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received9March2015

Receivedinrevisedform16September 2015

Accepted21September2015 Availableonline9October2015

JELclassiﬁcation: C13 G12 Keywords: Aggregatevolatility Thresholdregression ConditionalCAPM Range VIX

a

b

s

t

r

a

c

t

Weproposeavolatility-basedcapitalassetpricingmodel(V-CAPM)

inwhichassetbetaschangediscretelywithrespecttochanges

ininvestors’expectationsregardingnear-termaggregate

volatil-ity.Usinganovelmeasuretoproxyuncertaintyaboutexpected

changesinaggregatevolatility,i.e.monthlyrangeoftheVIXindex

(RVIX),we ﬁnd that portfoliobetas change signiﬁcantly when

uncertaintyaboutaggregatevolatilityexpectationsisbeyonda

cer-tainthresholdlevel.Duetochangesintheirmarketbetas,smalland

valuestocksareperceivedasriskierthantheirbigandgrowth

coun-terpartsinbadtimes,whenuncertaintyaboutaggregatevolatility

expectationsishigh.Theproposedmodelyieldsapositiveand

sig-niﬁcantmarketriskpremiumduringperiodswheninvestorsdonot

expectsigniﬁcantuncertaintyinnear-termaggregatevolatility.Our

ﬁndingssupportavolatility-basedtime-varyingriskexplanation.

∗ Correspondingauthor.Tel.:+33144054360.

E-mailaddresses:eser.arisoy@dauphine.fr(Y.E.Arısoy),asalih@bilkent.edu.tr(A.Altay-Salih),akdeniz@bilkent.edu.tr (L.Akdeniz).

1 _Tel:₊₉₀₃₁₂₂₉₀_2047. 2 _Tel:₊₉₀₃₁₂₂₉₀_2202.

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1. Introduction

The capitalassetpricing model(CAPM)assumesthat aﬁrm’sriskiness, which iscapturedby itsbeta,isconstantthroughtime.However,changesinbusinessconditions,technology,andtaste mightinduceshiftsintheinvestmentopportunitysetandinvestors’associatedrisk-returntradeoffs (Jagannathan&Wang,1996).Manystudiesmodelthevariationinbetasusingcontinuous approx-imationandthetheoreticalframeworkoftheconditionalCAPM1_._Yet,_despite_a_strong_theory_and considerableevidenceontimevariationinbetas,thereisnoconsensusonhowthisvariationshouldbe modelled.

Inthispaper,wemodelassetbetasneitherasstaticnorasacontinuousapproximationimpliedby conditionalmodels,insteadweassumethatassetbetaschangediscretelyintime2_._Our_approach follows thespirit of regime-switching models,which have been extensively used in modelling ﬁnancialtime-series3_. _More _{particularly,} _we _posit _that _investors _re-assess_ﬁrms’ _systematic _risk withrespecttoexpectedchangesinaggregateriskconditionsbasedontheirexpectations regard-inguncertaintyaboutfutureaggregatevolatility.Thereareseveralreasonswhyweassumebetas shouldchangewithrespecttouncertaintyaboutaggregatevolatilityexpectations.First,itiswell documentedthat both equity andaggregate volatility istime-varying4_. _Therefore,_an _asset pric-ingmodelthatincorporatestime-variationinaggregatevolatilitywouldnaturallyimplythatasset betasalsochangeaccordingly5,6_._Second,_time-varying_risk_literature_suggests_that_stocks_have dif-ferentexposurestomarketriskduringrecessionsandexpansions(Lettau&Ludvigson,2001;Petkova &Zhang, 2005). Given the fact that change in aggregate volatility is tightly linkedto business cycles,ourmodel isabletocapturethis linkbyconditioningchanges in assetbetasonchanges in aggregate volatility expectations7_. _Driven_by _the _fact _that _{option-implied}_volatility _measures aregood forecastsoffuture volatility, weconditiontime variation inbetas based onan option-impliedmeasure,whichsummarizesinvestors’expectationswithrespecttochangesinnear-term aggregatevolatility8_._Rather_than_using_macro_variables _as_in_previous_studies,_our_approach con-tributestotheliteraturebyproposinganovelconditioningvariable,whichhasaforward-looking

1_See_Harvey₍₁₉₈₉₎_,_Ferson_and_Harvey₍₁₉₉₁_,_1993,₁₉₉₉_),_Ferson_and_Korajczyk₍₁₉₉₅₎_,_Jagannathan_and_Wang₍₁₉₉₆₎_,_and PetkovaandZhang(2005).

2_The_intuition_behind_discrete_changes_in_betas_with_respect_to_two_different_regimes_similar_to_{downside-upside}_beta approachinAng,Cheng,andXing(2006a)whoshowthatassetbetaschangeduringdownsideandupsidemarketsanddownside riskispriced.Methodologically,ourapproachisalsorelatedtoMarkovchainregimeswitchingmodelsasinGuidolinand Timmermann(2008)andChen,Gerlach,andLin(2011),andoptimalchangepointapproachasinBollenandWhaley(2009), andPattenandRamadorai(2013).

3_See_Hamilton₍₁₉₈₉₎_,_Hamilton_and_Lin₍₁₉₉₆₎_,_and_Ang_and_Bekaert₍₂₀₀₂₎_for_details_and_applications_of_{regime-switching} modelsindifferentsettings.

4_For_theoretical_background_and_empirical_evidence_on_stochastic_volatility_of_equity_and_stock_market_returns,_see_Engle andBollerslev(1986),French,Schwert,andStambaugh(1987),Schwert(1989),EngleandNg(1993),CaninaandFiglewski (1993),Duffee(1995),Braun,Nelson,andSunier(1995),Andersen(1996),BollerslevandMikkelsen(1999)andBekaertandWu (2000).

5_Given_the_deﬁnition_of_beta,_i.e.,_ˇ i=

Cov (ri,rm)/Var (rm)

,anyuncertaintyininvestors’aggregatevolatilityexpectations (i.e.thedenominator)isexpectedtoaffectreturnsinthecross-sectionthroughbetasonthemarketportfolio.Hence,our approachisdifferentfromAngetal.(2006a),Ang,Hodrick,Xing,andZhang(2006b)whoconditionexpectedreturnsdirectly onaggregatevolatilityandwhomodelaggregatevolatilityasaseparateriskfactor.OurapproachisalsodifferentfromWang andMa(2014)whoexaminetheeffectofexcessvolatilityattheindividualstocklevel.Weinvestigatetheimplicationsof uncertaintyaboutaggregatevolatilityonbetasandonthecross-sectionofexpectedreturns.

6_Pollet_and_Wilson₍₂₀₁₀₎_show_that_increases_in_market_volatility_can_be_due_to_either_increases_in_average_volatilities_or averagecorrelations,orboth.Furthermore,Buraschi,Porchia,andTrojani(2010)andBuraschi,Trojani,andVedolin(2014)show thatinvestorshavehedgingdemandsagainstbothstochasticaggregatevolatilityriskandstochasticcorrelationrisk.Intheir model,bothriskfactorsstemasaresultofuncertainty(disagreementinbeliefsacrossagents)intheeconomy.Wedonottake adirectstandonthecorrelationstructureinthispaper.However,becauseourmeasureRVIXessentiallycapturesuncertainty inexpectedvolatilityofthemarketportfolio,itiscloselylinkedtobothsourcesofrisk.

7_See_Hsu_and_Li₍₂₀₀₉₎_who_document_{counter-cyclicality}_of_volatility_across_different_asset_classes.

8_For_the_predictive_ability_of_{option-implied}_volatility_measures_ranging_from_intra-day_forecasts_to_one-year_ahead_forecasts, andindifferentmarketssuchasforeignexchange,stock,bond,andoptionmarkets,seePoonandGranger(2005),Taylor,Yadav, andZhang(2010),Busch,Christensen,andNielsen(2011),HanandPark(2013)andBianconi,MacLachlan,andSammon(2015).

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featurebyconstructionandwhichmodelstimevariationinanasset’sriskinessinaparsimonious way9,10_.

Inparticular,weproposeavolatility-basedthresholdCAPM(V-CAPM)whereassetbetaschange withrespecttoinvestors’assessmentofaggregateriskconditions,proxiedbyuncertaintyabout mar-ket’sexpectationsregardingchangesinaggregatevolatility.ThecontributionoftheproposedV-CAPM isfourfold.First,weproposeanovelmeasuretoproxyexpectedchangesinaggregateriskconditions, i.e.rangeoftheVIXindex(RVIX)11_._VIX_is_inherently_a_{forward-looking}_volatility_measure_and_it revealsimportantinformationaboutinvestors’expectationsofneartermvolatilityinthemarket12_. DeﬁnedasthedifferencebetweenthemaximumandminimumleveloftheVIXindex,RVIXessentially capturesexpectedchangesinnear-termaggregatevolatility,orputdifferentlythedegreeof uncer-taintyinfutureaggregatevolatility.Inarecentpaper,Baltussen,Bekkum,andGrient(2015)show thatthevolatilityofvolatility(vol-of-vol)isanimportantfactorinthecross-sectionofstockreturns. Usingthevolatilityofoption-impliedvolatilityasameasureofuncertaintyaboutvolatility,they doc-umentthatstockswithhighvol-of-volunderperformstockswithlowvol-of-vol.Theauthorsargue thatthevolatilityofoption-impliedvolatilityisanintuitivemeasure,whichistightlyrelatedtothe literaturethatmodelsuncertaintyassecond-orderbeliefs.Inasimilarvein,RVIXcanbeinterpreted asameasurethatcapturesuncertaintyregardingfutureaggregatevolatility.Hence,ifinvestorshold secondorderbeliefsandcareaboutthisuncertainty,thenstocksorportfolioswithdifferent sensi-tivitiestochangesinuncertaintycouldhavedifferentrisk-returndynamics,whichcouldalsoimply differentmarketriskpremiumdynamicsattimesofincreaseduncertaintyaboutexpectedaggregate volatility.

Secondcontributionis ourapproach tomodellingtime variation in betas.In standard condi-tionalCAPMsetting,betaspracticallychangeateach pointintime,howeverthisapproach might havea tendency tooverstatethetime variation in betasand resultin estimates thatare highly volatile.OursettingdiffersfromthestandardconditionalCAPMmodelsbyallowingbetastochange onlywhenthedegreeofuncertainty aboutexpectedaggregatevolatilitymovesbeyonda certain thresholdlevel,admitting a discretevariation inbetas intwo distinctregimes.Third,ourmodel implicitlyallowsfortimevariationinaggregatevolatilitywhichisnotpossibleinthestaticCAPM setting.Byendogenouslyincorporatingchangesininvestors’expectationsaboutaggregate volatil-ityinadynamicway,weallowbetastocapturepotentialshiftsintheinvestmentopportunityset linkedtoexpectedchangesinaggregateriskconditions13_._Our_fourth_contribution_is_econometric. Weformallytestthehypothesisontheexistenceofarelationbetweenbetasanduncertaintyabout aggregatevolatilityusingHansen’s(2000)thresholdregressionmethodology,whichisintuitiveand fullysupportedbytheeconometrictheory14_._The_proposed_model_is_rich_in_its_predictions_and_offers

9_Among_the_most_widely_used_macro_conditioning_variables_in_the_literature_are_the_dividend_yield₍_Fama_&_French,₁₉₈₈_), defaultspread(Keim&Stambaugh,1986),termspread(Campbell,1987),shorttermtreasurybillrate(Fama&Schwert,1977) andlogconsumption-wealthratio(Lettau&Ludvigson,2001).

10_Our_study_is_also_related_to_the_recent_strand_of_literature_showing_that_the_difference_in_{option-implied}_volatility_measures hassigniﬁcantexplanatorypowerinthecross-sectionofstockreturns.AmongthemarestudiesbyBaliandHovakimian(2009), Bollerslev,Tauchen,andZhou(2009),CremersandWeinbaum(2010)andAtilgan,Bali,andDemirtas(2015).

11_We_examine_whether_using_other_conditioning_variables_documented_in_the_literature_(such_as_one_month_T-bill_rate, aggregatedividendyield,inflationrate,termspreadandcreditspread)resultinsignificantregimechangesinportfoliobetas asimpliedbythethresholdCAPMmodel.Noneoftheexaminedvariablesyieldsignificantregimeshiftsinportfoliobetasas strongasRVIX.

12_Often_referred _to_as_the_{“fear ¨or}_“market_{sentiment¨index,}_VIX_estimates_near-term_(roughly _next_30-day)_expected volatility by weighted-averaging the prices of puts and calls written onthe S&P 500 index over a range of strike prices.

13_We_also_investigate_whether_the_documented_regime_changes_in_betas_with_respect_to_uncertainty_about_aggregate_volatility expectationsislinkedtodownsideriskasinAngetal.(2006a)andBali,Demirtas,andLevy(2009).Usingupsideanddownside betasofAngetal.(2006a),wedonotﬁndastrongcorrelationbetweenhighuncertaintyvs.downsidebetasandlowuncertainty vs.upsidebetas.Furthermore,dividingthesampleperiodintomonthsthatcorrespondtohighvs.lowuncertaintyabout aggregatevolatilityandmonthsthatcorrespondtoupsideanddownsidemarkets,weﬁndmajordifferencesbetweenthe correspondingtimeperiods.

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avolatility-basedexplanationtosomeoftheempiricalassetpricinganomaliesdocumentedinthe literature.

UsingRVIXasproxyforuncertaintyaboutaggregatevolatilityexpectations,andportfoliossorted withrespecttomarketcapitalizationsandbook-to-marketratiosastestassets,ourresultscanbe sum-marizedasfollows15_._First,_using_the_modified_sup_LM_test_suggested_by_Hansen₍₁₉₉₆₎_,_we_document significanttimevariationinbetas.15outof22testportfolioshavesignificantbootstrapp-valuesat5% level16_._The_evidence_confirms_the_existence_of_aggregate_volatility_related_changes_in_betas_of_most portfolios,mostparticularlyfortheextremesizeandbook-to-marketportfoliosaswellasSMBand HMLportfolios.Theinitialresultssupportthehypothesisthatassetbetaschangediscretelyintime anduncertaintyaboutaggregatevolatilityexpectationsisakeydeterminantofinvestors’assessment aboutanasset’ssystematicrisk.

Next,wetestwhetherdifferentsizeandbook-to-marketportfolioshavedifferentbetasensitivities withrespecttoinvestors’expectationsaboutuncertaintyofaggregatevolatilityandriskconditions. Thethresholdestimatessuggestthatinvestorsoverwhelminglyupdatetheirbetariskassessments whenmonthlyrangeoftheVIXindexisbeyond9.33points17_._What_makes_the_results_further remark-ableisthedirectionofthisupdate.Lookingatthechangesinportfoliobetas,onecanseethatstocksin small(andvalue)portfolioshaveconsistentlyhigherbetasattimeswhenuncertaintyaboutexpected aggregatevolatilityishigh(i.e.whenRVIXinagivenmonthisabovethethresholdlevel).Onthe con-trary,theportfoliooflargestmarketcapitalizationstocks(andthegrowthportfolio)exhibitslower betasduringtheseuncertaintimes.Theincreaseinbetasismostpronouncedforthesmallestdecile, highestbook-to-marketdecile,SMBandHMLportfolios.

Frenchetal.(1987),andCampbellandHentschel(1992)documentthatperiodsofhighvolatility usuallycoincidewithdownwardmarketmoves.Furthermore,risk-averseinvestorsarereluctantto losewealthinperiodsofhighvolatilitybecauseitrepresentsadeteriorationininvestment opportu-nities,whichusuallycoincideswithperiodsoflowconsumption(recessions)18_._The_increase_in_betas ofsmallandvalueportfoliosimpliesthatstockswiththesecharacteristicsareperceivedtoberiskier attimesofincreaseduncertaintyaboutexpectedvolatility.ThisalsoholdsforSMBandHML port-folioswhosesensitivitiestomarketreturnsbecomehigherduringthosetimes.Investorsviewsmall andvalueﬁrmsriskierbecausetheirreturnscorrelatestronglywithmarketreturnsinepisodeswhen uncertaintyaboutexpectedaggregatevolatilityishigh.Ontheotherhand,returnsonbigandgrowth stockscorrelatelesswithmarketreturnsduringthosetimes.Ourresultsareconsistentwiththose ofLettauandLudvigson(2001)andPetkovaandZhang(2005)whoﬁndthatvalueandsmallstocks correlatemorewiththeconsumptiongrowth(marketreturns)duringbadtimesrelativetobigand growthstocks,whiletheoppositeholdsduringgoodtimes.Wearguethatinvestorsviewsmalland valuestocksriskierthantheirbigandgrowthcounterpartsbecausetheirreturnsaremuchmore sen-sitivetomarketriskattimesofincreaseduncertaintyaboutaggregatevolatilityandadversemarket conditions.

Totesttherobustnessoftheaboveresultsandtofurtherexaminetheeffectoftimevariation ininvestors’expectationsofneartermvolatilityonassetrisk-returndynamics,wenextcalculate Jensen’salphas and Sharperatios ofour test assets in thefull sample aswell asin two differ-entregimes(i.e.highandlowuncertaintyaboutvolatility)impliedbythethresholdlevelofRVIX estimated via theV-CAPM. It is well-documented that small and value stocks (and the associ-atedSMBand HMLstrategies) producesignificantly higheraveragereturns thantheirlargeand growthcounterparts. Looking at Jensen’s alphas and betas of differentsize and book-to-market portfolios,we confirmthepreviousfindingsthat thestaticCAPMis unabletooffera risk-based

15_We_also_use_an_{orthogonalized}_version_of_RVIX_(RVIXORTH)_in_order_to_ensure_that_the_results_are_not_driven_by_potential variablesthathavebeendocumentedtobeimportantintheliterature.RVIXORTHisdeﬁnedastheresidualtermobtainedfrom regressingRVIXonaggregatedividendyield,thedefaultspread,thetermspread,theshort-termtreasurybillrate,andtheVIX. TheresultsarerobusttouseofRVIXORTHastheconditioningvariable.

16₉₍₁₇₎_portfolios_exhibit_signiﬁcant_change_in_betas_at_1%_(10%)_level.

17_The_threshold_estimates_(which_could_practically_be_any_positive_real_number)_are_quite_stable_for_portfolios_that_exhibit signiﬁcantbetachanges(rangingfrom6.07to11.10),conﬁrmingtherobustnessofthechosenthresholdvariable,RVIX.

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explanationtoSMBandHMLreturndifferentials.Ontheotherhand,theanalysisofalphasimplied by the proposed V-CAPM helps us uncover an important aspect of size and value vs. growth puzzles. In particular,we document that size and value strategies yield signiﬁcantand positive risk-adjustedreturnsduringcalmtimeswhenuncertaintyaboutnear-termaggregatevolatilityis low.Ontheotherhand,thetrade-offforsizeandvaluestrategiesisthattheyhaveextremelybad (signiﬁcantandnegative)risk-adjustedreturnsattimesofhighuncertaintyaboutexpected volatil-ity.

PortfolioSharperatiosalsoofferasimilarvolatility-basedtime-varyingriskexplanationtosizeand valuevs.growthanomalies.Inperiodswhenuncertaintyaboutaggregatevolatilityislow,thestrategy inthesmallest(value)decileportfolioscommandhigherreward-to-variabilityratiosasopposedtothe biggest(growth)decileportfolios.However,inperiodsofhighuncertaintyaboutaggregatevolatility, investorsexperiencemuchworsereward-to-variabilityratiosforthesmallest(value)decileportfolio relativetobiggest(growth)decileportfolio.Theresultsconﬁrmourhypothesisthatmarket’s expec-tationofuncertaintyaboutaggregatevolatilityisanimportantdeterminantofinvestors’assessment ofriskandexpectedreturns.Changesinbetasandrisk-adjustedreturnsduringperiodswith differ-entlevelsofuncertaintyaboutaggregatevolatilityexpectationscontributetoourunderstandingof whysmallandvaluestocksonaverageearnhigherreturnsthantheirbigandgrowthcounterparts. Weshowthatinvestinginsmallandvaluestocksareriskystrategiesinperiodswhenthereishigh uncertaintyaboutexpectedaggregatevolatility,andthusinvestorsgetcompensatedfortheriskthat theyaretakingagainstthisuncertainty.

We finallytest thepricingimplicationsof theproposedV-CAPMbydividingthesample into periods of highand low uncertainty aboutexpected aggregate volatility and by estimating the betas and the corresponding risk premium in the cross-section. To avoid the problem of fac-tor structure related biases in the estimation procedure, we estimate the betas and the risk premia at the individual stock level rather than at the portfolio level. We start with examin-ingtherelationshipbetweenstocks’betasand futurereturns.To thatend, wefirstestimatethe beta loadings via monthly regressions using daily returns as in Ang et al. (2006a,b). Classify-ingbetasashighuncertainty(RVIX>9.33)vs.lowuncertainty(RVIX<9.33)betas,weformdecile portfolios each month by sorting individual stocks according to theirbetas. We then examine out-of-sampleaveragedecilereturnsforthefollowingmonthtoinvestigatewhetherstocks’beta exposures determined by uncertainty about aggregate volatility explain thecross-sectional dis-persionintheirnext-monthreturns.Univariateportfoliosortsindicatethatstocksinthehighest uncertainty(calm)betadecileunderperform(outperform)stocksinthelowestuncertainty(calm) beta decile by 0.72% (0.60%) per month. Furthermore, the differences in risk-adjusted returns (CAPM and FF 3-factor alphas) of portfolios with highest and lowest exposure to uncertainty betasarealsonegativeandstatisticallysignificant.Theresultsarerobusttousingvalue-weighted returns.

Wefinallyestimatethecorrespondingriskpremiaforuncertaintyandcalmbetasusingthe stan-dardFamaandMacBeth(1973)cross-sectionalregressionmethodology.Consistentwiththeprevious studies,wedocumentaninsignificantmarketriskpremiumthroughoutthefullsampleperiod, con-firmingtheinabilityofastaticversionofCAPMtoexplainthecross-sectionofstockreturns.Onthe otherhand,wedocumentapositiveandsignificantmarketriskpremiumduringcalmtimeswhen uncertaintyaboutaggregatevolatilityislow.Thisresultisrobusttotheinclusionofdifferentfactor exposuressuchasSMBbeta,HMLbetaandMOMbetaaswellasvariousfirmcharacteristicssuchas idiosyncraticvolatility,size,book-to-marketratio,andfirm-levelmomentum.Onthecontrary, dur-ingperiodsofhighuncertaintyaboutaggregatevolatility,themarketriskcommandsasignificantand negativepremium,howeveritssignificancedisappearswhendifferentportfolioexposuresandfirm characteristicsareincluded.Theresultsimplyamajorimprovementoverpricingrelativetostatic CAPMbyre-establishingapositivemarketriskpremiumduringcalmperiodswhenuncertaintyabout aggregatevolatilityisfairlylow.

Theremainderofthepaperisorganizedasfollows.Section2introducesthethresholdV-CAPM andtherelatedeconometricframework.Section3presentsdataandsomestylizedfacts.Section4 documentsempiricalﬁndingsfortime-seriesandcross-sectionaltestsoftheproposedV-CAPM.The ﬁnalsectionoffersconcludingremarks.

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2. ThethresholdCAPM

Tocapturetheeffectofuncertaintyaboutaggregatevolatilityexpectationsonmarketbeta,we startwiththefollowingconditionalCAPM:

E

ri,t+1

Zt

=˛i+ˇtE

rm,t+1|Zt

+εit+1, (1)

whereri,t+1istheexcessreturnonasseti,rm,t+1istheexcessreturnonthemarketportfolioandE

istheexpectationoperator.ˇtcapturestime-variationinmarketbetas,andZt istheconditioning

informationoninvestors’assessmentofnear-termaggregatevolatilityrisk.Usingmonthlyrangeof theVIXindexasaproxyforinvestors’informationsetforexpectedchangesinaggregatevolatility,we modeltime-variationinbetasasinFersonandHarvey(1999)19_:

ˇt=ˇ11{Zt≤}+ˇ21{Zt>}, (2)

where1_{}istheindicatorfunctionandisthethresholdparameterforaggregatevolatility.Combining Eqs.(1)and(2),wehavethefollowingthresholdvolatilityCAPM:

ri,t+1=

˛11{Zt≤} +˛21{Zt>}

+

ˇ11{Zt≤}+ˇ21{Zt>}

rm,t+1+εi,t+1, (3)

whereZtisthemonthlyrangeoftheVIXindex(RVIX)thatsummarizesinvestors’informationset

regardingtheevolutionofuncertaintyaboutnear-termaggregatevolatility. 2.1. Econometricmodel

Theobservedsampleis{rt+1,rm,t+1,Zt},t=1,...,T−1.Therandomvariablesrt,rm,t,andZtare

real-valued.ThethresholdvariableZtisassumedtohaveacontinuousdistribution20.Thethreshold

regressionhasthesameformatasinEq.(3),whichcanberewrittenas

rt+1=xt+1+ıxt+1() +et+1 (4)

wherext+1=rm,t+1,xt+1() =xt+11{zt≤},=ˇ2andı=ˇ1−ˇ2.

Theabovemodelcanfurtherbegeneralizedtothecasewhereonlyasubsetofparametersswitches betweentheregimesandtothecasewheresomeregressorsonlyenterinoneofthetworegimes. Also,takesvaluesinaboundedsubsetoftherealline,.Thisappliestothecaseofourconditioning variableRVIX,whichisboundedbelowbyzerobydeﬁnition.Weassumermt,Zt,andetarestrictly

stationaryergodicand-mixing21_. 2.2. Testingforathreshold

Weusetheheteroskedasticity-consistentLagrangeMultiplier(LM)testforathreshold,asinHansen (1996).WetestforthenullofH0:ı=0againstH1:␦ /= 0.Ifthenullisrejected,thisimpliesasigniﬁcant

changeinbetaswithrespecttolevelsaboveorbelowthresholdRVIX.

Forall∈ wehavethefollowingLMstatisticsforthenullofnothreshold: LM_T() =T

R ()ˆ

R ˆV∗

T() R

−1

R ()ˆ

,

19_See_Section₂_for_details_on_the_construction_of_the_conditioning_variable_RVIX. 20_See_Hansen₍₂₀₀₀₎_for_detailed_explanations_related_to_the_assumptions. 21_The_-mixing_{coefﬁcients}_satisfy

m1/2<∞.The-mixingassumptioncontrolsthedegreeoftimeseriesdependence andallowstheprocessestobeautocorrelatedandheteroskedastic,andissufﬁcientlyﬂexibletoembracemanynon-lineartime seriesprocesses,includingthresholdautoregressions.

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where, R= [0,I] , ˆ ()=

ˆ()_{, ˆı()}

₌

_T

t=1 x∗ t+1()x∗t+1()

−1

_T t=1 x∗ t+1()rt+1

, x∗ t+1()= [xt+1,xt+1()] , ˆ V∗ T()=MT()−1V˜T()MT()−1, MT()=_T1 T

t=1 x∗ t+1()x∗t+1(), ˜ VT()= 1_T T

t=1 x∗ t+1()x∗t+1()e˜2t+1,

andwhere ˜etisobtainedfromtherestrictedleastsquares.OnelimitationoftheLMtestisthelarge

samplelimitforthesup-LM,whichisnotnuisancefreebecausethethresholdisnotidentiﬁedunder thenullofno-thresholdeffect.Becauseofthisissue,Hansen(1996)suggestsabootstrapanalogofthe sup-LMtestandshowsthatthisbootstrapmethodyieldsasymptoticallycorrectp-values.Weusethe bootstrapanalogfollowingthestepsoutlinedinHansen(1996)andestimatetheunknownthreshold parameter,,asinHansen(2000).

3. Data

ThemarketandstockreturndataisfromCenterforResearchin SecurityPrices(CRSP) value-weightedmarketindexforallNYSE,AMEX,andNASDAQstocks.Therisk-freerateistheone-month T-BillrateobtainedfromIbbotsonAssociates.DataonVIXandVXOisobtainedfromChicagoBoard OptionsExchange’s(CBOE).ThesamplecoverstheperiodfromJanuary1986toDecember2012,witha totalof324months22_._The_test_portfolios_consist_of_stocks_sorted_according_to_their_market capitaliza-tions,andbook-to-marketratios.Moreprecisely,weuse10portfoliossortedaccordingtotheirmarket capitalizations,10portfoliossortedaccordingtotheirbook-to-marketratios,and2factorportfolios SMBandHML23_._For_{cross-sectional}_tests_performed_in_Section_4.3_,_we_use_the_CRSP_universe_covering allNYSE/AMEX/NASDAQcommonstockswithsharecodes10and11.

Inordertoproxyinvestors’expectationsabouttheevolutionofnear-termaggregatevolatility,we usethemonthlyrangeoftheVIXindex(RVIX).SimilartoChou(2005),wedeﬁneRVIXinagiven monthas:

RVIXt= Max

VIX

− Min

VIX

, = 1,2,...,T (5) wheredenotestradingdaysinagivenmonth,andtdenotesmonths.Takingthedifferencebetween themaximumandminimumlevelofVIXindexinagivenmonth,RVIXsummarizesinvestors’ expec-tationsregardingchangesinnear-termaggregatevolatility.

22_VIX_data_is_available_from_January₁₉₉₀_onwards._In_order_to_have_as_much_data_as_possible,_we_use_the_VXO_index_(which_is basedonS&P100indexoptions)fromJanuary1986toDecember1989,andtheVIXindexfromitsintroductioninJanuary1990 onwards.Theresultsremainunaffectedwhenwelimitthesampleperiodto1990–2012usingtheVIXindexonly,oromitting VIXandusingVXOthroughoutthe1986–2012period.

23_SMB_(Small_Minus_Big)_is_the_average_return_on_the_three_small_portfolios_minus_the_average_return_on_the_three_big portfolios,andHML(HighMinusLow)istheaveragereturnonthetwovalueportfoliosminustheaveragereturnonthetwo growthportfolios.

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Rangebasedvolatilitymeasureshavegainedrecentinterest,andtheyfarequitewellinpredicting futurevolatility24_._To_the_best_of_our_knowledge,_this _is_the_ﬁrst_study_to_propose_a _range_based measureoftheVIXindex25_._We_further_tested_a_battery_of_volatility_measures_ranging_from_statistical andhistoricalmeasuresofvolatilitysuchasstandarddeviationofreturns,squaredreturns,GARCH basedvolatilityestimatestoforward-lookingmeasuresofvolatilitysuchaschangeintheVIXindex andS&P500straddlereturns26_._The_proposed_RVIX_together_with_S&P₅₀₀_straddle_returns_are_the mostsuccessfulincapturingtime-variationinbetas.

WearguethatthesuccessofRVIXindetectingchangesinbetasisduetoitsabilitytocharacterize uncertaintyaboutfutureaggregatevolatilitymuchbetterthanalternativemeasures.Byusinga mea-surewhichessentiallycapturesvolatilityofoption-impliedaggregatevolatility,wehavetheadvantage ofﬁrstidentifyinginvestors’expectationsabouttheevolutionofnear-termaggregatevolatility(VIX), andsecondmeasuringthedegreeofuncertaintyinexpectedaggregatevolatilitycapturedbytherange ofthethisforward-lookingoption-impliedaggregatevolatilitymeasure(RVIX).Aggregatevolatility riskhasbeendocumentedtobeanimportantfactorthatdeterminesinvestors’risk-returntradeoff andtime-variationininvestmentopportunityset27_._Because_RVIX_is_essentially_a_proxy_for_the_degree ofuncertaintyregardingaggregatevolatilityexpectations,weexpectRVIXtobeastrongconditioning variablethatcapturesuncertaintyininvestors’informationsetregardingaggregatevolatilityrisk,and hencetohaveimplicationsregardingassetpricing,portfolioallocation,andstockreturnpredictability asanimportantconditioningvariable.

WefurthertestwhetherthechosenconditioningvariableRVIXiscorrelatedwithotherbusiness cyclemeasuresdocumentedintheliterature28_._Towards_that_end,_we_are_particularly_interested_in thedividendyield(DIV)oftheS&P500index,thedefaultspread(DEF)whichthespreadbetweenBAA andAAAratedcorporatebondyields,thetermspread(TERM)whichisthespreadbetween10-year, 1-yearU.S.governmentbondyieldsandtheshort-termtreasurybillrate(TB)andtheVIXindex,allof whichhavebeendocumentedasstrongpredictorsofbusinesscyclesandhencetheconditionalCAPM informationset29_._To_check_whether_our_results_are_not_affected_from_a_potential_correlation_with_the businesscyclevariablesdocumentedintheliterature,wecreateanorthogonalizedmeasureofRVIX (RVIXORTH),whichisdeﬁnedastheresidualtermfromthefollowingregression:

RVIX_t=˛+ˇMKTMKTt+ˇSMBDIVt+ˇDEFDEFt+ˇTERMTERMt+ˇTBTBt+ˇVIXVIXt+εt (6)

Table1reportsthesummarystatisticsofRVIX,themarketportfolio,aswellastheorthogonalized versionofRVIX(RVIXORTH),andmacrovariablesusedintheanalysis.Lookingatthemean(6.66), median(4.98),theminimum(0.92)andthemaximum(129.04)oftheRVIX,onecansaythatVIX index(expectationsofnear-termmarketvolatility)and itsrange(expectedchangesinnear-term aggregatevolatility)arequitestableanddonotmovesigniﬁcantlyinmostofthemonthsduringour sampleperiod.Withoutmuchsurprise,themaximumlevelofRVIXwasrecordedinOctober1987, wheretheVXOindexskyrocketedfromitsminimumvalueon3rdOctober1987of21.15pointsto itshistoricalmaximumof150.19pointsonblackMonday.Finally,similartothenegativecorrelation

24_See_Alizadeh,_Brandt,_and_Diebold₍₂₀₀₂₎_,_Chou₍₂₀₀₅₎_,_Brandt_and_Jones₍₂₀₀₆₎_,_Chou_and_Liu₍₂₀₁₀₎_,_Harris,_Stoja,_and Yilmaz(2011)andBannouh,Martens,andvanDijk(2013)forarticlesthatmotivatetheuseofrangebasedvolatilitymeasures indifferentsettings.

25_Previous_studies_such_as_Garman_and_Klass₍₁₉₈₀₎_and_Parkinson₍₁₉₈₀₎_as_well_as_many_others_use_the_logarithmic trans-formationofthestockpriceasameasureofstockvolatility.Althoughtheasymptoticpropertiesandforecastingpowerof price-basedrangemeasureshasbeenextensivelydocumented,asfarastheauthorsareawareof,thisistheﬁrstpaperthat appliestheconceptofrangetoimpliedvolatility.TohaveabetterunderstandingofthepropertiesofRVIX,wecheckthe predic-tiveabilityofRVIXtoforecastrealizedvariance(VAR)asdeﬁnedinWelchandGoyal(2008).Thepairwisecorrelationbetween RVIXinmontht-andVARinmontht+1is0.27.

26_The_reader_is_referred_to_Section_4.4_for_a_detailed_discussion_of_results.

27_See_Campbell₍₁₉₉₃₎_,_Chen₍₂₀₀₂₎_,_Ang_et_al._(2006a,b)_and_Barinov₍₂₀₁₂₎_for_papers_that_document_pricing_of_aggregate volatilityrisk.

28_See_Ang_and_Chen₍₂₀₀₂₎_and_Ang_and_Bekaert₍₂₀₀₂₎_for_studies_that_document_increase_in_correlations_during_recessions. 29_Dividend_yield_data_is_from_Robert_Shiller’s_website₍_{http://www.econ.yale.edu/∼shiller/data.htm}_),_government_and cor-poratebondyieldsarefromSt.LouisFedwebsite(https://research.stlouisfed.org/fred2/),andshort-termTreasurybillratesare fromKenFrench’sdatalibrary,(http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/datalibrary.html).

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Table1

Descriptivestatistics.

PanelA:summarystatistics

MKT RVIX RVIXORTH TERM DEF DIV TB Mean 0.89 6.66 0 1.43 1.00 0.02 0.31 Median 1.49 4.98 0.04 1.47 0.91 0.02 0.37 Maximum 12.88 129.04 84.04 3.40 3.38 0.39 0.79 Minimum −22.64 0.92 −16.67 −0.41 0.55 0.01 0 Std.Dev 4.61 8.36 6.11 1.07 0.40 0.01 0.20 Skewness −0.89 10.25 8.60 0.10 2.87 0.34 −0.11 Kurtosis 5.48 144.02 121.09 1.82 14.94 1.85 2.08 PanelB:correlations MKT 1 RVIX −0.41 1 RVIXORTH 0 0.73 1 TERM −0.04 0.09 0 1 DEF −0.08 0.28 0 0.29 1 DIV 0.02 0.08 0 0.13 0.32 1 TB 0.04 −0.05 0 −0.71 −0.34 0.36 1

Thistablereportsthedescriptivestatisticsformonthlyreturnsonthemarketportfolio(MKT),andmonthlyrangeofthe VIXindex(RVIX),aswellasorthogonalizedmeasureofRVIX(RVIXORTH),and4businesscyclerelatedmeasures.The mar-ketportfolioistheCRSPvalue-weightedindexforallNYSE,AMEX,andNASDAQstocks.RVIXisthedifferencebetween maximumandminimumlevelofVIXinagivenmonth,i.e.RVIXt=Max{VIX}−Min{VIX},=1,2,...,T,wheredenotes tradingdaysinagivenmonth,andtdenotesmonths.DIVisthedividendyieldoftheS&P500index.DEFisdeﬁnedasthe spreadbetweenBAAandAAAratedcorporatebondyieldandTERMisthespreadbetween10-yearand1-yearU.S. gov-ernmentbondyields.TBistheone-monthTreasuryBillrate.RVIXORTHistheresidualtermfromthefollowingregression: RVIXt=˛+ˇMKTMKTt+ˇSMBDIVt+ˇDEFDEFt+ˇTERMTERMt+ˇTBTBt+ˇVIXVIXt+εtThesamplecoverstheperiodfromJanuary 1986toDecember2012(324months).FortheperiodcoveringJanuary1986toDecember1989,VIXisreplacedbyVXOwhich isbasedonS&P100indexoptions.Allreturnﬁguresareinpercentages.

documentedinpreviousstudiesbetweentheVIXindexandmarketreturns,thecorrelationbetween

RVIXandthemarketis−0.41.

3.1. Stylizedfacts

Thissectiondocumentssomestylizedfactsaboutthechosenthresholdparameter,marketreturns,

andtheempiricallydocumentedsizeandvaluevs.growthanomalies.

First,lookingatFig.1,onecanseethattheproposedconditioningvariableRVIXindeedtracks

signiﬁcantnegativemarketmoves.Giventheempiricalevidencethatnegativemarketmoves are mostassociatedwithincreasesinaggregatevolatility,ournovelmeasureRVIXisessentiallycapableof providingarelationshipbetweentheevolutionofnear-termmarketvolatilityanddownwardmarket moves.

Next,weconductasimpleexercisetoexaminereturnsondifferentsizeandbook-to-market portfo-liosindifferentvolatilityregimesinmoredetail.UsingthresholdestimatesoftheRVIXindex,wedivide thesampleintotworegimes,whereregime1(2)representscalm(uncertain)monthsinwhichRVIX isbelow(above)theestimatedthresholdleveloftheassociatedportfolio.Thiswayofdecomposing returnsintocalmanduncertainmonthsgivesusinterestinginsightsregardinginvestor’ expecta-tionsaboutuncertaintyinnear-termaggregatevolatilityandportfolioreturndynamics.Forexample, lookingatcolumns5and10ofTable2,onecanseethatassetclasses,regardlessoftheirportfolio characteristics,losemuchmorewhenmarketvolatilityisexpectedtobehighlyvolatile.Thisisinline withHsuandLi(2009)whodocumentthatvolatileperiodscoincidewithbearmarkets.Ontheother hand,columns2,3,7and8documentthetypicalsizeandvaluevs.growthanomalies.More particu-larly,staticCAPMfailstoofferaclearandlinearrelationshipbetweenbetasandportfolioreturns,i.e. high(low)returnsarenotalwaysjustiﬁedbyhigh(low)CAPMbetas.However,lookingatcolumns 4,5,9,and10ofTable2,onecangaininterestinginsights.Forexample,incalmmonths(whenRVIX

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0 20 40 60 80 100 120 -.25 -.20 -.15 -.10 -.05 .00 .05 .10 .15 86 88 90 92 94 96 98 00 02 04 06 08 10 12 MARKET RVIX

Fig.1.Time-seriesofRVIXandmarketreturns.

Table2

Stylizedfactsaboutportfolioreturns.

Size Beta Full sample Regime1 (Calm) Regime2 (Volatile) B/M Beta Full sample Regime1 (Calm) Regime2 (Volatile) Small 1.0163 0.0097 0.0176 −0.0365 High 1.0557 0.0115 0.0187 −0.0298 Decile2 1.1604 0.0096 0.0170 −0.0330 Decile2 0.9864 0.0107 0.0160 −0.0203 Decile3 1.1434 0.0104 0.0173 −0.0297 Decile3 0.9716 0.0091 0.0149 −0.0239 Decile4 1.1262 0.0094 0.0158 −0.0271 Decile4 0.9512 0.0102 0.0147 −0.0156 Decile5 1.1380 0.0104 0.0164 −0.0240 Decile5 0.9467 0.0090 0.0135 −0.0167 Decile6 1.0713 0.0104 0.0157 −0.0213 Decile6 0.9029 0.0096 0.0143 −0.0179 Decile7 1.0673 0.0108 0.0164 −0.0205 Decile7 0.8487 0.0098 0.0153 −0.0221 Decile8 1.0795 0.0103 0.0156 −0.0203 Decile8 0.8445 0.0102 0.0141 −0.0124 Decile9 1.0101 0.0101 0.0148 −0.0170 Decile9 0.9228 0.0093 0.0129 −0.0109 Big 0.9449 0.0087 0.0118 −0.0096 Low 1.0501 0.0090 0.0116 −0.0066 SMB 0.0716 0.0010 0.0058 −0.0269 HML 0.0059 0.0026 0.0071 −0.0231 Market 0.0089 0.0129 −0.0143 Market 0.0089 0.0128 −0.0143 Thistablepresentsthereturnsonseveralportfoliosthathavebeenusedastestassetsinthisstudyandthemarketportfolio duringthefullsampleperiodfromJanuary1986throughDecember2012(324months)andtwodifferentvolatilityregimes. Sizerepresentsportfolioswhichcontainstockssortedwithrespecttotheirmarketcapitalizations.B/Mrepresentsportfolios whichcontainstockssortedwithrespecttotheirbook-to-marketratios.SMBisaportfoliothatislonginstocksinthesmallest decileandshortinstocksinthebiggestdecile.HMLisaportfoliothatislonginstockswhichareinthehighestB/Mdecileand shortinstockswhichareinthelowestB/Mdecile.

isbelowtheestimatedthreshold),onecanseealmostamonotonousdecreaseinreturnsgoingfrom

smallandvalueportfoliosthroughbigandgrowthportfolios.Theoppositeistrueforepisodeswhen

uncertaintyaboutaggregatevolatilityishigh,whensmallandvalueportfoliosbecometheworst

per-formers.Despitetheirhigheraveragereturnsrelativetobigandgrowthportfolios,smallandvalue

stockportfoliosbecometheworstperformersattimesofhighuncertaintyaboutexpectedaggregate

volatilitywhenthemarketisexpectedtodobadly30_._On_the_other_hand,_by_losing_less_than_the

mar-ketportfolio,bigandgrowthportfolioscanbeseenasrelativelysaferassetclassesduringuncertain periodsaboutaggregatevolatilityandmarketconditions.

Thepreliminaryﬁndingsinformallyconﬁrmourhypothesesthatsizeandbook-to-market portfo-lioshavedifferentsensitivitiestomarketriskduringperiodsofdifferentexpectationsregardingthe

30_This_also_holds_for_the_zero-cost_SMB_and_HML_portfolios,_which_earn_on_average₅₈_and₇₁_basis_points_per_month_during calmmarketconditions,butwhichbecomeextremelyriskystrategiesandlose269and231basispoints,respectively,during highexpectedvolatilityperiods.

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evolutionofaggregatevolatility.Thus,wepositthatanassetpricingmodelthatcorrectlytakesinto accountthisvolatility-basedtimevariationinriskandreturnsisexpectedtodobetterinpricingand inexplainingsizeandvaluevs.growthanomalies.

4. TestsofV-CAPM

Webeginbyexaminingwhethertherearestatisticallysignificantregimeshiftsinbetasdueto changesininvestors’expectationsregardinguncertaintyaboutaggregatevolatilityandmarket condi-tions.OurconditioningvariableisrangeoftheVIXindexandTable3reportstheassociatedbootstrap p-valuesforthesup-LMtest.Thenullhypothesisisthatthereisnosignificantregimeshiftin portfo-liobetas.Accordingtobootstrapp-valuespresentedinTable3,therearesignificantregimechanges inbetasofmostportfolios.Forportfoliossortedwithrespecttomarketcapitalizations,eightoutof tenexperiencesignificantchangesintheirbetasbetweenuncertainandcalmperiodsaboutexpected aggregatevolatility.Forportfoliossortedwithrespecttobook-to-marketratios,theevidenceindicates aregimeshiftinbetasofsevenoutoftenportfolios.SMBandHMLportfoliosalsoexhibit signifi-cantregimeshiftsinbetas.TheresultsarealsorobusttousingtheorthogonalizedversionofRVIX (RVIXORTH).TakingintoaccountthatthethresholdparameterRVIXcapturesinvestors’expectations aboutuncertaintyinnear-termvolatility,uncertaintyabouttheevolutionofaggregatevolatilityseems tobeanimportantdeterminantoftheirassessmentofaggregateriskconditionsandanasset’s sensitiv-itytooverallmarketrisk.Theresults,ifpersistent,offernewevidenceandanalternativeexplanation totheempiricallyobservedsizeandvaluevs.growthanomalies.

TheconditionalCAPM modelsmayhavea tendencytooverstatethetime variation,and asa result,continuousapproximationsofCAPMproducehighlyvolatilebetaestimates.Thisisfurther conﬁrmedwiththeevidencereportedinBraunetal.(1995),whouseabivariateEGARCHmodelto estimateconditionalbetasanddocumentweakevidenceoftimevariation.Ontheotherhand,our thresholdmethodologyusingRVIXasaconditioningvariablesuggeststhatportfoliobetasare sta-bleduringdifferentexpectedvolatilityregimes,howeverinvestorsupdatetheirbetaestimateswhen theirexpectationsregardingtheuncertaintyaboutnear-termvolatilitychangeconsiderably.

Table3

Bootstrapp-valuesfor10sizeandB/Mportfolios.

Size RVIX RVIXORTH B/M RVIX RVIXORTH Small 0.000*** _0.000*** _High _0.012** _0.000*** Decile2 0.000*** _0.024** _Decile2 _0.009*** _0.001*** Decile3 0.000*** _0.035** _Decile3 _0.013** _0.000*** Decile4 0.015** _0.042** _Decile4 _0.144 _0.148 Decile5 0.025*** _0.051* _Decile5 _0.029** _0.009** Decile6 0.140 0.044** _Decile6 _0.067* _0.022** Decile7 0.016** _0.003*** _Decile7 _0.008*** _0.030** Decile8 0.144 0.009*** _Decile8 _0.435 _0.642 Decile9 0.056* _0.018** _Decile9 _0.219 _0.167 Big 0.008*** _0.007*** _Low _0.000*** _0.000*** SMB 0.009*** _0.058** _HML _0.001*** _0.000*** Thistablereportsthebootstrapp-valuesofthemodiﬁedsup-LMtestsuggestedbyHansen(1996).Wetestthenullhypothesis ofnosigniﬁcantregimeshiftsinportfoliobetasduetochangesinthelevelofuncertaintyregardingaggregatevolatility expec-tations,capturedbyRVIX,aswellasanorthogonalizedmeasureofRVIX(RVIXORTH).RVIXORTHistheresidualtermfromthe followingregression:

RVIXt=˛+ˇMKTMKTt+ˇSMBDIVt+ˇDEFDEFt+ˇTERMTERMt+ˇTBTBt+ˇVIXVIXt+εt

Sizerepresentsportfolioswhichcontainstockssortedwithrespecttotheirmarketcapitalizations,andB/Mrepresentsportfolios whichcontainstockssortedwithrespecttotheirbook-to-marketratios,respectively.ThesampleperiodcoversJanuary1986 toDecember2012(324months).

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Table4

Thresholdestimatesfor10sizeand10book-to-marketportfolios.

PanelA:10sizeportfolios

CAPMbeta Betaforregime1 Betaforregime2 Thresholdestimate

Small 1.0163 0.9093 1.2350 8.80 Decile2 1.1604 1.0772 1.1662 8.42 Decile3 1.1434 1.0453 1.2060 9.33 Decile4 1.1262 1.0401 1.1874 9.33 Decile5 1.1380 1.0684 1.1884 9.33 Decile6 1.0713 1.0330 1.2516 17.69 Decile7 1.0673 1.0109 1.1067 9.33 Decile8 1.0795 1.0478 1.0919 6.07 Decile9 1.0101 0.9723 1.0190 9.33 Big 0.9449 0.9762 0.9223 9.33 SMB 0.0716 −0.0616 0.3026 9.33 PanelB:10B/Mportfolios

CAPMbeta Betaforregime1 Betaforregime2 Thresholdestimate

High 1.0557 0.9776 1.0831 9.58 Decile2 0.9864 0.9357 1.0513 9.58 Decile3 0.9716 0.9247 1.0606 9.58 Decile4 0.9512 0.9038 1.0399 9.58 Decile5 0.9467 0.8687 1.0648 11.10 Decile6 0.9029 0.8308 1.0127 11.10 Decile7 0.8487 0.7983 0.8906 10.92 Decile8 0.8445 0.7425 0.9266 15.00 Decile9 0.9228 0.8373 0.9944 10.78 Low 1.0501 1.1249 0.9892 9.33 HML 0.0059 −0.1514 0.0930 9.33

ThistablereportstheunconditionalCAPMbetas,thethresholdbetaestimateswithrespecttolowandhighvolatilityregimes, andtheirassociatedthresholdvolatilityestimates,proxiedbyS&P500at-the-moneystraddlereturns.PanelsAandBpresent resultsforportfoliossortedwithrespecttomarketcapitalizations,andbook-to-marketratios,respectively.SMBisaportfolio thatislonginstocksinthesmallestdecileandshortinstocksinthebiggestdecile.HMLisaportfoliothatislonginstocks whichareinthehighestB/MdecileandshortinstockswhichareinthelowestB/Mdecile.Thesamplecoverstheperiodfrom January1986toDecember2012(324months).Regime1(2)correspondstolow(high)uncertaintyaboutvolatilityregimes wheremonthlyRVIXislower(higher)thantheestimatedthresholdlevel.

4.1. Therelationbetweenaggregatevolatilityexpectationsandbeta

Havingdetectedsigniﬁcantregimeshiftsinbetasformostoftheportfolios,weproceedtotestthe

magnitudeofthischange,andestimateassetbetasandtheirassociatedthresholdparametersduring

uncertainandcalmperiodsaboutexpectedaggregatevolatility.Table4reportsthestaticCAPMbetas,

betasestimatedviatheV-CAPMincalm(regime1)andhighuncertainty(regime2)regimes,together withthethresholdestimateofRVIX,whichdeterminesthechangeinuncertaintyaboutaggregate volatilityexpectations,above(orbelow)whichinvestorsre-assessastock’sriskiness.

BeforegoingintodetailedanalysisofportfoliobetasTable4,lookingatthelastcolumnofPanelA, onecanseethattheestimatedthresholdlevelofRVIXisverystableacrosssizeportfolios,whichis estimatedat9.33in7ofthe11cases31_._Given_that_RVIX_can_take_on_any_positive_real_number,_this con-sistentlevelofthethresholdestimateafﬁrmstherobustnessofthethresholdestimationprocedure, theproposedmodel,andthechosenthresholdparameter,RVIX.WearguethatthestabilityofRVIX acrossportfoliossignalstothedegreeofuncertaintythatthemarketviewsascriticalregarding aggre-gatevolatilityexpectations.WhenrangeofVIXindexinagivenmonthisbelow9.33,theuncertainty

31_The_threshold_levels_of_17.69_and_6.07_for_deciles₆_and₈_might_seem_as_big_deviations_from_9.33_at_first_sight,_however, notethatthesearethetwoportfolioswherethesup-LMtestwasunabletodetectsignificantregimechanges.Thisisalsothe casefordecile8ofbook-to-marketsortedportfolios,whichisdetectedasaportfoliowithinsignificantregimeshiftandhasa relativelyhighthresholdestimateof15.00.

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inthemarketistolerableandbetasremainunaffected.However,whenRVIXismorethan9.33,this indicatesthatuncertaintyinthemarketregardingtheevolutionofexpectedaggregatevolatilityhas increased,andhenceinvestorsupdatetheirinformationsetandrisk-returndynamicswithrespectto thisinformation,whichisalsoreﬂectedinbetasaccordingly.

Next,adetailedanalysisofcolumns3and4ofPanelArevealsimportantinsightsabouthowthe riskinessofdifferentsizesortedportfolioschangesfromonevolatilityregimetotheother.Wenote signiﬁcantchangesinbetariskofsizesortedportfolios.Inparticular,betasofsmallstockportfolios increaseconsiderablyattimesofhighuncertaintyaboutexpectedaggregatevolatility.Furthermore, itisonlythebiggestdecileportfolio,whichexhibitsadecreaseinitsbetaduringhighuncertainty episodes.

Theabovefindingsimplythatinvestorsre-assesstheriskinessofsizesortedportfolioswhenrange oftheVIXindexisabove(orbelow)thethresholdlevelof9.33.Forexample,whenaggregatevolatility isexpectedtobevolatilesignificantly(i.e.whenRVIXisabovethethreshold),investorsre-estimate thebetaforthesmallestdecileportfolio,andupdateitfrom0.91incalmperiodsto1.24inuncertain periods.Similarly,theriskinessofthebiggestdecileportfoliochangeswhentheRVIXisabove(or below)thethresholdlevelof9.33.Morespecifically,thebetaforthebiggestportfoliodropsfrom0.98 incalmvolatilityperiodsto0.92inuncertainperiods.Furthermore,thebetadifferentialbetweenthe smallestandbiggestportfolios(SMB)increasesfrom−0.06inthelowexpectedvolatilityregimeto 0.30inthehighuncertaintyregimeaboutexpectedaggregatevolatility.

Ourﬁndingsimplythatthesensitivityofanasset’sreturnwithrespecttothelevelofuncertainty regardingexpectedaggregatevolatilityisanimportantdeterminantofanasset’sriskiness.Thishas clearimplicationsonpricingandportfolioallocation.Forexample,byhavingalowercovariancewith themarketattimesofhighuncertaintyaboutmarketvolatility,biggestdecileportfoliotendstolose lessthananyothersize-basedstrategyduringvolatileperiods.Alsogiventhatvolatileepisodes usu-allycoincidewithdownwardmarketmovesandrecessions,astrategyinvestedinthebiggestdecile portfolioappearstoberelativelylessriskyforrisk-averseinvestors,whoarereluctanttolosewealth duringthosetimes.Thisimpliesademandforbigstocks,thuspushingtheirpricesupandresulting inloweraveragereturns.Similarly,theriskofsmallstockportfoliosgoesupwhennear-term aggre-gatevolatilityisexpectedtobevolatile.Becauseuncertaintyandincreasesinaggregatevolatilityare mostlyassociatedwithbadmarketconditionsanddeteriorationsininvestorwealth,bycorrelating highlywiththemarketattimesofhighuncertaintyaboutaggregatevolatility,smallstocksareviewed asriskierattimeswhenextradollaroflossismuchmoreimportant.

PanelBofTable4offerssimilarresultsforportfoliossortedwithrespecttobook-to-marketratios. Valueportfolioshaveconsistentlyhigherbetasattimesofhighuncertaintyaboutexpectedvolatility, whereasitisonlythegrowthportfoliowhosebetadecreasesduringthosetimes.Theresultsindicate signiﬁcanttimevariationintheriskassessmentsofvalueandgrowthportfolioswithrespectinvestors’ expectationsaboutuncertaintyinnear-termaggregatevolatility.Investorsviewvaluestocksmuch riskierbecausetheyhaveahighercorrelationwiththemarketattimesofhighuncertaintyabout aggregatevolatility.Similarly,aportfoliostrategyingrowthstockstendstobelessriskyatthose times.TheresultsareinlinewithLettauandLudvigson(2001)andPetkovaandZhang(2005),whoalso documenttimevariationinriskinessandexpectedreturnsofvalueandgrowthstocks,inconditional CCAPMandconditionalCAPMsettings,respectively.

4.2. Therelationbetweenaggregatevolatilityexpectationsandrisk-adjustedreturns

Thedocumentedevidencesofarindicatesthatassetbetaschangesignificantlybetweendifferent volatilityregimes,dependingonwhetherinvestorsexpectsignificantuncertaintyaboutaggregate volatilityornot.Furthermore,theproposedV-CAPMrevealsadistinctivepatternregardingchangeof betariskamongdifferentassetclasses.Moreparticularly,smallmarketcapitalizationandhigh book-to-market(value)portfoliosbecomeriskierattimesofhighuncertaintyaboutaggregatevolatility expectations.Ontheotherhand,bigmarketcapitalizationandlowbook-to-market(growth)portfolios becomelessriskyatthosetimes.Thefindingsofourmodelofferapotentialremedytothestatic CAPManditsfailureinexplainingthewelldocumentedsizeandvaluevs.growthanomalies.Inorder toexaminetherobustnessoftheproposedvolatility-basedtime-varyingbetariskexplanation,and

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Table5

ComparisonofJensen’salphas.

PanelA:10Sizeportfolios

˛CAPM ˛V-CAPM,Regime1 ˛V-CAPM,Regime2 Small 0.0636(0.25) 0.5601(1.98**₎ _−2.1113_(−4.47***₎ Decile2 −0.0221(−0.11) 0.3322(1.37) −1.5210(−3.50***₎ Decile3 0.0680(0.41) 0.3908(1.96*₎ _−1.1887_(−2.85***₎ Decile4 −0.0177(−0.12) 0.2389(1.32) −0.9651(−2.48**₎ Decile5 0.1753(0.58) 0.2750(1.80*₎ _−0.6550_(−2.22**₎ Decile6 0.1089(0.94) 0.2543(2.00) −0.4421(−1.50) Decile7 0.1521(1.43) 0.3319(3.02***₎ _−0.5203_(−1.77*₎ Decile8 0.0938(0.92) 0.2158(1.94*₎ _−0.4439_(−1.86*₎ Decile9 0.1147(1.49) 0.1997(2.35**₎ _−0.2266_(−1.25) Big 0.0111(0.20) −0.0792(−1.27) 0.3373(2.37**₎ SMB −0.3088(−0.85) 0.2543(1.33) −2.6881(−4.48***₎ PanelB:10B/Mportfolios

˛CAPM ˛V-CAPM,Regime1 ˛V-CAPM,Regime2 High 0.2306(0.94) 0.5930(2.26**₎ _−1.4147_(−3.40***₎ Decile2 0.2241(1.40) 0.4556(2.34**₎ _−0.6294_(−2.27**₎ Decile3 0.1143(0.61) 0.4352(2.29**₎ _−1.1103_(−2.54**₎ Decile4 0.2190(1.33) 0.3595(1.75*₎ _−0.3378_(−1.14) Decile5 0.0443(0.35) 0.1756(1.16) −0.1842(−0.63) Decile6 0.1256(0.89) 0.3015(1.77*₎ _−0.4080_(−1.21) Decile7 0.1060(0.73) 0.3309(1.95*₎ _−0.7462_(−2.50**₎ Decile8 0.1566(1.58) 0.1791(1.75*₎ _0.1433_(0.49) Decile9 0.0547(0.62) 0.0167(0.15) 0.3516(1.61) Low −0.0222(−0.18) −0.2426(−1.88*₎ _0.7474_(2.66**₎ HML 0.0600(0.17) 0.4331(2.53**₎ _−2.4016_(−4.10***₎ ThistablereportsJensen’salphasfortheunconditionalCAPMandforthethresholdvolatilitymodel(V-CAPM)withrespectto lowandhighvolatilityregimes.ThesamplecoverstheperiodfromJanuary1986toDecember2012(324months).Regime1 (2)correspondstolow(high)uncertaintyaboutvolatilityregimeswheremonthlyRVIXislower(higher)thantheestimated thresholdlevel.Thenumbersinparenthesesdenotetheassociatedt-statisticswithNewey-Westcorrectedstandarderrors.***, **,and*denotesigniﬁcanceat1%,5%,and10%level,respectively.

toseewhetherinvestors’expectationsaboutuncertaintyinaggregatevolatilityhasasimilar

time-varyingeffectonrisk-adjustedreturns,wenextcompareJensen’salphasandSharperatioswithinthe

fullsample,andincalmandhighuncertaintyregimesdeterminedbythethresholdlevelofRVIX.

4.2.1. ComparisonofJensen’salphas

Itiswell-documentedthatsmallandvaluestocksonaverageproducesigniﬁcantlyhigherreturns

thantheirlargeandgrowthcounterparts32_._However,_looking_at_Jensen’s_alphas_and_betas_of_different

sizeandbook-to-marketportfoliosinTable5,weconﬁrmpreviousstudiesthatstaticCAPMisunable toofferarisk-basedexplanationtotheseabnormalreturns.Ontheotherhand,alphasimpliedby differentvolatilityregimeshelpusuncoveranimportantaspectofsizeandvaluevs.growthpuzzles, offeringavolatility-basedtime-varyingriskexplanation.

Weinitiallydocumentthatsizeandvaluestrategiespayoffattimesoflowexpectedvolatility yieldingsignificantandpositiverisk-adjustedreturns.However,thetrade-offforthesestrategiesis thattheyhaveextremelybad(significantandnegative)risk-adjustedreturnsattimesofhigh uncer-taintyaboutexpectedvolatility.Forexample,astrategyinvestedinthesmallestdecileportfolioearns anaveragerisk-adjustedreturnof56basispointsduringcalmmonths,whereasthesamestrategy yieldsarisk-adjustedreturnof-211basispointsinmonthswhenuncertaintyaboutvolatilityishigh. Similarly,astrategyinvestedinthehighestbook-to-market(value)portfolioearnsanaverage risk-adjustedreturnof59basispointsduringcalmmonths,butyieldsanaveragerisk-adjustedreturn 32_Although_excess_returns_on_small_stocks_over_big_stocks_have_been_disappearing_during_the_last_two_decades,_excess_returns onvaluestocksovergrowthstockshavebeensignificantlypersistentoveryears.

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of−142basispointsinmonthswhenuncertaintyaboutvolatilityishigh.SMBandHMLstrategies alsoyieldsimilarandsigniﬁcantrisk-adjustedreturnsovercalmanduncertaintyperiods.Onthe con-trary,althoughstrategiesinbiggestandlowestbook-to-market(growth)portfoliosdisappointtheir investorsincalmmonthswithaveragerisk-adjustedreturnsof8and24basispoints,respectively, theyyieldpositiveandsigniﬁcantrisk-adjustedreturnsinmonthswhenuncertaintyaboutvolatility ishigh(24and75basispoints,respectively).

4.2.2. ComparisonofSharperatios

Next,welookatanotherpopularmeasureofrisk-adjustedreturnproposedbySharpe(1966,1975). Sharperatioisacommonlyusedmeasuretotracktheperformanceofmutualfundsanditcanbeeasily appliedtomeasurethereward-to-variabilityofanyinvestmentassetorportfolio.Byscalinganasset’s excessreturntothestandarddeviationofexcessreturnsontheasset,itisanidealwayofmeasuring ofreward-to-variabilityofamanagedfundandthesensitivityofreturnsonaninvestmentclassora tradingstrategyperunitofrisktaken33_._The_measure_is_model_free,_hence_it_provides_an_indirect_test fortherobustnessofourchosenvolatilityparameterRVIX,aswewillcomparetheSharperatiosoftest assetswithinthewholesamplewiththoseobtainedintwodifferentvolatilityregimesdeterminedby RVIX.Analyzingreward-to-variabilityratiosindifferentvolatilityregimeswillgiveusfurtherinsight abouttherisk-returndynamicsofthetestassetswithrespectinvestors’expectationsregardingthe uncertaintyabouttheevolutionofnear-termvolatility.

LookingatPanelAofTable6,onecandetectnoclearpatterninSharperatiosofportfoliossorted withrespecttomarketcapitalizationswithinthefullsample.Onecanevensaythatduringthesample period,aninvestmentstrategybasedonstocksinthesmallestsizedecilecommandsalowerreward perunitofrisktakenasopposedtoastrategybasedonstocksinthehighestdecile,whichisnot consistentwitharationalrisk-basedexplanation.Ontheotherhand,whenwedecomposethesample intotwovolatilityregimesdeterminedbytheRVIX,weseedifferentrisk-returndynamicsacrosssize sortedportfoliosindifferentvolatilityregimes.Incalmperiods,thestrategyinthesmallestdecile port-foliocommandsahigherreward-to-variabilityratiocomparedtothebiggestdecileportfolio(0.2727 vs.0.2250),howeverinperiodswhenuncertaintyaboutvolatilityishigh,thesituationisreversed, investorsexperienceamuchworsereward-to-variabilityratioforthesmallestdecileportfolioagainst thebiggestdecile(−0.4513vs.−0.1727).ThisdifferentpatterninSharperatiosisalsoconsistentwith ourpreviousresultsdocumentingsigniﬁcantdifferencesinbetasandJensen’salphasofthose strate-giesandexplainswhyinvestorswouldwanttobecompensatedfortheextrariskthattheyaretaking byinvestinginsmallstocks.

WeobserveasimilarpatternfortheSharperatiosofportfoliossortedwithrespecttobook-to marketratios.AlthoughthereisnotasigniﬁcantinSharperatiosinthefullsample,wedocument thatvalueportfolioscommandhigher(lower)reward-to-variabilityratioscomparedtogrowth port-foliosincalm(uncertain)periods,offeringacoherentvolatility-basedrisk-returnexplanationtothe empiricallydocumentedvaluevs.growthanomaly.Theresultsconﬁrmourhypothesisthatmarket’s expectationofuncertaintyinaggregatevolatilityisanimportantdeterminantofinvestors’assessment ofrisk-returndynamics.Changesinbetasandrisk-adjustedreturnshelpusuncoverwhysmalland valuestocksonaverageearnhigherreturnsthantheirbigandgrowthcounterparts.Byconditioning assetreturnsusinganovelforward-lookingvolatilitymeasure(RVIX),whichsummarizesinvestors’ expectationsabouttheuncertaintyofnear-termaggregatevolatility,theproposedV-CAPMoffersa volatility-basedtime-varyingriskexplanationtothesizeandvaluevs.growthanomalies.

4.3. Cross-sectionalanalysis

Thetime-seriesanalysesattheportfoliolevelintheﬁrstpartindicatedistinctexposureofsize andbook-to-marketratiosortedportfoliostouncertaintyaboutaggregatevolatilityexpectations, whichmanifestsitselfwithsigniﬁcantchangesinportfoliobetasduringlowandhighuncertainty

33_The_excess_return_on_the_asset_can_be_on_any_benchmark_such_as_the_S&P₅₀₀_returns_or_the_risk-free_rate._As_in_most_studies, wechoosereturnsinexcessoftherisk-freeratetomeasureanasset’sexcessreturn

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Table6

ComparisonofSharperatios.

PanelA:10Sizeportfolios

Sfull full SRegime1 Regime1 SRegime2 Regime2 Small 0.1047 6.1800 0.2727 5.2697 −0.4513 8.6296 Decile2 0.0995 6.4755 0.2486 5.5158 −0.3761 9.4052 Decile3 0.1197 6.0513 0.2833 4.9713 −0.3423 9.3709 Decile4 0.1073 5.8601 0.2605 4.8034 −0.3200 9.2201 Decile5 0.1264 5.7654 0.2774 4.7388 −0.2899 9.1022 Decile6 0.1369 5.2880 0.2894 4.3103 −0.2692 8.4899 Decile7 0.1475 5.1879 0.3159 4.1494 −0.2799 8.4778 Decile8 0.1374 5.1926 0.2912 4.2296 −0.2718 8.3341 Decile9 0.1451 4.7889 0.2965 3.8871 −0.2493 7.7732 Big 0.1238 4.4706 0.2250 3.8149 −0.1727 6.9428 SMB −0.0664 3.2503 0.0570 3.2547 −0.5456 2.8644 PanelB:10B/Mportfolios

Sfull full SRegime1 Regime1 SRegime2 Regime2 High 0.1376 6.0808 0.3009 5.1259 −0.3601 8.9403 Decile2 0.1548 4.8706 0.3275 3.9062 −0.2871 7.8923 Decile3 0.1305 4.5919 0.3300 3.5221 −0.3430 7.6930 Decile4 0.1564 4.5171 0.3034 3.7634 −0.2560 7.0277 Decile5 0.1240 4.7423 0.2702 3.7813 −0.2403 7.9366 Decile6 0.1389 4.6370 0.2981 3.7196 −0.2668 7.6018 Decile7 0.1360 4.8830 0.3072 3.9252 −0.3128 7.8432 Decile8 0.1503 4.6758 0.2802 3.8612 −0.1958 7.5451 Decile9 0.1306 4.7558 0.2433 3.9492 −0.1732 7.6979 Low 0.1125 5.1623 0.1829 4.5820 −0.1185 7.5777 HML 0.0180 3.0837 0.0943 2.9435 −0.3368 3.5854 Thistablereportsportfolioex-postSharperatiosandstandarddeviationsforthefullsampleandtwosubsamplesrepresenting twodifferentvolatilityregimes.Regime1(2)correspondstolow(high)uncertaintyaboutvolatilityregimeswheremonthly RVIXislower(higher)thantheestimatedthresholdlevel.PanelsAandBpresentsresultsforportfoliossortedwithrespectto marketcapitalizations,andbook-to-marketratios,respectively.SMBisaportfoliothatislonginstocksinthesmallestdecile andshortinstocksinthebiggestdecile.HMLisaportfoliothatislonginstockswhichareinthehighestB/Mdecileandshort instockswhichareinthelowestB/Mdecile.ThesamplecoverstheperiodfromJanuary1986toDecember2012(324months).

periodsaboutaggregatevolatility.However,itisimportanttonotethatstockscanexhibitsigniﬁcant

cross-sectionalvariationwithineachportfolio.Thereforeeventhoughtime-seriesanalysisatthe

port-foliolevelpointtowardsRVIXbeingapotentiallyimportantconditioningvariableinexplainingstock

returns,thisexplanatorypowermightresultfromstocks’othercharacteristics.Inthenextsection,

weexaminewhethercross-sectionaldifferencesinbeta-returnrelationshipareattributabletoRVIX

attheindividualstocklevel,andwhetherbetasimpliedbyhighvs.lowuncertaintyaboutaggregate

volatilityisapricedriskfactorinthecross-section.

4.3.1. UnivariateportfoliosortsbasedonthresholdRVIXbetas

WestartwithexaminingwhetherstockbetasestimatedviathresholdlevelofRVIXcanpredict

thecross-sectionaldifferencesintheirreturns.FollowingAngetal.(2006a,b),weestimateEq.(3)

usingmonthlyregressionwindowswithdailydata.Inlinewithourpreviousresults,weusethemost commonlyobservedRVIXthresholdlevel toidentifylow(high)expectedvolatilityperiods.More particularly,weidentifymonthsinwhichthethresholdparameterRVIXisgreater(less)than9.33 asperiodsofhigh(low)uncertaintyaboutexpectedaggregatevolatility34_._We_denote correspond-ingbetasasˇUNC andˇCALM,respectively.ThesampleistheuniverseofCRSPstockscoveringall

34_We_further_tried_six_other_threshold_levels_of_RVIX_ranging_from_8.42_to_11.10,_which_coincide_with_the_next_commonly observedRVIXthresholdlevelsinourtestsafter9.33.Theresultsarerobusttodifferentthresholdlevelsfoundinportfoliolevel analyses.

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NYSE/AMEX/NASDAQcommonstockswithsharecodes10and11.Theregressionsareestimatedeach monthfromJanuary1986toDecember2010.

Wenextconductportfolio-levelanalysistoinvestigatecross-sectionalpredictivepowerofˇUNC

andˇCALMForeachmonth,fromFebruary1986toDecember2010,stocksaresortedintodecile

port-foliosbasedontheirˇUNCandˇCALM.Ourportfolioformationexerciseusesinformationavailableonly

asoftheformationdate.Henceitavoidspotentiallook-aheadbiasintheestimationofbetas.Decile 1(10)containsstockswiththelowest(highest)betas.Next-monthpost-rankingportfolioreturnsare calculatedusingbothequally-andvalue-weightedweightingschemes,andtheprocedureisrepeated eachmonth.Table7reportsnext-monthreturns,CAPMandFamaandFrench(1993)3-factoralphas ofˇUNCandˇCALM.sorteddeciles.

Univariateportfoliosortsindicateanalmostmonotoneandnegativerelationshipbetweenthe betasandnext-monthaveragereturnswhenuncertaintyaboutexpectedaggregatevolatilityishigh. PortfolioofstockswithlowestˇUNC(portfolio1)earns1.72%permonth,whereasreturnonthe

port-folioofstockswithhighestˇUNC(portfolio10)is1.16%permonth.Thespreadportfoliowhichislong

inthehighestˇUNCstocksandshortinthelowestˇUNCstocks(10−1)losesonaverage0.72%per

monthwithat-statisticof−2.92.Nextmonth’srisk-adjustedreturns(CAPMandFF3-factoralphas) aswellasvalue-weightedreturns(bothrawandrisk-adjusted)alsoimplyanegativebetanext-month stockreturnrelationshipduringperiodsofhighuncertaintyaboutexpectedaggregatevolatility,with alphasandrawreturnsofthespreadportfoliorangingfrom−0.49%to−0.73%withalmostallbutone beingsigniﬁcantat5%level.

Ontheotherhand,ourproposedmodelisabletoestablishapositivebetanext-monthstockreturn relationshipduringperiodsoflowuncertaintyaboutexpectedaggregatevolatility.Inparticular, port-folioofstockswithlowestˇCALM(portfolio1)earns1.84%permonth,whereasreturnontheportfolio

ofstockswithhighestˇCALM(portfolio10)is2.44%permonth.Thespreadportfoliowhichislong

inthehighestˇCALMstocksandshortinthelowestˇCALMstocksearnsonaverage0.60%permonth

whichissignificantat5%level.Althoughvalue-weightedreturnsfurtherconfirmapositivebetastock returnrelationshipatperiodsoflowuncertaintyaboutaggregatevolatility,nextmonth’srisk-adjusted returns(CAPMandFF3-factoralphas)ofthespreadportfolioarerelativelylowerandinsignificant. Overall,betasimpliedbythethresholdlevelofRVIXcaptureuncertaintyaboutexpectedaggregate volatilityindicatingthatcross-sectionofexpectedreturnsaresystematicallyrelatedtothelevelof uncertaintyaboutaggregatevolatility.

4.3.2. Fama–MacBeth(1973)testsforriskpremia

Thereisnowaconsensusontimevariationinmarketrisk.TheconditionalCAPMisanattemptto capturethisvariation.However,Ghysels(1998)showsthattheconditionalCAPMisunabletospecify timevariationaccurately,leadingtohigherpricingerrorscomparedtotheunconditionalCAPM.In viewofthesefindings,webelievethatitiscrucialtounderstandthetruedynamicsoftimevariationin betariskandincorporatethisdynamicsinthepricingmodel.Ourpreviousfindingsestablishthatbeta riskexhibitssignificantchangestriggeredbyshiftsininvestors’expectationsregardingtheevolution ofnear-termaggregatevolatility.Hence,thenextnaturalstepwouldbetoanalyzewhetherthisrisk ispricedinthecross-section.

WeemploystandardFamaandMacBeth(1973)two-passregressionmethodology.Thefullmodel tobetestedis rj_i,t=˛jt+ˇjMKT,i,tjMKT,t+ K

k=1 ˇj_k,i,tk k,t+εji,t, j=0,1,2 (7)

where’srepresentunconditionalpricesofriskforvariousloadingsandcharacteristics,andj=0,1, and2representfullsample,lowandhighuncertaintyaboutexpectedvolatility,respectively.Inline withourpreviousresults,weusethemostcommonlyobservedRVIXthresholdleveltoidentifylow (high)uncertaintyperiods.Moreparticularly,weidentifymonthsinwhichthethresholdparameter RVIXisgreater(less)than9.33asuncertain(calm)periodsaboutexpectedaggregatevolatility.

Intheﬁrstpass,betaloadingsareestimatedateachmonthusingdailyobservations.Inthesecond pass,across-sectionalregressionisruneachmonth,withbetaloadingsobtainedfromtheﬁrstpass