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Idiosyncratic Riskand the Cross-Section of Stock Returns

February 2011

Rene GarciaProfessor of Finane, EdhEc Business Soolan resear fellow at cIREQ an cIRANO

Daniel Mantilla-GarciaPd in Finane caniate, EdhEc Business Soolan Resear Assistant at EdhEc-Risk Institute

Lionel MartelliniProfessor of Finane, EdhEc Business Soolan Sientifi diretor of EdhEc-Risk Institute

with the support of



AbstractIiosynrati volatility as reeive onsierable attention in te reent finanial literature.

Weter average iiosynrati volatility as reently risen, weter it is a goo preitor for

aggregate market returns an weter it as a positive relationsip wit expete returns in te

ross-setion are still matters of ative ebate. We revisit tese questions from a novel perspetive,

by taking te ross-setional variane of stok returns as a measure of average iiosynrati

variane. Two key avantages of tis measure are its moel-free nature an its observability at

any frequeny, wi allows us to present new results on te properties of aily iiosynrativolatility series. Troug entral limit arguments, we formally sow tat te ross-setional

ispersion of stok returns an be regare as a onsistent an asymptotially effiient estimator

for iiosynrati volatility. We empirially onfirm tat te ross-setional measure provies a

very goo proxy for average iiosynrati risk as implie by stanar asset priing moels an

tat it preits well aggregate returns, espeially at te aily frequeny. Te preitability power

of iiosynrati risk is furter inrease wen aing a measure of ross-setional skewness

to te ross-setional variane fator. We finally provie eviene tat iiosynrati risk is a

positively reware risk fator.

We tank Miel crouy, Robert Engle, Stépane Gregoir, Abraam Lioui, harrison hong, Frééri

Smaja, Georges Tauen, Stepane Ty, Vijay Vaiyanatan, Volker Ziemann, as well as seminaran onferene partiipants at EdhEc Business Sool, te Finanial Eonometris conferene

(Toulouse), te Finanial Eonometris Worksop (SMU), te conferene in Asset an Risk

Management in te Aftermat of te Finanial crisis (hEc Lausanne), for useful isussions an

omments. Tis resear as benefite from te support of te cair "Prouits Struturés et

Prouits dérivés", Féération Banaire Française.

EdhEc is one of te top five business sools in Frane. Its reputation is built on te ig quality

of its faulty an te privilege relationsip wit professionals tat te sool as ultivate

sine its establisment in 1906. EdhEc Business Sool as eie to raw on its extensive

knowlege of te professional environment an as terefore fouse its resear on temes tat

satisfy te nees of professionals.

EdhEc pursues an ative resear poliy in te fiel of finane. EdhEc-Risk Institute arries out

numerous resear programmes in te areas of asset alloation an risk management in bot te

traitional an alternative investment universes.

copyrigt © 2011 EdhEc



3

1 - This amounts to imposing unit beta restrictions in an industry-market model.

2 - This is also the approach followed in Ang et al. (2006).

3 - Goyal and Santa-Clara (2003) argue infor mally that their measure can be interpreted as a measure of cross-sectional dispersion of stock returns, but do not establish a formal link between the two. In the

practitioners' literature (see DiBartolomeo (2006)), cross-sectional dispersion of returns is called variety and is used in risk management and performance analysis.

1. IntroductionTe reent finanial literature as pai onsierable attention to iiosynrati volatility. campbell

et al. (2001) an Malkiel an Xu (2002) oument tat iiosynrati volatility inrease over

time, wile Brant et al. (2009) sow tat tis tren ompletely reverse itself by 2007, falling

below pre-1990s levels an suggest tat te inrease in iiosynrati volatility troug te 1990s

was not a time tren but rater an "episoi penomenon". Bekaert et al. (2008) onfirm tat

tere is no tren bot for te Unite States an oter evelope ountries. A seon fat about

iiosynrati volatility is also a soure of ontention. Goyal an Santa-clara (2003) put forwartat iiosynrati volatility as foreasting power for future exess returns, wile Bali et al. (2005)

an Wei an Zang (2005) fin tat te positive relationsip is not robust to te sample osen.

Finally, wile some eonomi teories suggest tat iiosynrati volatility soul be positively

relate to expete returns, Ang et al. (2006) fin tat stoks wit ig iiosynrati volatility

ave low average returns.

An unerlying issue in all tese stuies is te measurement of iiosynrati volatility. campbell et

al. (2001) use a value-weigte sum of iniviual firm iiosynrati varianes, ompute as te

varianes of resiuals of ifferenes between iniviual firm returns an te return of an inustry

portfolio to wi te firm belongs.1 In aition to tis measure, Bekaert et al. (2008) use also teiniviual firm resiuals of a stanar Fama an Fren tree-fator moel to ompute a value-

weigte aggregate iiosynrati volatility.2

We revisit te issues regaring te ynamis an foreasting power of iiosynrati variane by using

instea te ross-setional ispersion of stok returns. Troug entral limit arguments, we provie

te formal onitions uner wi te ross-setional variane (cSV) of stok returns asymptotially

onverges towars te average iiosynrati variane.3 One key avantage of tis measure is obviously

its observability at any frequeny, wile te previous approaes ave use montly measures base

on time series of aily returns. A seon important feature is tat tis measure is moel-free, sine

we o not nee to obtain resiuals from a partiular moel to ompute it.

We onfirm empirially tat te ross-setional variane is an exellent proxy for te iiosynrati

variane obtaine from te cAPM or te Fama-Fren moels, as one in te previous literature.

correlations between te cSV measure an te moel-base measures estimate montly, are

always above 99%, weter we onsier equally-weigte or apitalization-weigte measures of

iiosynrati variane. We also estimate a regime-switing moel for cSV time series at bot aily

an montly frequenies an fin remarkably oerent results in terms of parameter estimates.

If we were to buil a aily series of moel-base iiosynrati variane, we will roll a winow of

one-mont of aily ata, wi will result in a very persistent time series. We onstrut su a

aily series but oul not fin any regimes. Tis reinfores te usefulness of te cSV to aptureiiosynrati volatility at ig frequeny.

Te regime-switing moel iniates learly tat te cSV is ounter-ylial, te ispersion of

returns being ig an quite variable wen eonomi growt subsies. We analyze furter te

relation between cSV an eonomi an finanial variables. In partiular, we fin tat tere exists

a substantial orrelation between te equal-weigte cSV an onsumption growt volatility.

Tis is onsistent wit Teongap (2010) wo provies strong eviene tat onsumption volatility

risk explains a ig perentage of te ross-setional ispersion in average stok returns for te

usual set of size an book-to-market portfolios tat ave been use in tests of asset priing

moels. In intertemporal asset priing moels of Bansal an Yaron (2004), Bollerslev et al. (2009)

an Bollerslev et al. (2009), onsumption growt volatility is a measure of eonomi unertainty,

wi is a prie risk fator tat affets returns, terefore proviing a rationale for te observe

orrelation between cSV an onsumption growt volatility.



On te ebate about preitability of aggregate returns by te iiosynrati variane, we first

verify empirially tat te cSV measure leas to te same onlusions tat oter stuies (in

partiular Goyal an Santa-clara (2003) an Bali et al. (2005)) ave reporte at te montly

frequeny. Ten, we report new results at te aily frequeny. Speifially, we sow tat te

preitive power of iiosynrati volatility is mu stronger bot quantitatively an statistially

at te aily frequeny tan at te montly frequeny. Tis relationsip is robust to te inlusion of

return variane an option-implie variane as aitional variables in te preitive regressions.

We fin tat te relation is mu stronger an stable aross perios between te equally- weigte

measure of aggregate iiosynrati volatility an te returns on te equally-weigte inex tan

for te market-ap-weigte equivalents. Eonomi soures of eterogeneity between firms,

as iverse as tey an be, are better reflete in an equally weigte measure, all oter tings

being equal. Tis argument is onsistent wit previous finings in Bali et al. (2005), wo argue

tat te relationsip between equal-weigte average iiosynrati risk an te market-ap

weigte inex on te sample ening in 1999:12 is mostly riven by small stoks trae in te

NASdAQ. Of ourse, wen te bubble burst, te market apitalization of ot-om small firms was

relatively more affete ausing te relationsip to break own in 2000 an 2001. Tis effet is

not prevalent in an equally-weigte inex, for wi te relationsip remains strong.

however, te frequeny at wi preitive regressions are run as an important impat on

te results, sine at lower frequenies we fin little eviene of preitability for te equally-

weigte measure of cSV . At quarterly an annual frequenies, we fin tat te apitalization-

weigte measure of cSV is a very strong preitor of te aggregate value-weigte returns.

Wen using cSV cW alone as a preitor we obtain remarkable R 2s of 4% an 26% at quarterly an

annual frequenies, respetively. Aing te implie variane brings te R 2s to almost 19% an

29%. In all tese preitability regressions, te sign of te cSV cW variable is negative. We relate

tese results to potential explanations in terms of missing fators, Guo an Savikas (2008), or

ispersion of investors' opinions, cao et al. (2005).

Finally, we unveil an asymmetry in te relationsip between iiosynrati variane an returns

an sow tat te preitive power of speifi risk is substantially inrease wen a ross-

setional measure for iiosynrati skewness is ae as explanatory variable. In fat, tis is

yet anoter key avantage of our measure tat it lens itself to straigtforwar extensions to

iger-orer moments.

Te statistial signifiane of te moments of te ross-setional istribution in tese preitive

regressions of future returns is not te same as te ross-setional priing of stoks or portfolios.

however, as empasize in Goyal an Santa-clara (2003), te two piees of eviene are relate.Using a Fama-MaBet proeure wit several sets of portfolios, we fin support for a positive an

signifiant prie of risk for te exposure to te iiosynrati variane risk. Teoretial rationalizations

of a positive relation between iiosynrati risk an expete returns an be foun in te asset priing

literature. Levy (1978), Merton (1987) an Malkiel an Xu (2002) priing moels relate stok returns

to teir beta wit te market an teir beta to market-wie measures of iiosynrati risk. In tese

moels, an important portion of investors' portfolios may iffer from te market. Teir olings may

be affete by orporate ompensation poliies, borrowing onstraints, eterogeneous beliefs an

inlue non-trae assets tat a bakgroun risk to teir trae portfolio eisions (e.g. uman

apital an private businesses). Tese teoretial preitions are also in line wit campbell et al.'s

(2001) argument tat investors oling a limite number of stoks oping to approximate a well-

iversifie portfolio woul en up being affete by anges in iiosynrati volatility just as mu

as by anges in market volatility. More reently, Guo an Savikas (2008) argue tat anges in

average iiosynrati volatility provie a proxy for anges in te investment opportunity set an

tat tis proxy is losely relate to te book-to-market fator.4

4 - Alternative explanations of the relation between idiosyncratic risk and return are the firm's assets' call-option interpretation by Merton (1974) where equity is a function of total volatility as i n Black and

Scholes (1973) as well as Barberis et al. (2001) prospect theory asset pricing model with loss aversion over (owned) individual stock's variance.



Ang et al. (2006) an Ang et al. (2009) fin results tat are opposite to our finings an to tese

teories sine stoks wit ig iiosynrati volatility ave low average returns but annot fully

rationalize tis result. however, huang et al. (2009) fin tat te negative sign in te relationsip

between iiosynrati variane an expete returns at te stok level beomes positive after

ontrolling for return reversals. Similarly, Fu (2009) ouments tat ig iiosynrati volatilities

of iniviual stoks are ontemporaneous wit ig returns, wi ten to reverse in te following

mont.

Te rest of te paper is organize as follows. In Setion 2, we provie a formal argument for

oosing te ross-setional variane of returns as a measure of average iiosynrati volatility,

explore its asymptoti an finite-istane properties, as well as te assumptions bein its use,

an ompare it to oter measures formerly selete in te literature. Setion 3 provies an

empirial implementation of te onept, again in omparison wit oter measures, by stuying

its time-series beavior, outlining te presene of regimes an a ounter-ylial property. In

Setion 4, we provie new results on te preitability of returns by iiosynrati volatility, an

we also exten te analysis to iiosynrati skewness. Setion 5 fouses on te analysis of te

ross-setional relationsip between iiosynrati risk an expete returns. Setion 6 onlues

an a tenial appenix ollets proofs an more formal erivations.

2. The Cross-sectional Variance as a Measure of Idiosyncratic VarianceLet N t be te total number of stoks in a given universe at ay t , an assume wit no loss of

generality a onitional single fator moel for exess stok returns.5 Tat is, we assume tat for

all i = 1,…, N t te return on stok i in exess of te risk-free rate an be written as:

(1)

were F t is te fator exess return at time t , it is te beta of stok i at time t , an ε it is te

resiual, wit E (ε it ) = 0 an ov (F t , ε it ) = 0. We assume tat te fator moel uner onsierationis a strit fator moel, tat is ov (ε it , ε jt ) = 0 for i ≠ j .6

Given T observations of te stok returns an te fator return, one an use te resiuals of te

regression to obtain a measure of te iiosynrati variane of asset i by:

An average measure of iiosynrati variane over te T observations (say a mont) an be

obtaine by averaging aross assets su iniviual iiosynrati variane estimates. Tis is te

approa tat as been followe by most relate papers wit observations of te returns at a

aily frequeny to ompute montly iiosynrati varianes. We propose instea to measure

at ea time t te ross-setional variane of observe stok returns. Using formal entral-

limit arguments, we sow tat, uner mil simplifying assumptions, tis ross-setional measureprovies a very goo approximation for average iiosynrati variane. In ontrast wit most

previous measures of average iiosynrati variane, te cSV offers two main avantages: it an

be ompute iretly from observe returns, wit no nee to estimate oter parameters su as

betas, an it is reaily available at any frequeny an for any universe of stoks.

2.1 Measuring the cross-sectional variance

To see tis, first let be a given weigt vetor proess. Te return on te portfolio efine

by te weigt vetor proess (w t ) is enote by an given by:

(2)

5

5 - Assuming a single factor structure is done for simplicity of exposure only and the results below can easily be extended to a multi-factor setting.

6 - This assumption is made in the single index or diagonal model of Sharpe (1963) and in the derivation of the APT in Ross (1976). It implies that all commonalities are explained by the factor model in

place. One should notice that the very denition of idiosyncratic risk relies precisely on the assumption of orthogonal residuals: assuming that the model is the "true" factor model implies that the "true"

idiosyncratic risk is the one measured with respect to that model, which in turn implies that no commonalities should be left after controlling for the common factor exposure.



We restrit our attention to non-trivial weigting semes, ruling out situations su tat te

portfolio is ompose by a single stok. We also restrit te weigts to be positive at every given

point in time. hene, a weigting seme (w t ) is a vetor proess wi satises 0 < w it < 1 ∀ i, t .

Te ross-setional variane measure is efine as follows.

Definition (CSV): Te cross-sectional variance measure uner te weigting seme (w t ),

enote by , is given by

(3)

A partiular ase of interest is te equally-weighted CSV (or EW cSV), enote by an

orresponing to te weigting seme w it = 1/N t ∀ i, t :

(4)

were is te return on te equally-weigte portfolio.

Anoter weigting seme of interest is te ap weigting seme. If we enote by it be

te market apitalization of stok i at te beginning of te mont orresponing to ay t ,

te total market apitalization an te return on te market apitalization-

weigte portfolio, te cap-weighted (CW) (or cW cSV) is efine as:

(5)

were

For any given weigting seme (in partiular EW or cW), te orresponing ross-setional

measure is reaily omputable at any frequeny from observe returns. Tis stans in ontrast

wit te previous approaes tat ave use montly measures base on time series regressions

on aily returns. Te seon important feature of te cSV is its moel-free nature, sine we o

not nee to speify a partiular fator moel to ompute it.7

2.2 A Formal Relationship between CSV and Idiosyncratic VarianceTe following proposition establises a formal link between cSV an iiosynrati variane. It

is an asymptoti result (N t -> 1) obtaine uner te assumptions of omogeneous betas an

resiual varianes aross stoks, i.e. Tese assumptions

will be relaxe below.

Proposition 1 (CSV as a proxy for idiosyncratic variance - asymptotic results):

Assume (omogenous beta assumption) an

(omogeneous resiual variane assumption), ten for any stritly positive weigting seme, we

ave tat:

almost surely. (6)

7 - While Goyal and Santa-Clara (2003) and Wei and Zhang (2005)consider the equally-weighted CSV in conjunction with other measures, they do not provide a thorough discussion about the conditions

under which it can be interpreted as a proxy for idiosyncratic variance nor their empirical validity in the data, as we provide in this paper.



7

Proof See Appenix A.

Tis result is important beause it raws a formal relationsip between te ynamis of te

ross-setional ispersion of realize returns an te ynamis of iiosynrati variane. Note

tat tis asymptoti result ols for any weigting seme tat satisfies

. Of ourse, at finite istane, ifferent weigting semes will generate

ifferent proxies for iiosynrati variane. In te empirial analysis tat follows, we sall

fous on te equally-weigte seme, wile also onsiering te ap-weigte seme

for omparison purposes. Formal justiation for our fous on te equally-weigte semeis provie in te next setion, were we sow tat te EW cSV is te best estimator for

iiosynrati variane witin te lass of cSV obtaine uner a stritly positive weigting

seme.

2.3 Properties of CSV as an Estimator for Idiosyncratic Variance

First, we erive in Proposition 2 te bias an te variane of te cSV as an estimator of iiosynrati

variane. Ten we stuy teir asymptoti limits as te number of firms grows large an onlue

tat te equally-weigte cSV is te best among all positively-weigte estimators.

Proposition 2 (Bias and variance of CSV) :Maintaining the homogeneous beta assumption an the homogeneous

resiual variane assumption , for any stritly positive weighting sheme,

we have that:

(7)

To analyze the variane of the cSV estimator, we further make the assumption of multi-variate

normal resiuals , where Σ ε enotes the variane-ovariane matrix of the resiuals.

Uner this aitional assumption, we obtain:

(8)

Proof See Appenix B for a proof in te sligtly more general ase wen te omogeneous speifi

variane assumption as been relaxe. hene te cSV is a biase estimator for iiosynrati

variane, wit a bias given by te multipliative fator , wi an be easily

orrete for sine it is available in expliit form. In te en, te bias an variane of te cSV

appear to be minimum for te EW seme, wi orrespons to taking w it = 1/N t at ea ate t .

It is easy to see, tat tis bias isappears an te variane tens to zero for te equally-weigte

seme wen te number of stoks grows large, as explaine in te following proposition.

Proposition 3 (Properties of the equally-weighted CSV)

The bias an variane of the EW cSV as an estimator for speifi variane isappear in the limit

of an inreasingly large number of stoks:

Proof See Appenix B for a proof in te sligtly more general ase wen te omogeneous

speifi variane assumption as been relaxe.

Te equally-weigte cSV tus appears to be a onsistent an asymptotially effiient estimator

for iiosynrati variane. As su, it is te best estimator in te lass of cSV estimators efine

uner any positive weigting seme, an it ominates in partiular te ap-weigte cSV as an

7



estimator for iiosynrati variane. If we relax te omogeneous resiual variane assumption,

we obtain tat:

hene, te assumption of omogeneous resiual varianes omes wit no loss of generality. In te

general ase wit non-omogeneous varianes, te cSV simply appears to be an asymptotially

unbiase estimator for te average iiosynrati variane of te stoks in te universe. We alsoave:

were te quantity is an upper boun for te iniviual iiosynrati varianes

(see Appenix B).

We now isuss te impat on tese results of relaxing te omogeneous beta assumption.

2.4 Relaxing the Homogeneity Assumption for Factor LoadingsRelaxing te omogeneous beta assumption involves a bias tat remains stritly positive even

for an innite number of stoks an an equal-weigting seme. We araterize tis bias in te

next proposition in orer to gauge its magnitue for given moels of returns.

Proposition 4 Bias of CSV as an estimator for average idiosyncratic variance in the presence

of heterogenous betas:

Relaxing the assumptions (homogeneous beta assumption) we have, for

any stritly positive weighting sheme:

(9)

where enotes the ross-setional variane of stok betas:

Proof See Appenix c.

Te first term in equation (9) represents te average iiosynrati variane of stoks

witin te universe uner onsieration. Te seon term is te negativebias tat was also present even in te presene omogeneous beta assumptions. If we fous on

te equally-weigte seme, te sum of tese two terms is equal to so

tat te bias isappears in te limit of an inreasingly large number of stoks. Te tir term

in equation (9) represents, on te oter an, an aitional (positive) bias for te

cSV as an estimator of average iiosynrati variane, wi is introue by te ross-setional

ispersion in betas, an wi oes not isappear in te limit of a large number of stoks.

Using te expliit expression provie ere, in setion 3.1 we iretly measure tis beta ispersion

bias using te cAPM an te Fama an Fren tree-fator moel as benmark fator moels.

As we will see, altoug te ross-setional ispersion of betas as a non-negligible magnitue,

one it is multiplie by te square of te return of te market portfolio its relative size wit

respet to te level of iiosynrati risk beomes very small. An extensive analysis of te cSV

in te empirial setion suggests tat te omogeneous beta assumption oes not represent a

material problem for te cSV as an estimator of iiosynrati variane as implie by stanar

asset priing moels (i.e. cAPM an Fama-Fren).



2.5 Competing Measures of Idiosyncratic Risk

In tis setion, we esribe measures tat ave been use in te literature, an wi will be use

for omparison purposes in subsequent setions of te paper. Te stanar approa onsists of

onsiering iiosynrati variane eiter relative to te cAPM an or to te Fama-Fren (FF)

moel (Fama an Fren (1993)):

(10)

were r it enotes te exess return at time t of stok i , XMKT is te exess return on te marketportfolio, SMB is te size fator an hML is te value fator. Te iiosynrati variane for asset

i is te variane of te resiuals of te regression, tat is, . To obtain an estimate for

average iiosynrati variane, Bekaert et al. (2008) an Wei an Zang (2006) use a market

apitalization weigting:

(11)

For omparison purposes we also look at te equally-weigte average of FF iiosynrati variane in

wat follows. An alternative approa to average (mostly ) iiosynrati risk estimation as beensuggeste by Goyal an Santa-clara (2003), wit a measure given by:

(12)

were r i is te return on stok i in ay an d t is te number of traing ays in mont t .8

campbell et al. (2001) propose yet an alternative measure of average iiosynrati variane, uner

a very partiular setting tat allows one to avoi running regressions ea perio.9

however,teir measure is not instantaneous sine a winow of ata is still neee to estimate iniviual

varianes. In wat follows, we o not repeat te analysis wit tis measure beause Bekaert et

al. (2008) ave sown tat it is very losely relate to te measure obtaine from stanar asset

priing moels. In partiular, Bekaert et al. (2008) fin tat te measure of campbell et al. (2001)

an te FF-base one ave a orrelation of 98% an sare most of te same strutural breaks.

3. Empirical ImplementationIn orer to perform an empirial analysis of our measure for iiosynrati risk, we ollet aily US

stok returns (ommon equity sares only) an teir market apitalization from cRSP ata base.Our longest sample runs from July 1963 to deember 2006. We also extrat te FF fators an te

one-mont Treasury bill from Kennet Fren web-site ata library for te same sample perio.

Ea mont, we rop stoks wit missing returns an wit non-positive market apitalization

at te beginning of te mont. Te number of firms varies between 377 an 7293, an remains

greater tan one tousan 75% of te time. Te maximum number of stoks is reae uring

te .om bubble. Ten, we estimate every mont te ap-weigte iiosynrati variane as in

equation (11), as well as te equal-weigte version.10 Similarly, we estimate te ap-weigte

an equal weigte average iiosynrati variane relative to te cAPM. We also estimate te GS

average variane measure as in equation (12) an its ap-weigte version. Finally, we estimate

on a aily basis te equal an ap-weigte versions of te cSV as in equations (4) an (5). In

orer to onstrut te montly series for our ross-setional measures, we estimate te average

of te aily series at te en of ea mont. For omparison purposes we also estimate te

FF-base average iiosynrati variane (EW an cW) on a aily basis using a rolling winow

sample of one mont. We annualize all figures in orer to ompare aily an montly measures.

98 - As in Goyal and Santa-Clara (2003), when the second term makes the estimate negative, it is ignored. This measure has been originally used in French et al. (1987).

9 - They assume that all betas are equal to one and substract industry returns in addition to market returns to control for risk.

10 - We use previous period market capitalization and assume it is constant within the month.



11 - This is measured as in equation (11) with just the market returns with both weighting schemes.

12 - As noted earlier, it would be straightforward to remove the impact of this bias by dividing the CSV measure by the factor , equal to in the EW case.

Following Bekaert et al. (2008), we fit a regime-switing moel to te montly an aily series

in orer to furter ompare te ierent measures. Last, we look at te relation between te cSV

measures of iiosynrati variane an selete eonomi an finanial variables.

3.1 Measuring the CSV bias

Some of te previous resear on iiosynrati volatility as been onute uner te assumption

of omogeneous betas aross stoks (see campbell et al. (2001) an Goyal an Santa-clara (2003)

in partiular). As illustrate in Proposition 4 an isusse in Appenix c, te presene of non-omogeneous betas introues a positive bias on te cSV as an estimator for average iiosynrati

variane, wi is given by te first term in equation (9). We now measure te impat of tis bias

wit respet to te cAPM as a benmark moel.

First, we ompute te bias for every mont in te sample using beta estimates for

ea stok wit bot te equal-weigte an te ap-weigte market returns. To gauge its

importane, we ivie it by te average iiosynrati variane, also measure wit respet to te

cAPM.11

Table 1 presents a summary of te istribution of te time series of ross-setional ispersionof betas, its prout wit te square return of te market portfolio (ene te bias itself) an

te proportion of tis bias wit respet to te average iiosynrati variane at te en of every

mont. Altoug te ross-setional ispersion of betas is sizable, one it is multiplie by te

square return of te market portfolio, te size of te bias remains relatively small. Te meian

of te istribution of , is 0.348% for te equal-weigte seme an 0.351% for te ap-

weigte measure, ompute over te wole sample (July 1963 to deember 2006). Te 97.5

quantiles are 3.24 an 3.47 respetively.

On te oter an, te formal isussion about te properties of te cSV as a measure of

iiosynrati variane on setion 2.4 also unovere te fat tat anoter bias (but negativein sign) oming from te cSV weigting seme onentration is also introue. Proposition 2

preits two properties about tis weigting bias: first, it soul be negative an minimal for an

equally-weigte seme. Seon, it soul be very small for a ig number of stoks. Te beta-

bias ten is more likely to ominate te onentration-bias wen using an equal-weigt seme.

Using te expliit expression for tis bias provie in Proposition 4 we estimate te proportion

of te size of tis weigts-onentration bias wit respet to te average iiosynrati varianes

implie by te cAPM.12 In te last line of te upper an lower panels of Table 1 we report

quantiles of te istribution of tis bias for bot weigting semes. Te orresponing meians

are 0.030% an 0.426% for te EW an cW semes respetively. Sine te bias is of opposite

sign to te beta ross-setional ispersion bias, we nee to assess te resulting overall bias.

We measure te total bias as te interept of a regression of te cSV on te average iiosynrati

variane estimate wit respet to te cAPM or te Fama-Fren tree-fator moel:

(13)

were w t refers to te weigting seme (equal-weigt or market-ap) an moel stans for

eiter te cAPM or te Fama-Fren tree-fator moel.

Table 2 reports summary statistis for regression (13). Te bias of te cSV measure wit respet

to stanar asset priing moels is small in magnitue for bot weigting semes (in te orer

of 10-5). Wile it remains statistially signifiant, we an safely onsier tat te impat of te

bias remains immaterial for any pratial purposes. Anoter interesting fining is te sign of



te bias. For te equal-weigte quantities, te sign of te bias is positive, wile it is negative

for te ap-weigte ones. Terefore, te beta bias ominates te weigting bias for equal-

weigte averages in bot moels. Tis is onsistent wit te preition mae by te teoretial

analysis regaring te relative impat of te weigting-bias for ifferent weigting semes.

Regaring te moel, te bias is larger wen te iiosynrati variane is measure wit respet

to te Fama-Fren moel instea of te cAPM for bot weigting semes, as expete, but its

magnitue remains negligible.

3.2 Comparison with Other Measures

In tis setion we ompare te cSV measure to te afore-mentione, more onventional, measures

of iiosynrati risk (i.e., FF-base, cAPM-base an GS). To obtain tese oter measures, we

nee to re-estimate te relevant fator moel using a rolling winow of one-mont wort of

aily ata to allow for time-variation in beta estimates (or total-variane variation for te GS).

In Table 3, we report summary statistis for te montly time series of annualize iiosynrati

varianes base on 516 observations from January 1964 to deember 2006.13

On te montly series, te annualize means of te equally-weigte cSV, FF-base an

cAPM-base measures are 38.4%, 38.3% an 38.7%, respetively, wile te EW GS variane is34.2%. Te stanar eviations are 8.5%, 8.6%, 8.7% for te cSV, FF-base an cAPM-base

measures an 7.0% for te GS measure. For te ap-weigte version, te cSV, FF an cAPM

iiosynrati variane measures ave an annualize mean of 8.5%, 7.6%, 8.0%, respetively

an te GS measure mean is 11.2%. Te stanar eviations are also loser for te cSV, FF an

cAPM measures tan for GS. Altoug GS argue tat teir measure funamentally onstitutes

a measure of iiosynrati risk, wit te iiosynrati omponent aounting for about 85% of

te total EW average measure, it is stritly speaking an average of total stok variane. Our cSV

measure is very lose to iiosynrati variane measures erive from traitional asset priing

moels, onfirming tat te assumption tat beta omogeneity is not a major problem.

Te ross-orrelation matrix reporte in Table 3 provies furter eviene on te loseness of

te cSV to te oter moel-base measures. correlations are very ig between cSV EW an

cAPM EW (99.93%) an FF EW (99.75%, as well as between cSV cW an cAPM cW (99.48%) an cSV cW

an FF cW (98.56%). Te ig orrelations between te cAPM an te FF measures (99.88% an

99.18% for EW an cW respetively) also iniate tat aing fators oes not rastially affet

te estimation of iiosynrati variane. correlations between te GS measures an te oter

measures are always smaller but remain lose to 90% wen onsiering te same weigting

seme. correlations between measures for ifferent weigting semes are mu lower,

irrespetive of te estimation meto, iniating tat te oie over te weigting seme

is funamentally important for estimating iiosynrati variane, as stresse in our teoretialanalysis in setion 2.

Table 4 provies mean an stanar-eviation estimates for te aily average iiosynrati variane

measures. Te mean of te EW cSV is 38.4%, pratially equal to te mean of EW iiosynrati

variane base on te FF moel. For te ap-weigte measures, te cSV as a sligtly iger

mean tan te FF-base one. For te cSV aily series, te stanar eviation is iger tan for

te FF-base measure for bot weigting semes. Tis is ue to te ifferent nature of te two

series. Te cSV only inlues information from te ross-setion of realize returns, wile te FF

iiosynrati variane is a persistent, overlapping, rolling-winow estimate. Ea aily estimate of

iiosynrati variane for te FF moel iffers from te previous one by only two observations out

of te approximately 21 traing ays inlue in a mont (te first an last ays).

Te smootness of te iiosynrati variane estimates obtaine wit te rolling-winow

metoology is illustrate in Figures 1 an 2, wi plot aily cSV an FF iiosynrati varianes

1113 - In this section of the paper, we start the sample period in January 1964 to allow for direct comparison with Bekaert et al. (2008). In the predictability section, we instead start the sample in July 1963.



for ea weigting seme respetively. It soul also be note tat te estimation of te FF-

measure is omputationally mu more expensive tan for te cSV measure, wi is base on

observable quantities.

Te lower panel of Table 4 presents ross-orrelations for te aily series of iiosynrati variane

measures. Altoug te oeffiients are smaller tan for te montly series, te relationsip

remains strong provie te omparison is one for te same weigting seme: 82.6% an

73.9% for EW an cW measures respetively. Te ifferene wit te montly series orrelationsmay again be explaine by te presene of te smoote estimation proeure inerent to te FF-

base measure. Overall, it appears tat te cSV measure is extremely lose to cAPM or FF-base

measures at te montly frequeny, wen te latter measures suffer from no partiular bias, an

tat te cSV measure appears to be a goo an instantaneous proxy for iiosynrati variane

at te aily frequeny, wen te stanar measures are subjet to artifiial smooting ue to

overlapping ata.

3.3 Extracting Regimes in Idiosyncratic Risk

Bekaert et al. (2008) t a Markov regime-switing moel wit a first-orer autoorrelation

struture (see hamilton (1989)) for te montly series of iiosynrati variane base on te FFmoel. In tis setion, we want to estimate tis moel wit our cSV measure bot at te montly

an aily frequenies. Wile we expet tat te t will be lose to Bekaert et al. (2008) for te

montly series given our previous results on te similarity of te series, we want to verify weter

su a moel provies a similar t for te aily series.

In tis moel, two regimes are inexe by a isrete state variable, s t , wi follows a Markov-

ain proess wit onstant transition probabilities. Let te urrent regime be inexe by i an

te past regime by j an x t be te original iiosynrati variane. In tis parsimonious moel, x t follows an AR(1) moel:

(14)

Te transition probabilities are enote by p = P [s t = 1|s t -1 = 1] an q = P [s t = 2|s t -1 = 2]). Te

moel involves a total of seven parameters, µ 1, µ 2, σ 1, σ 2, φ, p , q.

We first verify tat te cSV an te FF-base measures give te same results for te montly series.

Te estimation results for te montly series of te FF cW , cSV cW , FF EW an cSV EW are reporte in

te upper panel of Table 5. For orresponing weigting semes, te parameters in bot regimes

are similar between te two measures. For bot measures te low-mean, low-variane regime

presents a iger probability of remaining in te same state.

We ten fit te same moel to te aily time series an present te parameter estimates in te

lower panel of Table 5. It soul be stresse tat for our cSV measure, te parameter values

of te average level of iiosynrati variane µ in bot regimes are foun to be quite lose to

te values obtaine wit te montly series. Tis result suggests tat te proess observe at

te aily frequeny is not just a noisy series, but atually aptures te same unerlying proess

observe at te montly frequeny. Tis stans in sarp ontrast wit te FF-base measure,

for wi te maximum-likelioo estimation proeure oul not reognize two regimes wen

aily ata is use, as eviene by te fat tat te parameter values for te mean level of

iiosynrati variane are basially te same for te two regimes. Tis problem, ombine wit

an autoorrelation parameter very lose to one, is likely ause by te overlapping ata problem

present in te aily FF measure, wi orrespons to te smoothing effet mentione in te

previous setion.

In Figures 3 an 4 we plot te filtere probabilities (onitional on information up to time t ) of

remaining in state 1 (ig-mean an ig variane regime), as well as te montly cSV an FF



average iiosynrati variane time series for te cW an EW weigting semes, respetively.14

At te montly frequeny, our measure an te FF-base measure appear to be remarkably lose

for bot te equal-weigte an ap-weigte semes. Also, we fin tat te ates of regime

anges, marke by te filtere probabilities, are te same most of te times for te ap-weigte

an te equal-weigte measures.15 We also fin tat perios in te iger-mean an iger-

variane regime are more persistent for te equally-weigte measure ompare to te ap-

weigte measure (exept uring te te bubble perio).

Overall, our filtere probability series resembles losely te one presente in Bekaert et al. (2008)

for te ap-weigte FF an campbell et al. (2001) measures.16

Te sae areas in Figures 3 an 4, wi time stamp te NBER reession perios, iniate tat

te peaks in te probability of remaining in te ig-mean ig-variane regime oinie most

of te times wit te ontration perios. Terefore, te cSV measure is ounter-ylial, te

ispersion of returns being ig an quite variable wen eonomi growt subsies. In te next

setion, we want to analyze furter te relation between te cSV an oter eonomi an nanial

variables.

3.4 CSV Relation with Economic and Financial Variables

To put tis analysis in te proper ontext, we soul go bak to te very nature of iiosynrati

risk. In an asset priing moel, it represents te risk tat belongs speially to an iniviual firm,

after aounting for te soures of risk tat are ommon to all firms. In te previous setions,

we ave sown tat te ross-setional variane of returns provies a very goo measure of tis

iiosynrati risk, even if it ignores te risk exposures to te usual ommon risk fators su as

te market return or te Fama-Fren fators. Yet we onlue our time series analysis of cSV by

stressing its strong ounter-ylial beavior. To pursue tis analysis furter we nee terefore to

rely on equilibrium moels tat link returns to eonomi funamentals. Reently, Bansal an Yaron

(2004) ave revive onsumption-base asset priing moels by sowing tat two soures of long-run risk — expete onsumption growt an onsumption volatility as a measure of eonomi

unertainty — etermine asset returns. Furter, Teongap (2010) provies strong eviene tat

onsumption volatility risk explains a ig perentage of te ross-setional ispersion in average

stok returns for te usual set of size an book-to-market portfolios tat ave been use in tests

of asset priing moels. Anoter stran of literature base on te intertemporal cAPM or te

onitional cAPM as linke te ross-setion of expete returns to oter eonomi or finanial

variables su as te term sprea, efault sprea, implie or realize measures of aggregate returns

variane, an many oters.

Wile our cSV measure is base on te ross-setional ispersion of realize returns over tewole universe of trae stoks, as oppose to te ross-setional ispersion of average returns

of a limite number of size an book-to-market portfolios, te same teoretial impliations

soul prevail. Terefore, we present below a simple orrelation an grapial analysis of te

relation between te cSV an some of tese key variables. For te eonomi variables, we ose

onsumption-growt volatility as a measure of eonomi unertainty. Following Bansal an Yaron

(2004) an Teongap (2010), we filter onsumption-growt volatility wit a GARch moel. For

onsumption, we use FREd's personal onsumption expenitures of non-urables an servies

montly series, ivie by te onsumer prie inex an te population values to obtain a per-

apita real onsumption series. We ten ompute its growt rate from July 1963 to 2006.17 Te

seon eonomi variable we onsier is inflation volatility, wi we filter also wit a GARch

proess.18 For te finanial variables we use Wel an Goyal's (2008)'s ata for orporate bon

13

14 - These are estimates of the transition probabilities conditional to information up to time t given all sample data.

15 - One notable exception is the regime change of 1980 : 05, which is present for the cap-weighted measure and absent for the equally-weighted one.

16 - The small diff erence might come from the fact that Bekaert et al. (2008) t a model with two different autocorrelation coefficients (one for each regime) as opposed to one. However, they find the two

coefficients to be fundamentally equal in both regimes, which supports using a more parsimonious model.

17 - The series IDs at the FRED's webpage are, PCEND and PCES for "Personal Consumption Expenditures: Nondurable Goods" and "Personal Consumption Expenditures: Services", CPIAUCNS for "Consumer

Price Index for All Urban Consumers: All Items" and POP for "Total Population: All Ages including Armed Forces Overseas". Bansal and Yaron (2004) used the Bureau of Economic Analysis data available at

www.bea.gov/national/consumer spending.htm on real per-capita annual consumption growth of non-durables and services for the period 1929 to 1998. The series is longer but is available only at annual

and quarterly frequencies.

18 - For space considerations, we do not report parameter estimates for the two AR(1)-Garch(1,1) we estimate. They are available upon request from the authors.



yiels on BAA an AAA-rate bons, long-term government bon yiel an 3-mont T-bill rate

to estimate te reit sprea an term sprea (as te ifferene between te first an te seon

rate in bot ases).19 In Table 6 we report te orrelations between te equally-weigte an

ap-weigte measures of ross-setional variane an te five eonomi an finanial variables

uring te 1990-2006 perio. We also explore some potential asymmetries by omputing te

cSV EW for te positive an negative returns.

Te igest orrelation (0.401) is obtaine between onsumption growt volatility an teequally-weigte measure cSV EW . In Figure 5 we plot te two series for te perio 1990 to

2006. Wile te cSV series is mu noisier tan onsumption-growt volatility, te oinient

movements between te two series are quite remarkable. After a ig volatile perio just before

2000, bot series sow a marke ownwar tren after te turn of te entury. A reasonable

explanation for tis strong orrelation is to tink about a ommon fator (aggregate eonomi

unertainty) affeting te iiosynrati variane of ea seurity. Aggregating over all seurities

will make te cSV a funtion of eonomi unertainty. In intertemporal asset priing moels

of Bansal an Yaron (2004), Bollerslev, Sizova, an Tauen (2009) an Bollerslev, Tauen, an

Zou (2009), eonomi unertainty is a prie risk fator tat affets returns, terefore proviing

a funamental rationale for te observe orrelation between cSV an onsumption growtvolatility. Tis suggests tat cSV soul appear to be prie wen a Fama-MaBet proeure

is applie to a set of portfolios. We explore tis issue in Setion 5. Te orrelation of te ap-

weigte cSV wit onsumption growt volatility is not as ig (0.241) sine it puts more weigt

on large ap seurities, wi are in general less affete by eonomi unertainty. Looking at

te split between cSV EW+ an cSV EW-, we see tat te orrelation is iger for te cSV wen

onitioning on te negative returns (0.346). Tis suggests tat return ispersion in bear perios

is relatively more affete by eonomi unertainty.

Te next most igly negatively orrelate variable is inflation volatility (-0.367). Sine 1998,

inflation volatility seems to ave been on an upwar tren, wile te ross-setional variane of returns as been sarply elining. Tis is learly apparent in Figure 6. In te presene of iger

inflation unertainty, investors will move towars alloating more to stoks relative to bons in teir

portfolios, generating a general inrease in stok returns tat reues teir ross-setional variane.

Te T-bill rate is also relatively igly orrelate wit cSV EW (0.302). In te type of equilibrium

moels we ave referre to, te risk-free rate, proxie ere by te T-bill, will be a funtion of

onsumption growt volatility, ene its positive relation wit te ross-setional variane.

For te finanial variables (reit sprea an term sprea), it is interesting to note tat te iger

orrelations are wit te ap-weigte measures of te ross-setional variane. Te signs are

intuitive. creit risk affets ifferently iniviual firm returns an terefore tens to inrease cSV,wile a pervasive term sprea risk will reue ispersion by being ommon to many seurities ue

to a move of investors away from bons into te stok market.

Given tat te ross-setional variane is signifiantly linke wit eonomi an finanial fators

tat ave been sown to preit returns, we explore in te next setion te preitive power

of cSV for aggregate returns at various frequenies, espeially at aily frequenies, sine our

measure of iiosynrati variane allows us to measure cSV at any frequeny witout any artifiial

smooting effet. Tis is a main avantage over oter metos of reovering tis iiosynrati

variane.

4. New Evidence on the Predictability of the Market ReturnTere is an ongoing ebate on te preitive power of average iiosynrati variane for average

(or aggregate) stok market returns. Goyal an Santa-clara (2003) fin a signifiantly positive

19 - Data available at Amit Goy al's webpage: http://www.bus.emory.edu/AGoyal/Research.html



relationsip between te equal-weigte average iiosynrati stok variane an te ap-

weigte portfolio returns for te perio 1963:07 to 1999:12. Tey fin tat teir measure of

average iiosynrati (in fat total) variane as a signifiant relationsip wit next mont return

on te ap-weigte portfolio. Te regression in GS is as follows:

(15)

were orrespons to . In a subsequent analysis, Bali et al . (2005) argue tattis relationsip isappeare for te extene sample 1963:07 to 2001:12, an attribute te

relationsip observe in GS to ig-te-bubble-type stoks (i.e., stoks trae on te NASdAQ)

an a liquiity premium. In a similar way, Wei an Zang (2005) fin tat te signifiane of te

relationsip foun by GS isappeare for teir sample 1963:07 to 2002:12 an argue tat te

presumably temporary result of GS was riven mainly by te ata in te 1990s. Wei an Zang

(2005) ritiize te fat tat GS looke at te relationsip between an equally-weigte average

stok variane an te return on a ap-weigte average stok return, as oppose to an equally-

weigte portfolio return. Moreover, bot Bali et al. (2005) an Wei an Zang (2005) fin no

signifiant relationsip between te ap-weigte measures an te ap-weigte portfolio

return in all tree sample perios (ening in 1999, 2001 an 2002, respetively).

4.1 Monthly Evidence

In tis setion we onfirm existing results an exten tem in a number of imensions, inluing

a longer sample perio. Te first panel in Table 7 presents te preitability regression of equally-

weigte variane measures on te ap-weigte return as in Goyal an Santa-clara (2003) an

Bali et al. (2005) for teir sample perios, as well as te extene sample up to 2006:12. Te

regression is as in equation 15, were orrespons to te EW cAPM-base measure an te

cSV.20 For omparison purposes we start te sample perio in tis setion in 1963:07, as in Goyal

an Santa-clara (2003), Bali et al. (2005) an Wei an Zang (2005).

For te montly series, we onfirm tat tere is a signifiant positive relationsip in te first

sample, an also tat it weakens for te subsequent extene samples.21 Te Newey an West

(1987) autoorrelation orrete t-stat for 12 lags of te oeffiient of bot cSV an te cAPM-

base measures goes from 3:5 for te first sample perio own to 0:9 for te largest sample.

consequently, te ajuste R2 goes from 1:3% own to 0:04%. Tis result onfirms te finings

of Bali et al. (2005) an Wei an Zang (2005) for te furter extene sample. In setion 4.4 we

propose a possible explanation for tis puzzling result.

In te seon panel of Table 7 we present te results of te regression between te equally-

weigte average return wit te lagge equally-weigte iiosynrati variane measure, asgiven by:

(16)

were is taken as te cAPM-base average iiosynrati variane or as te cSV measure. In

ontrast wit te former regression, te relationsip is foun to be signifiantly positive for te

tree sample perios for bot measures.22

In te tir panel of Table 7 we present te results for te tree sample perios of te one-

mont-aea preitive regression of te ap-weigte market portfolio using te ap-weigte

iiosynrati variane return as a preitor. In tis ase, te beta of te iiosynrati variane is

not signifiant for all tree sample perios. Tis result onfirms te finings of Bali et al. (2005)

an Wei an Zang (2005) for te extene sample.

15

20 - As explained before, the monthly CSV is the average of its daily values during the month.

21 - We found a similar result using the GS measure of equally-weighed average variance. We do not present these regression results for the sake of brevity given that they generate a similar picture, which

has also been confirmed in Bali et al. (2005) and Wei and Zhang (2005).

22 - Wei and Zhang (2005) find a significantly positive relation between the equal-weighted GS measure and the equal-weighted market return for the initial sample. They also test the robustness of the

relation by using an equally-weighted cross-sectional variance of monthly returns. They found a significantly positive coefficient for predicting the equal-weighted portfolio return mainly for the long

samples starting in 1928 but not for the sample going from 1963 to 2002. Note that our cross-sectional measures dier. Ours is an average of the daily cross-sectional variances over the month. Theirs is the

cross-sectional variance of the returns computed over the month.



4.2 New Predictability Evidence at Daily Frequency

Prevailing measures use in te literature require a sample of past ata to estimate aitional

parameters, onstraining existing eviene to te montly estimations. Fu (2009) fins tat ig

iiosynrati volatilities of iniviual stoks are ontemporaneous wit ig returns, wi ten

to reverse in te following mont. huang et al. (2009) fin tat te negative relationsip between

iiosynrati variane an expete returns at te stok level unovere in Ang et al. (2006) an

Ang et al. (2009) beomes positive after ontrolling for te return reversals. Tis provies aitional

motivation for looking at te preitability relation at a iger frequeny tan te montly basis.Using te cSV as a proxy for aggregate iiosynrati variane allows us to ek tis relationsip at

te aggregate (market) level in a more iret way (witout aving to ontrol for reversals). Taking

avantage of te instantaneous nature of te cSV, we run te same preitability regression (16)

on te one-ay-aea portfolio return using te average iiosynrati variane.

Te upper panel of Table 8 sows tat at a aily basis, tis relationsip is mu stronger, wit

(Newey-West orrete) t-stats of oeffiients for te average iiosynrati variane aross te

tree samples ranging between 4 an 4:7.

In te lower panel of Table 8 we report te results for te one-ay-aea preitive regression onte ap-weigte pairs (cSV an market return) for wi we fin te relation also to be positive

an signifiant, but wit a mu more obvious eterioration of te t-stat of te ap-weigte

iiosynrati variane oeffiient, going from about 5:91 in te first sample own to 1:97 for

te longest sample. For tis reason an for brevity, we now fous on te relationsip between

aggregate iiosynrati risk an te equal weigte market return.23

4.3 Interpretation of Predictability results

Given tis eviene on te preitability of average aggregate returns by iiosynrati risk, a

natural question to ask woul be: wy oes te relationsip between te equal-weigte measure

an te ap-weigte iffer aross ifferent sample perios?

Wei an Zang (2005), Bali et al. (2005) argue tat te relationsip between iiosynrati risk an

te market inex first foun by Goyal an Santa-clara (2003) on te sample ening in 1999:12

was riven by small stoks trae in te NASdAQ an te ata oming from te ot-om bubble

perio. Altoug we onfirm teir empirial finings for our sample perio, we isagree wit

teir onlusion tat te relationsip between average iiosynrati risk an expete returns

isappeare sine te en of te ot-om bubble. Even toug it appears lear tat NASdAQ

ompanies playe an important role in te relationsip of te equal-weigte average iiosynrati

variane wit te average market-apitalization expete return uring te en of te 1990s,

wi (obviously) weakene after te burst of te bubble, we fin tat te relationsip betweenaverage iiosynrati risk an future average market returns is robust to oies of te sample

perio, provie tat aequate weigting semes an orizons are osen to test tis inter-

temporal relationsip.

Te transitory relationsip between te equal-weigte average iiosynrati variane an

te ap-weigte market inex observe up to te en of te 1990, an be explaine by te

eterogeneous an transitory nature of te omitte soures of risk apture by iiosynrati risk

an its relation wit te inflate valuation of several NASdAQ ompanies uring tat perio.24

Some intuition bein te far more robust relationsip between te equally-weigte average

iiosynrati variane an te equally-weigte portfolio omes preisely from te logi of

stanar asset priing teory. As isusse in te introution, tere are multiple reasons for

wi average iiosynrati risk soul be relate to average returns, ue to te eterogeneous

23 - The corresponding results using a market cap-weighted scheme can be obtained from the authors upon request.

24 - The strongest omitted factors in that period (call it the irrational.com factor), partially captured by the equally weighted idiosyncratic variance, started to be increasingly represented in the market-

cap index, due to the suddenly-higher market capitalization of precisely the group of companies carrying this temporarily strong omitted factor. The posterior reversal of the situation ( i.e., the burst of the

bubble) subsequently explains the sharp fade in the relationship between the average idiosyncratic variance and the market- cap portfolio, precisely due to the posterior sudden deterioration of the market

capitalization of most stocks carr ying this irrational.com factor, and hence notably reducing their representation in the market-capitalization index.



soures tat may ompose iiosynrati risk. Aoring to te cAPM, only systemati risk soul

explain future returns. however, if uring a ertain perio of time tere are anomalies of any

kin (prie omitte risk fators) tat, presumably, are not proportionally reflete in te urrent

market apitalization of te ompanies arrying tese fators, ten te omitte soures of risk

are more likely to explain te returns of a portfolio were all kins of firms are represente in a

similar manner, su as te EW as oppose to a portfolio were big ompanies are proportionally

better represente tan smaller ones.

Along tese lines, Ponti (2006) argues tat iiosynrati risk is te largest oling ost borne by

rational arbitrageurs in teir pursuit of mispriing opportunities. Tis teory implies tat te urrent

level of iiosynrati risk soul preit returns sine it soul measure te amount of urrent

mispriing opportunities present in te market. Assuming tat te same mispriing opportunities

isappear in te long run, it appears more likely to observe tis relationsip between iiosynrati

variane an returns over very sort orizons. Moreover, all tings being equal, large-ap stoks

are less likely to present misspriing an ene te preitability implie by tis teory woul

be more likely to be present on te equal-weigte inex return rater tan te ap-weigte

inex return, as we observe in preitive regressions at aily an montly orizons.25 Te sign of

te relationsip is not preite by Ponti's teory in general, beause it epens on weter teaverage (equal or ap-weigte) portfolio is over- or uner-prie (it preits a positive sign for

unerprie stoks an a negative sign for overprie stoks).

4.4 Robustness Checks

In tis setion, we test furter explore te relationsip oumente in te former setion in

several imensions. We first want to plae te return preitability by iiosynrati variane in

te ontext of te literature of te risk-return trae-off. Most of te literature on tis topi is

base on a linear regression between return an volatility. We want to see if inluing te return

variane in te regression anges te preitability results. Seon, we test te robustness of

te relationsip in te presene of an option implie volatility measure. Tir, we furter testte preitability relationsip at quarterly an annual orizons. Finally, we look for te potential

asymmetry in te relationsip between iiosynrati variane an future average returns, wen

te ross-setional variane is split in two an is ompute for returns above or below te mean.

Su an asymmetry often exists for positive an negative returns in te volatility moeling of

finanial time series. Te reporte presene of asymmetries will provie us wit a motivation

for extening te ross-setional ispersion measure to te tir moment an fin tis measure

is relate wit average iiosynrati skewness an as strong preitive power of te average

market return.

4.4.1 Inlusion of Return VarianeIn orer to ek weter te relationsip between te market portfolio expete return an te

aggregate level of iiosynrati variane (wi we oument at te montly an aily frequeny)

is robust to te inlusion of te variane of te market portfolio, we run te following joint

regression:

(17)

We also run te univariate regression:

(18)

For te montly estimations of we use te realize sample variane over te mont

(from aily returns). For aily estimations we fitte an AR(1)-EGARch(1,1) moel on te overall

sample.26 In te first two panels of Table 9, we report regression results at te montly an aily

1725 - It is well known that large cap stocks are more liquid than small-cap stocks, which implies a higher number of people trading them and usually a higher number of analysts looking at them. Together with

less constraints to short-selling, we expect a higher price efficiency for large cap stocks.

26 - Using the overall sample to estimate the parameters would only give the portfolio variance an advantage to predict future returns. However, from the results we see that even when using such for looking

estimates for , the significance of the CSV remains strong.



frequeny of bot (17) an (18). In te latter univariate regression, te variane of te equally-

weigte portfolio returns oes not appear to be signifiant in explaining te average future

returns at te montly an aily frequenies.

In te regression from equation (17), te oeffiient of , υ , is negative an non-

signifiant at te montly frequeny. At te aily frequeny, te oeffiient υ was still foun to

be negative an (marginally) signifiant. Te signifiane of te cSV oeffiient remains vali for

bot montly an aily frequenies, an if anyting improves sligtly after te inlusion of teequally-weigte portfolio variane.

Te latter two panels of Table 9 present te regression results at te montly an aily frequeny

of bot (17) an (18) but using te ap-weigte inex an cSV equivalents. Te relationsip

at te aily orizon beomes non-signifiant after te inlusion of te realize variane of te

market-ap-weigte inex. At te montly orizon te relationsip remains non-signifiant.

In Table 10, were we report te quarterly an annual preitability wit an witout te market

variane, we onfirm tat te equally-weigte ross-setional variane oes not foreast future

average returns at low frequenies. however, for te ap-weigte measure of cSV, we observepreitability over te perio 1963 to 2006 wen it is joine wit market variane. Te sign is

negative wile te market variane enters wit a positive sign as preite by te benmark

risk-return trae-offff.27

One fair remark on te results of te preitability regressions is tat te relationsip using equal-

weigte measures only ols at sorter orizons (i.e., aily an montly). however, tis result

is in line wit Ponti's (2006) interpretation of iiosynrati risk as a barrier for arbitrageurs an

wit te eviene presente by Fu (2009) at te stok level, wo fins tat ig iiosynrati

volatilities of iniviual stoks are ontemporaneous wit ig returns, wi ten to reverse in

te following mont.

4.4.2 Inlusion of Market Realize Variane an Implie Variane

Oter measures of variane ave been use in trying to link market returns to a measure of market

risk. Implie variane (VIX 2 ) as been use as a forwar-looking measure of market variane in

aition to realize variane (te sum of square returns at iger frequeny tan te targete

frequeny for te measure of variane).28 We use tese measures in Table 11 along wit bot cSV

measures for aily an montly preitability. We repeat te exerise in Table 12 for quarterly an

annual frequenies. For tese regressions we start te sample in 1990 for ata availability for te

implie volatility.

Results are similar to te ones in te previous setion wit market variane. For cSV EW , we observe

preitability at ig-frequeny but not at low frequeny, wile it is te opposite for cSV cW . For

te aily estimates wit cSV EW , we fin a R 2 of almost 5% wen we inlue all tree measures of

variane, an all oeffiients are signifiant. But te remarkable result, unoumente until now

to our knowlege, is te very ig R 2 obtaine at quarterly an annual frequenies for te cSV cW

measure. Wen using cSV cW alone as a preitor we obtain R 2 s of 4% an 26% at quarterly an

annual frequenies, respetively. Aing te implie variane brings te R 2 s to almost 19% an

29%. If instea one uses te realize variane instea of implie variane te R 2 s are lose to 11%

an 34%. In all tese preitability regressions, te sign of te cSV cW variable is negative. Guo

an Savikas (2008) argue tat average iiosynrati volatility is negatively relate to future stok

market returns possibly beause of its negative orrelation wit te aggregate book-to-market

ratio.29 If iiosynrati volatility is measure from a cAPM moel ten it will apture te missing

book-to-market fator. Tis explanation runs ounter to our previous finings regaring te very

27 - See also Guo and Savickas (2008) for similar results.

28 - For example, for the monthly variance, one will sum the daily squared returns, while for the daily variance, it is customary to use five-minute or one-minute squared returns.

29 - The argument starts by considering average idiosyncratic volatility as a proxy for changes in the opportunity set related related to technological shocks. They argue that technological innovations have

two effects on the firm's stock price: they tend to increase the level of the firm's stock price because of growth options and they also tend to increase the volatility of the firm's stock price because of the

uncertainty about which firms will benefit from the new opportunities. The final argument is to say that the book-to-market ratio captures these investment opportunities.



ig orrelation between te measures of iiosynrati volatility base on te cAPM an te

Fama-Fren moels. Te two series were almost iential. A more appealing explanation may be

to tink of ross-setional variane as a measure of ispersion of returns refleting te ispersion

of opinions among market partiipants. Te negative sign of tis relationsip at quarterly an

annual orizons in te presene of market variane as te seon preitor (an also at montly

orizons in te presene of implie variane as te seon preitor) is onsistent wit te moel

of cao et al. (2005), in wi ispersion of opinions among investors is positively relate to stok

market volatility but negatively relate to onitional exess stok market returns. Furtermore,one may argue tat ifferenes of opinions forge temselves over a perio of time an ene tis

effet is more likely to be present at orizons longer tan a ay.

More generally, we may interpret te cSV as measuring te eging terms in an intertemporal

cAPM moel. In tis regar, it is interesting to see tat te positive risk-return trae-off at te

aggregate level, i.e., te relationsip between market volatility an expete returns, beomes

signifiant only wen taking into aount te presene of te omitte fators as apture by te

cSV. It is also interesting to note tat te interations of te cSV wit te realize variane of te

market take plae at longer orizons (quarterly an annual), wile its interations wit implie

variane (VIX 2 ) ten to be more important at sorter orizons.

4.4.3 Asymmetry in te cross-setional distribution of Returns

We now explore for a potential asymmetry in te relationsip between iiosynrati variane

an future average returns, wen te ross-setional variane is split in two an is ompute for

returns above or below te mean.

Tis asymmetry may be te result of te leverage effet put forwar by Blak (1976) sine we

are onsiering iniviual firms in te ross setion. We also mentione in an earlier setion tat

onsumption volatility risk affets ifferently small an large firms or value an growt firms.

Terefore, we explore i ) weter te preitability power is te same for te cSV of returns tote left an right of te enter of te returns' istribution, ii ) weter te relationsip is riven

by one of te sies an iii ) weter te relationsip wit bot sies woul ave te same sign on

teir oeffiient. In orer to o tis, we efine te as te ross-setional variane of te

returns to te rigt of te ross-setional istribution (i.e., meaning te ross-setion istribution

tat inlues all stoks su tat ) an onversely efine te as te ross-

setional variane of te returns to te left of te ross-setional istribution (i.e., meaning te

ross-setion istribution tat inlues all stoks su tat ). Ten we run te following

regression:

(19)

Table 13 presents te results of regression (19) for aily, montly, quarterly an annual estimates,

an sows a ouple of interestingfinings. First, splitting te cSV into rigt an left sies of te

ross-setional istribution mae te ajuste R 2 of te preitive regression jump from 0:8% to

1:17% on montly ata an from 0:6% to 1:36% on aily ata. Seon, tere is an asymmetri

relationsip between te cSV of te returns to te rigt an left of te ross-setional istribution

an te expete market return: te oeffiient of te is positive wile te one of

is negative in bot aily an montly regressions. however, te oeffiients (of bot rigt an

left cSVs) are signifiant only on te aily regression. Te summary statistis of te preitive

regression on te ap-weigte inex using te equivalent ap-weigte cSV measures, isplaye

in te lower panel of Table 13, are qualitatively similar to te results on te equal-weigte

measures.

Tese finings suggest tat a measure of asymmetry of te ross-setional istribution woul

be relevant in te ontext of exploring te relationsip between market expete returns an

19



aggregate iiosynrati risk. Anoter key avantage of te cSV measure is tat it an be easily

extene to iger-orer moments. We onsier below te skewness of te ross-setional

istribution of returns an assess its preitive power for future returns. To te best of our

knowlege, tis aitional fator, wi appears as a natural extension of te cSV for measuring

iiosynrati risk,30 is entirely new in tis ontext.31 We follow Kim an Wite (2004) an use a

quantile-base estimate (see Bowley (1920)), generalize by hinkley (1975), as a robust measure

of te skewness of te ross-setional istribution of returns:32

(20)

for any α 1 between 0 an 0.5 an Q 2 = F -1 (0.5). Te Bowley oeffiient of skewness is a speial

ase of hinkley's oeffiient wen α 1 = 0.25 an satisfies te Groenevel an Meeen's (1984)

properties for reasonable skewness oeffiients. It as upper an lower bouns {-1, 1}.

In Table 14, we report te results of preitive regressions at te aily an montly frequenies

were we a te robust measure of te ross-setional skewness to te equally-weigte cSV.

Te first observation is tat te cSV oeffiients are very lose to te values estimate wit tecSV as te only regressor (0.4 for te aily frequeny an 0.25 for te montly one). Te t-stats are

also almost iential to te ones foun in te cSV regressions. however, skewness appears to be

a major ontributor to te preitability of returns sine te R 2 inreases signifiantly ompare

to te regressions wit cSV alone. At te aily frequeny, te ajuste R 2 inreases to a value of

5.8%. At te montly frequeny, it is still 4.6%. Tis large inrease in preitability wen aing

skewness suggests tat maroeonomi or aggregate finanial soks affet asymmetrially te

istribution of returns.

5. Is Average Idiosyncratic Risk Priced?Aoring to Merton's IcAPM, a fator tat preits stok returns in te ross setion soul also

preit aggregate market returns (see campbell (1993)). By te reverse argument, motivate by

te preitability power of (equal-weigte) ross-setional variane on te average return in te

market, we explore in tis setion weter te cSV EW , interprete as a risk fator, is reware an

ommans a premium in te ross-setion.

5.1 CSV Quintiles' Premium

Using aily exess returns every mont we run te following regression for ea stok i:33

(21)

At te en of every mont in te sample, we sort stoks using te cSV EW fator loaing, β sv , an

form equally-weigte an ap-weigte quintile portfolios. We alulate te average return

uring te overall perio for ea quintile an te average return ifferene ( i.e., premium) -

between te first quintile an ea of te oter four quintiles.

Te results for te equally-weigte quintile portfolios are isplaye in te upper panel of Table

15 an in te lower panel for te ap-weigte quintiles. As we an see from tis table, all premia

are signifiantly ifferent from zero an eonomially meaningful. Te ifferene between te

first quintile (te one wit iger sensitivity) an te seon, tir an fourt quintiles, is aroun

an annualize 30%, wile te ifferene wit te fift quintile is aroun 15%. Tis result suggests

tat te relationsip of te cSV an stok returns migt not be best esribe in te simple linear

form, wi is in line wit te asymmetri effet foun in setion 4.4, wit te quantities cSV +

an cSV -.

30 - We show formally in an appendix available upon request from the authors that there is a link between idiosyncratic skewness and the skewness of the cross-sectional distribution of returns.

31 - At the stock level, Kapadia (2009) uses cross-sectional skewness to explain the puzzling nding in Ang et al. (2006) that stocks with high idiosyncratic volatility have low subsequent returns.

32 - The usual non-robust skewness measure of the cross-section of returns is highly noisy compared to the proposed robust measure, especially at the daily frequency. 33 - We use stocks with non-missing

values during the current month.



5.2 Fama-MacBeth Procedure

In orer to use te stanar set of assets in te asset priing literature, we extrat aily returns

ata from Kennet Fren ata library on teir 100 (10x10) an 25 (5x5) size/book-to-market

portfolios for te perio July 1963 to deember 2006. Ten we run every alenar mont te

following regression for ea portfolio:34

(22)

Using te reore fator loaing, β (montly) time series, we run te following ross-setionalregression every mont on te next mont's exess returns an reor te oeffiients:

(23)

We finally test weter te average γ oeffiients are statistially ifferent from zero. In orer to

take into aount possible serial orrelation in te oeffiients, we ompute te t-statisti using

Newey an West (1987) stanar errors wit four lags (same number of lags as in Ang et al. (2009)).

We use four sets of assets: 100 (10x10) size/book-to-market equally-weigte portfolios an

ap-weigte weigte portfolios, an 25 (5x5) size/book-to-market equally-weigte an ap-

weigte portfolios. For ea of tem, we use te cSV EW as te fourt risk fator. Te first twopanels of Table 16 present te orresponing Fama-MaBet regression results. Te table isplays

te annualize oeffiients an stanar errors (multiplie by 12 from te original montly

values), as well as teir orresponing autoorrelation-orrete t-stat an te average R 2 . We

fin te γ oeffiient for cSV EW to be positive an signifiant wen we use te 100 an 25 size/

book-to-market Fama-Fren equally-weigte portfolios. however, it is not signifiant wen

we use te 25 market-ap-weigte portfolios an marginally signifiant for te 100 market

ap-weigte portfolios (altoug positive in bot ases). Tis later result, again, is not entirely

surprising onsiering tat te ross-setional variation in returns is reue troug te market-

apitalization ajustment.

6. ConclusionIn tis paper we formally introue an instantaneous ross-setional ispersion measure as a

proxy for aggregate iiosynrati risk tat as te istint avantage of being reaily omputable

at any frequeny, wit no nee to estimate oter parameters. It is terefore a moel-free measure

of iiosynrati risk. We extensively sow ow tis measure is relate to previous proxies of

iiosynrati variane, su as te Goyal an Santa-clara (2003) measure an measures relative

to te lassi Fama an Fren (1993) an cAPM moels, wi ave been previously sown to

be very lose to te campbell et al. (2001) proxy as well. We onfirm previous finings of Goyal

an Santa-clara (2003), Bali et al. (2005) an Wei an Zang (2005) on te montly preitabilityregressions for te extene sample perio using our ross-setional measure an more stanar

measures of iiosynrati variane. We fin tat te results are robust aross tese measures.

Tanks to te instantaneous nature of our measure, we are able to exten to aily ata te

eviene on te preitability power of iiosynrati variane on te future market portfolio

return. We provie a statistial argument to support te oie of an equally-weigte measure

of average iiosynrati variane as oppose to a market-ap-weigte one an explain wy

bot empirially an teoretially su a measure soul foreast better te equal-weigte

market return. We also sowe tat tis ross-setional measure isplays a sizable orrelation

wit eonomi unertainty, as measure by onsumption growt volatility, an wit several

eonomi an finanial variables. One aitional avantage of our measure is tat it generalizes

in a straigtforwar manner to iger moments an we sowe tat te asymmetry of te ross-

setional istribution is a very goo preitor for future returns. We leave for furter resear

an exaustive analysis of te properties of te skewness of ross-setional return istribution

as a measure of average iiosynrati skewness. We also leave for furter resear an empirial

analysis of te cSV measure using international ata.21

34 - As before, XMKT stands for excess market return, SMB and HML are the size and book-to-market Fama-French factors, also directly extracted from Kenneth French data library.



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23



A. Proof of Proposition 1consier te fator moel eomposition

an

Uner te omogeneous betas assumption, we ave

(24)

an terefore

so tat

Noting tat

we finally ave:

We now argue tat te term onverges to 0 for inreasingly large numbers of stoks.

To sow tis, we nee to use a reent result by cuzik (1995) regaring te Marinkiewz−Zygmun

strong law of large numbers for weigte sums of i.i.. variables:

almost surely (25)

wen {X ,X N ,N ≥ 1} is a sequene of i.i.. ranom variables wit E (X ) = 0, E |X | < ∞ an {aNi , 1 ≤ i

≤ N ,N ≥ 1} is an array of onstants uniformly boune satisfying35

(26)

35 - See Theorem 1.1, particular case of Cuzick (1995).



here we take an . For te result (25) to ol aNit nees to be uniformly

boune an to satisfy onition (26). We terefore restrit our attention to non-trivial weigting

semes, ruling out te situation su tat te inex is ompose by a single stok. Please note

tat tis onition togeter wit te fat tat implies N t > 1 an also restrit te

weigts to be (stritly) positive at every given point in time. hene, a weigting seme (w t ), is

efine as a vetor proess wi satisfies 0 < w it < 1 ∀ i , t . Tis onition seems reasonable sine

our fous is to measure iiosynrati risk in te market.

By efinition, te weigting seme w it an aNit is uniformly boune by N t an te following

onition ols,

0 < w it < 1 ∀ i , t (27)

Multiplying by N t , we get

wi implies tat onition (26) ols. Tus, for a positive weigting seme from te strong law

of large numbers for weigte sums of i.i.. variables, it follows tat:

,

Using similar arguments, an te omogeneous iiosynrati seon moment assumption,

, we obtain tat for a stritly positive weigting seme, w t , an i.i.. ε i ,

almost surely

Using tese results, we finally ave tat:

almost surely.

B. Properties of the CSV EstimatorB.1 Bias of the CSV Estimator

Uner te fator moel eomposition (1) an equation (2) an using te omogeneous beta

assumption, we ave:

(28)

Replaing result (28) in equation (3) we ave as before:

(29)

25



By efinition of a strit fator moel, for i ≠ j , an Applying te

expetation operator in equation (29) we get:

(30)

Te seon term in (30) implies tat te cSV woul ten to unerestimate te average iiosynrati

variane. consiering te equal-weigte seme were , (29) simplifies into

an we obtain:

B.2 Variance of the CSV Estimator

Let w t an ε t be olumn vetors of te weigting seme an resiuals respetively anΩt = w t w' t ,Λt = iag (w t ), N t ×N t matries, an enote Σ ε te variane ovariane matrix of te resiuals,

wi is iagonal for a strit fator moel.

For a finite number of stoks in te ase were F t ≠ r (wt) , we ave from equation (29):

.

Letting Q t = Λt − Ωt , cSV t an be written in matrix form, as follows:

(31)

Using te quarati struture of te cSV an assuming normal resiuals, we ave (see for instane

Kaman (1999)):36

(32)

Uner te assumption of a strit fator moel, i.e. , equation (31) simplifies

to:

(33)

Assuming an upper boun for te iniviual iiosynrati varianes, enote as equation (33)

yiels te following inequality (replaing ea variane for its upper boun)

(34)

Wen w t = 1/N t , equation (34) simplifies to

(35)

36 - The operator tr stands for the trace of a matrix, which is the sum of the diagonal terms.



For a large number of stoks,

(36)

C. Relaxing the Assumption of Homogeneous Betas

Te assumption tat β it = β t for all i is obviously a simplisti one an is one only for exposurepurposes. Starting wit te single fator eomposition on te efinition of te cSV we ave:

After simple rearrangement of terms we get:

Applying te expetation operator an assuming a strit fator moel, te last expression

simplifies so as to yiel:

Uner an equal-weigting seme, we finally ave:

27



D. Tables and FiguresTable 1: Estimates of the biases due to the cross-sectional dispersion of betas and weight

concentration: Tis table ontains a summary of te istribution of te following time series: te

ross-setional ispersion of betas , estimate wit respet to te cAPM at te en of every

mont using aily returns; te average iiosynrati variane wit respet to te cAPM; te

prout of te average return of te market portfolio square, , an te beta ispersion, ;

te proportion of te prout to an te proportion of to . Te upper

panel orrespons to te equal-weigt seme (cSV EW ) an te lower panel to te market-apweigting (cSV cW ). All figures are aily. Te perio is July 1963 to deember 2006.

Table 2: Total bias associated with CSV: Tis table reports te output summary of te regression

, were represents montly estimates of te

weigte average iiosynrati variane estimate using te orresponing moel (eiter cAPM

or FF). Te average an te cSV are ompute wit eiter te ap-weigte seme (cW) or teequal-weigte one (EW). Te perio is July 1963 to deember 2006.



Table 3: Comparison of monthly measures of idiosyncratic variance: Te upper panel of tis

table ontains te annualize mean an stanar eviation of te montly time series for te

cSV, te average iiosynrati variane base on te cAPM an te Fama-Fren moels an te

average Goyal an Santa-clara (2003) variane measure as in equations (4), (11) an (12) using

bot EW an cW weigting semes. Te lower panel presents te ross-orrelation matrix among

tese variables. Te perio is January 1964 to deember 2006.

Table 4: Comparison of daily measures of idiosyncratic variance: Te upper panel of tis tableontains te annualize mean an stanar eviation of te aily time series for te cSV an

te average iiosynrati variane base on te Fama-Fren moel as in equations (4) an (11)

using bot weigting semes. Te lower panel presents te ross-orrelation matrix among tese

variables. Te perio is January 1964 to deember 2006.

Table 5: Regime-switching parameters: Tis table ontains te parameter estimates of te

Markov regime-switing moel speifie in equation 14 for te cSV an te average iiosynrati

variane base on te FF moel as in equations (4) an (11) using bot, equalweigte an ap-

weigte semes. Te upper panel orrespons to montly estimates an te lower panel to

aily estimates. µ i is te average level of te variable on regime i , σ i is te stanar eviation

level of te variable on regime i , φ is te autoorrelation oeffiient, p an q are te probabilities

of remaining in regimes 1 an 2 orresponingly. Te perio is January 1964 to deember 2006.

29



Table 6: Correlations between the monthly series of several measures of cross-sectional

variance and economic variables. Te sample perio is January 1990 to deember 2006.

Table 7: Predictability Regression on the CRSP broad market portfolio using the CSV and CAPM-

idvol equal-weighted and cap-weighted measures: Te first panel of tis table presents te

results of a one-mont-aea preitive regression of te exess ap-weigte montly portfolio

returns, enote by r cW , on te montly lagge equal-weigte average iiosynrati variane

an cSV EW for tree sample perios. Te seon panel presents te results of a one-mont-aea

preitive regression of te exess equal-weigte montly portfolio returns, enote by r EW , on

te montly lagge equal-weigte average iiosynrati variane an cSV EW . Te tir panel

presents te results of a one-mont-aea preitive regression of te exess ap-weigtemontly portfolio returns, enote by r cW , on te montly lagge ap-weigte iiosynrati

variane an cSV cW . cAPM w is te average iiosynrati variane erive from cAPM, estimate

using one mont of aily ata an cSV w is te average of te aily ross-setional variane over

ea mont. Te interept, te regression oeffiient of te orresponing lagge iiosynrati

variane, te stanar errors, enote by St, te Newey-West orrete t-stats an te ajuste

oeffiient of etermination, enote by , are reporte. Te sample perios are 1963:08

—1999:12, 1963:08 — 2001:12 an 1963:08 — 2006:12.



Table 8: Daily predictability Regression on CRSP broad market portfolio with average

idiosyncratic variance measures: Te upper panel presents te results of a one-ay-aea

preitive regression of te exess equal-weigte aily portfolio returns, enote by r EW , on te

aily lagge equal-weigte ross-setional variane enote as cSV EW estimate as in equation

(4) for tree sample perios. Te lower panel presents te results of te preitive regression on

te ap-weigte market portfolio using te ap-weigte cSV. Te interept, te regression

oeffiient orresponing to te cSV, te stanar error of te regression oeffiients, enote

by St, te Newey-West orrete t-stats (30 lags) an te ajuste oeffiient of etermination,enote by , are reporte. Te sample perios are 1963:07 to 1999:12, 1963:07 to 2001:12 an

1963:07 to 2006:12.

Table 9: Daily and monthly predictability with and without market volatility: Tis table presents

summary statistis for tree preitive regressions. Te first two panels orrespon to aily an

montly preitions on te equal-weigte broa market inex using equal-weigte cSV an

te realize variane of te equal-weigte broa market inex an te two lower panels teir

ounterparts using wit te ap-weigte inex an cSV. In ea panel, te first row orrespons

to te regression te seon row to an te tir

one to . Te sample perio is July 1963 to deember 2006

31



Table 10: Quarterly and annual predictability with and without market variance: Tis table

presents summary statistis for tree preitive regressions. Te first two panels orrespon to

quarterly an annual preitions on te equal-weigte broa market inex using equal-weigte

cSV an te realize variane of te equal-weigte broa market inex an te two lower panels

teir ounterparts wit te ap-weigte inex an cSV. In ea panel, te first row orrespons

to te regression te seon row to an te tir

one to . Te sample perio is July 1963 to deember 2006



Table 11: Daily predictability with and without market realized variance and implied variance

(V I X 2): Te first two panels orrespon to aily an montly preitions of te equal-weigte

broa market inex using equal-weigte cSV, te realize variane of te equal-weigte broa

market inex an te square VIX inex. Te two lower panels feature similar regressions wit

te ap-weigte ounterparts. In ea panel, we report regression results for ea preitor

iniviually an for ombinations of te tree orresponing preitors over te perio July 1990

to deember 2006

33



Table 12: Quarterly and annual predictability with and without market realized variance and

implied variance (V I X 2): Te first two panels orrespon to aily an montly preitions of te

equal-weigte broa market inex using equal-weigte cSV, te realize variane of te equal-

weigte broa market inex an te square VIX inex. Te two lower panels feature similar

regressions wit te ap-weigte ounterparts. In ea panel, we report regression results for

ea preitor iniviually an for ombinations of te tree orresponing preitors over te

perio July 1990 to deember 2006



Table 13: Predictability regression on CRSP broad market index with right and left CSV

measures: Te upper panel presents te results of a one-ay, one-mont, one quarter an

one -year-aea preitive regressions of te exess equal-weigte portfolio returns, enote by

, on te aily or montly (orresponingly) lagge equal-weigte ross-setional variane

of te returns to te rigt (iger tan) of te ross-setional istribution mean (wi is atually

) enote as cSV + an te ross-setional variane of te returns to te left (lower tan)

te mean of te ross-setional istribution , enote as cSV −. Te lower panel presents te

results of te preitive regressions on te ap-weigte market inex using te ap-weigtecSV measures as preitors. Te interept, te regression oeffiients orresponing to te cSV +

an cSV −, te stanar error of te regression oeffiients, enote by St, te Newey-West

orrete t-stats an te ajuste oeffiient of etermination, enote by , are reporte. Te

sample perio is 1963:07 to 2006:12.

35



Table 14: Daily and Monthly predictability with skewness for : Tis table presents te

results of one-ay an one-mont aea preitive regressions of te exess equal-weigte

aily portfolio returns, enote by . Te first explanatory variable is lagge estimate of te

equal-weigte cSV estimate as in equation (4); Te seon explanatory variable is te robust

estimate of skewness estimate as in equations (20). Te interept, te orresponing regression

oeffiients togeter wit teir Newey-West auto-orrelation orrete t-stats (wit 30 lags for

aily an 12 lags for montly) an stanar errors are reporte. enotes ajuste oeffiient

of etermination. Te regression is reporte for te main sample perio from 1963:07 to 2006:12.

Table 15: Quintiles premium. Te upper panel present te results for equal-weigte quintile

portfolios an te lower panel on ap-weigte quintile portfolios. Te first olumn presents

te (aritmeti) average return annualize on quintiles forme at te en of every mont on te

cSV EW's oeffiient estimate wit one mont of aily returns. Te seon olumn presents te

average return ifferene of te first quintile wit every oter quintile. Te tir olumn presents

te p-value of te test of te ifferene to be signiantly positive. Te sample perio is July 1963

to deember 2006.



Table 16: Tis table isplays te average values, stanar errors an Newey-West orrete t-stats

for te oeffiients in te Fama-MaBet regression run every mont in te sample, using te 3

Fama-Fren fators an cSV EW on 100 an 25 size/book2market Fama-Fren equally-weigte

(first two panels) an ap-weigte (last two panels) portfolios. It also isplays te average R 2

aross subsamples of Fama-MaBet regressions. Te sample perio is July 1963 to deember

2006.

37



Figure 1: Annualize ap-weigte iiosynrati varianes, aily estimation: Te wite line is te

time series of te ap-weigte iiosynrati variane wit respet to te FF moel estimate

aily as in equation 11. Te arker line sows te time series of te ap-weigte version of cSV

estimate aily as in equation 5. Te sample perio is January 1964 to deember 2006.

Figure 2: Annualize equally-weigte iiosynrati varianes, aily estimation: Te wite line

is te time series of te equal-weigte average iiosynrati variane wit respet to te FF

moel estimate aily similar to equation 11. Te arker line sows te time series of te cSV EW

estimate aily as in equation 4. Te sample perio is January 1964 to deember 2006.

38



Figure 3: Smoote probabilities an annualize ap-weigte cSV montly estimation: Te

re line plots te smoote (onitional) probability of te cSV cW being in te igmean ig-

variane regime of a Markov regime-switing moel speifie in equation 14. Te blue line sows

te montly time series of te cSV cW estimate at te en of ea mont as te average of te

aily estimations (as in equation 4) uring te mont. Te sae areas are te NBER reessions.

Te sample perio is July 1963 to deember 2006.

Figure 4: Smoote probabilities an annualize equal-weigte cSV montly estimation: Te

re line plots te smoote (onitional) probability of te cSV EW

being in te igmean ig-variane regime of a Markov regime-switing moel speifie in equation 14. Te blue line sows

te montly time series of te cSV EW estimate at te en of ea mont as te average of te

aily estimations (as in equation 4) uring te mont. Te sae areas are te NBER reessions.

Te sample perio is July 1963 to deember 2006.



Figure 5: Montly time series of cSV EW on te rigt-an axis an onsumption volatility on te

left-an axis. Te sample perio is January 1990 to deember 2006.

Figure 6: Montly time series of cSV EW on te rigt-an axis an inflation volatility on te left-

an axis. Te sample perio is January 1990 to deember 2006.

edhec working paper idiosyncratic risk f

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