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ModelInsight
PredictingRiskatShortHorizonsACaseStudyfortheUSE4DModel
JoseMenchero,
Andrei
Morozov
and
Andrea
Pasqua
January2013
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ContentsIntroduction.................................................................................3
Evaluating the Accuracy of Risk Forecasts.......................
3
Bias Statistics............................................................................................................4
Mean Rolling Absolute Deviation (MRAD)......................................................4
Adjusted MRAD.......................................................................................................7
Cross-sectional Bias Statistics.............................................................................7
Q-statistics.................................................................................................................7
The USE4 Model and Data Set.......................................... 10
Estimating Volatility
................................................................
11
Exponentially Weighted Moving Averages (EWMA).................................11
GARCH(1,1)..........................................................................................................13
Volatility Regime Adjustment (VRA)...............................................................13
Empirical Results.................................................................... 16
Setting the Correlation Half-Life.......................................... 22
Conclusion...............................................................................24
References...............................................................................25
Client Service Information is Available 24 Hours a Day............................26
Notice and Disclaimer.........................................................................................26
About MSCI............................................................................................................26
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IntroductionEquityfactormodelsareafairlyrecentinvention.BarrRosenbergpioneeredtheuseofmultifactorrisk
modelsasarobustwaytoestimatetheassetcovariancematrix(1974).In1975hefoundedBarra,which
developedthefirstcommerciallyavailableriskmodelforUSequities,dubbedUSE1.
Initially,theUSE1Modelwasestimatedfromquarterlydata.Later,asdatabecomemorewidely
available,theobservationfrequencywasincreasedtomonthly.Formanyyears,usingmonthly
observationstoestimatethefactorcovariancematrixwasstandardpractice.Forinstance,theBarra
USE3Model,releasedin1998,usedmonthlyfactorreturnswithahalflifeof90monthsinthe
estimationprocess.
TheInternetBubblepresentedaseriouschallengeformodelsestimatedwithmonthlydata.Thecruxof
theproblemwasthatvolatilitychangesweretoorapidandextremetobereliablycapturedusinglow
frequencyobservations.Inresponsetothischallenge,BarraresearchersdevelopedtheUSE3SModel,
whichuseddailyfactorreturnsforestimatingthecovariancematrix.Thehigherfrequencyof
observationsallowedthemodeltoadaptmorerapidlytochanginglevelsofvolatility.
Althoughthe
USE3S
Model
employed
daily
factor
returns,
it
maintained
aprediction
horizon
of
one
month.Thiswasaccomplishedbyexplicitlyaccountingfortheeffectsofserialcorrelationinfactor
returns,whichcancausesignificantdeviationsfromthefamiliarsquarerootoftimescaling.For
instance,thereturnstotheMomentumfactortypicallyexhibitpositiveserialcorrelation.Thismakesthe
factorsignificantlymorevolatileatthemonthlyhorizonthanwouldbesuggestedbyapplyingsquare
rootoftimescalingtodailyvolatility.
Formanyinstitutionalinvestors,however,therelevanthorizonmaybemuchshorterthanonemonth.
Inthispaper,wefocusononedaypredictionhorizons.Onepossibleapproachforpredictingriskatone
dayhorizonswouldbetotaketheforecastsfromamonthlymodelandsimplyapplysquarerootoftime
scalingtobringthepredictionhorizontoasingleday.However,therearetwoshortcomingswiththis
approach.First,themonthlymodeliscalibratedforalongerhorizonandwillnothavetheappropriate
responsivenessfor
aone
day
forecast.
Second,
the
serial
correlation
adjustments
that
were
used
to
provideaccurateforecastsatamonthlyhorizonnowrepresentsourcesoferrorataonedayhorizon.In
otherwords,iftheobservationfrequencyissynchronizedwiththepredictionhorizon,thenserial
correlationadjustmentsshouldnotbeincorporated.
Inthispaper,wehighlightsomeofthemodelingissuesthatmustbeaddressedwhenconstructinga
modelwithaonedaypredictionhorizon.Centraltothischallengeistheidentificationofareliable
metrictoevaluatetheaccuracyofriskforecasts.
EvaluatingtheAccuracyofRiskForecastsBuildingasoundriskmodelfirstrequiresareliablemeansofevaluatingtheaccuracyofriskforecasts.
Thisprovidesanessentialguideforsettingmodelparametersandevaluatingperformance.Ideally,our
measureofforecastingaccuracywillprovidebothtimeandportfolioresolution,meaningthatthe
measurecanbeusedtoevaluateriskforecastsforasingleportfolioacrosstime,orforacollectionof
portfolioswithinasingletimeperiod.
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BiasStatisticsOnecommonlyusedmeasuretoevaluatetheaccuracyofriskforecastsisthebiasstatistic,which
conceptuallyrepresentstheratioofrealizedrisktoforecastrisk.Tocomputethebiasstatisticfora
portfolio n ,wefirstusetheriskmodeltopredicttheportfoliovolatility nt atthestartofeveryperiod
t.We
then
observe
the
out
of
sample
return
of
the
portfolio
ntR
overthe
subsequent
period.
The
standardizedreturnisdefinedby
ntnt
nt
Rz
, (1)
andexpressestheportfolioreturnasazscore.Thebiasstatisticisgivenbythestandarddeviationof
standardizedreturns,
2
1
1
1
T
n nt n
t
B z zT
, (2)
whereT isthenumberofdaysinthetestingwindow.
Conceptually,thebiasstatisticmeasureswhethertheriskforecastswereaccurate,onaverage,fora
singleportfolioacrosstime.Foraccurateforecasts,weexpecttherealizedbiasstatistictobecloseto1.
However,duetosamplingerror,thebiasstatisticwillneverbeexactly1evenforperfectrisk
forecasts.Instead,itiscustomarytoidentifyaconfidenceinterval.Assumingnormallydistributed
returnsandperfectforecasts,the95percentconfidenceintervalisapproximately1 2 / T .
Twoattractivefeaturesofthebiasstatisticarethatitissimpletointerpretandprovidesportfolio
resolution.Unfortunately,itdoesnotprovidetimeresolution,meaningthatitpossibletounderpredict
risk
for
some
sub
periods
and
over
predict
it
for
others
while
nonetheless
obtaining
a
bias
statistic
close
to1.Inotherwords,thebiasstatisticmayallowcancellationoferrorsacrosstime.Thisbecomes
especiallyproblematicoverlongsampleperiodsencompassingmanyyearsandmultiplemarket
regimes.Ariskmodelusermustbeconfidentthatvolatilityforecastsarereliableforallmarketregimes
notjustonaverage.
MeanRollingAbsoluteDeviation(MRAD)Onewaytopartiallymitigatethetimeresolutionproblemistodividethelongsampleperiodinto
smallersubperiods,andtocomputethebiasstatisticsoverthesesubperiods.Forinstance,ifweselect
12daysasthelengthofoursubperiod,weobtain,
11 21
11n nt n
t
B z z
, (3)
where denotesthestartofthe12daysubperiod.Thewindowisthenrolledforwardonedayatatimeuntilreachingtheendoftheentiresample.
TheconventionalMeanRollingAbsoluteDeviation,orMRAD,isdefinedasthemeanabsolutedeviation
ofthebiasstatisticsfromtheiridealvalueof1,
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1
111
n
n
MRAD BN T
, (4)
whereN isthenumberofportfoliosand ( 11)T isthetotalnumberof(overlapping)12daysub
periods.By
virtue
of
the
absolute
value,
MRAD
penalizes
both
under
prediction
and
over
prediction
of
risk.Assumingperfectforecasts,normallydistributedreturns,and12daysubperiods,itmaybeshown
thattheexpectedvalueofMRADisapproximately0.17.
Realfinancialdata,ofcourse,tendtohavefattails.InFigure1,weplottheexpectedvalueofMRAD
versuskurtosisforperfectriskforecasts.Theseresultsweregeneratedviasimulationbydrawing12
randomlygeneratedreturnsfromafattaileddistributionwithstandarddeviationequalto1.Thefat
tailswereobtainedusingastudenttdistributionwiththeappropriatenumberofdegreesoffreedom.
Weseethatasthekurtosisrisestoevenmodestlevels,theexpectedMRADforperfectforecasts
increasessignificantly.
Figure 1: MRAD versus kurtosis for perfect risk forecasts. Kurtosis levels were varied using a student t-distribution with the appropriate number of degrees of freedom.
Kurtosis
3 4 5 6 7 8 9 10
MRAD
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
WhiletheMRADmeasurehelpsmitigatecancelationoferrorsoverthelongrun,itisnotsuitableforrisk
modelcalibration,asitmayspuriouslyfavoroverlyresponsiveforecasts.Thiscanbestbeseenbya
simulationexperiment.Wegeneratedonemillionsimulatedreturnsfromastandardnormal
distribution.Wethenestimatedthevolatilityusingexponentiallyweightedaverageswithagivenhalf
lifeparameter.Sincethesimulatedreturnsarestationary(i.e.,theirdistributiondoesnotchangeover
time),thetrueoptimalhalflifeisinfinite,asthiseliminatessamplingerrorandhencerecoversthe
perfectriskforecast.
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InFigure2,wepresenttheresultsofthesimulationstudy.Whereasthetrueoptimalhalflifeisinfinite,
Figure2spuriouslypointstoanoptimalhalflifeofonlysixdays.Furthermore,theminimumMRADis
approximately0.16,wellbelowthetheoreticallowerboundof0.17forperfectforecasts.Howcanthese
puzzlingresultsbeexplained?
Figure 2: Conventional MRAD versus EWMA volatility half-life. Returns were generated using a standardnormal distribution. The conventional MRAD displays a spurious minimum at a six-day half-life, and exhibitsa value below the theoretical lower bound for perfect forecasts.
VolatilityHalfLife(Days)
0 5 10 15 20 25 30 35 40
ConventionalM
RAD
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
SpuriousMinimum
ThisconundrumisresolvedbyrecognizingthatthebiasstatisticandbyextensiontheMRADisnot
atrueoutofsamplemeasure.Thismaybesurprisingatfirstglance,sincethebiasstatisticiscomputed
usingoutofsamplezscores.Nevertheless,becauseriskforecastsareupdateddaily,thebiasstatistic
usesinformationinthevolatilityforecaststhatwasnotknownatthestartofthe12dayperiod.
Thatthistendstofavoroverlyresponsiveriskforecastsmaybeseenbythefollowingthought
experiment.Assumeforsimplicitythatreturnsaredrawnfromastandardnormaldistribution.Suppose
that,by
chance,
the
portfolio
experiences
several
large
standardized
returns.
In
this
case,
the
12
day
biasstatisticislikelytoexceed1.However,byoverreactingtotheeventanddramaticallyincreasingrisk
forecasts,thesubsequentzscoreswillbeartificiallyreduced.Althoughtheriskforecastsfollowingthe
eventarebiasedupward,therealizedbiasstatisticiscloserto1.Conversely,astringofsmallreturnsis
likelytoleadtoabiasstatisticlessthan1.Byreducingriskforecasts,thesubsequentstandardized
returnswillbeartificiallyinflated,againleadingtoabiasstatisticcloserto1.Inotherwords,excessive
responsivenessproducesnoisyandinaccurateriskforecasts,althoughtherealizedbiasstatisticsmaybe
deceptivelycloseto1.
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AdjustedMRADAstraightforwardwaytoremovetheinsampleeffectistosimplyuseriskforecaststakenfromthestart
ofthe12dayperiod,andtoholdtheseforecastsconstantoverthenext12days.Thisleadstothe
adjustedMRADmeasure.AnadvantageoftheadjustedMRADmeasureisthatitusesnoinformationin
thevolatilityforecaststhatwasnotknownatthestartofthe12daysubperiod,andtherefore
representsatrue
out
of
sample
measure.
A
disadvantage
of
the
adjusted
MRAD
is
that
it
requires
using
somewhatstaleforecasts.
CrossSectionalBiasStatisticsAnothermeasureofforecastingaccuracyisthecrosssectionalbiasstatistic,
2
1
1
1
N
t nt t
n
B z zN
. (5)
Thisis
similar
to
the
time
series
bias
statistic
of
Equation
2,
except
that
now
the
standard
deviation
is
computedacrossportfoliosforasingleday t.Sincethecrosssectionalbiasstatisticusesnoinformation
forvolatilityforecaststhatwasunknownatthestartofthesubperiod,itrepresentsatrueoutof
samplemeasure.
Conceptually,thecrosssectionalbiasstatisticmeasureswhethertheriskmodelforecastswere
accurate,onaverage,foracollectionofportfoliosonasingleday.Itcannot,however,answerwhether
theriskmodelforecastswereaccurateforindividualportfolios.Inotherwords,thecrosssectionalbias
statisticprovidestimeresolution,butnotportfolioresolution.Consequently,itispossibletoobtaina
crosssectionalbiasstatisticcloseto1,evenifriskforecastsforindividualportfolioswerepoor.In
particular,thismayoccurifoverpredictionerrorsforsomeportfoliosarecanceledbyunderprediction
errorsforothers.
QstatisticsPatton(2011)describesmeasuresofforecastaccuracyintermsoflossfunctions.Hedefinesaloss
functionasrobustiftherankingofanytwovolatilityforecastsbyexpectedlossisthesamewhether
therankingisdoneusingthetruevariance(unobservable)orsomeunbiasedvarianceproxy(e.g.,
squaredreturn).OneexampleofarobustlossfunctionistheQstatistic,definedforportfolio n andtime tas
2 2lnnt nt nt Q z z . (6)
Pattonfurther
shows
that
the
Q
statistic
is
the
unique
loss
function
(up
to
trivial
additive
and
multiplicativeconstants)thatdependssolelyonstandardizedreturns(i.e.,zscores).ThismakestheQ
statisticidealforevaluatingriskmodelaccuracy,becauseitplaceseveryobservationonanequalfooting
(whetherthevolatilityishighorlow).
Anotherkeypropertyofrobustlossfunctionsisthattheyareminimizedinexpectationwhenthe
predictedvolatilityequalsthetruevolatility.Forotherlossfunctions,thisisnottrue.Thatis,abiased
volatilityforecastcanminimizethelossfunction.Thiswouldobviouslybeproblematicforriskmodel
calibrationpurposes.TheMRADisanexampleofalossfunctionthatisnotrobust.
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AlsonotethattheQstatisticclearlysatisfiesbothtimeandportfolioresolution.Thatis,itisnotbased
onaveraging,soitisnotpossibletooffsetanoverforecastingerrorforoneobservationwithanunder
forecastingerrorforadifferentobservation.Ofcourse,theQstatisticcanbeaveragedeitheracross
timeoracrossportfoliostoobtainthedesiredresolution.
Intuitively,wecanthinkoftheQstatisticasbeingcomprisedoftwopenaltyfunctions.Thefirstterm,2
ntz ,becomeslargewhenriskforecastsaretoolowandthereforerepresentsanunderforecasting
penaltyfunction.Thesecondterm, 2ln ntz ,representstheoverforecastingpenaltyfunctionanddominateswhenriskforecastsaretoohigh.
ItisimportanttounderstandtheexpectedvalueoftheQstatisticforperfectriskforecasts.Ifreturns
arenormallydistributed,theexpectedvalueoftheQstatisticisapproximately2.27.AswiththeMRAD,
however,itisarathersensitivefunctionofkurtosis.InFigure3weplottheexpectedvalueoftheQ
statisticversuskurtosis,assumingperfectriskforecasts.
Figure 3: Q-statistic versus kurtosis for perfect risk forecasts. Kurtosis levels were varied using a student t-
distribution with the appropriate number of degrees of freedom.
Kurtosis
3 4 5 6 7 8 9 10
Q
statistic
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
ItisalsointerestingtoconsiderhowbiasesinvolatilityforecastsaffectchangesintheQstatistic.
Assumethat
returns
are
drawn
from
astandard
normal
distribution
(i.e.,
with
true
volatility
of
1).
Let
thevolatilityforecastsbedenotedby P .Onemayshowthattheexpectedlossrelativetoperfect
forecastsisgivenby
21
2ln 1P
P
E Q
. (7)
ThisfunctionisplottedinFigure4.Ifthepredictedvolatilityisequaltothetruevolatility 1P ,thentheexpectedlossiszero.Notealsothatthelossfunctionisasymmetric.Thatis,itpenalizesunder
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predictionofriskmoreheavilythanoverpredictionofrisk.Forinstance,ifthepredictedvolatilityishalf
thetruevolatility,thentheincreaseinQstatisticisapproximately1.61.However,ifthepredicted
volatilityisdoublethetruevolatility,thentheincreaseinQstatisticisonlyabout0.64.Thisisanother
attractivefeatureoftheQstatistic,sincemostinvestmentmanagersregardunderpredictionofriskas
moreproblematicthanoverpredictionofrisk.
Figure 4: Increase in Q-statistic versus predicted volatility when the true volatility is 1. Results were obtainedassuming a normal distribution. The asymmetry indicates that the Q-statistic penalizes under-forecasting ofrisk more heavily than over-forecasting of risk.
PredictedVolatility
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Q
0.0
0.5
1.0
1.5
2.0
ItisinstructivetorepeatthesimulationexerciseofFigure2,exceptnowusingtheadjustedMRADand
Qstatistics.TheresultsarepresentedinFigure5.Weseethatneitherofthesemeasuresexhibitsa
spuriousminimum,thuscorrectlypointingtoaninfiniteoptimalhalflife.
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Figure 5: Q-statistic and adjusted MRAD versus EWMA volatility half-life. Returns were generated using astandard normal distribution. Neither quantity exhibits a spurious minimum, thus correctly pointing to aninfinite half-life.
VolatilityHalfLife(Days)
0 5 10 15 20 25 30 35 40
Q
2.2
2.3
2.4
2.5
2.6
MRAD
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
MeanQ
MRAD(Adjusted)
TheUSE4ModelandDataSetTheempiricaldatasetusedforthisstudyconsistsofthehistoryofdailyfactorreturnsfromtheBarraUS
EquityRiskModel(USE4).Thehistoryofdailyfactorreturnsstartsin1993,whilethevolatilityforecasts
inourempiricalstudybeginonJuly19,1995.Thedatasetcontains4,328tradingdays,endingon
September21,2012.
TheUSE4Modelisavailableinlonghorizon(USE4L)andshorthorizon(USE4S)versions.Bothmodels
sharethesamefactorstructureandfactorreturns,butdifferintheirresponsiveness.TheUSE4SModel
isdesignedtoprovidethemostaccurateforecastsataonemonthpredictionhorizon.TheUSE4LModel
is
tailored
for
longer
term
investors
who
are
willing
to
trade
some
degree
of
forecasting
accuracy
for
greaterstabilityintheriskforecasts.WiththelaunchofUSE4D,theUSE4Modelisnowavailableina
thirdversionfordailyhorizonforecasts.Again,theUSE4DModelsharesthesamefactorstructureand
factorreturnsasthetwootherUSE4Modelvariants.
Here,webrieflyreviewsomehighlightsoftheUSE4Model.EmpiricalresultscanbefoundinYang,
Menchero,Orr,andWang(2011),whilemethodologydetailsaredescribedinMenchero,Orr,andWang
(2011).TheUSE4Modelcontains73factors,comprisedof: (a)theCountryfactor,whichisthe
regressionintercept,(b)60industryfactors,and(c)12stylefactors.Factorreturnsareestimatedby
performingdailycrosssectionalregressionsofstockreturnsagainstthestartofdayfactorexposures.
TheestimationuniverseistheMSCIUSAIMI,abroadindexrepresentingtheUSmarket.Themodel
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employssquarerootofmarketcapitalizationastheregressionweights.Theexactcolinearitybetween
industryfactorexposuresandtheCountryfactorisremovedbyconstrainingtheindustryfactorreturns
tobecapweightedmeanzeroeveryperiod.
AsdescribedbyMenchero(2010),theestimatedfactorreturnsmaybeinterpretedasthereturnsof
factormimickingportfolios.TheCountryfactorportfolioessentiallyrepresentsthecapweighted
estimationuniverse(i.e.,MSCIUSAIMI).Theindustryfactorportfoliosaredollarneutralandcapture
theperformanceoftheindustrynetofthemarketandotherstyles.Thestylefactorportfoliosaredollar
neutralandhaveunittiltontheparticularstyle,withzeroexposuretoindustriesandotherstyles.
EstimatingVolatilityThesimplestwaytoestimatevolatilityistocomputethesamplestandarddeviationoveratrailing
window.Ifstockreturndistributionswerestationary,thenusingthemaximumsamplesizeandequally
weightingeveryobservationwouldminimizesamplingerrorandhenceproducethemostaccurate
forecast.
Stockreturndistributions,however,arenotstationary.Eventsthatoccurredtenyearsagohavelittleto
dowithcurrentvolatilitylevels.Therefore,toreflectcurrentmarketconditions,wemustgivemore
weighttorecentobservations.Thecrucialquestionishowmuchmore?Ontheonehand,ifwegivetoo
muchweighttorecentobservations,thenwebaseourestimatesonveryfewdatapointsandweare
stronglypenalizedwithsamplingerror.Bycontrast,ifwegivetoomuchweighttodistantobservations,
thenweareharmedbyincorporatingstaledataintoourvolatilityestimates.Makinggoodriskforecasts
forrealfinancialdatarequiresanoptimaltradeoffbetweenthesetwoeffects.
ExponentiallyWeightedMovingAverages(EWMA)
A
simple,
yet
effective,
technique
for
attaching
more
weight
to
recent
observations
is
exponential
weightedmovingaverages(EWMA).Inthisapproach,thevarianceiswrittenasaweightedaverageof
laggedsquaredreturns.
2 2
t m t m
m
w r
, (8)
wheretheweights mw sumto1,anddecreasebyfixedproportioneveryperiod.Morespecifically,
1m mw w
,where isthedecayfactor.TheEWMAestimatecanbeconvenientlyrewrittenin
recursiveformasaweightedaveragebetweenyesterdayssquaredreturnandyesterdaysvariance
forecast,
2 2 21 11t t tr . (9)
Therelativeweightweplaceonyesterdaysvarianceforecastdeterminestheresponsivenessofthe
modelandisrelatedtotheEWMAvolatilityhalflifeparameterasfollows,
ln 0.5
lnHL
. (10)
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TheEWMAvolatilityhalflifeisacriticalmodelparameter.Toguideusinselectinganappropriatevalue,
wecomputetheaverageadjustedMRADandQstatisticforUSE4dailyfactorreturns,withtheaverages
takenoverallfactorsandtimeperiods.InFigure6,weplotthesequantitiesversustheEWMAvolatility
halflife.UnlikethesimulatedresultsinFigure5,weseethatrealfinancialdatadoexhibitaclear
minimum.Figure6showsthattheoptimalvolatilityhalflifebyboththeQstatisticandtheadjusted
MRADis
about
21
trading
days,
or
one
month.
Reassuringly,
both
out
of
sample
statistics
point
to
the
sameoptimalhalflife.Notethata21dayvolatilityhalflifecorrespondstoadecayfactorof0.97
accordingtoEquation10.
Figure 6: Q-statistic, adjusted MRAD, and conventional MRAD versus volatility half-life for EWMA forecastsusing USE4 daily factor returns. The optimal half-life is about one month (21 trading days) by either the Q-statistic or the adjusted MRAD. The conventional MRAD spuriously points to an optimal half-life of about fivetrading days.
VolatilityHalfLife(Days)
0 20 40 60 80 100
Q
2.35
2.40
2.45
2.50
2.55
2.60
MRAD
0.2000
0.2200
0.2400
0.2600
0.2800
0.3000
MeanQ
MRAD(Conventional)
MRAD(Adjusted)
Forcomparisonpurposes,wealsopresenttheconventionalMRADinFigure6.Notethatthismeasure
suggestsanoptimalhalflifeofjustfivedays.AlsoobservethattheminimumMRADiswellbelowthe
minimumvalueoftheadjustedMRAD.Theseresultsillustratethistimeusingrealfinancialdata
thesameeffectseeninthesimulatedresultsofFigure2.Thatis,insampleeffectsspuriouslypointto
excessivelyshorthalflifeparameters,whichinturnproducedeceptivelyattractiveMRADvalues.
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GARCH(1,1)ThemethodofGeneralizedAutoRegressiveConditionalHeteroskedasticty,orGARCH,wasdevelopedby
Engle(1982),Bollerslev(1986)andothersinthe1980s.Bynow,therearemanyvariationsofGARCH.An
excellentsurveyofthesetechniquesisprovidedbyBauwens,Laurent,andRombouts(2006)
Inthis
paper,
we
focus
on
the
most
widespread
variant
of
GARCH,
namely
GARCH(1,1).
In
this
approach,
varianceestimatesaregivenby
2 2 2 2
1 1t t tr
. (11)
Thisexpressespredictedvarianceasaweightedaverageofthreeterms:(a)yesterdayssquaredreturn,
withweight,(b)yesterdaysvarianceforecast,withweight ,and(c)alongrunvariance,with
weight .Thestabilityconditionrequiresthattheweightssumto1,
1 . (12)
Wedetermine
the
parameters
using
maximum
likelihood
estimation
with
atwo
year
look
back
window.
WeestimateGARCHparametersseparatelyforeveryfactorinthemodel.
GARCHhasseveralconceptuallyappealingattributes.First,themodelprovidesformeanreversionin
volatilityforecaststhroughthelongrunvarianceterminEquation11.Also,GARCHreplicatesthe
volatilityclusteringoftenfoundinfinancialtimeseries.Finally,bysettingthe parametertozero,
GARCH(1,1)encompassesEWMAasaspeciallimitingcase.
VolatilityRegimeAdjustment(VRA)TheUSE4DModelutilizestheVolatilityRegimeAdjustment(VRA)techniquetoestimatefactor
volatilities.ThistechniquewasfirstintroducedwiththelaunchoftheBarraUSEquityModel(USE4),as
describedby
Menchero,
Orr,
and
Wang
(2011).
The
central
concept
behind
this
approach
is
to
use
cross
sectionalobservationstocalibratethemodeltocurrentvolatilitylevels.
Riskmodelsbaseforecastsonhistoricalobservations.However,sincevolatilitylevelsvaryacrosstime,
thismayleadtocertainbiasesintheriskforecasts.Forinstance,enteringaperiodoffinancialcrisis,
volatilitylevelstendtorise.Inthiscase,riskmodelsusethelowervolatilitypasttopredictthehigher
volatilityfuture,thuscausingatendencytounderpredictriskduringtheseperiods.Conversely,
immediatelyfollowingaperiodoffinancialturmoil,therearemanyextremeeventsintheestimation
window,whichcreatesatendencytooverpredictriskatsuchtimes.Sinceriskmodelsarenotcrystal
balls,thereisnowaytocompletelyeliminatethesebiases.TheVRAtechnique,however,isdesignedto
mitigatethesebiasesbyusingcrosssectionalobservationstocalibratethemodeltocurrentvolatility
levels.
ThefirststepintheVRAtechniqueistoestimatefactorvolatilitiesusingEWMA,asinEquation8.This
estimateischaracterizedbythevolatilityhalflifeparameter,definedbyEquation10.Next,wecompute
thecrosssectionalbiasstatistic,asinEquation5.Thecomputationisperformedoverthezscoresof
dailyfactorreturns,usingtheEWMAvolatilityforecastsfromthefirststep.Wecanthinkofthecross
sectionalbiasstatisticasprovidinganinstantaneousmeasureofriskforecastingbias.Thefactor
volatilitymultiplier,attime t,isdefinedintermsofthetrailingcrosssectionalbiasstatistics,
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2F
t m t m
m
u B
, (13)
where mu istheexponentialweightcharacterizedbytheVRAhalflifeparameter.
Ifthefactorvolatilitymultiplierisgreaterthan1,itsaystheoriginalEWMAestimatesweretoolow,and
riskforecastsshouldbeadjustedupward.Conversely,when Ft
islessthan1,EWMAestimateswere
toohighandriskforecastsshouldbeadjusteddownward.Theadjustedvolatilityforecastforfactor kattime tisgivenby
F
kt t kt . (14)
Thisadjustmentisdesignedtoremovetheaveragebias,withallfactorvolatilitiesscaledbythesame
proportion.
VolatilityestimationwiththeVRAtechniquethereforerequirestwohalflifeparameters:theEWMA
volatilityhalf
life
defined
in
Equation
10,
and
the
VRA
half
life
used
in
Equation
13.
To
find
the
optimal
valuesforthesetwoparameters,weperformasearchfortheglobalminimumQstatisticoverthetwo
dimensionalspace.InFigure7,weplotcurvesofQstatistics(averagedacrossfactorsandtimeperiods)
versustheVRAhalflifeforseveralvaluesoftheEWMAvolatilityhalflife.Wefindthatthecombination
thatminimizestheQstatisticisa42dayEWMAvolatilityhalflifeinconjunctionwithatwodayVRA
halflife.
Figure 7: Q-statistic curves versus VRA half-life for three values of EWMA volatility half-life. The Q-statistic isminimized at a VRA half-life of two days and an EWMA volatility half-life of 42 days.
VolatilityRegimeHalfLife(days)
0 5 10 15 20
Q
2.41
2.42
2.43
2.44
21day
VOL
HL
84dayVOLHL
42dayVOLHL
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Atfirstglance,atwodayVRAhalflifemayseemsurprisinglyshort.Recall,however,thatitisprimarily
thepenaltyofsamplingerrorthatimposeslimitsontheresponsivenessofthemodel.IntheEWMA
approach,wherevolatilitiesareestimatedonefactoratatime,samplingerrorisrelativelylarge.Inthe
VRAapproach,bycontrast,welargelymitigatesamplingerrorbyvirtueofusing73crosssectional
observationseveryday.
AlthoughatwodayVRAhalflifeisoptimalintermsofforecastingaccuracy,itmayleadtoratherlarge
dailyfluctuationsinpredictedvolatilities.Toreducesuchvariability,theUSE4DModelcombinesthe42
dayEWMAvolatilityhalflifewithafourdayVRAhalflife.Thisprovidesconsiderablymorestabilityin
volatilityforecasts,whileproducingonlyamodestincreaseintheQstatistic,asevidentfromFigure7.
Thefinancialcrisisperiodof2008and2009affordsaprimeexampletoillustratethebehaviorofthe
VRAtechnique.InFigure8,weplotthefactorvolatilitymultiplier Ft
togetherwiththefactorcross
sectionalvolatility(CSV),definedas
21t kt
k
CSV f K
. (15)
Figure 8: Factor volatility multiplier and factor cross-sectional volatility (CSV) for the USE4D model. Resultswere smoothed using 10-day rolling windows.
Year
2008 2009 2010
FactorVolatilityMult
iplier
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
FactorCSV(percentd
aily)
0
1
2
3
4
5
6Factor
Volatility
Multiplier
Factor
CSV
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InJanuary2008,weseethatfactorCSVspikedfrom60bpstoabout150bpsperday.Thefactor
volatilitymultiplierrespondedquicklytotheincreasedvolatilitylevels,reachingamaximumvalueof1.7
duringthemonth.ThenextbigspikeinfactorCSVcameatthestartofSeptember2008,whenlevels
stoodatabout100bpsperday.Overthenexttwomonths,theselevelshadmorethandoubledto250
bpsperday.Again,thefactorvolatilitymultiplierquicklydetectedtheincreasedvolatilitylevels,hitting
apeak
of
1.8
in
October
2008.
BytheendofApril,2009,theworstofthefinancialstormwasover,andfactorvolatilitylevelsstoodat
about150bpsperday.Thenexteightmonthssawastrongseculardeclineinvolatilitylevels,withfactor
CSVhittingalowofabout50bpsperdaybytheendof2009.Overthistimeofrapidlydeclining
volatility,thefactorvolatilitymultiplierwaswellbelow1,hittinglowsofabout0.6duringthisperiod.
ThisexampledemonstratestheeffectivenessoftheVRAtechniqueforreducingvolatilityforecastsin
theaftermathofafinancialcrisis.
EmpiricalResultsInthissection,wecompareforecastsforsixdifferentsetsofvolatilityestimates.Thedatasetusedfor
ourstudyisthefullhistoryofUSE4dailyfactorreturns,asdescribedpreviously.Thefirstsetofvolatility
estimatesistakendirectlyfromtheUSE4Dmodel,whichusesavolatilityhalflifeof42daysandaVRA
halflifeoffourdays.Next,weconsiderahighlyresponsiveEWMAforecast,withvolatilityhalflifeof
fourdays(i.e.,equaltotheVRAhalflifeofUSE4D).WerefertothisastheEWMA(4)forecast.Wealso
considertheEWMA(21)forecast,whichusestheoptimalvolatilityhalflifeof21days,asseeninFigure
6.ThelastEWMAforecastweconsiderisEWMA(42),whichusesthesamevolatilityhalflifeasUSE4D.
WealsoreportresultsfortheGARCH(1,1)model,describedabove.Finally,weexaminetheUSE4S
model,withvolatilityforecastsscaledtoaonedayhorizon.
Beforepresentingresultsonforecastingaccuracy,weconsiderthestabilityofthevolatilityforecasts.To
investigate
this,
we
first
compute
for
factor
k
and
day
t
the
absolute
change
in
volatility
forecast
over
thepreviousday,
, , 1
,
, 1
k t k t
k t
k t
v
. (16)
Wethendefinetheforecastvariabilityastheaverageoverallfactorsandalltimeperiods,
,
,
1k t
k t
v vKT
. (17)
Thevariability
for
each
volatility
forecast
is
reported
in
Table
1.
The
variability
ranged
from
alow
of
63
bpsforUSE4S,toahighof7.96percentforEWMA(4).TheUSE4DModelhadanintermediatevariability
at2.82percent.
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Table 1: Comparison for six different volatility estimates. Averages were computed across all 73 factors and4328 trading days. Differences in Q-statistics are relative to the USE4D model.
Variability Conventional Adjusted Q Statistic
Model (Percent) MRAD MRAD Q Statistic (Difference)
USE4D 2.82 0.2124 0.2478 2.4182 0.0000
EWMA(4) 7.96 0.2059 0.3092 2.5231 0.1049
EWMA(21) 1.66 0.2239 0.2562 2.4455 0.0274
EWMA(42) 0.85 0.2403 0.2590 2.4564 0.0382
GARCH(1,1) 4.33 0.2147 0.2622 2.4486 0.0304
USE4S 0.63 0.2507 0.2647 2.4631 0.0450
InTable1,wealsoreportMRADandQstatisticsforthesixvolatilityforecasts.Bytheconventional
MRADmeasure,
EWMA(4)
has
the
lowest
score.
However,
as
we
have
seen,
the
conventional
MRAD
stronglyfavorsoverlyresponsiveforecastswhichcanmaketheMRADappearartificiallylow.Thetrue
outofsamplemeasures(adjustedMRADandQstatistic)paintadifferentpictureofforecasting
accuracy.
ByeithertheadjustedMRADortheQstatistic,theUSE4Dmodelproducedthemostaccuratevolatility
forecasts.WealsoreportinTable1thedifferenceinaverageQstatisticrelativetotheUSE4DModel.
Theaveragewascomputedoverall73factorsandtheentiresampleperiodof4,328tradingdays.The
secondmostaccurateforecastswereprovidedbytheEWMA(21)model,whichhadaQstatistic0.0274
aboveUSE4D.TheleastaccurateforecastswerefortheEWMA(4)model,whichscoredaQstatistic
0.1049abovetheUSE4Dmodel.
ItisausefulexercisetotranslatedifferencesinQstatisticsintoforecastingerrors.Thisisdifficulttodo
precisely,sinceitdependsontheforecasterrorsofeachestimateaswellasthereturndistributionsofthetestportfolios.Nevertheless,aroughsensecanbegleanedfromEquation7andFigure4.Wesee
thatadifferenceinQstatisticof0.10translatesroughlyintoapredictedvolatilityof 0.81P (i.e.,19
percentunderforecastingbias)or 1.28P (28percentoverprediction).Similarly,aQstatistic
differenceof0.03correspondsroughlytoforecastsof 0.89P (11percentunderforecast)or
1.14P (14percentoverforecast).
Itisalsoimportanttoconsiderthestatisticalsignificanceofthesefindings.Onewaytogaugestatistical
significanceistocountthenumberoffactorsforwhichUSE4DproducedlowerQstatisticsthanthe
otherforecasts.ComparedwithGARCH(1,1),theUSE4DmodelproducedlowerQstatisticsfor71ofthe
73factors.
Relative
to
EWMA(42)
or
USE4S,
the
USE4D
model
outperformed
for
72
of
the
73
factors.
Finally,versusEWMA(4)orEWMA(21),theUSE4Dmodeloutperformedforallfactors.Thelikelihoodof
theseresultsoccurringbymerechanceisexceedinglysmall.Thus,wecanconcludewithhighstatistical
confidencethattheUSE4Dmodelprovidedmoreaccurateriskforecaststhantheotherapproaches.
Although,onaverage,theUSE4DModelproducedmoreaccurateforecastsoverthefullsampleperiod,
itisimportanttoinvestigatethepersistenceoftheseresultsacrosstime.Forinstance,ifthe
outperformancewasdueprimarilytoaspecificsubperiod(suchasthe2008/2009financialcrisis),it
maycallintoquestiontherobustnessoftheVRAmethodology.Toexplorethispossibility,wedefinethe
cumulativeQincrement,
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1 1
1 K A Bkt kt
k t
Q Q QKT
, (18)
where Akt
Q istheQstatisticforforecast A ,factor k,attime t,and Bkt
Q isthecorrespondingquantity
forforecastB .Wecanthinkof Q askeepingarunningtallyofthedifferenceinaccuracy
betweenforecasts A andB .Attheendofthesampleperiod, T ,thecumulativeQincrementbecomessimplytheaveragedifferenceinQstatisticbetweenthetwoforecasts.HereweletforecastB
representtheUSE4Dmodel,whereasforecast A isusedtorepresentthealternative.
InFigure9weplotthecumulativeQincrementforallmodelsrelativetoUSE4D.Notethatallofthelines
haveapersistentupwardslope,indicatingthattheUSE4Dmodelconsistentlyprovidedmoreaccurate
forecastsacrossallmarketregimes.
Figure 9: Cumulative Q increment (relative to USE4D) versus time for five sets of volatility forecasts. Thepersistent upward slope indicates that the USE4D consistently provided more accurate forecasts acrossdifferent market regimes.
Year
1995 1997 1999 2001 2003 2005 2007 2009 2011 2013
CumulativeQ
0.00
0.02
0.04
0.06
0.08
0.10
0.12
EWMA(4)
EWMA(21)
EWMA(42)
GARCH(1,1)
USE4S
Inordertodevelopgreaterintuitionforhowtheseforecastscompare,itisusefultocomparetime
seriesplotsofthevolatilityforecastsforspecificfactors.
InFigure10,weplotthepredictedvolatilityoftheUSE4CountryfactorcomputedwithEWMA(42)and
USE4D,overtheperiod20072010.Weseethatoverthelastfourmonthsof2008,theUSE4DModel
predictedsignificantlyhighervolatilityfortheCountryfactorthantheEWMA(42)forecast.
Conversely,forthelasteightmonthsof2009,theEWMA(42)forecastsweresignificantlyhigher.These
resultsareconsistentwiththegreaterresponsivenessofUSE4DrelativetoEWMA(42).
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Figure 10: Predicted volatility of USE4 Country factor for USE4D and EWMA(42) forecasts.
Year
2007 2008 2009 2010
Vo
latility(Percent)
0
10
20
30
40
50
60
70
80
90
USE4D
EWMA(42)
InFigure
11
we
plot
the
volatility
of
the
Earnings
Yield
factor
for
GARCH(1,1)
and
USE4D
over
the
time
period20072010.Thiswasafactortowhichmanyquantitativeinvestorshadpositiveexposureover
thisperiod.
PriortotheQuantMeltdownofAugust2007,weseethattheGARCH(1,1)forecastswereverystable
andaboutatthesamelevelasUSE4Dforecasts.DuringtheheightoftheQuantMeltdown,the
GARCH(1,1)forecastswereabove12percent,versuseightpercentforUSE4D.
Forthenexttwoyears,whiletheQuantMeltdownremainedintheestimationwindow,theGARCH(1,1)
forecastswereveryjumpy.InAugust2009,astheeventexitstheestimationwindow,weseethatthe
GARCH(1,1)forecastssuddenlybecomemorestable.
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Figure 11: Predicted volatility of USE4 Earnings Yield factor for USE4D and GARCH(1,1) forecasts.
Year
2007 2008 2009 2010
Vo
latility(Percent)
0
2
4
6
8
10
12
14
16
USE4D
GARCH(1,1)
Aswe
have
seen,
the
Q
statistic
is
composed
of
an
over
forecasting
and
an
under
forecasting
penalty
function.Forperfectriskforecasts,theexpectedvalueoftheunderforecastingpenaltyfunctionis
exactly1.Assumingnormaldistributionandperfectriskforecasts,theexpectedvalueoftheover
forecastingpenaltyfunctionis1.27.Plottingthesetwopenaltyfunctionsversustimeandcomparing
themtotheiridealvaluescanyieldimportantinsightintowhetherthevolatilityforecastsweretoohigh
ortoolow.
InFigure12,weplotthemeanQstatistic,togetherwiththeoverforecastingandunderforecasting
penaltyfunctions,averagedacrossall73factors,forEWMA(42)volatilityforecasts.Theplotcoversthe
financialcrisisperiodof2008and2009,andlinesweresmoothedusing10dayrollingaverages.TheQ
statisticexhibitsseveraldistinctpeaksoverthecourseof2008,withthelargestsuchpeakoccurringin
October2008.Weseethateachofthesepeaksin2008wascausedbyaspikeintheunderforecasting
penaltyfunction,
implying
that
the
EWMA(42)
forecast
was
not
responsive
enough
during
this
period.
WecanevenusetheQstatistictoestimatethemagnitudeofthebias.InOctober2008,theQstatistic
wasatabout3.9,orabout1.6abovetheidealposition.FromFigure4,weseethistranslatesroughly
intoa50percentunderpredictionofrisk.Bycontrast,weseethatoverthelasteightmonthsof2009,
therelativelyhighvalueoftheQstatisticwasdominatedbytheoverforecastingpenaltyfunction.In
thiscase,theEWMA(42)modeldidnotadaptquicklyenoughtoreducedvolatilitylevelsinthewakeof
thefinancialcrisis.
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Figure 12: Plot ofQ-statistic together with over-forecasting and under-forecasting penalty functions for theEWMA(42) forecast. The quantities represent averages across factors. Lines were smoothed using 10-dayrolling windows. The EWMA(42) forecast generally under-forecasts for much of 2008 and over-forecasts formost of 2009.
Year
2008 2009 2010
QStatistic
0
1
2
3
4
5
QStatistic
UnderforecastPenalty
OverforecastPenalty
InFigure13weplotfortheUSE4DforecaststhemeanQstatistic,averagedacrossallfactors,together
withoverforecastingandunderforecastingpenaltyfunctions.FromFigure13,itisclearlyevidentthat
theQstatisticforUSE4Dwasmuchclosertotheidealvalueof2.27thanthecorrespondingplotin
Figure12forEWMA(42).Furthermore,weseethattheoverforecastingandunderforecastingpenalty
functionsneverdeviatedtoofarfromtheiridealpositions.
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Figure 13: Plot ofQ-statistic together with over-forecasting and under-forecasting penalty functions for theUSE4D forecast. The quantities represent averages across factors. Lines were smoothed using 10-day rollingwindows. Compared to EWMA(42) forecasts, the USE4D Q-statistics are much closer to their ideal values.Furthermore, the over-forecasting and under-forecasting penalties stay relatively close to their ideal values,indicating that the model adapts well to changing volatility levels.
Year
2008 2009 2010
Q
Statistic
0
1
2
3
4
5
QStatistic
UnderforecastPenalty
OverforecastPenalty
SettingtheCorrelationHalfLifeThusfar,wehaveonlyconsideredvolatilityforecasts.Inbuildingafactorcovariancematrix,however,
wemustalsoestimatetheoffdiagonalcovariances.Thesearegivenbytheproductofthefactor
volatilitiesandthefactorcorrelations.Thecorrelations,inturn,areestimatedusingEWMAwitha
specifiedhalflifeparameter.
Thecorrelationhalflifeisacrucialmodelparameter.Ifitistoolong,theremaybeapenaltybyusing
staledatainourcorrelationestimates.Ifthecorrelationhalflifeistooshort,thenthecovariancematrix
maybecomenoisyandillconditioned,leadingtopoorperformanceandunderestimationofriskof
optimizedportfolios,
as
described
by
Menchero,
Wang,
and
Orr
(2011).
Onewaytodeterminetheoptimalcorrelationhalflifeistocomputethevaluethatproducesthebest
outofsampleperformanceofoptimizedportfolios.Tostudythis,weconstructtheminimumriskfully
investedfactorportfolioatthestartofeveryday.TheCountryfactorexposureandthesumofindustry
factorexposuresareconstrainedtoequal1.InFigure14weplottheoutofsamplerealizedvolatility
overthefullsampleperiodasafunctionofcorrelationhalflife.Weseethat200daysappearsoptimal,
asthisminimizestheoutofsamplerealizedvolatility.Ifthecorrelationhalflifeistoolong,wedoseea
penaltyinoutofsamplevolatilitybyusingstaledata.Note,however,thatthispenaltyisrelatively
modest:8.6percentat500daysversus8.3percentat200days.Bycontrast,whenthecorrelationhalf
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lifebecomestooshort,thereisasteeppenaltyinoutofsamplevolatility.Inthiscase,modelestimates
areharmedbytoomuchnoiseinthecorrelationestimates.
Figure 14: Out-of-sample volatility for minimum-risk fully invested factor portfolio versus correlation half-life.
The out-of-sample period consisted of 4328 trading days. The volatility was minimized near a correlation half-life of 200 days.
CorrelationHalfLife(Days)
0 100 200 300 400 500
Volatility(Annualized)
8.0
8.5
9.0
9.5
10.0
10.5
MinRiskFullyInvested
Thereareseveralotherpossiblewaysofselectingthecorrelationhalflifeparameter.Forinstance,one
maystudythehalflifeparameterthatminimizestheoutofsamplevolatilitiesofstyleoptimized
portfolios.Anotherwaytosetthecorrelationhalflifeistoinvestigatethevaluethatproducesthebest
betaforecasts.Yetanotherapproachistodeterminewhichcorrelationhalflifeproducesthemost
accurateriskforecastsforfactorpairportfolios.
TheresultsoftheseinvestigationswerequalitativelysimilartotheresultsofFigure14.Namely,the
optimalcorrelationhalflifeiscloseto200days.Ifthecorrelationhalflifeistoolong,wefindaweak
penalty,butobserveamuchlargerpenaltyifthecorrelationhalflifeistooshort.Basedonthese
studies,weselecta200daycorrelationhalflifefortheUSE4Dmodel.
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ConclusionWehaveinvestigatedtherelativeaccuracyofvariousvolatilityforecastsoveraonedayprediction
horizon.WeexaminedseveralEWMAforecasts,aGARCH(1,1)model,theUSE4SModel(scaledtoaone
dayhorizon),andfinallytheUSE4DModel.WefoundthattheUSE4DModelprovidedthemostaccurate
forecastsamong
all
models
considered.
Furthermore,
the
outperformance
was
consistent
across
factors
aswellaspersistentacrosstime.
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Noneofthe Informationconstitutes anoffertosell(orasolicitationofanoffertobuy),anysecurity,financialproductorotherinvestmentvehicleoranytradingstrategy.Youcannoinvestinanindex.
MSCIsindirectwhollyownedsubsidiaryInstitutionalShareholder Services,Inc.(ISS)isaRegisteredInvestmentAdviserundertheInvestment AdvisersActof1940. Exceptwithrespectoanyapplicable productsorservicesfrom ISS(includingapplicableproductsorservicesfromMSCIESGResearch Information,whichareprovidedby ISS),neitherMSCInoranyof it
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neitherMSCInoranyofitsproductsorservicesisintendedtoconstitute investmentadviceorarecommendation tomake(orrefrainfrommaking)anykindofinvestmentdecisionand
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AboutMSCIMSCI Inc. isa leadingprovider of investmentdecisionsupport tools to investorsglobally, including assetmanagers, banks, hedge funds andpension funds.MSC
productsandservicesincludeindices,portfolioriskandperformanceanalytics,andgovernancetools.
Thecompanysflagshipproductofferingsare:theMSCIindiceswithclosetoUSD7trillionestimatedtobebenchmarkedtothemonaworldwidebasis1;Barramulti
assetclassfactormodels,portfolioriskandperformanceanalytics;RiskMetricsmultiassetclassmarketandcreditriskanalytics;IPDrealestateinformation,indices
andanalytics;MSCIESG(environmental,socialandgovernance)Researchscreening,analysisandratings;ISSgovernanceresearchandoutsourcedproxyvotingand
reporting services; FEA valuation models and risk management software for the energy and commodities markets; and CFRA forensic accounting risk research
legal/regulatoryriskassessment,andduediligence.MSCIisheadquarteredinNewYork,withresearchandcommercialofficesaroundtheworld.
1AsofMarch31,2012,aspublishedbyeVestment, LipperandBloomberginSeptember2012 Jan 2013
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