barra predicting risk at short horizons

7/28/2019 Barra Predicting Risk at Short Horizons

1/26

msci.com

ModelInsight

PredictingRiskatShortHorizonsACaseStudyfortheUSE4DModel

JoseMenchero,

Andrei

Morozov

and

Andrea

Pasqua

[email protected]

[email protected]

[email protected]

January2013


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MSCI AnalyticsResearch msci.com2013MSCIInc.Allrightsreserved.Pleaserefertothedisclaimerattheendofthisdocument.

ModelInsighPredictingRiskatShortHorizons(USE4D

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.


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.


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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January2013

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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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January2013

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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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References

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Engle.1982.AutoregressiveConditionalHeteroskedasticitywithEstimatesofVarianceUKInflation.

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Menchero.2010.TheCharacteristicsofFactorPortfolios.JournalofPerformanceMeasurement,vol.

15,no.1(Fall):5262.

Menchero,OrrandWang.2011.TheBarraUSEquityModel(USE4).MethodologyNotes.

Menchero,Wang,andOrr.2012.ImprovingRiskForecastsforOptimizedPortfolios,FinancialAnalysts

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FinancialandQuantitativeAnalysis,vol.9,263274.

Yang,Menchero,OrrandWang.2011.TheBarraUSEquityModel(USE4).EmpiricalNotes.


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January2013

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