2014-033

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Research DivisionFederal Reserve Bank of St. Louis

Working Paper Series

Can risk explain the profitability of technical trading in

currency markets?

Yuliya IvanovaChris Neely

David Rapach

and

Paul Weller

Working Paper 2014-033Ahttp://research.stlouisfed.org/wp/2014/2014-033.pdf

October 2014

FEDERAL RESERVE BANK OF ST. LOUISResearch Division

P.O. Box 442

St. Louis, MO 63166

______________________________________________________________________________________

The views expressed are those of the individual authors and do not necessarily reflect official positions of

the Federal Reserve Bank of St. Louis, the Federal Reserve System, or the Board of Governors.

Federal Reserve Bank of St. Louis Working Papers are preliminary materials circulated to stimulate

discussion and critical comment. References in publications to Federal Reserve Bank of St. Louis Working

Papers (other than an acknowledgment that the writer has had access to unpublished material) should be

cleared with the author or authors.

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Can

risk

explain

the

profitability

of

technical

trading

in

currency

markets?

YuliyaIvanova*

ChrisNeely

DavidRapach

Paul

Weller

October30,2014

JELCodes: F31,G11,G12,G14

Keywords:Exchangerate;Technicalanalysis;Technicaltrading;Efficient

markets

hypothesis;

Risk;

Stochastic

Discount

Factor;

Adaptive

markets

hypothesis;Carrytrade;

*Ph.D. Candidate, Department of Finance, University of Iowa, Iowa city, IA 52245; e-mail: [email protected]; phone: +1-319-335-0926; fax: +1-319-335-3690. Assistant Vice President, Federal Reserve Bank of St. Louis, St. Louis, MO 63166; e-mail:[email protected]; phone: +1-314-444-8568; fax: +1-314-444-8731.

Professor of Economics,

Saint Louis University, St. Louis, MO 63108; e-mail: [email protected]; phone:

+1-314-977-3601; fax: +1-314-997-1478.John F. Murray Professor of Finance Emeritus, University of Iowa, Iowa city, IA 52245; e-mail: [email protected]; phone: +1-319-335-0948; fax: +1-319-335-3690.

Chris Neely is the corresponding author. We thank seminar participants at the Federal Reserve Bank of St.Louis for helpful comments. The usual disclaimer applies. The views expressed in this paper are those of

the authors and do not reflect those of the Federal Reserve Bank of St. Louis or the Federal Reserve

System.

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AbstractItisarobustfindingthattechnicaltradingrulesappliedtoforeignexchangemarkets

haveearnedsubstantialexcessreturnsoverlongperiodsoftime.However,theapproachto

riskadjustmenthas typicallybeen rathercursory,andhas tended to focuson theCAPM.

Weexaminethereturnstoasetofdynamictradingrulesandlookattheexplanatorypower

ofawiderangeofmodels:CAPM,quadraticCAPM,CCAPM,Carharts4factormodel,an

extendedCCAPMwithdurableconsumption,LustigVerdelhan(LV)factors,volatilityand

skewness.Althoughskewnesshassomemodestexplanatorypowerfortheobservedexcess

returns, no model can plausibly account for the very strong evidence in favor of the

profitabilityof technical analysis in the foreign exchangemarket.We conclude that thesefindings strengthen the case for consideringmodels incorporating cognitivebias and the

processesoflearningandadaptation,asexemplifiedintheAdaptiveMarketsHypothesis.

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1

1.

Introduction

It is a stylized fact that excess returns for currencyrelated trading strategies,

especially the carry trade, areweakly correlatedwith traditional risk factors, such as the

CAPMsequitymarketfactor.Tobettermeasureabnormalreturnsincurrencymarketsand

assess market efficiency, recent studies propose a variety of risk factors for carry trade

portfolios. These risk factors include consumption growth (Lustig and Verdelhan, 2007),

crash risk (Brunnermeier, Nagel, and Pedersen, 2008), a forward premium slope factor

(Lustig,Roussanov, andVerdelhan, 2011), economic size (Hassan, 2013),global exchange

rate volatility (Menkhoff, Sarno, Schmeling, and Schrimpf, 2012), and liquidity (Mancini,

Ranaldo,andWrampelmeyer,2013).

Theserecentlyproposedcurrencyriskfactorsusefullyexplainthereturnstoacross

sectionofcarrytradeportfolios.Nevertheless,theeconomiccaseforthesefactorswouldbe

morecompellingiftheycouldalsoexplainexcessreturnsforinvestmentstrategiesbeyond

thecarrytrade(Burnside,2011).Suchexplanatoryabilitywouldallaydataminingconcerns

andbetterestablishtheeconomicrelevanceofthenewlyproposedcurrencyriskfactors.In

this spirit, the present paper investigates the ability of recently proposed currency risk

factorstoaccountfortheexcessreturnsoftechnicaltradingstrategies.Technicaltradinghas

received less attention than the carry tradedespitebeing as successful,welldocumented

and mysterious. Technical analysis (or trend following) is very popular among currency

market participants (Menkhoff and Taylor, 2007), and a sizable literature indicates that

traditional risk factors fail to explain the profitability of a variety of technical strategies.

Fromthisstandpoint,theabilityofnewlyproposedcurrencyriskfactorstoaccountforthe

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2

profitabilityof technicalstrategiesprovidesan informativetestofthefactorsrelevance.If

newriskfactorsfailtoadequatelyaccountforthebehavioroftechnicalstrategies,thenthese

factorsarelessappealingeconomicallyandthesearchformorerobustcurrencyriskfactors

shouldcontinue.

Weinvestigatetheabilityofabroadarrayofcurrencyriskfactorstoexplainexcess

returnsfor the technicalportfoliosdeveloped inNeelyandWeller (2013).Theseportfolios

arebased on a variety of popular technical indicators and provide a realistic picture of

returns for trendfollowing practitioners. We consider the following models: CAPM,

quadraticCAPM,CCAPM,Carharts4factormodel,anextendedCCAPMwithdurable

consumption, LustigVerdelhen (LV) factors, volatility and skewness. In a nutshell, our

resultsshowthatrecentlyproposedcurrencyriskfactorshavelittleexplanatorypowerfor

technicalportfolioreturns.Thenewriskfactorsidentifiedintherecentliteraturethusdonot

appearrelevantforanimportantclassofportfoliosinthecurrencyspace.Wehighlightthe

dimensionsalongwhich thenew risk factors fail toaccount for thebehaviorof technical

portfolios. The inadequacies of extant currency risk factors highlight the challenges in

explainingtechnicalportfolioreturns.

The restof thepaper isorganizedas follows.We firstdescribe theconstructionof

currencyportfolios.We thendescribe thedifferent currency risk factors thatwe consider

andeconometricmethodology.Ourempiricalresultsfollow.

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3

2.

Trading

Rules

and

Data

2.1TradingRules

Thegoalofourpaperistoexaminewhetherrecentadvancesinriskadjustmentcan

explain the seemingly very strong performance of traditional technical trading rules in

foreign exchange markets. We would like our trading rules to represent those that the

academic literature has investigated but also to be chosen dynamically, in an ex ante

fashion, to mimic the backtesting habits of practitioners and also to exploit changing

patternsinadaptivemarkets.Inordertodoso,webasicallyfollowNeelyandWeller(2013)

who dynamically constructed portfolio strategies from an underlying pool of frequently

studied rules 7 filter rules, 3moving average rules, 3momentum rules, and 3 channel

ruleson19dollarand21crossexchangerates.5There isonenotabledifferencebetween

therulesusedinthispaperandthoseinNeelyandWeller(2013):toisolatethedeterminants

oftechnicaltradingrules,thepresentpaperdoesnotincludecarrytradereturnsamongthe

rules.

Menkhoffetal.(2012b)havepreviouslystudiedtheriskadjustmentofcertaintypes

ofcurrencymomentumstrategies. Onemightbeconcerned that the rulesstudiedhere

technicalruleswouldbeverysimilartothemomentumrulesstudiedbyMenkhoffetal.

(2012b). But Menkhoff et al. (2012b) empirically evaluate this concern and argue that

technicalrulesandmomentumrulesarequitedifferent.

Allofthebilateralrulesborrowinonecurrencyandlendintheother.Thustheyall

5Dooley and Shafer (1984) and Sweeney (1986) look at filter rules; Levich and Thomas (1993) look at filter

and moving average rules; Jegadeesh and Titman (1993) consider momentum rules in equities, citing Bernard

(1984) on the topic; and Taylor (1994) tests channel rules, for example.

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4

produceexcessreturns.Wewillfirstdescribethetypesoftradingrulesbeforedetailingthe

dynamicrebalancingprocedureforcurrencytradingstrategies.

Afilterrulegeneratesabuysignalforaforeigncurrencywhentheexchangerate

(domesticpriceofforeigncurrency)hasrisenbymorethanypercentaboveitsmostrecent

low. It generates a sell signalwhen the exchange ratehas fallenbymore than the same

percentagefromitsmostrecenthigh.Thus,

(1)

where is an indicator variable that takes the value +1 for a long position in foreign

currencyand1 fora shortposition.nt is themost recent localminimumof andxt the

most recent localmaximum.The seven filter ruleshave filter sizes (y)of0.005,0.01,0.02,

0.03,0.04,0.05,and0.1.

Amovingaveragerulegeneratesabuysignalwhenashorthorizonmovingaverage

ofpastexchangeratescrossesalonghorizonmovingaveragefrombelow.Itgeneratesasell

signalwhen the shortmoving average crosses the longmoving average from above.We

denotetheserulesbyMA(S,L),whereSandLarethenumberofdaysintheshortandlong

moving averages, respectively. The moving average rules are MA(1, 5), MA(5, 20), and

MA(1, 200). Thus, MA(1, 5) compares the current exchange rate with its 5day moving

averageandrecordsabuy(sell)signaliftheexchangerateiscurrentlyabove(below)its5

daymovingaverage.

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5

Ourmomentumrules takea long (short)position inanexchangeratewhen then

daycumulativereturnispositive(negative).Weconsiderwindowsof5,20and60daysfor

themomentumrules.

Achannelruletakesalong(short)positioniftheexchangerateexceeds(islessthan)

themaximum (minimum)over thepreviousndaysplus (minus) thebandof inaction (x).

Wesetntobe5,10,and20,andxtobe0.001forallrules.Thus,

(2)

We apply these 16 bilateral rules 7 filter rules, 3 moving average rules, 3

momentumrules,and3channelrulesto19dollarand21crossexchangerates, listed in

Table1.TheseriesfortheDEMwassplicedwiththatfortheEURafterJanuary1,1999.For

simplicitywerefertothisseriesthroughoutastheEUR.Notallexchangeratesaretradable

throughout the sample. Table 1 details the dates on which we permit trading in each

exchangerate.

Inanystudyoftradingperformanceespeciallywhenusingexoticcurrenciesone

mustpay closeattention to transaction costs.Rulesand strategies thatmay appear tobe

profitablewhensuchcostsareignoredturnoutnottobeoncetheappropriateadjustments

have been made. We follow the methods in Neely and Weller (2013) and calculate

transactionscostsusinghistoricalestimatesforsuchcostsinthedistantpastandfractionsof

Bloomberg spreads after those were available. Neely and Weller (2013) detail these

calculations.

otherwise

1,...,minif

1,...,maxif

1

1

21

21

1

xSSSS

xSSSS

z

zntttt

ntttt

t

t

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6

2.2DynamicTradingStrategies

We would like to construct dynamic strategies to mimic the actions of foreign

exchange traders who backtest potential rules on historical data to determine trading

strategies. Accurately modeling potential trading returns provides the most realistic

environmentforassessingwhetherriskadjustmentexplainssuchreturns.

Therefore,weconstructdynamictradingstrategiesasfollows:

1. Ateachpoint in the sample,weapply the16bilateral rules toallavailable

exchangerates,calculatingthehistoricalreturnstatisticsforeachexchangeraterulepairat

eachpoint.There isamaximumof(16*40=)640exchangeraterulesonanygivenday,but

missingdataforsomeexchangeratesoftenleavefewerthanhalfthatnumberofcurrency

rulepairs.

2. Starting 500 days into the sample, we evaluate the Sharpe ratios of all

exchangeraterulepairswithatleast250daysofdatasincethebeginningoftherespective

samples.WethensorttheraterulepairsbytheirexpostSharperatios,rankingtheraterule

pairsbySharperatiofrom1to640.Wethenmeasuretheperformanceofthestrategiesover

thenext20days.

3. Every 20business days, we evaluate, sort and rank all available raterule

pairsusingthecompletesampleofdataavailabletothatpoint.Thus,thereturnsonthetop

ranked strategy pair willbe generatedby a given trading rule applied to a particular

exchangerateforaminimumof20days,atwhichpointitmay(ormaynot)bereplacedby

anotherruleappliedtothesameoradifferentcurrency.

Werefertotheseriesofsortedpairsasdynamicstrategies.Thatis,theseriesofthe

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7

bestexante ruleexchangeratepairs isknownasstrategy1, thesecondbestexante rule

exchangeratepairsisknownasstrategy2,etc.

Although we select the raterule pairs for the dynamic strategies based upon

historicalperformance,asdescribedabove,weevaluatethestrategiesperformanceoverthe

periodsaftertheyareselected. Thatis,allreturnperformancestatisticsinthispaperarefor

strategiesthatwerechosenexanteandarethusimplementable.

2.3Currencyportfolios

As is customary in related literature,we examine the riskadjustment of technical

tradingrulesinthefollowingway:Usingstrategies1to300touseastestassets,weform12

equallyweightedportfoliosof25strategiesperportfolio.Thusportfoliop1attimetconsists

ofthe25currencyrulepairswithSharperatiosranked1to25.Portfoliop2consistsofthe25

currencyrule pairs with Sharpe ratios ranked 26 to 50, and so on. The makeup of the

portfolios of currencyrule pairs may change from period to period with ex post Sharpe

ratiorankings.6

Iftechnicalanalysishaspredictivevalue,onewouldexpectthehigherranked(lower

numbered) ex ante strategies to also do better ex post than the lower ranked (higher

number) strategies. Figure 1 confirms a positive relation between ex ante and ex post

performance over the sample. The figure shows the spread in excess return, standard

deviationandSharperatiosoverthe12portfolios.Thetoppanelshowsthatall12portfolios

6Alternatively, one could examine riskadjusted returns to a portfolio of the top N strategies. Althoughwe omit the full results for brevity, we also riskadjust the return to four such portfolios, using equal andex ante, meanvariance optimal weights and N = 10, 50. We will note results from these exercises whereappropriate.

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8

havepositiveexcessreturns,generallydecliningasonegoesfromp1(4.61%perannum)to

p12(1.08%perannum). Themiddlepanelshowsthatthehighrankedportfoliosalsotend

tohavemorevolatile returns, though the relation isnotassteepas for returns.The third

panelconfirmsthis:expostSharperatiosarehigherfor theportfolioswithhigherexante

rankings,rangingfrom0.92forp1to0.29forp12.

3. Theoreticalframeworkforriskadjustment

3.1GeneralFramework

Toprovideageneralframeworkwithinwhichtomeasureriskexposureweneedto

characterize equilibrium in the foreign exchange market. We assume the existence of a

representative investorbased in theUS,and introducea stochasticdiscount factor (SDF),

that prices payoffs in dollars.7It represents a marginal rate of substitutionbetween

presentandfutureconsumptionindifferentstatesoftheworld.Thefirstorderconditions

forutilitymaximizationsubjecttoan intertemporalbudgetconstraintimplythatanyasset

returnmustsatisfy

|=1 (3)

wheredenotes the informationavailable to the investor at time t.Varyingassumptions

about the content ofproduce thedifferent categories ofmarket efficiency advancedby

Fama. Since we are modeling the risk exposure of technical trading rules we will be

primarilyconcernedwithweakform

efficiency inwhich the information setcontainsonly

pastprices.

7 Although we motivate the SDF framework with a representative investor, much weaker assumptions are

sufficient. In particular, absence of arbitrage implies the existence of a SDF framework, as in equation (3).

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9

Equation(3)impliesthattheriskfreeassetreturnisgivenby

= | (4)

Using(3),(4)andthedefinitionofcovariance,itfollowsthat

|= ,|| . (5)

That is, the excess return to any asset or trading strategy will be proportional to the

covarianceoftheassetreturnwiththeSDF.

The implication for technical trading strategies that take positions in foreign

currencies based on past prices is then straightforward. Positive excess returns are

consistentwithmarketefficiencyonlyiftheexcessreturnalsocovariesnegativelywiththe

SDF. This then raises the question of how to model the SDF and how to test whether

equation(5)explainsreturns.

Therearepotentiallyseveralways inwhichonecould test theextent towhich the

SDFframeworkcanexplainexcessreturnstothetradingrules. Themostdirectwouldbeto

model theSDF,, in (3)witha specificutility functionandcalibratedparametersand

test whether the errors from (3) are mean zero. Alternatively, one could estimate the

parameters of with some nonlinear optimization method, such as the generalized

method of moments (GMM), and test the overidentifying restrictions. Finally, one could

linearize theSDF,,with aTaylor series expansion, estimate a linear time seriesor a

returnbetamodelandevaluatewhether therisk factorsexplain theexpected returns.The

nextsubsectionsdetailthosetestingprocedures.

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10

3.2TestingaCalibratedSDF

Initially,wewillfollowLustigandVerdelhan(2007)inusinganextendedversionof

the CCAPM to adjust returns for risk. They use the model of Yogo (2006) in which a

representative agent has EpsteinZin preferences over durable consumption and

nondurableconsumption.Utilityisgivenby

1 ,

(6)

whereisthetimediscountfactor,isameasureofriskaversion,istheelasticity

ofintertemporalsubstitutioninconsumption,and 1 1 1 .Theoneperiodutilityfunctionisgivenby

, 1 (7)

where is the weight on durable consumption and is the elasticity of substitution

between durable and nondurable consumption. Yogo (2006) shows that the stochastic

discountfactortakestheform

, (8)

Where 1

and,isthereturnonthemarketportfolio.

Thismodel, theEpsteinZindurable consumptionCAPM (EZDCAPM)nests two

othermodelsofinterest,thedurableconsumptionCAPM(DCAPM)andtheCCAPM.The

DCAPMholds ifwe impose the restriction 1 .TheCCAPMholds if inadditionwe

impose .Thestochasticdiscountfactorin(8)satisfiesthefamiliarEulerequationin(3).

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11

Toinitiallyassesstheseconsumptionbasedmodelsofrisk,wecarryoutacalibration

exercise similar to that in Lustig and Verdelhan (2007). We choose parameter values

identified in Yogo (2006): 0.023, 0.802, 0.700. Then we use sample data on

durableandnondurableconsumptionand thereturnon themarketportfolio togenerate

pricing errors = 0, whereR is the excess return to portfolio pi, and i

1, ,12.8The coefficient of relative risk aversion is chosen to minimize the sum of

squaredpricingerrorsintheEZDCAPM.Table2presentstheseresults.Allmodelsclearly

perform very poorly; theis negative in every case.9The maximum Sharpe ratios and

priceofriskdiffersubstantiallyfromthoseinTable4ofLustigandVerdelhen(2007).Since

their testassetsare currencyportfolios sortedaccording to interestdifferentialwewould

expect thesenumbers tobesimilar.Theportfolioswith thehighest returnshavenegative

betas(p1hasabetaof 1.97).Thisimpliesthattheportfolioreturncovariespositivelywith

M.SinceMishighinbadtimeswhenmarginalutilityishighandconsumptionislow,these

portfoliosactasconsumptionhedgesandwouldbeexpectedtoearnrelativelylowreturns.

3.3LinearFactorModels

ResearcherstypicallylinearizetheSDFwithafirstorderTaylorapproximationand

thenassessthemodelsfitofthedatawiththatlinearsystem. Althoughitisnotclearhow

well the linearmodel approximates the SDF, thisproceduremakes estimation somewhat

easierandisconsistentwiththeliterature.

Therefore,weconsidertheclassoflinearSDFsthattaketheform

8Recall that portfolios p1 to p12 each consist of 25 currency-rule pairs, ranked every 20 days by ex-ante Sharpe

ratio. P1 contains strategies 1 to 25; p2 contains strategies 26 to 50 and so on.9The R2s are negative because the models predictions covary negatively with the mean excess returns of the

test assets.

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12

(9)

whereis a scalar, is a 1vector of parameters andis a 1vector of demeaned

factors that explain assetprice returns.The constantin (9) isnot identifiedandwe can

normalizeittounity.10WehypothesizethattheSDFpricesportfoliosofexcesstradingrule

returns,,inwhichcaseequation(3)impliesthat

| 0 (10)

Theunconditionalversionof(10)is

1 0 (11)

fromwhichitfollowsthat

(12)

whereis the factor covariancematrix, is a 1vectorof coefficients in a

regressionofonand isa 1vectoroffactorriskpremia.

The model (12) is a returnbeta representation. It implies that an assets expected

return isproportional to itscovariancewith therisk factors. Thebetasaredefinedas the

timeseriesregressioncoefficients

1, ,, 1, , (13)

whereis the nondemeaned factor at time t. In the special case that the factors,, are

excessreturns,thentheintercepts()inthetimeseriesrepresentation(13)arezero.Wecan

seethisbyfirstnotingthattheexpectationofthefactormustsatisfy(12)becausewehave

assumedthatthefactorisalsoareturn.

10For any pair, {a,b}, such that 0, any {ca, cb} where cis a real constant would also satisfythe equation because 0. Therefore, only the ratio b/amatters and one can normalize ato 1or to any other constant.

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13

(14)

wherethesecondequalityfollowsbecausemustequalonebecausethefactorcovaries

perfectlywithitself.Second,wetakeexpectationsin(13)andsolvefortheconstant

0 (15)

Thesecondequalityin(15)uses(12)and(14).

Forthesecases inwhichthefactor isitselfanexcessreturne.g.,theCAPMone

cantestthemodelbyregressingasetofNexcessreturnstotestassetsonthefactor,asin

(13),andthendirectlytestingwhethertheconstants()arezero. Iftheconstantsarenon

zero,thenthemodelfailstoexplaintheexcessreturntothetestassets.

For testsofmoregeneralsetsof factors,FamaMacBeth (1973)suggesta twostage

procedurethatfirstestimatesthesforeachtestassetwiththetimeseriesregression(13).11

Thesecondstagethenestimatesthefactorprices fromacrosssectionalregressionofmean

excessreturnsfromthetestassetsonthebetas.

, (16)

where is the regression coefficient tobe estimated ands are the pricing errors. The

model impliesnoconstant in (16)butone isoften includedwiththereasoning that itwill

pickupestimationerrorintherisklessrate.Alargevalueof,orasignificantchangeinthe

fitofthemodelwithaconstantindicatesapoorfit(Burnside(2011)).Foragivensetoftest

assets,thevariationofthebetasin(14)determinestheprecisionoftheestimatedfactorrisk

11Fama-MacBeth (1973) originally used rolling regressions to estimate the s and cross-sectional regressions at

each point in time to estimate a and for each time period, then using averages of those estimates to getoverall estimates. The time series of and estimates could then be used to estimate standard errors for theoverall estimates that correct for cross-sectional correlation.

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premia, . If thebetas do not vary sufficiently, then is not identified and the test is

inconclusive.

ThestandarderrorsinthesecondstageoftheoriginalFamaMacBethproceduredo

not account for the fact that the regressors () are generated regressors. Shanken (1992)

suggestsacorrectiontoaccountforthisissue.OnecanuseGMMtosimultaneouslyestimate

both (13) and (16), obtaining the identical point estimates as the 2stage procedurebut

properlyaccounting for crosssectional correlation,heteroskedasticityand theuncertainty

aboutin the covariancematrix of theparameters (seeCochrane, 2005 chapter 10).The

momentrestrictionsaregivenby

, 0, 0

, 0 for 1,2,and 1,2, (17)

Inourempiricalexercises,weestimatethebetareturnmodelswithGMMandpresentthree

setsofstandarderrorsOLS,ShankenandGMMintheinterestofcomparison.

4. Results

4.1CAPMmodelsofthereturnstop1throughp12

Figure1showedthattheexpostSharperatiosofthetechnicalstrategiesvariedwith

their ex ante rank. That is, past returns tend to predict future returns. Does the risk

adjustmentaffecttheexpectedcrosssectionofreturns?

AsabenchmarkwefirstlookatwhethertheCAPMcanexplaintheexcessreturnsto

the12portfolios,P1P12,whichconsistofstrategies125,2650,276300,respectively.The

model has a single factor, the market excess return, and so we consider the following

regressionequationforeachportfolio:

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15

, , (18)

whereistheexcessreturntothedynamicportfoliostrategyandisthemarketexcess

return.Themodelrequiresthattheintercept,alpha,notbesignificantlydifferentfromzero.

Apositivealphawouldimplythatthemodelfailstoexplainthereturn.

PanelAofTable3 shows the results for regression (13) for the12portfolios. The

riskadjustmentleavesthemeanreturn(alphas)essentiallyunchangedforall12portfolios.

PortfolioP1,forexample,hasahighlysignificantmonthlyalphaof0.39,or4.68percentper

annum.Thehighlysignificantalphas indicate that themarket factorcannotexplain the

excessreturnstothetechnicalportfoliosandthenegativebetasindicatethatthereturnsare

notevenpositivelycorrelatedwiththemarketreturns.WeconcludethattheCAPMfailsto

explaintheexcessreturnsofthedynamicportfoliostrategies.

WeturntothequadraticCAPMinTable3.

, , , (19)

Because the squared excess return is not the return to a tradable portfolio, we cannot

interpret theconstant in the timeseriesregressionasariskadjustedreturn.Here too, the

coefficients on the market (are all negative, which would tend to indicate that the

marketriskdidnotexplaintheforexportfolioreturns,butthequadratictermsusuallyhave

significantlypositivecoefficients( thatsuggestthatequitymarketvolatilityinfluences

dynamic technical portfolio returns. Menkhoff et al. (2012a) document a similar

phenomenon forcarry tradereturns.Further, therightmostcolumnsshow that these

coefficientsjointly differ from zero and from each other, which potentially allows us to

identifythepriceofriskforthequadraticCAPMfactor.

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16

TheQuadCAPM columnsofPanelB show thataddingaconstant to thecross

sectional regression raises the R2 from 0.11 to 0.18 but the constant is not statistically

significantwithGMMstandarderrors.Thepriceofriskforthequadraticmarketreturnis

0.31 and statistically significant. These statistics indicate that the quadratic CAPM might

partiallyexplainsomeoftheexcessreturnsofthedynamicportfoliostrategies.

WenextexamineCarharts(1997)fourfactorextensionofthethreefactormodelof

FamaandFrench(1993)wheretheriskfactorsaretheexcessreturnontheU.S.stockmarket

(),thesizefactor(),thevaluefactor()andthemomentumfactor().12

, , , , (20)

Panel A of Table 3 shows that the alphas for the top portfolios are positive and

highlysignificantandthecoefficientsontheregressorsaregenerallynegative(,and

)orinsignificant().TherightmostcolumnsofPanelAshowthatonecannotreject

the nulls of no variation in any of the fourbeta vectors. This indicates that one cannot

conclusively identify the price of risk for these factors. Perhapsbecause of this lack of

identification,thepricesoffactorriskestimatedbycrosssectionalregressionforbothSMB

andHMLarenegative,whichisinconsistentwithestimatesderivedfromthesamplemean.

Asnotedabove,whenthefactorsaretradableexcessreturnsthenfactormeansareequalto

pricesofrisk.The factormeans inoursampleare0.58% (),0.24% (),0.29% ()

and 0.68% ( ). There is no evidence that the fourfactor model explains the excess

12 Fama and French (1993) showed that 3-factors, market return, firm size and book-to-market ratios very

effectively explained the returns of certain test assets. The factors used in equation (15) are returns to zero-

investment portfolios that are simultaneously long/short in stocks that are in the highest/lowest quantiles of thesorted distribution. For example, the small-minus-big (SMB) portfolio takes a long position in small firms and a

short position in large firms.

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returnstothedynamicportfoliostrategies.

4.2Consumptionbasedmodelsofthereturnstop1throughp12

Wenowexaminewhetherconsumptionbasedmodelsofassetpricingexplainthe

returnstothe12portfoliosofdynamicstrategies.TheCCAPMrelatesassetreturnstothe

realconsumptiongrowthofariskaverserepresentativeagent,asinequation(8).

Wefirstconsiderthreevariationsofthelinearapproximationofthefactormodelin

(8),theCCAPM,DCAPMandEZDCAPM(Yogo(2006)).

, , , , (21)

whereandare lognondurableanddurableconsumptiongrowth respectively,and

,isthelogreturnonthemarketportfolio.Thelinearapproximationforthemostgeneral

of thesenestedmodels,theEZDCAPM,usesnondurablesplusservices,durablesandthe

marketexcessreturnasfactors.Onecanestimatethefactorprices,,andportfoliobetas,i

withthebetarepresentationgivenbyequations(13)and(16).

, , , ,, , (22)

, , , , (23)

where , , , , ,is the factor covariance matrix and is

thecovariancematrixbetweenthetestreturnsandthefactors.TheDCAPMandCCAPM

restrict, andthepair,{, ,,}toequalzero,respectively.13

Table4,PanelApresentstheresultsof the timesseriesregressions.Allmodels,C

CAPM,DCAPMandEZDCAPMperformpoorly.FortheCCAPMthebeta forportfolio

p1issignificantatthe10percentlevelbuthasthewrong(negative)sign.Mostofthebetas

13One could infer the utility function parameter values from the estimates of the coefficients on the factors in

the linear model, as in Lustig and Verdelhan (2007), see equation (4) in that paper.

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for the DCAPM and EZDCAPM are either negative or insignificant. The rightmost

columnsofPanelA show that thebetavectors allvary significantly from zeroand from

eachother,permittingidentificationofthepricesofrisk.

PanelBshowstheresultsofthecrosssectionalregressionsforeachmodelestimated

withandwithoutaconstant.IncludingconstantsappearstochangethefitoftheCCAPM

andDCAPMmodels:Theconstantsarenotsignificantattheonesided10percentleveland

change the R2s from negative to positive levels. The negative R2s are damning: They

indicatethatasimpleconstantwouldexplaintheexpectedreturnsbetterthanthemodel.

TheEZDCAPMmodelappearstofitbetterthantherestrictedmodels.Theconstant

isnotsignificantandtheR2issizable,at0.61anddoesnotchangemuchwiththeadditionof

the constant.Theprice of risk fornondurable consumption in theEZDCAPMmodel is

statisticallysignificantbutnegative,2.24percent.Thisnegativepriceofriskisimplausible

becausetheorypredictsthatitshouldbepositivewhenthecoefficientofriskaversion 1,

andtheelasticityofsubstitutioninconsumption 1.Yogo(2007)estimates tobe0.210

fortheEZDCAPMwithequityassets.

We find in all cases that the coefficient of risk aversion is estimated to be

significantlynegative.The reason for thiscanbe seenclearly in thecaseof theCCAPM.

Equation (5) implies that , ; the Fundamental Theorem of Asset Pricing

requirestobepositiveandsoitfollowsthatifanassethasapositiveexpectedexcess

return itmust covarynegativelywith theSDF.TheSDF (M), in turn, ismarginalutility,

which will covary negatively with consumption. In the case of the CCAPM where the

singlefactorisconsumptiongrowth,thismeansthatthemodelpredictsthatreturnstothe

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higher ranking dynamic portfolio strategies should be positively correlated with

consumptiongrowth. In fact,we find thereverse: thedynamicportfoliostrategies tend to

performwellinstateswhenconsumptiongrowthislow,andthusprovideahedgeagainst

consumption risk. Tobe consistent with the model such strategies should earn negative

excessreturns.

Figure2displaystheactualmeanexcessreturntoeachportfolio()versusthe

predicted return ( for the crosssectional regressions that include a constant.

Intriguingly, there does appear tobe a positive relationshipbetween the predicted and

actualreturnforallthreemodels,CCAPM,DCAPMandEZDCAPM. Recall,however,that

the prices of risk are implausibly signed and that the crosssection regression should

excludetheconstant.

If one excludes the constant from the regression, the actual returns exceed the

predicted return for all but one return for the CCAPM and DCAPM cases (Figure 3).

Although the lower leftpanelofFigure3appears to illustrateanicepositive relationship

betweenEZDCAPMpredictionsandactual returns, this relation ismisleading: It ignores

thenegativeR2sandnegativepricesofnondurableconsumptionrisk.

4.3Foreignexchangebasedmodelsofthetechnicalreturns

Consumptionbasedmodelshavegenerallyfailedtoexplainriskadjustedreturnsin

severalcontextssotheresultsinTable4comeasnosurprise.Thisfailureofconsumption

basedmodelshas ledresearchersto lookforotherriskfactorsthatmightproxyforfuture

investment opportunities. In the context of stock returns it hasbecome commonplace to

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work with factors that are the returns to various stock portfolios (see Fama and French

(1993)andCarhart(1997)).

Lustig,RoussanovandVerdelhan (2011)have recentlyapplied thisgeneral idea to

the foreignexchangemarket.They formcurrencyportfolioson thebasisof interest rates.

Portfolio 1 contains those currencies with the lowest interest rates, portfolio 6 those

currencieswith thehighest interest rates.Theportfoliosare rebalancedat theendofeach

month.FromtheseportfoliosLustigetal.(2011)constructtworiskfactors.Thefirstfactor,

whichtheydenoteRX,istheaveragecurrencyexcessreturntogoingshortinthedollarand

longinthebasketofsixforeigncurrencyportfolios.Thesecondfactor,HMLFX,isthereturn

to a strategy thatborrows low interest rate currencies (portfolio 1) and invests in high

interestratecurrencies(portfolio6),inotherwordsacarrytrade.14

Canthesefactorsexplainthecrosssectionalvariationinexpectedreturnsacrossthe

12technicalportfoliosthatweresortedonpastSharperatio?Weexaminethisquestionwith

theGMMestimationofthebetareturnmodelsusingtheRXfactorandtheHMLFX factoras

riskfactors.Thisprovidesuswithestimatesofthefactorriskpremia(and)and

the parameters of the model. Because the factors are tradable we can test the modelby

comparing theestimatesof the riskpremiawith the factormeans.We reject themodel if

theydiffersignificantly.

Panel A of Table 5 shows that thebetas on the RX factor are small and always

insignificant.However,foreightofthedynamicportfoliosthebetasontheHMLFXfactorare

significantlynegativeatthetenpercentlevelandallofthepointestimatesarenegative.The

14RX and HMLFXare very similar to the first two principal components of the returns to the 6 portfolios.

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signsofthebetassuggestthatthereturnstothetechnicaltradingrulestendtobehighwhen

thecarry tradereturnsare low.TheHMLFXbetas forportfoliosp1,p3,p4,andp5arenot

significant, however, and the alphas for the top five portfolios are often significant and

always higher than the unadjusted mean returns. The fact that the constants (i.e., risk

adjusted returns) are always higher than the unadjusted mean returns indicates that

accounting for HMLFX and RX risk actually deepens the puzzle of the profitability of

technical trading rules. These results make it seem unlikely that any risk factor that

explainedthecarrytradecouldalsoexplainthetechnicalreturns.

Tests from the time series regression in Panel A, far right columns, indicate that

neithersetofbetasvariessufficientlytoidentifythepricesofrisk. Nevertheless,wepresent

resultsforthecrosssectionalregressioninPanelBofTable5. Thoseresultsshowthatthe

priceofriskfortheHMLFX factoris2.08andishighlysignificant,andtheR2is0.35(Table5,

PanelB).Buttheconstantintheregressionis0.22andisalsohighlysignificant.Themeanof

theHMLFXis0.69%.Whentheconstantisexcluded,theR2becomesnegative,however.We

concludethattheRXandHMLFXfactorscannotexplainthereturnstothedynamictechnical

portfolios.

Menkhoff,Sarno,SchmelingandSchrimpf(2012a)findthatglobalforeignexchange

volatilityisanimportantfactorinexplainingcarrytradereturns.Toinvestigatethisfactors

explanatory power for our technical returns, we estimate a global volatility factor in a

mannerverysimilarto thatofMenkhoff,Sarno,SchmelingandSchrimpf(2012a).Wefirst

calculatethemonthlyreturnvarianceforeachoftheavailableexchangeratesateachmonth

inour sampleand then calculateaglobal foreign exchangevolatility factor from the first

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principalcomponentofthemonthlyvariances.Vol1 istheseriesof innovationsofanAR1

process fit to thisprincipal componentwhileVol2 is the firstdifference in thisprincipal

component. We then estimate abeta representation using these volatility factors and a

dollarexposurefactorthattheynoteisverysimilartotheLustig,RoussanovandVerdelhan

(2011)RX factor.Menkhoff, Sarno, Schmeling and Schrimpf (2012a)denote this theDOL

factorintheirwork.

Table6displaystheresultsfromtheGMMestimationofthebetasystem. PanelAof

Table6shows that thebetasonRXarenotjointlystatisticallysignificantnorsignificantly

differentfromeachother. ThusthepriceofRXriskisnotidentified. Incontrast,betason

thevolatilityfactors(Vol1andVol2)arepositiveandhighlysignificantbutdonotappearto

captureanyofthecrosssectionalspread inreturns.That is,there isnoobviouspatternto

thebetasfromthelow tohighrankingportfolios.

Turning to Panel B of Table 6, when one excludes the constant from the cross

sectionalequation,thepricesofriskonVOL1andVOL2aresmallandinsignificantandthe

R2s of the crosssectional regressions are negative. In addition, the constant terms are

significant. In other words, the volatility factor picks up time series variation in returns

whichiscommontoallportfolios,butthemodeldoesnotexplainthecrosssectionalspread

in returns, or the level of returns for portfolios p1 and p2. There is no evidence for a

negativepriceofvolatilityrisk.

Researchers have also explored skewness as a risk factor for carry trade returns

(Rafferty(2012))andthecrosssectionofequityreturns(Amaya,Christoffersen,Jacobs,and

Vasquez (2011). To investigatewhetherexposure toskewnesscangenerate thereturns to

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the technical trading rules, we form a skewness factora tradable portfolio that is long

(short) currencies in thehighest (lowest) skewnessquintile in agivenmonthand then

estimatethemodelsbetarepresentation.Table7presentsthoseresults.

Panel A of Table 7 shows that the skewness betas are all positive and highly

significant.Onecannot,however,rejectthehypothesisthattheyareallequaltoeachother,

precludingstrong identificationofthepricesofrisk. InPanelB,theconstant inthecross

sectionalregressionismarginallysignificantandtheexclusionofthisconstantreducesthe

R2fromahealthy0.25toaverymodest0.09.Withouttheconstant,thepriceofriskis0.82

andstatisticallysignificant.These results suggest thatexposure to skewnessdoesaccount

forsometechnicaltradingrulereturns,althoughitsexplanatorypowerisfairlylimited.

5. DiscussionandConclusion

Researchershave longdocumented theprofitabilityof technicalanalysis in foreign

exchangerates.StudiesthatfoundpositiveresultsincludePoole(1967),DooleyandShafer

(1984), Sweeney (1986), Levich and Thomas (1993), Neely, Weller and Dittmar (1997),

Genay(1999),Lee,GleasonandMathur(2001)andMartin(2001)).

Despitesuchasubstantialrecordofdocumentedgains,thereasonsforthissuccess

remain mysterious. Neely (2002) appears to rule out the central bank intervention

explanation suggestedbyLeBaron (1999).To investigate thepossibilityofdata snooping,

dataminingandpublicationbias,Neely,WellerandUlrich(2009)analyzetheperformance

ofrulesintrueoutofsampleteststhatoccurlongafteranimportantstudy.Theyconclude

thatdatasnooping,dataminingandpublicationbiasareunlikelyexplanationsbutthatthe

adaptivemarketshypothesisisplausible.

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It remainspossible, however, that technical trading rule profitability results from

exposuretosomesortofrisk.Recently,severalauthorshaveconsideredmoderntechniques

for riskadjustmentof thecarry tradeormomentum rules in foreignexchangemarkets. If

these factors explain the returns to other trading strategies, they become much more

interesting economically.The goal of thispaperhasbeen to apply abroad range of risk

adjustment techniques to determine whether there is any evidence that exposure to risk

plausiblyexplainstheprofitabilityoftechnicalanalysisintheforeignexchangemarket.

We examine many types of risk adjustment models, including the CAPM, a four

factormodel,severalconsumptionbasedmodels,andfactorsmotivatedbythecarrytrade

puzzle, volatility measures and skewness. Although skewness might have some limited

explanatory power, no model of risk adjustment can plausibly explain the very robust

findingsoftheprofitabilityoftechnicalanalysisintheforeignexchangemarket.

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Table1:Datadescription

Notes:The tabledepicts the21exchangeratesversus theUSDand19nonUSDcrossratesused in

oursamplealongwiththestartingandendingdatesofthesamples,numberoftradingdates,average

transactioncost,andstandarddeviationofannualizedlogreturns.

Currency Group Country

Currency abbreviation

versus the USD

# of trading

obs

Trading start

date

Trading end

date Mean TC

STD of Annualized

FX Return

Advanced Australia AUD 9008 4/7/1976 12/31/2012 3.1 11.6

Advanced Canada CAD 9344 1/2/1975 12/31/2012 2.9 6.7

Advanced Euro Area EUR 9717 4/3/1973 12/31/2012 3.0 10.6Advanced Japan JPY 9599 4/3/1973 12/28/2012 3.0 10.5

Advanced New Zealand NZD 6027 8/3/1987 12/31/2012 3.9 12.4

Advanced Norway NOK 6515 1/2/1986 12/31/2012 3.4 11.6

Advanced Sweden SEK 7278 1/3/1983 12/28/2012 3.3 11.4

Advanced Switzerland CHF 9697 4/3/1973 12/31/2012 3.1 12.0

Advanced UK GBP 9338 1/2/1975 12/31/2012 2.9 9.9

Dev. Europe Czech Republic CZK 5049 1/5/1993 12/31/2012 5.2 12.4

Dev. Europe Hungary HUF 4466 1/2/1995 12/28/2012 10.3 14.3

Dev. Europe Hungary/Switzerland HUF_CHF 4165 1/3/1996 12/28/2012 10.5 12.0

Dev. Europe Poland PLN 3918 2/24/1997 12/31/2012 7.1 14.6

Dev. Europe Russia RUB 3055 8/1/2000 12/28/2012 3.6 7.4

Dev. Europe Turkey TRY 2769 1/2/2002 12/31/2012 12.9 15.4

Latin America Brazil BRL 3330 5/3/1999 12/31/2012 6.0 16.8

Latin America Chile CLP 4359 6/1/1995 12/28/2012 5.9 9.5

Latin America Japan/Mexico JPY_MXN 3887 1/4/1996 12/28/2012 4.6 16.9Latin America Mexico MXN 4220 1/4/1996 12/31/2012 4.6 10.5

Latin America Peru PEN 4252 4/1/1996 12/31/2012 5.3 5.0

Other Israel ILS 3750 7/20/1998 12/31/2012 8.1 7.8

Other Israel/Euro Area ILS_EUR 2552 1/2/2003 12/31/2012 8.5 10.2

Other South Africa ZAR 4394 4/3/1995 12/31/2012 8.7 16.4

Other Taiwan TWD 3605 1/5/1998 12/28/2012 5.0 5.3

Adv. Cross Rates Switzerland/UK CHF_GBP 9169 1/3/1975 12/31/2012 3.0 9.8

Adv. Cross Rates Australia/UK AUD_GBP 8920 4/7/1976 12/31/2012 3.2 12.4

Adv. Cross Rates Canada/UK CAD_GBP 9217 1/2/1975 12/31/2012 3.0 10.3

Adv. Cross Rates Japan/UK JPY_GBP 8982 1/2/1975 12/28/2012 3.0 12.2

Adv. Cross Rates Euro Area/UK EUR_GBP 9187 1/2/1975 12/31/2012 3.0 8.1

Adv. Cross Rates Australia/Switzerland AUD_CHF 8848 4/7/1976 12/31/2012 3.2 14.4

Adv. Cross Rates Canada/Switzerland CAD_CHF 9150 1/3/1975 12/31/2012 3.1 12.4

Adv. Cross Rates Japan/Switzerland JPY_CHF 9338 4/3/1973 12/28/2012 3.2 11.7

Adv. Cross Rates Euro Area/Switzerland EUR_CHF 9602 4/3/1973 12/31/2012 3.2 5.9Adv. Cross Rates Canada/Australia CAD_AUD 8894 4/7/1976 12/31/2012 3.2 10.3

Adv. Cross Rates Japan/Australia JPY_AUD 8633 4/7/1976 12/28/2012 3.2 15.3

Adv. Cross Rates Euro Area/Australia EUR_AUD 8861 4/7/1976 12/31/2012 3.2 12.8

Adv. Cross Rates Japan/Canada JPY_CAD 8968 1/2/1975 12/28/2012 3.1 12.7

Adv. Cross Rates Euro Area/Canada EUR_CAD 9158 1/2/1975 12/31/2012 3.1 10.7

Adv. Cross Rates Japan/Euro Area JPY_EUR 9347 4/3/1973 12/28/2012 3.1 11.3

Adv. Cross Rates New Zealand/Australia NZD_AUD 5943 8/3/1987 12/31/2012 3.9 8.7

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Table2: Calibration

CCAPM DCAPM EZCCAPM EZDCAPM

stdT[M]/ET[M] 0.93 1.22 0.92 1.22

varT[M]/ET[M] 0.60 0.61 0.59 0.61

MAE(in%) 1.32 0.96 1.33 0.96

R2 1.50 0.13 1.50 0.13

Notes:Thesampleis19782010(annualdata).ThetableshowstheresultsofestimatinganEuler

equation, as in equation (3), with 12 dynamic technical trading strategies as test assets. The

returnsare those toportfoliosp1 top12,asdescribed in the text.The first tworowsreport the

maximumSharperatio(row1,stdT[M]/ET[M])andthepriceofrisk(row2,varT[M]/ET[M]).The

lasttworowsreportthemeanabsolutepricingerror(inpercentagepoints)andtheR2.FollowingYogo (2006), we fixed sigma () at 0.023 (EZCCAPM and EZDCAPM), alpha () at 0.802

(DCAPMandEZDCAPM),delta()at0.98,andrho()at0.700(DCAPM,EZDCAPM).Gamma

()isfixedat41.16tominimizethemeansquaredpricingerrorintheEZDCAPM.

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Table3:ResultsforCAPM,QuadraticCAPMandCarhartmodelforportfoliosp1p12

p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11CAPM

Constant 0.39 *** 0.25 *** 0.21 *** 0.13 * 0.20 *** 0.07 0.14 ** 0.14 ** 0.12 * 0.13 ** 0.11

Rm 0.04 * 0.05 ** 0.04 * 0.04 * 0.03 0.05 ** 0.05 *** 0.04 * 0.05 *** 0.04 ** 0.04

R2

0.01 0.02 0.01 0.02 0.01 0.02 0.03 0.02 0.03 0.03 0.02

Quad.CAPM

Constant 0.26 *** 0.17 ** 0.12 * 0.05 0.14 ** 0.00 0.09 0.07 0.05 0.09 0.02

Rm 0.03 0.04 * 0.03 0.04 * 0.02 0.04 ** 0.05 *** 0.03 0.05 *** 0.04 *** 0.03

Rm2 0.65 *** 0.39 0.46 0.38 0.25 0.35 0.23 0.32 0.33 0.20 0.60

R2

0.04 0.03 0.03 0.03 0.02 0.04 0.03 0.03 0.04 0.03 0.05

CarhartConstant 0.41 *** 0.27 *** 0.21 *** 0.13 * 0.17 *** 0.07 0.13 * 0.13 ** 0.11 0.12 ** 0.08

Rm 0.04 * 0.05 ** 0.03 0.04 ** 0.02 0.04 * 0.03 0.03 0.04 ** 0.03 0.02

SMB 0.05 * 0.04 * 0.04 0.02 0.02 0.03 0.03 * 0.03 0.04 * 0.04 * 0.03

HML 0.06 * 0.03 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.02

UMD 0.03 * 0.02 0.02 0.02 0.03 ** 0.02 0.02 0.02 0.02 * 0.03 ** 0.03

R2

0.04 0.03 0.02 0.02 0.02 0.04 0.03 0.03 0.05 0.04 0.03

Mean R 0.37 0.23 0.19 0.11 0.18 0.05 0.11 0.12 0.09 0.11 0.09

Panel A: Time Series Regressions

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Rm 3.34 3.29 2.99 0.61 3.16 2.27

(2.13) * (2.59) ** (1.92) * (0.47) (2.23) * (1.57)

[1.70] [2.08] * [1.44] [0.37] [1.09] [0.70]{1.44} {2.02} * {1.26} {0.33} {1.06} {0.62}

Rm2 0.17 0.31

(1.70) (3.88) ***

[1.31] [3.10] **

{1.31} {3.10} **

SMB 1.64 2.41

(1.01) (2.04) *

[0.49] [0.91]

{0.44} {0.90}

HML 3.76 3.50

(3.76) *** (3.72) ***

[1.83] [1.67]

{1.92} * {1.70}

UMD 4.01 5.37

(2.08) * (3.58) ***

[1.01] [1.60]

{1.08} {1.60}

Constant 0.28 0.18 0.09(3.11) ** (2.00) * (0.82)

[2.55] ** [1.50] [0.38]

{2.33} ** {1.29} {0.35}

R2

0.05 0.16 0.18 0.11 0.73 0.72

Panel B: Cross Sectional Regressions

CarhartQuad. CAPMCAPM

Notes: Monthly data 06/1977 12/2012. The table shows the results of estimating a beta

representationmodelofdynamic technical trading strategiesonequity factors.Factorsare the

excessreturnontheU.S.stockmarket(),thesizefactor(SMB),thevaluefactor()andthemomentum factor (). tstatisticsare indicated inparentheses. ( ) tstatisticsbasedonOLSstandarderrors. []tstatisticsbasedonShankenstandarderrors. {}tstatisticsbasedonGMM

standarderrors. Significancelevels: ***1%,**5%,*10%.

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Table4:ResultsforCCAPM,DCAPMandEZDCAPMmodelforportfoliosp1p12

p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11

CCAPM

Constant 5.95 *** 3.36 ** 2.56 1.14 2.14 0.89 1.18 2.39 ** 1.12 0.91 0.29

Nondurables 0.99 ** 0.44 0.29 0.09 0.09 0.05 0.19 0.67 0.11 0.20 0.46

R2

0.11 0.02 0.01 0.00 0.00 0.00 0.00 0.07 0.00 0.01 0.03

DCAPM

Constant 4.75 *** 4.66 *** 2.93 3.22 ** 3.90 1.37 2.78 3.76 * 1.83 1.30 0.33

Nondurables 1.55 *** 0.16 0.12 1.05 * 0.90 0.17 0.92 0.04 0.22 0.37 0.48

Durables 0.56 0.61 * 0.17 0.97 ** 0.82 0.23 0.74 * 0.64 0.33 0.18 0.02

R2 0.14 0.07 0.01 0.09 0.09 0.01 0.08 0.14 0.02 0.01 0.03

EZDCAPM `

Constant 5.73 *** 5.46 ** 3.56 4.82 *** 5.24 ** 2.32 3.52 4.35 ** 2.04 1.81 1.49

Nondurables 1.33 ** 0.34 0.02 1.41 * 1.20 0.38 1.09 0.10 0.27 0.49 0.74

Durables 0.37 0.76 0.29 1.29 ** 1.08 0.41 0.89 0.75 0.37 0.28 0.25

Market 0.07 0.05 0.04 0.11 ** 0.09 ** 0.06 0.05 0.04 0.01 0.04 0.08

R2

0.20 0.12 0.04 0.23 0.21 0.07 0.12 0.17 0.02 0.03 0.15

Mean R 4.57 2.76 2.16 1.26 2.27 0.81 1.44 1.46 0.97 1.19 0.94


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Notes:Annualdata19782010.The tableshowstheresultsofestimatingabetarepresentation

modelofdynamic technical trading strategieson consumption factors.Nondurables () andDurables()arelognondurable(plusservices)anddurableconsumptiongrowthrespectively,and Market ( ,) is the log return on the market portfolio. tstatistics are indicated inparentheses. ( ) t statisticsbased on OLS standard errors. [ ] t statisticsbased on Shanken

standarderrors. {}t statisticsbasedonGMMstandarderrors. Significancelevels: ***1%,**5%,

*10%.

Nondurables 1.95 3.09 1.90 3.20 2.03 2.24

(4.08) *** (5.28) *** (4.19) *** (5.19) *** (4.26) *** (5.14) ***

[2.79] ** [2.55] ** [2.94] ** [2.20] * [2.43] ** [2.60] * *

{3.08} ** {2.76} ** {3.29} *** {2.02} * {2.44} ** {2.65} **

Durables 1.76 4.77 1.61 1.89

(2.66) ** (3.72) *** (2.48) ** (2.83) **

[1.85] * [1.51] [1.43] [1.39]

{2.06} * {1.31} {1.43} {1.46}

Market 19.32 26.95

(2.63) ** (3.28) ***

[1.40] [1.47]

{1.56} {1.21}

Constant 1.43 1.48 0.58

(2.42) ** (2.68) ** (0.97)

[1.49] [1.67] [0.49]

{1.54} {1.68} {0.41}

Parameters

84.32 133.94 82.33 139.40 91.89 102.72

(4.08) *** (5.28) *** (4.19) *** (5.17) *** (4.27) *** (5.23) ***

[2.79] ** [2.55] ** [2.93] ** [2.19] * [2.41] ** [2.61] * *

{3.08} ** {2.76} ** {3.28} *** {2.01} * {2.45} ** {2.66} **

0.09 0.11

(3.28) ** (3.32) ***

[1.99] * [1.61]

{2.14} * {1.32}

0.06 0.50 0.01 0.12

(0.26) (2.35) ** (0.06) (0.54)

[0.18] [1.00] [0.06] [0.26]

{0.19} {0.87} {0.04} {0.26}

R2

0.50 1.10 0.50 0.45 0.65 0.61

Panel B: Cross Sectional Regressions

EZDCAPMDCAPMCCAPM

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Table5:ResultsforLVmodelforportfoliosp1p12

p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11

Constant 0.35 *** 0.20 *** 0.17 ** 0.06 0.12 0.04 0.10 0.09 0.12 0.15 * 0.11

Rx 0.02 0.03 0.04 0.02 0.02 0.02 0.01 0.02 0.06 0.04 0.02

HMLfx 0.03 0.07 * 0.05 0.05 0.03 0.08 ** 0.09 ** 0.07 * 0.09 ** 0.10 ** 0.09

R2

0.00 0.02 0.01 0.01 0.01 0.03 0.04 0.02 0.05 0.06 0.04

Mean R 0.32 0.15 0.12 0.02 0.10 0.02 0.04 0.04 0.04 0.07 0.04


Constant 0.22

(2.66) **

[2.04] *

{1.84} *

Rx 0.29 1.73

(0.53) (2.32) **

[0.41] [1.78]

{0.39} {1.68}

HMLfx 2.08 0.32

(2.96) ** (0.49)

[2.29] ** [0.38]

{1.78} {0.31}

R2

0.35 0.24

Panel B: Cross Sectional Regression

Note: Monthly data 11/ 1983 12/2012. The table shows the results of estimating abeta representation mod

strategiesoncurrencymarket factors.Rx theaveragecurrencyexcessreturn togoingshort in thedollarand

currencyportfolios.HMLFX thereturntoastrategythatborrowslowinterestratecurrenciesandinvestsinhigh

words a carry trade. tstatistics are indicated inparentheses. ( ) t statisticsbasedonOLS standard errors. [

standarderrors. {}t statisticsbasedonGMMstandarderrors. Significancelevels: ***1%,**5%,*10%.

Table6:Resultsforvolatilityfactorsforportfoliosp1p12

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p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11

Constant 0.29 *** 0.12 ** 0.11 * 0.01 0.08 0.05 0.01 0.03 0.02 0.04 0.02

Rx 0.01 0.03 0.03 0.01 0.01 0.01 0.01 0.02 0.05 0.04 0.02

VOL1 2.33 ** 2.15 ** 1.71 2.51 ** 1.70 ** 2.93 *** 1.94 ** 1.75 * 2.60 ** 2.63 ** 2.01

R2

0.03 0.03 0.02 0.05 0.03 0.07 0.03 0.03 0.06 0.07 0.04

Constant 0.32 *** 0.15 ** 0.13 ** 0.02 0.10 0.02 0.04 0.05 0.05 0.08 0.05

Rx 0.01 0.03 0.03 0.01 0.01 0.01 0.02 0.02 0.05 0.03 0.02

VOL2 1.86 ** 1.92 ** 1.67 * 2.39 *** 1.73 *** 2.83 *** 2.22 *** 1.62 ** 2.72 *** 2.64 *** 1.77 **

R2

0.02 0.03 0.03 0.05 0.03 0.08 0.05 0.03 0.08 0.09 0.03

Mean R 0.32 0.15 0.12 0.02 0.10 0.02 0.04 0.04 0.04 0.07 0.04


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Constant 0.19 0.28

(2.21) * (3.09) **

[2.03] * [2.37] **

{1.91} * {2.42} **

Rx 0.02 0.22 0.55 0.64

(0.04) (0.39) (0.96) (1.10)

[0.04] [0.38] [0.74] [1.04]

{0.03} {0.41} {0.68} {1.06}

VOL1 0.05 0.03

(1.90) * (1.24)

[1.75] [1.20]

{1.62} {1.35}

VOL2 0.10 0.03

(3.25) *** (1.04)

[2.52] ** [0.99]

{2.22} * {1.01}

R2

0.06 0.11 0.22 0.17


Note: Monthly data 11/ 1983 12/2012. The table shows the results of estimating abeta representation mod

strategiesoncurrencyvolatilityfactors.Rxtheaveragecurrencyexcessreturntogoingshort inthedollarand

currency portfolios. VOL1 volatility innovations measuredby the residuals from AR(1). VOL2 volatility

difference. tstatisticsareindicatedinparentheses.()t statisticsbasedonOLSstandarderrors. []t statisticsba

{}t statisticsbasedonGMMstandarderrors. Significancelevels: ***1%,**5%,*10%.

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Table7:Resultsforskewnessfactorforportfoliosp1p12

p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11

Constant 0.73 *** 0.78 *** 0.86 *** 0.91 *** 0.71 *** 0.93 *** 0.87 *** 0.91 *** 0.96 *** 0.91 *** 0.85 *

SKEW 0.20 *** 0.20 *** 0.19 *** 0.19 *** 0.16 *** 0.18 *** 0.18 *** 0.19 *** 0.19 *** 0.19 *** 0.17 *

R2

0.18 0.19 0.22 0.22 0.19 0.22 0.21 0.26 0.25 0.27 0.21

Mean R 0.37 0.23 0.19 0.11 0.18 0.05 0.11 0.12 0.09 0.11 0.09


Co nstant 0.6 4

(2.78) **

[1.78]

{1.83} *

SKEW 4.31 0.82

(3.45) *** (2.73) **

[2.16] * [2.73] **

{2.33} ** {3.15} ***

R2

0.25 0.09


Note: Monthly data 06/1977 12/2012. The table shows the results of estimating abeta representation mod

strategiesoncurrencyvolatilityfactors.SKEWreturnofaportfoliothatislongcurrenciesinthehighestskew

currencies in the lowest (negative) skewnessquintile foragivenmonth. tstatisticsare indicated inparenthes

standarderrors. []t statisticsbasedonShankenstandarderrors. {}t statisticsbasedonGMMstandarderror5%,*10%.

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Figure1:Return,standarddeviationandSharperatiostatisticsfromsortedportfoliosp1

top12

Notes: Thefigureshowsstatisticsfromthe12dynamicallyconstructed,technicaltrading

portfoliosinforeignexchangemarkets,basedonproceduresinNeelyandWeller(2013).

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Figure2:Actualmeanannualreturnsvs.predictedreturnsforportfoliosp1throughp12

(crosssectionalregressionwithaconstant)19782010

Notes: Thefigureshowsscatterplotsofactualversuspredictedreturnsfromthe12dynamically

constructed,technicaltradingportfoliosinforeignexchangemarkets,basedonproceduresinNeelyandWeller(2013). Thepredictedreturnsareconstructedfromthebetarepresentationof

thelinearconsumptionmodelswithaconstantinthecrosssectionalequation.

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1

2

3

4

5

6

7 8

9

10

11

12

Predicted R

ActualR

CCAPM

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1

2

3

4

5

6

7 8

9

10

11

12

Predicted R

ActualR

DCAPM

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1

2

3

4

5

6

7 8

9

10

11

12

Predicted R

ActualR

EZ-DCAPM

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Figure3:Actualmeanannualreturnsvs.predictedreturnsforportfoliosp1throughp12

(crosssectionalregressionwithoutaconstant)19782010

Notes: Thefigureshowsscatterplotsofactualversuspredictedreturnsfromthe12dynamically

constructed,technicaltradingportfoliosinforeignexchangemarkets,basedonproceduresin

NeelyandWeller(2013). Thepredictedreturnsareconstructedfromthebetarepresentationof

thelinearconsumptionmodelswithoutaconstantinthecrosssectionalequation.

-2 -1 0 1 2 3 4 5-2

-1

0

1

2

3

4

51

2

3

4

5

6

7 8

910

11

12

Predicted R

ActualR

CCAPM

-2 -1 0 1 2 3 4-2

-1

0

1

2

3

4

5

1

2

3

4

5

6

7 8

910

11

12

Predicted R

ActualR

EZ-DCAPM

-2 -1 0 1 2 3 4 5-2

-1

0

1

2

3

4

51

2

3

4

5

6

7 8

910

11

12

Predicted R

ActualR

DCAPM

Documents

2014-033