exchange rate forecasting and the performance of currency...

Received: 21 October 2016 Revised: 30 January 2018 Accepted: 31 January 2018

DOI: 10.1002/for.2518

R E S E A R C H A R T I C L E

Exchange rate forecasting and the performance ofcurrency portfolios

Jesus Crespo Cuaresma1,2,3,4 Ines Fortin5 Jaroslava Hlouskova5,6,7

1Department of Economics, ViennaUniversity of Economics and Business(WU), Vienna, Austria2Wittgenstein Centre for Demography andGlobal Human Capital (WIC), Vienna,Austria3World Population Program, InternationalInstitute for Applied Systems Analysis(IIASA), Laxenburg, Austria4Austrian Institute of Economic Research(WIFO), Vienna, Austria5Research Group Macroeconomics andEconomic Policy, Institute for AdvancedStudies, Vienna, Austria6Department of Economics, ThompsonRivers University, Kamloops, BC, Canada7Ecosystems Services and Management,International Institute for AppliedSystems Analysis, Laxenburg, Austria

CorrespondenceJesus Crespo Cuaresma, Department ofEconomics, Vienna University ofEconomics and Business (WU), 1020Vienna, Austria.Email: [email protected]

Funding informationAustrian Central Bank, Grant/AwardNumber: 16250

Abstract

We examine the potential gains of using exchange rate forecast models andforecast combination methods in the management of currency portfolios forthree exchange rates: the euro versus the US dollar, the British pound, and theJapanese yen. We use a battery of econometric specifications to evaluate whetheroptimal currency portfolios implied by trading strategies based on exchangerate forecasts outperform single currencies and the equally weighted portfo-lio. We assess the differences in profitability of optimal currency portfolios fordifferent types of investor preferences, two trading strategies, mean squarederror-based composite forecasts, and different forecast horizons. Our resultsindicate that there are clear benefits of integrating exchange rate forecasts fromstate-of-the-art econometric models in currency portfolios. These benefits varyacross investor preferences and prediction horizons but are rather similar acrosstrading strategies.

KEYWORDScurrency portfolios, exchange rate forecasting, profitability, trading strategies

1 INTRODUCTION

Foreign exchange risk is omnipresent in internationalportfolio diversification, but forecasting exchange rates iswell known to be a difficult task. Since the seminal workby Meese and Rogoff (1983), which shows that economet-ric specifications based on macroeconomic fundamentalsare unable to outperform simple random walk forecasts

at short time horizons (up to 1 year), a large number ofstudies have proposed models aimed at providing accu-rate out-of-sample predictions of spot exchange rates (see,among others, Berkowitz & Giorgianni, 2001; Boudoukh,Richardson, & Whitelaw, 2008; Cheung, Chinn, & Pascual,2005; Chinn & Meese, 1995; Kilian, 1999; MacDonald &Taylor, 1994; Mark, 1995; Mark & Sul, 2001). In parallel,a literature has emerged which examines empirically the

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Copyright © 2018 The Authors Journal of Forecasting Published by John Wiley & Sons, Ltd.

Journal of Forecasting. 2018;1–22. wileyonlinelibrary.com/journal/for 1

https://doi.org/10.1002/for.2518

http://orcid.org/0000-0003-3244-6560

2 CRESPO CUARESMA ET AL.

potential profitability of technical trading rules based onexchange rate predictions (see Menkhoff & Taylor, 2007,for a review). Although the random walk specification hasnaturally emerged as the benchmark to beat in terms ofout-of-sample predictive accuracy, it is not clear that itwill also yield the most profitable trading strategy. Port-folio managers are expected to be more concerned withprofitability than with out-of-sample accuracy.

Our study aims at addressing how the joint modeling ofexchange rates and fundamentals provide economic valuein terms of improving currency portfolio performance.We therefore contribute to the long-standing literature onthe use of exchange rate models based on fundamentalsfor forecasting, taking a new perspective in the evalua-tion of different econometric specifications. We provide anevaluation framework where we take the perspective of acurrency portfolio manager (investor) who follows trad-ing strategies based on exchange rate forecasts and whosemain goal is to maximize (risk-adjusted) profits, under cer-tain types of preferences. Our currency portfolio managerconsiders the exchange rates of the euro against the USdollar (USD), the British pound (GBP), and the Japaneseyen (JPY), and for each of these three exchange rates cre-ates a “single asset.” The returns of this asset are impliedby a certain trading strategy that is based on exchange rateforecasts. The optimal portfolio is then made up of thesethree single assets according to the manager's—or someinvestor's—preferences.

The two primary research questions in our study arethe following. First, does the information on exchangerate fundamentals provide valuable information to con-struct optimal currency portfolio that outperform simplebenchmark portfolios, and thus is there a value added inengaging in active portfolio management—or can the port-folio manager achieve the same (risk-adjusted) profit byjust investing in some simpler assets (benchmark portfo-lios)? As simpler assets we consider the single assets ofwhich the optimal portfolio consists as well as the equallyweighted portfolio based on forecasts from the modelbased on macroeconomic fundamentals as well as on ran-dom walk predictions. This research question links ourwork to the large literature on the statistical and economicevaluation of exchange rate forecasts (see Abbate & Mar-cellino, 2018; Della Corte, Sarno, & Tsiakas, 2009; Rossi,2013) and provides a novel evaluation context that goesbeyond the existing methods based on forecast errors anddirectional change statistics.

Relating to the first question, there is some empiricalevidence indicating that simple portfolios, like equallyweighted portfolios, are not necessarily outperformed(e.g., in terms of the Sharpe ratio) by more complex port-folios (see DeMiguel, Garlappi, & Uppal, 2009; Jacobs,Müller, & Weber, 2014). The existing evidence in the liter-

ature, however, relates to equity markets (DeMiguel et al.2009) and equity, bond and commodity markets (Jacobset al. 2014), and it is not obvious that these findingscarry over to foreign exchange markets. Our study con-tributes to enlarge this body of empirical evidence byconcentrating on foreign exchange markets. In order tocompare the different currency portfolios, we employ anumber of (risk-adjusted) performance measures, includ-ing the Omega measure, the Sharpe ratio, and the Sortinoratio. We consider all the multivariate time series modelsand the methods of forecast combinations entertained inCostantini, Crespo Cuaresma, and Houskova (2014, 2016)to generate exchange rate forecasts.1

We consider two different trading strategies in construct-ing the single assets. The first one is the simple “buylow, sell high” trading strategy described, for example,in Gençay (1998), where the trading signal is based onthe spot exchange rate and its forecast. The second oneis based on exploiting the forward rate unbiased expec-tation hypothesis, using forward contracts, and is similarto the carry trade strategy used, for example, in Burn-side, Eichenbaum, and Rebelo (2008). In this case thetrading signal is based on the forward exchange rate andthe exchange rate forecast. In order to assess the perfor-mance of optimal currency portfolios versus benchmarkportfolios (single assets, equally weighted portfolio), weuse a data-snooping bias-free test, which is based on anextensive bootstrap procedure. By employing this test weensure that the performance superiority of certain opti-mal portfolios—if any—is systematic and not merely dueto luck. The test identifies which optimal portfolios sig-nificantly beat the benchmark portfolio in terms of cer-tain risk-adjusted performance measures. In addition, weassess the values of optimal portfolios with respect tobenchmark portfolios by computing break-even transac-tion costs (see Della Corte et al., 2009; Della Corte &Tsiakas, 2013).

Returns implied by trading strategies have also beeninvestigated in other exchange rate studies. Burnside et al.(2008), for example, examine the returns implied by thecarry trade strategy, which determines to sell (buy) a cur-rency forward when it trades at a forward premium (dis-count). This trading strategy is similar to our second trad-ing strategy. The authors apply the carry trade strategy toindividual currencies as well as to an equally weightedportfolio of 23 currencies and find that constructing aportfolio improves the performance of the carry trade strat-egy substantially: the Sharpe ratio of the equally weightedcarry trade strategy is more than 50% higher than the

1See also Crespo Cuaresma and Hlouskova (2005), Crespo Cuaresma(2007), Costantini and Pappalardo (2010), and Costantini and Kunst(2011).

CRESPO CUARESMA ET AL. 3

median Sharpe ratio across currency-specific carry tradestrategies. Unlike our study, Burnside et al. (2008) donot test the statistical significance of the portfolio out-performance with respect to the single-currency-basedasset. While they use a simple, equally weighted portfolio,we take the investor's preferences into account explic-itly and optimize the portfolio according to these pref-erences. Another fundamental difference with respect totheir work is that we include exchange rate forecastsin the definition of our trading strategies with the aimof improving their performance. Burnside et al. (2008),on the other hand, use bid-and-ask spot and forwardexchange rates. In a related paper, Burnside et al. (2011)examine carry trade and momentum strategies for singlecurrencies and equally weighted currency portfolios, andreview possible explanations for the profitability of thesestrategies.

Recently, the work by Barroso and Santa-Clara (2015)aims at maximizing the expected return of a portfolio in theforward exchange market, given preferences described bythe power utility and using the parametric portfolio poli-cies approach of Brandt, Santa-Clara, and Valkanov (2009).Papaioannou, Portes, and Siourounis (2006) assess how theintroduction of the euro as a new currency has potentiallychanged the optimal composition of portfolios of foreignexchange reserves and how the optimal holdings compareto actual reserve portfolios held by central banks.

The economic value of using model-based exchangerate forecasts to construct optimal portfolios has severaldimensions. First, there is economic value which can beexclusively attributed to portfolio optimization and canbe assessed by comparing the performance of optimalportfolios with the performance of the equally weightedportfolio based on composite forecasts. Second, we canconsider the economic value of both portfolio optimizationand exchange rate forecasting, which can be assessed bycomparing the performance of optimal portfolios with theperformance of the equally weighted portfolio based on therandom walk. Third, we can assess the exclusive economicvalue of exchange rate forecasting based on fundamen-tals by comparing the performance of benchmark portfo-lios based on composite forecasts with that of benchmarkportfolios based on the random walk.2 Our main resultsimply that a positive economic value can be observedfor forecast horizons of 1, 3, and 6 months, indepen-dently of which one of the three dimensions of economicvalue we consider. This applies for the mean–variance andconditional-value-at-risk optimal portfolios with respectto the equally weighted portfolio. For a forecast hori-zon of 12 months, however, we only observe a posi-tive economic value of portfolio optimization but not of

2This issue is the main focus of Costantini et al. (2016).

forecasting, and also not of portfolio optimization andforecasting.

The paper is structured as follows. In Section 2 wepresent the analytical framework required for exchangerate forecasting and portfolio optimization. First we intro-duce the individual exchange rate forecast models andforecast combinations methods that we use to generate theexchange rate predictions. Then we describe how we com-pute the best forecasts (composite forecasts) and presentthe different types of preferences and the correspondingoptimization problems. We conclude Section 2 by describ-ing the risk-adjusted performance measures we consider,as well as the data-snooping bias-free test for equal per-formance and the calculation of break-even transactioncosts. Section 3 discusses the empirical results and Section4 concludes.

2 ANALYTICAL FRAMEWORK:EXCHANGE RATE FORECASTINGAND OPTIMAL CURRENCYPORTFOLIOS

2.1 Exchange rate specificationsWe start by describing the modeling framework used toobtain ensembles of exchange rate predictions that canbe used to construct currency portfolios. The class ofspecifications we entertain in order to obtain forecastsof the exchange rate can be conceptualized in the con-text of the so-called monetary model of exchange ratesoriginally developed in the work by, for example, Frenkel(1976), Dornbusch (1976), or Hooper and Morton (1982).The monetary model of exchange rate determination hasbeen often used as a theoretical framework to createexchange rate predictions based on macroeconomic fun-damentals (see Costantini et al., 2016; Crespo Cuaresma& Hlouskova, 2005). Starting with standard Cagan moneydemand equations for the domestic and foreign econ-omy, the monetary model of exchange rate formationassumes that purchasing power parity acts as a long-runequilibrium and thus leads to a relationship between theexchange rate on the one hand and money supply, inter-est rates, and income levels in the two economies on theother hand.

Consider a money demand equation in the domesticeconomy given by

mt − 𝑝t = 𝛼𝑦t + 𝛽it, (1)

where mt is the nominal money demand (in logs), ptdenotes the price level (in logs), yt is a measure of income(in logs) and it is the interest rate. Assuming a similar spec-ification in the foreign economy, where asterisks denotethe corresponding parameters and variables, we can write


the real money demand equation as

m∗t − 𝑝

∗t = 𝛼∗𝑦∗t + 𝛽

∗i∗t . (2)

The long-run equilibrium condition of the monetarymodel is given by the purchasing power parity (PPP) con-dition, equating the nominal exchange rate (St) to the pricedifferential between the two economies:

logSt = 𝑝t − 𝑝∗t = mt − m∗t − 𝛼𝑦t + 𝛼∗𝑦∗t − 𝛽it + 𝛽∗i∗t . (3)

This specification calls for the use of money sup-ply, income, and interest rates as potential covariatesin models aimed at assessing exchange rate dynam-ics. If, in addition, the uncovered interest rate parity(UIP) is assumed, together with identical interest ratesemi-elasticities in both economies, the resulting specifi-cation will not include interest rates as a potential deter-minant of exchange rate movements.

Given this modeling framework, the relationshipbetween the exchange rate and its determinants tends tobe routinely specified empirically in the form of vectorautoregressive (VAR) and vector error correction (VEC)models. Defining the vector zt, which contains the (log)exchange rate, the (log) money supply in the domesticand foreign economy, the (log) production index in thedomestic and foreign economy and the respective short-and long-term interest rates, this implies that its dynamicsare given by

zt = Φ0 +𝑝∑

l=1Φlzt−l + 𝜖t, 𝜖t ∼ n.i.d.(0,Σ𝜖), (4)

where Φl for l = 1, … , p are matrices of coefficients andΦ0 is a vector of constants. Alternatively, if the variablesof the model are linked by one or more cointegration rela-tionships which act as a long-run attractor of the data, thespecification given by Equation 4 can be written in vectorerror correction (VEC) form:

Δzt = Θ0 + 𝛿𝜓 ′zt−1 +𝑝−1∑l=1

ΘlΔzt−l + 𝜖t. (5)

In this specification the cointegration relationships aregiven by 𝜓 ′zt and 𝛿 quantifies the speed of adjustment tothe long-run equilibrium. Alternatively, if the variables inzt are unit-root nonstationary but no cointegration rela-tionship exists among them, a VAR model in first differ-ences (DVAR) would be the appropriate representation,which amounts to the model given in Equation 5 with𝛿 = 0. Alternatively to modeling each exchange rateindividually, a large vector autoregressive structure wherefundamentals for all pairs of countries and their respec-tive exchange rates are included could also be chosen asa modeling framework, but the extremely large number

of parameters involved in such a model would make theestimation difficult.

Myriads of studies have used VAR and VEC modelsbased on macroeconomic variables as specifications forexchange rate predictions, with no robust evidence forimproved forecasting ability as compared to the randomwalk in the short term but often better results for pre-dictions in longer horizons, in particular for VEC speci-fications. There is some agreement in the literature thatforecasts generated by models which explicitly address thepotential existence of long-run equilibria in the form ofcointegration relationships (VEC specifications) tend tosystematically outperform forecasts generated by the naiverandom walk model in terms of the mean square errorat horizons of around one year (see for example Mac-Donald & Taylor, 1994). This result is, however, far frombeing homogeneous across currencies and time periods. Itshould be noted that our contribution uses performancemeasures that do not correspond to those utilized in theexchange rate forecasting literature hitherto, and as suchthe existing results may not be a perfect reference forcomparison.

We entertain several types of vector autoregressiveand vector error correction models as specifications forexchange rate forecasting. On the one hand, we differen-tiate between restricted and unrestricted models depend-ing on whether the foreign and domestic covariates areincluded as individual variables in the model or as a sin-gle covariate measuring the domestic–foreign difference.We refer to models containing the latter as restricted mod-els (r-VAR, r-DVAR, r-VEC), whereas the models basedon separated domestic and foreign variables are labeledunrestricted models (VAR, DVAR, VEC). We also considersubset-VAR models, where statistically insignificant lags ofthe variables are omitted, and label them s-VAR, s-DVAR,rs-VAR, and rs-DVAR.

In terms of estimation method, we consider multivariatemodels estimated using standard frequentist methods andBayesian VARs. Bayesian DVARs are estimated using thestandard Minnesota prior (see Doan, Litterman, & Sims,1984; Litterman, 1986). The lag length of all multivariatemodel specifications under consideration is selected usingthe Akaike information criterion (AIC) for potential laglengths ranging from 1 to 12 lags. For the VEC models,selection of the lag length and the number of cointegrationrelationships is carried out simultaneously using the AIC.Table 1 lists the 12 individual forecast models used.

2.2 Forecast combinationsThe set of methods used to create forecast combinationsfrom individual multivariate time series models is simi-lar to that in Costantini et al. (2016). Let S𝑗t be the spot


TABLE 1 Individual forecast models

VAR(p) Vector autoregression in levels based on domestic and foreign variables with p lagsDVAR(p) Vector autoregression in first differences based on domestic and foreign variables with p lagsVEC(c, p) Vector error correction model based on domestic and foreign variables with c cointegration relationshipsr-VAR(p) Restricted VAR, based on differences between domestic and foreign variablesr-DVAR(p) Restricted DVAR, based on differences between domestic and foreign variablesr-VEC(c, p) Restricted VEC, based on differences between domestic and foreign variables with c cointegration relationshipss-VAR(p) Subset vector autoregression in levels based on domestic and foreign variables with p lagss-DVAR(p) Subset vector autoregression in first differences based on domestic and foreign variables with p lagsrs-VAR(p) Restricted subset VAR, based on differences between domestic and foreign variablesrs-DVAR(p) Restricted subset DVAR, based on differences between domestic and foreign variablesBDVAR(p) Bayesian vector autoregression in first differences based on domestic and foreign variablesr-BDVAR(p) Bayesian vector autoregression in first differences based on differences between domestic and foreign variables

exchange rate of euros (EUR) per foreign currency unit(FCU) of currency j, that is, EUR/FCU, at time t, andlet S𝑗m,t+h|t be the exchange rate forecast of euros per for-eign currency unit of currency j obtained using model m,m = 1, … ,M, for time t + h conditional on the informa-tion available at time t (i.e., h is the forecast horizon). Inthe following we drop superscript j to keep the expositionsimpler. The combinations of forecasts entertained in thisstudy, Sc,t+h|t, take the form of a linear combination of thepredictions of individual specifications:

Sc,t+h|t = whc,0t +

M∑m=1

whc,mtSm,t+h|t, (6)

where c is the combination method, M is the num-ber of individual forecasts and the weights are given by{wh

c,mt}Mm=0.

Since several combination methods require statisticsbased on a hold-out sample where the relative predictiveability of models is assessed, let us introduce here somenotation on subsample limits: T0 is used to denote the firstobservation of the available sample, the interval (T1, T2) isused as a hold-out sample to obtain weights for those meth-ods where such a subsample is required, and T3 is the lastavailable observation. The sample given by (T2, T3) is theproper out-of-sample period used to compare the differentmethods.

In order to pool the forecasts of individual specifica-tions, we consider a large number of combination methodsproposed in the literature. These are the same methodsthat have been recently used in Costantini et al. (2016) toevaluate exchange rate predictability:3

• Mean, trimmed mean, median. For the mean prediction,wh

mean,0t = 0 and whmean,mt = 1

Min Equation 6. The

trimmed mean uses whtrim,0t = 0 and wh

trim,mt = 0 for theindividual models that generate the smallest and largest

3Costantini et al. (2016) provide a more detailed discussion of theseforecast averaging methods.

forecasts, while whtrim,mt = 1

M−2for the remaining indi-

vidual models. For the median combination method,Sc,t+h|t = median{Sm,t+h|t}M

m=1 is used (see Costantini &Pappalardo, 2010).

• Ordinary least squares (OLS) combination. The weightsof this method coincide with the estimated coefficientsobtained by regressing actual exchange rates on a con-stant and corresponding exchange rate forecast. In ourapplication we use a rolling window over the hold-outsample. Granger and Ramanathan (1984) provide moredetails on this simple forecast pooling methodology.

• Combination based on principal components (PC). Thismethod allows us to overcome multicollinearity of pre-dictions across models by reducing them to a few prin-cipal components (factors). The method is identicalto the OLS combining method by replacing forecastswith their principal components. In our application, wechoose the number of principal components using thevariance proportion criterion, which selects the small-est number of principal components such that a certainfraction of variance is explained. We set the proportionto 80%.4

• Combination based on the discount mean square forecasterrors (DMSFE). Following Stock and Watson (2004),the weights in Equation 6 depend inversely on thehistorical forecasting performance of individual mod-els, wh

DMSFE,m,t = WMSE−1mth∕

∑Ml=1 WMSE−1

lth , where

WMSEmth =∑t

t=T1+h−1 𝜃T−h−t(St − Sm,t|t−h

)2, for t = T2−h, … ,T3 − h, m = 1, … ,M, wh

DMSFE,0,t = 0, and 𝜃 isa discount factor. In the empirical application, we use𝜃 = 0.95.

• Combination based on hit/success rates (HR). Themethod uses the proportion of correctly predicted

4More details on the method are provided in Hlouskova and Wagner(2013), where the principal components augmented regressions are usedin the context of the empirical analysis of economic growth differentialsacross countries. Except for Costantini et al. (2016), we are not awareof the existence of any study using this approach in the context of theexchange rate forecasts.


directions of exchange rate changes of model mto the number of all correctly predicted direc-tions of exchange rate changes by the models used,wh

HR,mt =∑t

t=T1+h−1 DAm,th∕∑M

l=1

(∑t𝑗=T1+h−1 DAl,th

),

where t = T2 − h, … ,T3 − h and the index of direc-tional accuracy is given by DAm,th = I

(sgn(St − St−h

)= sgn

(Sm,t|t−h − St−h)

), where I(·) is the indicator

function.• Combination based on the exponential of hit/success

rates (EHR) (Bacchini, Ciammola, Iannaccone,& Marini, 2010). The weights in this methodare wh

EHR,mt = exp(∑t

t=T1+h−1(DAm,th − 1))∕∑M

l=1 exp(∑t

t=T1+h−1(DAl,th − 1))

, where t = T2−h, … ,

T3 − h.• Combination based on the economic evaluation

of directional forecasts (EEDF). The weights inthis method capture the ability of models to pre-dict the direction of change of the exchange rate,while taking into account the magnitude of therealized change and are thus given by wh

EEDF,mt =∑tt=T1+h−1 DVm,th∕

∑Ml=1

(∑tt=T1+h−1 DVl,th

), where

t = T2 − h, … ,T3 − h and DVm,th = |St − St−h|DAm,th.• Combination based on predictive Bayesian model aver-

aging (BMA). The weights used are based on thecorresponding posterior model probabilities basedon out-of-sample (rather than in-sample) fit. See,for example, Raftery, Madigan, and Hoeting (1997),Carriero, Kapetanios, and Marcellino (2009), CrespoCuaresma (2007), and Feldkircher (2012).

We create weights based on comparing log-predictivescores of the different models in a hold-out subsam-ple, as this prediction error statistic is routinely used inmethodological comparisons involving BMA (see, e.g.,Fernandez, Ley, & Steel, 2001). The weights are thusgiven by

whBMA,mt =

∑t

t=T1+h−1exp

{−[

12

log(

2𝜋��2m,t|t−h

)+ 1

2

(St−Sm,t|t−h

��m,t|t−h

)2]}∑M

l=1

∑t

t=T1+h−1exp

{−[

12

log(

2𝜋��2l,t|t−h

)+ 1

2

(St−Sl,t|t−h

��l,t|t−h

)2]} , (7)

where ��2m,t|t−h is the estimated variance of the

corresponding prediction for model m, Sm,t|t−h, andt = T2 − h, … ,T3 − h.

• Combinations based on frequentist model aver-aging (FMA) (see Claeskens & Hjort, 2008;2003). The weights are calculated as wh

FMA,mt =

exp(− 1

2ICmt

)∕∑M

l=1 exp(− 1

2IClt

), where ICmt stands

for an information criterion of model m and t is the lasttime point of the data over which models are estimated.

We use combinations of forecasts based on the AIC,Schwarz criterion (BIC), and Hannan–Quinn criterion(HQ). The weights corresponding to the BIC can beinterpreted as an approximation to the posterior modelprobabilities in BMA.5

A list of all forecast combination methods we use can befound in Table 6 in the supporting information Appendix.

2.3 Predictive accuracyExchange rate forecasts are evaluated using the traditionalmean square error (MSE).6 We obtain the standard squareerror SEm,t,h =

(Sm,t|t−h − St

)2 by a rolling-window estima-tion; that is, we keep the estimation sample size constant(equal to T1 − T0) as we re-estimate the models, thusmoving the window that defines the sample used to esti-mate the model parameters. The MSE for each model andforecast combination method is thus calculated over theout-of-sample period for a given forecast horizon h andaggregated as7 MSEmh = 1∕(T3 − T2 + 1)

∑T3−T2𝑗=0 SEm,T2+𝑗,h.

In addition, we compute composite forecasts based onthe MSE of predictions from all models and combinationmethods over a certain period. In particular, for this tech-nique at each time point t we choose the model or forecastcombination method (and thus also the forecast for timepoint t + h) with the minimum MSE over a certain timewindow ending at time point t, SMSE,l

t+h|t = SmMSElth ,t+h|t, where

mMSElth = arg minm

∑t𝑗=l SEm,𝑗,h. Time point l, such that T2 ≤

l ≤ t, defines the beginning of the window over which theperformance is evaluated. The evaluation window is [l, t],where l ≤ t ≤ T3. For our empirical results we use l = t−12;that is, the model or forecast combination method with thesmallest MSE over the last 12 months is chosen.

5See Raftery et al. (1997) and Sala-i-Martin, Doppelhofer, & Miller (2004).6In a previous version of this paper we also use profit measures such as thedirectional value, statistics that capture the economic value of directionalforecasts and returns with respect to the two trading strategies (see CrespoCuaresma, Fortin, & Hlouskova, 2017). However, we restrict ourselves tothe MSE here.7The term model and thus also its abbreviation m is used in a broadersense that includes both individual models and forecasting rules andmethods (like forecast combinations and composite forecasts).


2.4 Trading strategiesWe use two trading strategies in order to define the threesingle assets (three assets for each trading strategy) that theinvestor can select from. Trading strategy 1 (TS1) is basedon buying the foreign currency if its price is forecast to riseand selling it when its price is forecast to fall (“buy low,sell high strategy”), and trading strategy 2 (TS2) exploitsthe forward rate unbiased expectation hypothesis and isrelated to the so-called carry trade strategy (“carry tradebased strategy”).

Trading strategy 1 is a simple “buy low, sell high” tradingstrategy as described in Gençay (1998), where the sell-ing/buying signal is based on the current exchange rate.Forecast upward movements of the exchange rate withrespect to the actual value (positive returns) are executedas long positions, while forecast downward movements(negative returns) are executed as short positions. For eachexchange rate model and forecast combination method mand forecast horizon h the trading strategy 1 is defined bythe following trading signal, 𝑦𝑗m,TS1

th , and (discrete) return,r𝑗m,TS1

t+h,h :

𝑦𝑗m,TS1th =

{1, if S𝑗m

t+h|t < S𝑗t (one FCU of currency 𝑗 is sold at t and bought at t + h)−1, if S𝑗m

t+h|t > S𝑗t (one FCU of currency 𝑗 is bought at t and sold at t + h),(8)

r𝑗m,TS1t+h,h =

⎧⎪⎪⎨⎪⎪⎩1S𝑗t

(S𝑗t − S𝑗t+h

)= 1 − S𝑗t+h

S𝑗t, if 𝑦

𝑗m,TS1th = 1

1S𝑗t

(S𝑗t+h − S𝑗t

)= S𝑗t+h

S𝑗t− 1, if 𝑦

𝑗m,TS1th = −1,

Trading strategy 2 is based on exploiting the forward rateunbiased expectation hypothesis. In perfect markets, theforward exchange rate is an unbiased predictor of the cor-responding future spot exchange rate. If this hypothesisdoes not hold, a trading strategy based on exchange rateforecasts may earn positive trading profits. This tradingstrategy thus depends on whether the exchange rate fore-cast is above or below the forward rate. The trading signal,𝑦𝑗m,TS2th , and return, r𝑗m,TS2

t+h,h , are defined as follows:

𝑦𝑗m,TS2th =

{1, if S𝑗m

t+h|t < F𝑗t+h|t (one FCU of currency 𝑗 is sold forward at t and bought at t + h)−1, if S𝑗m

t+h|t > F𝑗t+h|t (one FCU of currency 𝑗 is bought forward at t and sold at t + h),(9)

r𝑗m,TS2t+h,h =

⎧⎪⎨⎪⎩1

F𝑗t+h|t(

F𝑗t+h|t − S𝑗t+h

)= 1 − S𝑗t+h

F𝑗t+h|t , if 𝑦𝑗m,TS2th = 1

1F𝑗t+h|t

(S𝑗t+h − F𝑗t+h|t

)= S𝑗t+h

F𝑗t+h|t − 1, if 𝑦𝑗m,TS2th = −1,

where F𝑗t+h|t is the forward exchange rate (EUR/FCU) attime t with respect to currency j, maturing at time t + h.

2.5 Optimal portfoliosIn order to assess whether exchange rate forecasts basedon macroeconomic fundamentals improve the profitabil-ity of currency portfolios, we investigate the performanceof (optimal) currency portfolios of returns implied by twostrategies described above, which exploit the potential pre-dictability of exchange rate changes. The optimal portfolioconsists of returns implied by a certain trading strategyapplied to the three foreign exchange rates EUR/USD,EUR/GBP, and EUR/JPY. We refer to these individ-ual returns as (single) assets based on the EUR/USD,EUR/GBP, and EUR/JPY exchange rates, respectively.In building optimal portfolios, investors behave accord-ing to particular preferences. We model the followingtypes of preferences: mean–variance (MV), conditionalvalue-at-risk (CVaR), linear, linear loss aversion (LLA),and quadratic loss aversion (QLA). As benchmark portfo-lios, relative to which the optimal portfolios are evaluated,

we consider both (i) the single assets based on individualexchange rates from which the optimal portfolios arecom-posed, as well as (ii) equally weighted (EW) portfolios.8

Consider an investor who dynamically (e.g., on amonthly basis) re-balances her portfolio. Let rTS

th =(rUSD,TS

th , rGBP,TSth , rJPY,TS

th

)′, where rUSD,TSth is the return at time

t implied by trading strategy TS and exchange rate fore-casts of the EUR/USD for horizon h. Similarly, rGBP,TS

th isthe return based on EUR/GBP exchange rate forecasts,

8A portfolio with equal weights was also investigated in Burnside et al.(2008) and Burnside, Eichenbaum, and Rebelo (2011), using the carrytrade strategy (exploiting the forward rate unbiased expectation hypothe-sis) and the momentum strategy (stipulating to sell when it was profitableto sell before).


and rJPY,TSth is the return based on EUR/JPY exchange

rate forecasts. Let xTSth =

(xUSD,TS

th , xGBP,TSth , xJPY,TS

th

)′, wherexi,TS

th denotes the proportion of wealth invested at timet in trading strategy TS, whose returns are implied byMSE-based composite forecasts of exchange rate i for hori-zon h. Returns are constructed with respect to compositeforecasts based on the traditional MSE. This means thatforecasting models (or forecast combination methods) arechosen such that the MSE is the smallest over a span of 12months, a period that appears to be appropriate in order tocapture changing market conditions.9

In our application the returns are available from Jan-uary 2005 until January 2016 at a monthly frequency,and the optimization exercises are performed for a 3-yearrolling window; that is, we optimize over 36 observations.The evaluation period is therefore January 2008 to Jan-uary 2016 (97 observations). The portfolio optimizationexercises are carried out under the following preferenceschemes:

• Mean–variance (MV) preferences. We consider investorsthat minimize the variance of their portfolio:

minxTS

th

{(xTS

th

)′ΣTSt+h|txTS

th |0 ≤ xTSth ≤ 1, 1′xTS

th = 1}, (10)

where 0 = (0, 0, 0)′ , 1 = (1, 1, 1)′ , and ΣTSt+h|t is the esti-

mate of the (3 × 3)-dimensional conditional covariancematrix of returns implied by the trading strategy TS andmodel (or forecast combination method or compositeforecasts) m.10

• Conditional value-at-risk (CVaR) preferences. We con-sider an investor that maximizes the conditional expec-tation of the left tail of portfolio return distribution suchthat portfolio returns do not exceed the 𝛽 's quantile ofof portfolio return, that is:

max(xTS

th ,𝛼𝛽)

{E

((xTS

th

)′rTSth

)|||| (xTSth

)′rTSth ≤ 𝛼𝛽,

0 ≤ xTSth ≤ 1, 1′xTS

th = 1},

(11)

where 𝛽 ∈ (0, 1) and 𝛼𝛽 is the 𝛽-quantile of the portfolioreturn. Problem 11 is equivalent to11

max(xTS

th ,𝛼𝛽)

{𝛼𝛽 −

1t𝛽

t∑l=1

[𝛼𝛽 −

(xTS

th

)′rTSlh

]+ ||||||0 ≤ xTS

th ≤ 1, 1′xTSth = 1

},

(12)

9In a related analysis using similar returns, we also examined longer andshorter time periods to determine the composite forecast. First we lookedat the performance results based on using the total period up to time t,which were usually worse than those based on a period of 12 months.Then we experimented with a period of 6 months, which in most casesresulted in similar or lower performance measures.10For a seminal presentation of the mean–variance model, see Markowitz(1952).11See Rockafellar and Uryasev (2000) for more details.

where [t]+ denotes the maximum of 0 and t. In ourapplication we take 𝛽 = 0.05.

• Linear preferences. Investors with linear utilityfunctions maximize the expected return of theirportfolio:

maxxTS

th

{E

((xTS

th

)′rTSth

) |||0 ≤ xTSth ≤ 1, 1′xTS

th = 1}.

(13)We denote this investor the “linear” investor.

• Linear loss aversion (LLA) preferences. Loss aversion,which is a central finding of prospect theory (see Kah-neman & Tversky, 1979) describes the fact that peopleare more sensitive to losses than to gains, relative to agiven reference point ��t. The simplest form of such lossaversion is linear loss aversion, where the marginal util-ity of gains and losses is fixed.12 Linear loss aversionpreferences can be modeled as

maxxTS

th

{E

((xTS

th

)′rTSth − 𝜆

[��t −

(xTS

th

)′rTSth

]+)|||||0 ≤ xTS

th ≤ 1, 1′xTSth = 1

},

(14)

where 𝜆 > 0 is the loss aversion (or penalty) param-eter. Under the given utility, investors face a trade-offbetween return on the one hand and shortfall belowthe reference point on the other hand. Interpreted dif-ferently, the utility function contains an asymmetric ordownside risk measure, where losses are weighted dif-ferently from gains. In our application we take the zeroreturn as the reference point; that is, ��t = 0, and 𝜆 =1.25, 5.

• Quadratic loss aversion (QLA) preferences. Underquadratic loss aversion preferences, large losses arepunished more severely than under linear loss aversionpreferences.13 The quadratic loss aversion preferencescan be modeled as

maxxTS

th

{E

((xTS

th

)′rTSth − 𝜆

([��t −

(xTS

th

)′rTSth

]+)2)||||||

0 ≤ xTSth ≤ 1, 1′xTS

th = 1},

(15)where the notation is the same as in the LLA pref-erences; that is, in our application we take again thezero return as the reference point, ��t = 0, and 𝜆 =1.25, 5. The problems given by Equations 14 and 15 areequivalent to higher-dimensional linear programmingproblems (see Fortin & Hlouskova, 2015).

12The optimal ass et allocation decision under linear loss aversion hasbeen extensively studied; see, for example, Gomes (2005), He and Zhou(2011), and Fortin and Hlouskova (2011).13The penalty on losses under quadratic loss aversion is also referred toas quadratic shortfall.


2.6 Performance2.6.1 Performance measuresThe main performance measures we consider are themean return, the Omega measure, the Sharpe ratio (SR)and the Sortino ratio (SoR), where the last three mea-sures are risk adjusted and in that sense reflect betterthe overall considerations of a typical investor (who isnot only interested in the portfolio return but also inthe portfolio risk). These measures are, among others,reported in our empirical results. The Omega measureis the upside potential of the return with respect to itsdownside potential relative to the zero return, Omega =∑n

i=1 |max{ri, 0}|∕∑ni=1 |min{ri, 0}|, where i = 1 corre-

sponds to January 2008 and i = n to January 2016. Alarger ratio indicates that the asset provides the chanceof more gains relative to losses. The Omega measure isa risk-adjusted performance measure which considers allmoments of the return distribution, whereas the Sharpeand Sortino ratios only consider the first two moments (amodified version of the second moment in the case of theSortino ratio).14 The Sharpe ratio is the mean divided bythe volatility of the return, the Sortino ratio is a modifiedversion of the Sharpe ratio which uses downside volatilitywith respect to the zero return (instead of standard devia-tion) as the denominator, i.e., SR = r∕𝜎 and SoR = r∕𝜎D,where r and 𝜎 are the mean and the standard deviationof rt calculated with respect to the sample January 2008through January 2016, and 𝜎D is the downside volatil-ity (with respect to the zero return) calculated as 𝜎D =√

1n

∑ni=1 (min{ri, 0})2, where i = 1 corresponds to January

2008 and i = n to January 2016.15 The natural bench-mark return for our application appears to be a zero return,reflecting that the investor does not take any position inthe foreign exchange market. In both cases a larger ratioindicates a higher return per unit of risk, so usually higherfigures denote a better performance. This does not apply,however, when the mean return is negative.

2.6.2 Bootstrap test for equalperformanceIn order to assess whether the performance superiority ofcertain (optimal) portfolios is systematic and not due toluck, we perform the bootstrap stepwise multiple superior

14The formal definition of the Omega measure is ∫ ∞tr (1−F(x))dx∫ tr−∞ F(x)dx

=E(X−tr|X>tr)P(X>tr)E(tr−X|X≤tr)P(X≤tr)

, where F(·) is the cumulative distribution function ofreturns and tr is the target return, which in our case is zero.15In our empirical results we also present a performance measurelabeled “downside volatility ratio” that gives the proportion of down-side volatility to the downside and upside volatility 𝜎U, where 𝜎U =√

1n

∑ni=1 (max{ri, 0})2, and thus the ratio is calculated as 𝜎D

𝜎D+𝜎U.

predictive ability test (stepM-SPA) by Hsu, Hsu, and Kuan(2010) for the comparison of optimal portfolio perfor-mance with respect to the benchmark models. The testis based on the bootstrap method of Politis and Romano(1994), the stepwise test of multiple check by Romano andWolf (2005) and the test for superior predictive ability ofHansen (2005).16

The following relative performance measures, dTSopt,th,

opt = MV, CVaR, linear, LLA, and QLA with 𝜆 = 1.25, 5,t = January 2008 to January 2016, h = 1, 3, 6, 12, are com-puted and the tests are defined based alternatively on dTS

opt,th= rTS

opt,th − rTSB,th and dTS

opt,th = OmegaTSopt,th − OmegaTS

B,th.As benchmark portfolios, for which the corresponding

measures (denoted by subindex B) are calculated, we con-sider the single assets based on the returns implied by acertain trading strategy which is applied to single exchangerates, as well as the equally weighted portfolio (EW). Thesingle assets and the equally weighted portfolio are eitherbased on composite forecasts or on the random walk. Notethat the performance measure related to Omega is definedas Omegat = max{rt, 0}∕( 1

n

∑ni=1 |min{ri, 0}|), where we

skip indices and superscripts to simplify the notation. Notethat the bootstrap test cannot be performed for the Sharpeand the Sortino ratios, as for negative values it is not truethat larger values are associated with a better performance(the test involves the ordering of calculated statistics).17

The bootstrap stepM-SPA test is a comprehensive testacross all portfolio optimization models under consider-ation and directly quantifies the effect of data snoopingby testing the null hypothesis that the performance of thebest model is no better than the performance of the bench-mark model. For the seven optimization models opt =MV,CVaR, linear, LLA with 𝜆 = 1.25, 5, and QLA with 𝜆 =1.25, 5; we consider the test of Hopt

0 ∶ E

(dTS

opt,th ≤ 0)

against HoptA ∶ E

(dTS

opt,th > 0)

. In our empirical applicationwe use the output of the test to identify those optimal port-folios that outperform the benchmark portfolio at a certainsignificance level.

2.6.3 Break-even transaction costsThe break-even transaction cost, 𝜏be, is that constant (pro-portional) transaction cost that makes the net gain (net oftransaction cost) of the optimal portfolio equal to the netgain of the benchmark portfolio18:

16For more details on the test, see Hsu et al. (2010).17For example, with a higher volatility the Sharpe ratio increases insteadof decreases, given that the mean is negative.18Break-even transaction costs are also considered in Han (2006), DellaCorte et al. (2009), and Della Corte and Tsiakas (2013), for example. Ourcomputations are based on the transaction value (over the total periodof the investment strategy) whereas their computations are based on(one-period) returns.


𝜏be =

∑T

t=2

(V opt

t − V optt−1

)−∑T

t=2

(V B

t − V Bt−1

)∑T

t=2V opt

t

∑n

i=1

|||||xopti,t −

xopti,t−1

(1+ropt

i,t

)1+ropt

𝑝,t

||||| −∑T

t=2V B

t

∑m

i=1

|||||xBi,t −

xBi,t−1

(1+rB

i,t

)1+rB

𝑝,t

|||||(16)

where Vt is the value of the (optimal/benchmark) portfo-lio at time t, xi,t is the weight of asset i at time t in the(optimal/benchmark) portfolio, ri,t is the return of asset iat time t in the (optimal/benchmark) portfolio, and rp,t isthe return of the (optimal/benchmark) portfolio at time t.Superscripts opt and B denote the optimal and the bench-mark portfolios, respectively. We assume that the initialinvestment in both portfolios is the same; that is, V opt

1 =V B

1 .19 We compare all optimal portfolios (MV, CVaR, linear,LLA𝜆=1.25, LLA𝜆=5, QLA𝜆=1.25, and QLA𝜆=5) with the threesingle assets and the EW portfolio, based on compositeforecasts and on the random walk, respectively.

If the actual transaction cost is lower than thebreak-even transaction cost then the optimal portfolioachieves a larger gain than the benchmark portfolio, aftercontrolling for transaction costs.20 So larger break-eventransaction costs indicate that the optimal portfolio out-performs the benchmark portfolio more easily, whilenegative break-even transaction costs suggest that theoptimal portfolio does not outperform the benchmarkportfolio. For the discussion of our empirical results weuse a value of 0.1% for the actual transaction cost.21

3 EMPIRICAL RESULTS

3.1 Data, estimation, and predictionsWe base our empirical analysis on monthly data span-ning the period from January 1980 until January 2016 forthe EUR/USD, EUR/GBP, and EUR/JPY exchange rates.The beginning of the sample is thus T0 = January 1980;the beginning of the hold-out forecasting sample for indi-vidual models used in order to obtain weights based onpredictive accuracy is given by T1 = January 2000. The

19The break-even transaction cost is not sensitive with respect to thestarting value.20In fact this is only true if the trading cost for the optimal portfoliois larger than the trading cost for the benchmark portfolio (i.e., if thedenominator in Equation 16 is positive), which is always true in our appli-cations. Note that it is often a typical feature of the benchmark portfoliothat the involved amount of trading is rather small.21Marquering and Verbeek (2004), for example, consider three levels oftransaction costs, 0.1%, 0.5%, and 1%, to represent low, medium, and highcosts. They consider equity and bond transactions, however, while weconsider currency transactions. The latter usually involve lower trans-action costs. For example, the average bid–ask spread related to themiddle price over our sample period ranges from 0.02% for the EUR/USDexchange rate to 0.06% for the EUR/JPY exchange rate. So a transactioncost of 0.1% seems to be a quite high level in the foreign exchange market.

beginning of the actual out-of-sample forecasting sampleis T2 = January 2005, and the end of the data sample isT3 = January 2016. All data are obtained from ThomsonReuters Datastream.22

Given the large set of statistics computed for the analy-sis, we start by describing how the results are depicted inthe tables. We first report results on the performance ofoptimal currency portfolios constructed by different typesof investors, namely MV, CVaR, linear, linear loss averse(LLA𝜆=1.25, LLA𝜆=5), and quadratic loss averse (QLA𝜆=1.25,QLA𝜆=5) investors (the first seven data columns in thetables of results). The optimal portfolios consist of threesingle assets, the returns of which are implied by a certaintrading strategy (from the two trading strategies describedabove) with respect to MSE-based composite forecasts ofthe EUR against the USD, GBP, and JPY. In addition, wepresent results on how these optimal portfolios compare tobenchmark portfolios, which include the three single assetsthat compose the portfolio as well as the equally weightedportfolio (the following four data columns in the tablesof results). We also consider as benchmark portfolios thethree single assets which are based on the simple randomwalk (RW) model to forecast exchange rates (instead ofon composite forecasts) as well as the equally weightedportfolio of these three assets (the last four data columns).

The tables are structured in five horizontal blocks. Thefirst block presents the four main performance measureson which we base our analysis: the mean return, theOmega measure, the Sharpe ratio, and the Sortino ratio.For two of them, the mean and the Omega measure, wealso performed the bootstrap-based stepM-SPA test of Hsuet al. (2010). The second block shows additional statisticaldescriptives, namely median, volatility, downside volatil-ity, downside volatility ratio, CVaR, skewness, and kurto-sis. The third block presents break-even transaction costsfor the optimal portfolios, first in relation to the bench-mark portfolios based on composite forecasts (where theevaluation criterion is the MSE over the last 12 months)and then in relation to the benchmark portfolios basedon the random walk. For the discussion of our empiricalresults we use a value of 0.1% for the actual transactioncost. The fourth block shows the realized returns over thelast 5, 3, and 1 years, and the last block gives the meanportfolio allocations.

22Details on the sources for all variables used are given in the Appendix.


FIGURE 1 MSE-based best forecast models for 1-month-ahead forecasts. The figure shows the best forecast models, that is, the modelsgenerating the composite forecasts (left) and the distribution of the best models, that is, the number of times different forecast models areselected over the total period, in percent (right). From top to bottom the figure presents best models for the EUR/USD, the EUR/GBP, and theEUR/JPY [Colour figure can be viewed at wileyonlinelibrary.com]

The subindices in the first two rows of the first block ofthe benchmark portfolios show the results of the bootstraptest. Their values indicate how many optimal portfoliosoutperform (in terms of the mean return and the Omegameasure) that specific benchmark portfolio. If no subindexis present, no null hypothesis is rejected; that is, the bench-mark portfolio is not outperformed by any of the particularoptimal portfolios used. If there is only one subindex, itsvalue indicates the number of optimal portfolios that out-perform the benchmark portfolio at the 10% significancelevel. In the case of two subindices, the first one indi-cates the number of optimal portfolios that outperform thebenchmark portfolio at the 5% significance level and thesecond one at the 10% significance level. In general, state-ments relating to the bootstrap test will be made using the10% significance level.

The results are reported for forecast horizons of 1 and3 months, for both trading strategies, and for compos-ite exchange rate forecasts based on the minimum mean

square error over the last twelve months. Additionalresults for forecast horizons of 6 and 12 months are notdiscussed in detail but we point out the main differenceswith respect to shorter forecast horizons, with the corre-sponding tables presented in the supporting informationAppendix. To allow for a better reading flow, we sometimessimply state USD when we mean the single asset based onthe returns implied by the EUR/USD exchange rate (fore-casts) for a given trading strategy (analogously for GBP andJPY). To simplify the notation, we sometimes skip the word“portfolio” and just refer to MV instead of MV portfolio(analogously for CVaR, linear, LLA, and QLA).

3.2 A snapshot example: From compositeforecasts to single asset returns to theoptimal portfolioTo get a better picture of the process leading up to the opti-mal portfolio choice, we describe the required steps for aconcrete example, where we consider a forecast horizon of

http://wileyonlinelibrary.com


FIGURE 2 Optimal MV portfolio, single assets and EW portfoliobased on a 1-month forecast horizon and TS1. The figure showsindices (with a starting value of 100) based on the returns of theoptimal mean–variance portfolio, the single assets, and the equallyweighted portfolio for trading strategy 1 and a forecast horizon of 1month

1 month and a mean–variance investor. We use MV invest-ing because it is a popular way of selecting assets and,in addition, the MV portfolio often achieves the highestperformance measures. We start by presenting the mod-els and forecast combination methods that are chosen asthe best ones, based on the minimum MSE over the last12 months, for each exchange rate (MSE-based compositeforecasts). We then show the returns implied by these com-posite forecasts and trading strategy 1, for the single assets,the equally weighted portfolio, and the optimal portfolio.Finally we look at the resulting optimal portfolio weights.

The left part of Figure 1 shows how the best forecastmodels change over the period January 2005 to January2016, while the right part presents aggregated informationon the number of times (percent) a given forecast modelis chosen over time. From top to bottom we present theresults based on the EUR/USD, the EUR/GBP, and theEUR/JPY exchange rates. For each case, the list of bestmodels includes 15 forecast models/forecast combinationsfrom the total list of 25, but not all these 15 models coincideacross currencies. Even though it is visible how modelschange, and sometimes very quickly (although most of thetime best models remain the best for a while), some appearto be chosen quite frequently. The principal components(PC) method, for example, is the model most often selectedas the best one for the USD (23%) and for the GBP (26%).On the other hand, it appears only once as the best modelfor the JPY.23 The random walk is chosen as the best fore-cast model only 2% of the time for the USD, while it isselected as the best model 11% of the time for both the GBPand the JPY.

23For a forecast horizon of 3 months, however, the PC is picked up moreoften as the best forecast model for the JPY, namely 5% of the time.

If one is interested in how often single models are cho-sen as opposed to forecast combination methods, rathersurprisingly, single models appear particularly powerful inforecasting. For the USD, single models are chosen 65% ofthe time and for the JPY even 97% of the time. For the GBP,on the other hand, single models are selected less often,namely 41% of the time. The results for the USD and theJPY (approximately) carry over to a forecast horizon of 3months, while the results for the GBP are reversed. So fora forecast horizon of 3 months, individual models beat theforecast combination methods in all three currencies. Thisis also true for longer forecast horizons.24

Figure 2 plots the indices corresponding to the cumu-lative returns of the optimal MV portfolio together withthose of the three single assets that make up the portfo-lio and the equally weighted portfolio for trading strategy1 and a forecast horizon of 1 month. The starting val-ues of the indices are set to 100. The graph shows thatover the period ranging from January 2008 to January2016 the total return is highest for the single asset basedon the GPB, followed by the optimal MV portfolio andthe equally weighted portfolio. The USD shows a returnwhich is barely positive and the JPY shows a clearly neg-ative return. The mean returns are in line with the totalreturn one can see in the graph. The GBP achieves thehighest mean return (5.2%), followed by the MV optimalportfolio (3.9%) and the equally weighted portfolio (1.9%).If one takes into account risk considerations and hencelooks at, for example, the Sharpe ratio, then the MV port-folio outperforms all other assets as well as the equallyweighted portfolio. This is due to the volatility, which isclearly larger for the GBP than for the optimal portfolio.Descriptive statistics relating to the returns shown in thegraph, including the mean, the volatility and the Sharperatio, are presented in Table 2. The visual impression thatthe optimal portfolio beats the single assets based on theUSD and the JPY is confirmed by the result of the boot-strap test, which concludes that the MV optimal portfoliooutperforms both the USD and the JPY at the 10% signif-icance level in terms of the Omega measure. The graphshows that the total return of the MV portfolio exceeds thatof the equally weighted portfolio, that of the USD, and thatof the JPY. Taking transaction costs into account, this isstill true (provided that a transaction cost of 0.1%, or 1%, isassumed).

Figure 3 depicts the optimal weights of the single assetsbased on the three exchange rates for the mean–variance(MV) and the conditional value-at-risk (CVaR) investors.We still consider trading strategy 1 and a 1-month forecast

24More precisely, individual models are chosen as best models 63% (56%)of the time for the USD, 64% (59%) for the GBP, and 89% (89%) of the timefor the JPY, for a forecast horizon of 6 (12) months.


TAB

LE2

Opt

imal

curr

ency

port

folio

s:ou

t-of-s

ampl

eev

alua

tion

and

com

paris

onw

ithbe

nchm

ark

port

folio

s(TS

1,h=

1).

MV

CVa

R,𝛽

Line

arLL

A,𝜆

QLA

,𝜆U

SDG

BP

JPY

EWU

SDG

BP

JPY

EW0.

051.

255.

001.

255.

00R

WR

WR

WR

WPe

rfor

man

cem

easu

res

Mea

n3.

863.

66−

1.16

0.11

1.45

−0.

86−

0.92

0.84

5.16

−0.

361.

86−

2.75

−0.

750.

41−

1.04

2

Om

ega

1.55

1.54

0.93

1.01

1.17

0.95

0.94

1.06

2,2

1.53

0.98

1,2

1.27

0.83

2,3

0.94

21.

022

0.89

2,2

Shar

pera

tio0.

150.

16−

0.03

0.00

0.06

−0.

02−

0.02

0.02

0.14

−0.

010.

08−

0.07

−0.

020.

01−

0.05

Sort

ino

ratio

0.30

0.29

−0.

040.

010.

10−

0.03

−0.

040.

030.

30−

0.01

0.14

−0.

09−

0.03

0.01

−0.

07A

dditi

onal

desc

riptiv

esta

tistic

sM

edia

n2.

022.

48−

1.13

0.04

0.31

−1.

13−

2.01

4.65

1.27

0.78

1.85

−2.

32−

4.66

2.48

−1.

29Vo

latil

ity7.

086.

5512

.53

8.80

6.95

12.3

411

.18

11.6

710

.08

14.6

06.

2911

.64

10.1

914

.60

6.56

Dow

n.vo

l.3.

673.

588.

145.

364.

347.

857.

028.

294.

849.

643.

908.

676.

219.

824.

60D

own.

vol.

ratio

0.37

0.39

0.46

0.44

0.45

0.45

0.45

0.50

0.35

0.47

0.44

0.53

0.44

0.48

0.50

CVa

R,𝛽=

0.05

−32

.73

−32

.99

−60

.49

−43

.41

−35

.81

−57

.31

−50

.88

−62

.63

−37

.79

−65

.27

−35

.69

−64

.02

−50

.12

−65

.28

−36

.92

Skew

ness

1.18

0.79

0.98

1.71

0.48

1.10

1.36

−0.

132.

080.

630.

80−

0.05

1.86

0.49

0.48

Kur

tosi

s6.

114.

616.

6911

.49

3.39

6.84

8.84

3.72

12.2

44.

586.

843.

7013

.19

4.54

4.49

Brea

k-ev

entr

ansa

ctio

nco

stsU

SD4.

524.

49−

1.56

−0.

080.

74−

1.24

−0.

78G

BP−

1.98

−2.

13−

6.42

−2.

72−

2.56

−5.

78−

3.93

JPY

6.09

6.09

−0.

390.

561.

53−

0.15

−0.

01EW

4.05

4.00

−4.

10−

1.01

−0.

29−

3.45

−2.

07Br

eak-

even

tran

sact

ion

costs

(RW

)U

SD7.

937.

960.

981.

312.

471.

130.

88G

BP5.

485.

47−

0.84

0.31

1.23

−0.

57−

0.31

JPY

5.27

5.26

−1.

000.

231.

12−

0.72

−0.

41EW

8.63

8.70

−0.

820.

501.

66−

0.47

−0.

19Re

aliz

edre

turn

Last

5ye

ars

1.51

1.67

−4.

080.

450.

89−

3.56

−2.

901.

422.

17−

3.89

0.23

0.45

−4.

95−

3.29

−2.

28La

st3

year

s2.

162.

49−

5.20

0.42

1.33

−4.

94−

4.10

6.94

0.88

−4.

601.

240.

04−

3.25

−0.

63−

1.05

Last

year

8.59

8.91

3.50

7.49

8.10

3.50

1.87

3.50

11.7

05.

687.

173.

50−

2.25

2.87

1.52

Mea

nal

loca

tion

USD

27.3

018

.42

13.4

011

.86

18.5

313

.40

11.0

810

00

033

.33

100

00

33.3

3G

BP48

.52

57.8

656

.70

60.6

156

.58

57.7

064

.19

010

00

33.3

30

100

033

.33

JPY

24.1

923

.72

29.9

027

.53

24.8

928

.90

24.7

30

010

033

.33

00

100

33.3

3

Not

e.Th

eta

ble

repo

rtsa

nnua

lsta

tistic

sofa

mon

thly

real

loca

ted

optim

alcu

rren

cypo

rtfo

lioan

dm

ean

optim

alw

eigh

ts,b

ased

onan

optim

izat

ion

perio

dof

36m

onth

s(ro

lling

win

dow

),tr

adin

gst

rate

gy1

and

a1-

mon

thfo

reca

stho

rizon

.The

eval

uatio

npe

riod

cove

rsJa

nuar

y20

08to

Janu

ary

2016

.Sta

tistic

sar

eca

lcul

ated

onth

eba

sis

ofm

onth

lyre

turn

san

dth

enan

nual

ized

assu

min

gdi

scre

teco

mpo

undi

ng.T

hesa

me

stat

istic

sar

ere

port

edfo

rth

ebe

nchm

ark

port

folio

sba

sed

onco

mpo

site

fore

cast

s(i.

e.,t

hesi

ngle

asse

tsof

whi

chth

epo

rtfo

lios

are

cons

truc

ted

and

the

equa

llyw

eigh

ted

port

folio

)and

for

the

benc

hmar

kpo

rtfo

lios

base

don

the

rand

omw

alk.

The

subi

ndic

essh

owth

ere

sults

ofth

ebo

otst

rap

test

.The

irva

lues

indi

cate

how

man

yop

timal

port

folio

sout

perf

orm

(inte

rmso

fthe

resp

ectiv

epe

rfor

man

cem

easu

re)t

hats

peci

ficbe

nchm

ark

port

folio

.If

nosu

bind

exis

pres

ent,

the

benc

hmar

kpo

rtfo

liois

noto

utpe

rfor

med

byan

yof

the

optim

alpo

rtfo

lios.

Ifth

ere

ison

lyon

esu

bind

ex,i

tsva

lue

indi

cate

sthe

num

bero

fopt

imal

port

folio

sout

perf

orm

ing

the

benc

hmar

kpo

rtfo

lioat

the

10%

sign

ifica

nce

leve

l.In

the

case

oftw

osu

bind

ices

,the

first

one

indi

cate

sthe

num

bero

fopt

imal

port

folio

sout

perf

orm

ing

the

benc

hmar

kpo

rtfo

lioat

the

5%si

gnifi

canc

ele

vela

ndth

ese

cond

one

atth

e10

%si

gnifi

canc

ele

vel.

Retu

rns,

wei

ghts

,and

tran

sact

ion

cost

sare

give

nin

perc

ent.


FIGURE 3 Optimal portfolio weights based on a 1-month forecast horizon and TS1. The figure shows the optimal weights of the threesingle assets in the mean–variance optimal portfolio (left) and in the conditional value-at-risk optimal portfolio (right) for trading strategy 1and a forecast horizon of 1 month

horizon. On average, the optimal weights in the MV port-folio are 27.3%, 48.5%, and 24.2% for the USD, the GBP,and the JPY. The CVaR portfolio puts even more weight tothe GBP (57.9%) on average, thereby reducing the optimalweight in the USD (18.4%), while the optimal weight in theJPY (23.7%) remains basically the same. The two graphsshow how the optimal weights evolve over time. We cansee that the MV optimal weights change more smoothlyover time than the CVaR optimal weights, which remainrather constant over relatively long periods of time andthen jump to new weights.

3.3 Results for the forecast horizon of 1monthThe results relating to a forecast horizon of 1 month arepresented in Table 2 for trading strategy 1 and in Table 3for trading strategy 2.

3.3.1 Single assets and portfoliocompositionThe best-performing single asset based on composite fore-casts is the GBP, for both trading strategies. The worstperforming single asset, for both trading strategies, is theJPY. Even though the ordering of the single assets is thesame under TS1 and TS2, the mean returns (and other per-formance measures) are quite different across the tradingstrategies. The GBP, for example, is extremely strong underboth trading strategies and shows a mean return of 5.2%(TS1) and 6.1% (TS2), respectively. The JPY, on the otherhand, shows only a slightly negative mean return (−0.4%)under TS1, while it performs considerably worse underTS2 (−3.6%).25 Note that the volatilities of the single assetsare not in line with what one would expect according tothe means: The largest mean is actually coupled with thesmallest volatility and vice versa. The described proper-

25The corresponding Omega measures are 1.53 (TS1) and 1.65 (TS2) forthe best single asset GBP, and 0.98 (TS1) and 0.82 (TS2) for the worstsingle asset JPY.

ties of the GBP and the JPY are also reflected in the sevenoptimal portfolios: Under each trading strategy, the GBPhas a large weight in all optimal portfolios (approximately50% or well above), whereas the JPY shows a considerablysmaller weight.

The ordering of single assets based on the random walkis different. In particular, best (worst) assets are not thesame any more across the two trading strategies. Thebest-performing single asset based on the random walkis the JPY under TS1, and the USD under TS2. Theworst-performing single asset, on the other hand, is theUSD under TS1 and the JPY under TS2. The JPY changesfrom the best-performing single asset under TS1 to theworst-performing single asset under TS2.

Note that the USD based on the random walk outper-forms the USD based on the composite forecast under TS2(in terms of the mean return, the Omega measure, and theSharpe and Sortino ratios), where the difference in prof-itability is quite substantial. Also the JPY based on therandom walk outperforms its counterpart based on thecomposite forecast.26 In all other cases the assets based onthe random walk perform worse than their counterpartsbased on the composite forecast (GBP under both tradingstrategies, USD under TS1, JPY under TS2). So in mostcases there is a positive economic value of forecast models.However, the results might indicate that the USD and theJPY are harder to predict than the GBP.

3.3.2 Bootstrap analysisBenchmark portfolios based on composite forecasts. Con-sidering trading strategy 1, the CVaR and MV portfoliosoutperform the single assets based on the USD and theJPY in terms of the Omega measure. Applying tradingstrategy 2, again the single assets based on the USD andthe JPY are outperformed and, in addition, the equallyweighted portfolio (in terms of the Omega measure). While

26In fact, the JPY changes from the worst-performing asset based oncomposite forecasts to the best-performing asset based on the randomwalk.


TAB

LE3

Opt

imal

curr

ency

port

folio

s:ou

t-of-s

ampl

eev

alua

tion

and

com

paris

onw

ithbe

nchm

ark

port

folio

s(TS

2,h=

1).

MV

CVa

R,𝛽

Line

arLL

A,𝜆

QLA

,𝜆U

SDG

BP

JPY

EWU

SDG

BP

JPY

EW0.

051.

255.

001.

255.

00R

WR

WR

WR

WPe

rfor

man

cem

easu

res

Mea

n3.

963.

843.

470.

671.

543.

573.

201.

766.

06−

3.56

1.35

3.93

−0.

25−

5.25

2−

0.59

Om

ega

1.54

1.52

1.31

1.07

1.18

1.33

1.29

1.12

21.

650.

825,

61.

182

1.29

0.98

2,2

0.75

3,7

0.94

2,2

Shar

pera

tio0.

150.

150.

100.

030.

060.

100.

090.

040.

17−

0.07

0.06

0.10

−0.

01−

0.11

−0.

02So

rtin

ora

tio0.

260.

260.

160.

040.

090.

170.

150.

060.

36−

0.10

0.08

0.16

−0.

01−

0.13

−0.

03A

dditi

onal

desc

riptiv

esta

tistic

sM

edia

n3.

212.

184.

280.

951.

964.

963.

565.

212.

64−

0.99

3.55

2.50

−0.

54−

2.26

2.06

Vola

tility

7.44

7.23

10.3

57.

687.

3510

.27

10.2

111

.67

10.0

314

.52

6.80

11.6

310

.18

14.4

78.

33D

own.

vol.

4.25

4.22

6.03

5.04

4.82

6.00

5.94

8.17

4.74

10.5

44.

667.

066.

5311

.50

5.86

Dow

n.vo

l.ra

tio0.

410.

420.

420.

470.

470.

420.

420.

500.

350.

510.

490.

430.

460.

570.

50C

VaR,𝛽=

0.05

−40

.14

−39

.44

−50

.28

−43

.09

−43

.84

−50

.28

−50

.28

−62

.70

−37

.58

−68

.00

−43

.29

−56

.92

−52

.10

−75

.28

−51

.13

Skew

ness

0.76

0.43

1.32

0.49

0.23

1.33

1.40

−0.

152.

060.

400.

150.

411.

61−

0.51

0.40

Kur

tosi

s6.

514.

1910

.33

4.39

4.71

10.6

110

.85

3.80

12.3

84.

646.

943.

6313

.23

4.32

6.78

Brea

k-ev

entr

ansa

ctio

nco

stsU

SD3.

523.

831.

45−

0.26

0.27

1.29

1.04

GBP

−3.

26−

3.59

−2.

91−

3.22

−3.

38−

2.35

−2.

45JP

Y8.

779.

584.

832.

043.

094.

113.

74EW

4.91

5.61

1.65

−0.

430.

161.

411.

10Br

eak-

even

tran

sact

ion

costs

(RW

)U

SD0.

860.

92−

0.26

−1.

42−

1.16

−0.

14−

0.33

GBP

4.97

5.42

2.39

0.38

1.05

2.07

1.78

JPY

9.94

10.8

65.

592.

553.

734.

744.

34EW

7.70

8.75

3.33

0.68

1.56

2.76

2.39

Real

ized

retu

rnLa

st5

year

s3.

804.

853.

272.

682.

173.

353.

212.

884.

46−

2.99

1.77

3.06

2.28

−3.

770.

73La

st3

year

s5.

255.

582.

693.

613.

632.

832.

8711

.19

4.67

−4.

563.

856.

144.

003.

224.

64La

stye

ar7.

827.

963.

416.

417.

632.

553.

184.

109.

505.

706.

674.

10−

1.51

1.55

1.57

Mea

nal

loca

tion

USD

25.0

614

.68

11.3

411

.35

15.2

710

.20

8.73

100

00

33.3

310

00

033

.33

GBP

51.6

170

.74

75.2

670

.36

64.0

376

.13

77.0

60

100

033

.33

010

00

33.3

3JP

Y23

.33

14.5

813

.40

18.2

920

.70

13.6

714

.21

00

100

33.3

30

010

033

.33

Not

e.Th

eta

ble

repo

rtsa

nnua

lsta

tistic

sofa

mon

thly

real

loca

ted

optim

alcu

rren

cypo

rtfo

lioan

dm

ean

optim

alw

eigh

ts,b

ased

onan

optim

izat

ion

perio

dof

36m

onth

s(ro

lling

win

dow

),tr

adin

gst

rate

gy2

and

a1-

mon

thfo

reca

stho

rizon

.The

eval

uatio

npe

riod

cove

rsJa

nuar

y20

08to

Janu

ary

2016

.Sta

tistic

sar

eca

lcul

ated

onth

eba

sis

ofm

onth

lyre

turn

san

dth

enan

nual

ized

assu

min

gdi

scre

teco

mpo

undi

ng.T

hesa

me

stat

istic

sar

ere

port

edfo

rth

ebe

nchm

ark

port

folio

sba

sed

onco

mpo

site

fore

cast

s(i.

e.,t

hesi

ngle

asse

tsof

whi

chth

epo

rtfo

lios

are

cons

truc

ted

and

the

equa

llyw

eigh

ted

port

folio

)and

for

the

benc

hmar

kpo

rtfo

lios

base

don

the

rand

omw

alk.

The

subi

ndic

essh

owth

ere

sults

ofth

ebo

otst

rap

test

.The

irva

lues

indi

cate

how

man

yop

timal

port

folio

sout

perf

orm

(inte

rmso

fthe

resp

ectiv

epe

rfor

man

cem

easu

re)t

hats

peci

ficbe

nchm

ark

port

folio

.If

nosu

bind

exis

pres

ent,

the

benc

hmar

kpo

rtfo

liois

noto

utpe

rfor

med

byan

yof

the

optim

alpo

rtfo

lios.

Ifth

ere

ison

lyon

esu

bind

ex,i

tsva

lue

indi

cate

sthe

num

bero

fopt

imal

port

folio

sout

perf

orm

ing

the

benc

hmar

kpo

rtfo

lioat

the

10%

sign

ifica

nce

leve

l.In

the

case

oftw

osu

bind

ices

,the

first

one

indi

cate

sthe

num

bero

fopt

imal

port

folio

sout

perf

orm

ing

the

benc

hmar

kpo

rtfo

lioat

the

5%si

gnifi

canc

ele

vela

ndth

ese

cond

one

atth

e10

%si

gnifi

canc

ele

vel.

Retu

rns,

wei

ghts

,and

tran

sact

ion

cost

sare

give

nin

perc

ent.


the USD and the EW portfolio are again outperformed bythe MV and CVaR portfolios, the JPY is now outperformedby all optimal portfolios except the LLA𝜆=1.25 portfolio.The equally weighted portfolio is only outperformed underTS2. Regarding the performance based on the mean, noneof the benchmark portfolios is significantly outperformed,neither for TS1 nor for TS2.

Benchmark portfolios based on the random walk. Wenow observe more rejections than in the case of bench-mark portfolios based on composite forecasts, as we expect.Under TS1 all RW-based benchmark portfolios are outper-formed in terms of the Omega measure by the MV andCVaR portfolios. The USD is additionally outperformed byanother optimal portfolio. Under TS2 all RW-based bench-mark portfolios except the USD are outperformed in termsof the Omega measure by the MV and CVaR portfolios.The JPY is in fact outperformed by all optimal portfolios.In terms of the mean only the equally weighted portfoliois outperformed under TS1, and the JPY is outperformedunder TS2 (both by the MV and CVaR portfolios). Note thatunder both trading strategies the equally weighted portfo-lio based on the random walk shows a clearly worse perfor-mance than the corresponding equally weighted portfoliobased on composite forecasts. Again, this indicates a posi-tive economic value of forecast models.

3.3.3 Optimal portfolios across investorsand trading strategiesComparison across investors. For a given trading strategy,the best performance is basically always achieved by theMV portfolio. There is only a single instance when this isnot true, namely for trading strategy 1 when the Sharperatio is considered. In this case the CVaR optimal portfolioyields the best performance. The second-best optimal port-folio is always the CVaR portfolio (or the MV portfolio inthe only case when the CVaR portfolio performs the best).In fact, the MV and CVaR portfolios show a quite similarperformance in terms of all four performance measures,under both trading strategies.

Comparison across trading strategies. For a given type ofinvestor, optimal portfolios based on trading strategy 2 per-form better than those based on trading strategy 1, exceptfor MV and CVaR investors. For these types of investors theperformance results under both trading strategies are quitesimilar, but actually trading strategy 1 slightly outperformstrading strategy 2 (when considering the Omega measureand the Sortino ratio).

3.3.4 Break-even transaction costsTaking transaction costs into account, the MV and CVaRoptimal portfolios outperform all benchmark portfolios

(except the GBP single asset based on composite forecasts)under both trading strategies. In fact, in most cases whenthe mean returns of optimal portfolios exceed the meanreturns of benchmark portfolios, the optimal portfoliosstill outperform benchmark portfolios after controlling fortransaction costs. Under trading strategy 2, for example, alloptimal portfolios outperform the equally weighted port-folio, after taking transaction costs into account, except theLLA𝜆=1.25 investor whose benchmark is the EW portfoliobased on composite forecasts.

3.4 Results for the forecast horizon of 3monthsThe results relating to a forecast horizon of 3 months arepresented in Table 4 for trading strategy 1 and in Table 5for trading strategy 2.

3.4.1 Single assets and portfoliocompositionFor both trading strategies, the best-performing singleasset based on composite forecasts is the GBP (in terms ofthe mean return, the Omega measure, the Sortino ratio,and the Sharpe ratio). The worst-performing single assetfor TS1 and TS2 is the JPY. These results are the same asthose for a forecast horizon of 1 month. Even though theordering of the three single assets is the same under bothtrading strategies, the performance (of a given asset) is dif-ferent across these strategies (as in the case of the 1-monthtime horizon). For example, the mean of the USD is 0.7%under TS1, while it is basically zero (0.1%) under TS2. Alsofor the JPY the mean is larger under TS1 (−2.2%) thanunder TS2 (−7.1%). The GBP shows very similar meansunder the two trading strategies (2.2% and 2.4%). Again,the volatilities are not in line with what one would expectaccording to the means: the single asset with the largestmean has the lowest volatility and vice versa.

The ordering of single assets based on the random walkis different, as for the 1-month forecast horizon. Thebest-performing single asset is the USD under TS1, and theUSD and GBP under TS2.27 The worst-performing singleasset is the JPY under both trading strategies.

Under trading strategy 1 it is always true that the sin-gle assets based on the composite forecast show a betterperformance than their counterparts based on the randomwalk (in terms of all four performance measures). Undertrading strategy 2 this statement is always true except forthe JPY, where the RW-based single asset outperforms the

27The performance of the USD and GBP under TS2 is extremely simi-lar. While the mean is slightly larger for the GBP, the Omega measure isslightly larger for the USD. The Sharpe and Sortino ratios are the samefor both assets.


TAB

LE4

Opt

imal

curr

ency

port

folio

s:ou

t-of-s

ampl

eev

alua

tion

and

com

paris

onw

ithbe

nchm

ark

port

folio

s(TS

1,h=

3).

MV

CVa

R,𝛽

Line

arLL

A,𝜆

QLA

,𝜆U

SDG

BP

JPY

EWU

SDG

BP

JPY

EW0.

051.

255.

001.

255.

00R

WR

WR

WR

WPe

rfor

man

cem

easu

res

Mea

n1.

561.

35−

1.49

0.49

1.28

−1.

13−

0.37

0.71

2.24

−2.

210.

231

−0.

14−

0.26

−3.

13−

1.18

Om

ega

1.38

1.28

0.85

1.08

1.25

0.88

0.95

1.09

1.42

0.82

1,3

1.05

1,1

0.98

0.96

1,3

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single asset based on the composite forecast. These resultsmay indicate that the JPY is harder to predict than the USDand the GBP.

3.4.2 Bootstrap analysisBenchmark portfolios based on composite forecasts. Foreach trading strategy, the equally weighted portfolio aswell as the JPY are outperformed by optimal portfolios (interms of the Omega measure). Under TS1, however, onlythe MV portfolio outperforms the EW portfolio, and theMV, the CVaR, and the LLA𝜆=5 portfolios outperform theJPY, while under TS2 (nearly) all optimal portfolios out-perform the benchmark portfolios.28 These results indicatethat investors should in fact engage in active optimal port-folio management rather than just follow the naive equallyweighted portfolio approach. Note that for the 1-monthahead forecast horizon and TS1, the equally weightedportfolio was not outperformed, which implies that thegains from building optimal currency portfolios based onexchange rate forecasts (with respect to the naive EWportfolio) appear to be sensitive to the forecast horizonconsidered.

Benchmark portfolios based on the random walk. Boththe EW portfolio and the JPY are outperformed by opti-mal portfolios, under TS1 and TS2, in terms of the Omegameasure. However, while under TS1 only the MV, CVaR,and LLA𝜆=5 portfolios outperform the two benchmarkportfolios, under TS2 nearly all optimal portfolios (exceptthe linear one) outperform these benchmark portfolios.Further, the GBP is outperformed by the MV, the CVaR,and the LLA𝜆=5 under TS1. For the mean, only the JPYunder TS2 is outperformed, namely by all optimal port-folios except the linear one. Note that under both tradingstrategies the RW-based equally weighted portfolio showsa worse performance than the equally weighted portfoliobased on composite forecasts (in terms of all four per-formance measures). Under TS1, the difference is quitesubstantial whereas under TS2 the difference is rathersmall. This again confirms the positive economic value offorecast models.

3.4.3 Optimal portfolios across investorsand trading strategiesComparison across investors. Under TS1 the best perfor-mance is achieved by MV and CVaR portfolios (in terms ofthe mean, the Omega measure, the Sharpe ratio, and theSortino ratio)—similar to the 1-month forecast time hori-zon. Under TS2, however, the LLA𝜆=5 and CVaR portfoliosperform best.

28More precisely, all except the linear and the QLA𝜆=1.25, for the EWbenchmark, and all for the JPY.

Comparison across trading strategies. For the MV, CVaRand linear investors, trading strategy 1 yields a higherperformance than trading strategy 2 (in terms of all fourperformance measures). For all other types of investors,trading strategy 2 delivers better performance measures.

3.4.4 Break-even transaction costsTaking transaction costs into account, the MV and CVaRoptimal portfolios outperform all benchmark portfolios(except the GBP single asset based on composite fore-casts) under both trading strategies. This is completelysimilar to the case of the 1-month forecast horizon. Again,in most cases when the mean returns of optimal portfo-lios exceed the mean returns of benchmark portfolios, theoptimal portfolios still outperform benchmark portfoliosafter controlling for transaction costs. Under trading strat-egy 2, for example, all optimal portfolios outperform theequally weighted portfolio (based on composite forecastsand based on the random walk), after taking transactioncosts into account, except for the linear investor.

3.5 Main results for the forecast horizonsof 6 and 12 monthsA crucial observation with respect to the forecast horizonin general is that the performance of the single assets, aswell as of the optimal portfolios, seems to decrease withthe forecast horizon. For horizons of 6 and 12 months themean returns (of both benchmark and optimal portfolios)are in fact mostly close to zero or negative. Based on thebootstrap results, the equally weighted portfolio and theJPY (both based on the composite forecasts) are mostlyoutperformed, for both trading strategies. In most cases,this is true in terms of the Omega measure and the mean.

Regarding the types of investors, we observe that with anincreasing forecast horizon the group of investors achiev-ing the best or second-best performance seems to bewidening to include (in addition to the MV and CVaRinvestors) also the linear, LLA𝜆=5, and QLA investors.Although for a forecast horizon of 6 months the MV andCVaR portfolios are still the best or second-best perform-ing optimal portfolios in most cases, the linear and QLAinvestors seem to be taking over for a forecast horizon of12 months.

Note that the number of cases when the performance ofbenchmark portfolios (single assets and equally weightedportfolio) based on the random walk exceeds the perfor-mance of benchmark portfolios based on the compositeforecasts increases with a forecast horizon of 12 months.Although all benchmark portfolios based on the randomwalk are outperformed by their counterparts based oncomposite forecasts for the 6-month forecast horizon(under both trading strategies and in terms of all four


performance measures), nearly all benchmark portfoliosbased on the random walk in fact outperform their coun-terparts based on composite forecasts for the 12-monthforecast horizon.29

Taking transaction costs into account, the performanceof optimal portfolios for a forecast horizon of 6 months israther similar to the case of 1-month and 3-month fore-cast horizons. For a forecast horizon of 12 months thereare clearly fewer cases when optimal portfolios outper-form benchmark portfolios. In fact, under both tradingstrategies, none of the optimal portfolios outperforms theequally weighted portfolio based on the random walk.

3.6 Summary and stylized factsThe performance of benchmark portfolios (single assetsand the equally weighted portfolio) based on compositeforecasts is better than the performance of benchmarkportfolios based on the random walk, for forecast horizonsof 1, 3, and 6 months and both trading strategies, with onlyvery few exceptions.30 Thus the economic value of usingexchange rate forecast models rather than the naive ran-dom walk forecasts is pronounced for forecast horizons ofup to 6 months. This is not really the case for a 12-monthforecast horizon.

Investors with MV and CVaR preferences perform better,in terms of the mean return (also adjusted for transac-tion costs), the Omega measure, the Sortino ratio, andthe Sharpe ratio, under both trading strategies, and forforecast horizons of 1, 3 and 6 months, than investorsthat choose the naive equally weighted portfolio based oncomposite forecasts. This observation is even more pro-nounced when the benchmark is the equally weightedportfolio based on the random walk. In this case nearly alloptimal portfolios outperform the equally weighted port-folio. This indicates that exchange rate forecasting andportfolio optimization imply a positive economic valuefor MV and CVaR investors, when the benchmark is theequally weighted portfolio and the forecast horizon doesnot exceed 6 months.

In most cases when optimal portfolios show largermean returns than benchmark portfolios without transac-tion costs, they outperform the benchmark portfolios alsoafter controlling for transaction costs. In addition, if onebelieves that non-naive exchange rate forecasts providepositive economic value then break-even transaction costswith respect to benchmarks based on the random walkshould exceed break-even transaction costs with respect to

29This is in contrast to the stylized fact that non-naive exchange rateforecasts beat the random walk, if at all, at longer time horizons.30These are the JPY under trading strategy 1 for a 1-month forecast hori-zon, the USD under trading strategy 2 for a 1-month forecast horizon, andthe JPY under trading strategy 2 for a 3-month forecast horizon.

benchmarks based on composite forecasts. For a forecasthorizon of 12 months, however, this is mostly not true; thatis, break-even transaction costs are often larger when cal-culated with respect to the benchmark portfolios based oncomposite forecasts than when calculated with respect tothe benchmark portfolios based on the naive random walk.This is another example of the positive economic value ofexchange rate forecast models (for forecast horizons of upto 6 months).

The difference in performance between the two tradingstrategies is rather small.31 Regarding the average optimalweights, the highest proportion for forecast horizons of 1,3 and 6 months and for both trading strategies is allocatedto the single asset based on the EUR/GBP. This is alsothe best-performing single asset, the one with the lowestvolatility, and the only one which performs systematicallybetter when based on the composite forecasts than whenbased on the naive random walk forecasts, that is, for allforecast horizons of 1, 3, and 6 months. These results implya particularly large economic value of applying exchangerate models to the EUR/GBP exchange rate

4 CONCLUSION

Using a comprehensive set of exchange rate models, weanalyze whether investors in the foreign exchange mar-kets, whose goal is to maximize risk-adjusted profits,should engage in active portfolio management rather thanjust follow the naive equally weighted portfolio approachor simply invest in one of the assets composing the portfo-lio. These assets are defined through returns implied by atrading strategy based on exchange rate forecasts. In gen-eral, we find that the benchmark portfolios based on therandom walk, that is, the benchmark portfolios which donot make use of any (nontrivial) exchange rate forecasts,are more often outperformed by optimal portfolios thanbenchmark portfolios based on composite forecasts up toa forecast horizon of 6 months. This indicates indirectlythat the use of exchange rate forecasts adds to improvingthe (risk-adjusted) profitability of the benchmark portfo-lios we consider, that is, of the single assets and the equallyweighted portfolio.

31Clear exceptions to this rule are the linear and the quadratic loss-averseinvestors with a forecast horizon of 1 month, where the mean returnunder trading strategy 2 is by over 4 percentage points larger than themean return under trading strategy 1. Otherwise, the difference in meanreturns across the two strategies is smaller than 0.6 percentage points fora forecast horizon of 1 month and smaller than 0.8 percentage points fora forecast horizon of 3 months. For longer forecast horizons (6 and 12months) trading strategy 1 uniformly outperforms trading strategy 2 (interms of all performance measures), but note that in most of these casesthe mean returns are close to zero or negative.


Our results suggest that the performance of optimalportfolios and of benchmark portfolios decreases with anincreasing forecast horizon. With respect to the optimalweights, the highest proportion in optimal portfolios forforecast horizons of 1, 3, and 6 months is allocated to thesingle asset based on the EUR/GBP. This is the only assetwhich performs systematically better when based on com-posite forecasts than when based on the naive randomwalk forecasts. These findings might suggest a particu-larly large economic value of applying exchange rate mod-els (and forecast combination methods) to the EUR/GBPexchange rate. The overall difference between the “buylow, sell high” strategy and the carry trade strategy israther small.

ACKNOWLEDGEMENTS

The authors would like to thank an anonymous refereefor very helpful comments on an earlier draft of the paper.Ines Fortin and Jaroslava Hlouskova gratefully acknowl-edge financial support from Oesterreichische National-bank (Anniversary Fund, Grant No. 16250).

ORCID

Jesus Crespo Cuaresma http://orcid.org/0000-0003-3244-6560

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SUPPORTING INFORMATION

Additional Supporting Information may be found onlinein the supporting information tab for this article.

How to cite this article: Crespo CuaresmaJ, Fortin I, Hlouskova J. Exchange rate fore-casting and the performance of currencyportfolios. Journal of Forecasting. 2018;1–22.https://doi.org/10.1002/for.2518

APPENDIX : DATA DESCRIPTION ANDSOURCES

All time series have monthly periodicity (January 1980to January 2016) and have been extracted from ThomsonReuters Datastream. The spot (forward) exchange rates aremid exchange rates of the WM/Reuters closing spot (for-ward) rates. The variables used for the Euro area, Japan,UK, and USA are as follows:

• Money supply: M1 aggregate, indexed 1990:1 = 100.Seasonally unadjusted.

• Output: Industrial production index 1990:1 = 100.• Short-term interest rate: 3-month interbank offered

rate.• Long-term interest rate: 10-year government bond yield.

The values of the variables used for the euro area beforethe introduction of the euro are as calculated and providedby Thomson Reuters Datastream.



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