Working Paper Series

Faculty of Finance

No. 15

Currency Portfolios, Returns and Asset Pricing Tests

Philippe Dupuy, Jessica James, Ian W. Marsh

Currency portfolios, returns and asset pricing tests

Philippe Dupuy Jessica James Ian W. MarshGrenoble EM Cass Business School Cass Business School

philippe.dupuy@grenoble-em.com

jess@zetie.net i.marsh@city.ac.uk

Risk premia may justify the return to the FX carry trade but the identity of the risk factorsis still open to debate. We show that one may obtain very different results in terms of riskadjusted returns and the results of asset pricing tests depending upon the design of the carry tradeportfolios. In particular, both investors and academics may want to consider non-equally weightedand/or non-diversified portfolios that account for the dispersion of currencies’ expected returns.We study five portfolio designs on data from 22 currencies covering the period 1984-2013. We alsoprovide simulation-based evidence confirming our conclusions.

JEL Classification numbers: F31, G12

Keywords: Exchange Rate, Carry trade, Portfolio, Risk premium

1

1 Introduction

Buying high yielding currencies and selling low yielding currencies has been profitable on average:

this is the popular foreign exchange carry trade strategy (Figure 1). Like any conventional asset, it

gives access to a yield (the short term interest rate differential of the high yielding currencies versus

the low yielding ones) but exhibits price volatility (changes in the value of the exchange rate). In

theory, rational investors should favor the portfolio of carry trade positions exhibiting the largest

ratio of expected return to risk (e.g., Markowitz, 1952 and 1956). However, casual evidence show

that both investors and academics often prefer to use more heuristic allocation rules as reported

by Benartzi and Thaler (2001). They form portfolios of currencies which are equally weighted

instead of optimized. For instance, in the business side JPMorgan’s IncomeFX and IncomeEM,

Deutsche Bank’s Harvest investable indices, UBS V10 FX carry and HSBC Global FX carry index

are equally weighted indices while only Barclays Intelligent Carry Index or Credit Suisse’s Rolling

Optimized Carry Indices depart from them. In the Academic literature, the vast majority of the

papers also adopt the simple equally weighted version of the carry trade. Either researchers form

equally weighted portfolios of currency pairs like in Brunnermeier et al. (2009) and Jorda and

Taylor (2012) or of dynamically discount-ordered currencies as in Clarida et al. (2009), Darvas

(2009), Ang and Chen (2010), Lustig, Roussanov and Verdelhan (2011), Burnside (2012), Menkhoff

et al. (2012), Dobrynskaya (2014), Lettau et al. (2014), Atanasov and Nitschka (2014) and Dupuy

(2015) among others.1

Often, equally weighted portfolios produce payoffs which exhibit better risk return characteris-

tics than more complex approaches.2 They do so when the rationale for forming equally weighted

portfolios - that is, considering that all the assets have similar expected returns, correlation co-

efficients and variances - produces smaller estimation errors than any other set of expectations.3

Further, if these errors are imperfectly correlated in the cross section, they may be reduced by

mixing a large number of assets (e.g., Blume, 1970; Ross, 1976; Fama and French, 1993). For

instance, Fama and French build portfolios of at least 25 equities. In the literature on foreign

1Baroso and Santa Clara (2013) and Della Corte et al. (2008) use optimized, hence non equally weighted,

portfolios.2The results to Markowitz’s approach are very sensitive to estimation errors in the parameters. These errors

may offset the expected gain from optimal diversification (e.g., DeMiguel et al., 2009; Jacobs, Muller and Weber,

2014). There exist alternative methods to Markowitz approach such as portfolio resampling (e.g., Michaud, 1989) or

robust asset allocation (e.g., Tutuncu and Koenig, 2004) but out-of-sample performance is not superior to traditional

approaches (e.g., Scherer, 2007a and 2007b).3The equally weighted approach has an obvious lack of risk monitoring. Maillard et al. (2010) propose the use of

equally weighted risk contribution portfolios in which, thanks to risk budgeting, no asset contributes more than its

peers to the total risk of the portfolio. This approach still relies on estimates of the variance-covariance matrix.

2

exchange, the size of the portfolios is limited by the small number of currencies in which to invest.

For instance, Burnside (2012) and Menkhoff et al. (2012) build portfolios of five currencies each

while Lustig, Roussanov and Verdelhan (2011) build portfolios of six currencies. This number falls

to three in Christiansen et al. (2011) while Brunnermeier et al. (2009) and Bakshi-Panayotov

(2013) test portfolios with just one to three currencies. In the literature, choosing the number of

currencies per portfolio appears purely empirical since none of these papers provide a rationale for

their portfolio construction. It is the same on the business side where HSBC build portfolios of

three to five currencies while Deutsche Bank limits this number to three currencies without really

justifying their choices.

In this paper we study alternative ways to build portfolios of carry trade positions, and reveal

the impact the choice among these can have on portfolio performance and on the results of standard

asset pricing tests. We demonstrate that higher (lower) mean returns from investing in a currency

are associated with larger (smaller) forward discounts. However, this relationship is very weak for

middle range currencies. We show that these middle range currencies are not usually attractive

potential investments since transactions costs outweigh the expected return (ie the carry): they

are not ex ante profitable. However, once these have been removed, the remaining ex ante attrac-

tive currencies exhibit a near-monotonic relationship between return and forward discount. We

also confirm the results of Brunnermeier et al. (2009), and show that currencies with comparable

discounts tend to vary together.

Taken as a whole, our findings explain why non-equally weighted and/or non-diversified port-

folios tend to outperform the more popular equally weighted strategy. The best risk-adjusted

performance comes from taking a long position in the currency with the largest discount, and

shorting the currency with the smallest discount. Enlarging the long portfolio to include curren-

cies with lower discounts reduces the portfolio return, especially if the added currencies are not

ex ante attractive. Since the gains from diversification are limited, these reductions in return are

not sufficiently offset by risk reductions. If any enlargement is undertaken then portfolio weights

should be slanted towards the currencies with the more extreme discounts to maximise the returns.

This is an important result for the business side as few of the main players in the market offer

non-equally weighted and/or non-diversified portfolios.

From an academic perspective we show that different designs of the test portfolios can yield

significantly different results in asset pricing tests. In particular, we demonstrate that the con-

clusions of some recent studies relating the cross-section of the currencies to currency-based risk

3

factors (e.g., Lustig, Roussanov and Verdelhan, 2011; Menkhoff et al., 2012; Burnside, 2012; Do-

brynskaya, 2014; Dupuy, 2015 among others) might be impacted. Asset pricing tests are usually

carried out on equally weighted portfolios of currencies. For a given number of currencies, there is

freedom to choose the number of portfolios formed (denoted p) with a researcher facing a tradeoff

between having a sufficiently high number of portfolios to perform cross-sectional tests and having

relatively undiversified portfolios. Our results from performing asset price tests show that the way

one selects the number of portfolios p and the way the currencies are combined into the p portfolios

can have direct consequences on the dispersion of the betas, on the precision of the estimates of

risk premia and, finally, on the conclusions reached based on the asset pricing tests. Especially,

while the papers mentioned above tend to reject the concurrent factors to their own, in our sample,

using the correct portfolio construction for each factor, we might accept them all. This observation

seems to indicate that these currency-based factors are all informative about investors’ SDF.

The explanations for our results carry over from the previous section. Currencies that are not

ex ante attractive, due to transactions costs that are high relative to the discount, simply add

noise to the asset pricing relationship and the way they are packaged in portfolios may change the

results of any statistical tests. Together, ex ante profitable currencies exhibit a strongly mono-

tonic relationship between their β to the pricing factor and returns. However, combining them

into portfolios may compress the range of both returns and βs making it harder to identify a

significant relationship. Therefore, one has to be particularly careful when building the portfo-

lios. In this context, researchers may want to perform asset pricing tests on individual ex ante

attractive currencies with ex ante unattractive currencies combined into one diversified portfolio.

This packaging of the primitive assets is economically interesting because it points directly to the

phenomenon of interest. Also, we show, that it may offer new opportunities to accept the risk

premia story when other designs reject it. However, this is not a fail-safe approach and it is im-

portant to keep in mind that tests may have to be carried out also on the other possible p-portfolios.

The rest of paper is organized as follows: in section two we present the data and we review

several ways to design carry trade portfolios. We study the returns to these portfolios and justify

why the portfolios designed with maximum amount of information are, on average, more profitable

than equally weighted portfolios with normalized bets, even accounting for risk and tail risk. We

provide robustness tests and some basic conclusions for currency portfolio implementation for the

business side. In section three, we show that the conclusions of academic research on the carry

trade can be dependent on the way one builds the test assets. Especially, we show that the SDF-

GMM methodology implemented in the seminal work of Lustig, Roussanov and Verdelhan (2011)

4

can produce significantly different results for alternative designs of the carry trade portfolios. Fi-

nally, in section four we discuss results from some Monte Carlo experiments which confirm our

main findings. We conclude in the fifth section.

2 The Returns to the Foreign Exchange Carry Trade

This section presents several alternative ways to build portfolios of currencies to invest in the carry

trade. To some extent, they are all simplification of the portfolio selection process of Markowitz

(1952) and (1956). We review them from the standard optimization algorithm which defines non-

equally weighted portfolios through to the most simple adhoc rule which specifies equally weighted

portfolios. As we go through the different designs of the portfolios we provide justifications for

construction.

2.1 BUILDING CURRENCY PORTFOLIOS

Currency excess returns. We use s to denote the log of the spot exchange rate of the quotation

of the Foreign Currencies (USD/FCU) and f for the log of the forward exchange rate. We compute

the periodic (i.e. weekly, monthly and quarterly) excess return for holding any foreign currency c

as: Rct+1 = f ct −sct+1 with Rct+1 the log periodic excess return of currency c observed in t+1, f ct the

log forward rate of currency c observed at time t and sct+1, the log spot exchange rate observed at

time t+1. At the end of each period t (i.e. week, month and quarter), we rank the currencies from

low to high interest rates currencies on the basis of their forward discount (f-s) observed at that

time.4 The currency with the lowest forward discount receives the rank 1 while the currency with

higher forward discount receives the rank N. The carry trade consists of buying (selling) forward

high (low) yielding foreign currencies at time t and selling (buying) them back at the prevailing

spot exchange rate at t + 1. Portfolios are rebalanced at the end of every period. Accounting for

bid-ask spread transaction costs (with prices denoted b or a as appropriate), investing in the carry

4Sorting on forward discounts or on interest rate is the same as covered interest parity holds closely at daily

frequencies (e.g., Akram et al., 2008)

5

trade is profitable if:5

Rlongt+1 = f bt − sat+1 > 0 if the investor buys the FCU

Rshortt+1 = −fat + sbt+1 > 0 if the investor sells the FCU

(1)

Portfolio design. If agents think 1/sbt+1 and 1/sat+1 are martingales6 then the decision rule

is the following:

Buy the FCU if E(f bt /sat+1) = f bt /s

at > 1

Sell the FCU if E(fat /sbt+1) = fat /s

bt < 1

(2)

This is what Burnside et al. (2007) label the rationale for currency speculation. It precisely

defines the set of currencies which are ex-ante profitable. Bid-ask spreads for major currencies

are small and so one might think that accounting for transactions costs is not important. In our

sample, the average bid ask spread of the forward quotes is around 0.14%. For the G10 currencies,

this number falls to 0.09% with a minimum at 0.01% but for certain emerging currencies, it jumps

to 0.58% notably due to severe episodes of crises. This means that, on average, a currency is only

ex-ante profitable if the annualized interest rates differential with the US is around 1.7% for a one

month investment horizon (1% for G10 currencies but almost 7% for certain emerging ones).

Then, the traders’ problem for investing in the carry trade is given by:

xt =

Max

∑ni=1wi(f

bt /s

at )−

∑i

∑j wiwjVij if f bt /s

at > 1

Min∑n

i=1wi(fat /s

bt) +

∑i

∑j wiwjVij if fat /s

bt < 1

(3)

with n the number of currencies, wi the portfolio weights and Vij , the variance-covariance ma-

5We study the payoff for the strategy of buying the currencies with a forward discount and selling the currencies

with a forward premium. The payoff for this strategy differs by a factor (1 + rUS) from the strategy that borrows

funds in a low-interest-rate currency and lends them in a high-interest-rate currency.6Martingale: Et(1/s

bt+1) = 1/sbt and Et(1/s

at+1) = 1/sat

6

trix.

Solving Equation (3) defines non-equally weighted portfolios of ex-ante profitable currencies.

Of course, these portfolios tend to overweight currencies exhibiting high returns but low risk. The

carry trade strategy consists of investing in the portfolio combining long and short positions as

defined by Equation (3).

However, Markowitz’ approach has well known drawbacks. In particular it is very sensitive

to estimation errors in the parameters. To overcome this problem, investors often rely on simple

assumptions in the parameters and their structure. Below, we list five types of portfolio designs

justified by these simplifications. The mathematical formalization can be found in Appendix A.

1. The no-arbitrage condition implies that the cross-section of assets’ risk should match expected

returns: the higher the expected return (i.e. the forward discount), the higher the standard-

deviation (or the beta to the global currency market portfolio in a single factor model). In

such a case, traders might consider the relative weights of the currencies in the portfolio to

be somewhat in line with their relative expected returns and/or risks. In forming portfolios

in this way investors use all the information conveyed by the dispersion of the signals (f −s).

We call this design Carrysize. This way of building portfolios has received recent support as

Lustig, Roussanov and Verdelhan (2011) show that ranking currencies according to (f − s)

is similar to ranking currencies according to their β to a global currency portfolio.

2. In another representation satisfying the no-arbitrage condition, traders might consider that

the currencies have similar expected returns, variances and correlations. As a consequence,

they may choose to form equally weighted portfolios containing the set of all ex-ante profitable

currencies because nothing helps to discriminate between them. We call this design Carryba

because the time-varying transaction costs - the bid-ask spreads - define the set of investable

currencies for every period.

3. Investors might consider that the correlations across the currencies are equal to one and the

variances are similar but not the expected returns which is a violation of the no-arbitrage

condition. In this case, there is no particular benefit to diversification and the investor might

choose to invest in an equally weighted portfolio of the two extreme currencies: the highest

yielding currency and the lowest yielding currency. We call this strategy Carrymax.

4. If investors think transaction costs are insignificant and assets have similar moments, they

further simplify their representation. They do not discriminate between ex-ante profitable

7

and non-profitable currencies and they might invest in equally weighted portfolios containing

all the currencies. We call this strategy Carryall.

5. Casual evidence shows that limiting the investment to the extreme currencies produces higher

return for lower risk, even when transaction costs near zero. This is why most banks offering

investable indexes (such as the Deutsche Bank’s Harvest) limit the set of investment to the

n extreme currencies. We call this design Carryn.

The return to currency speculation. For all these designs, the returns to the portfolio of

long positions and the portfolio of short positions are:

Rlongt+1 = 1

(nl+ns)

∑nli=1wi ∗ rit+1

Rshortt+1 = 1(nl+ns)

∑nsj=1wj ∗ r

jt+1

(4)

with nl the number of currencies in which the investor has a long position and ns the number of

currencies in which she has a short position. Finally, with wi > 0 and wj < 0, the return of the

carry trade strategy for the 5 types of portfolio designs is the sum Rlongt and Rshortt . Of course,

for Carrymax and Carryn, nl = ns. But for Carryall, Carryba and Carrysize, the number of

currencies and/or the sum of the weights might differ between the two portfolios. This is not a

problem per se since each position is already a fully financed long-short position. The results are

presented for a total position standardized to 1 dollar for each period (sum of the long and the

absolute of the short positions7).

In the next section, we examine the risk-return statistics of the five designs of the portfolio of carry

trade. For Carryn, we choose to report the results for n = 3. The risk-adjusted ratios for n = 4

and n = 5 are very similar but then they deteriorate quickly when n becomes larger than 5.

2.2 DATA ANALYSIS

We use data collected by Barclays and Reuters and available on Datastream. We also complement

our dataset with data from the Bloomberg database when necessary. They cover the period from

December 1984 to May 2013. To avoid the possible impact of small, illiquid currencies which might

suffer from measurement error (e.g. Cochrane, 2005), we limit our dataset to the ones used by

Deutsche Bank, the largest player in the foreign exchange market, for its global carry trade strategy

7Alternatively, we could have presented the results for a strategy investing one dollar in the portfolio of long

positions and minus one dollar in the portfolio of short positions.

8

named Global Currency Harvest. Our dataset contains 22 different currencies: Australia, Brazil,

Canada, Czech Republic, Denmark, Euro Area, Germany, Hong Kong, Hungary, Japan, Mexico,

New Zealand, Norway, Russia, Singapore, South Africa, South Korea, Sweden, Switzerland, Thai-

land, Turkey and the UK8. The Euro series starts in January 1999 replacing the German DEM,

hence the maximum number of currencies in the largest dataset is 21. While forward rates may be

available for a larger basket of currencies, there would have been virtually no liquidity in many of

them (2013 BIS Triennial Survey). All currencies are quoted as the number of Foreign Currency

Unit (FCU) per dollar. We start from daily data including the spot exchange rates, the one-week,

one-month and three-month forward exchange rates with bid and ask rates for each observation.

We convert the daily data into weekly, monthly and quarterly data by sampling the daily data on

every Friday for the weekly series and every last open day of each month and quarter for monthly

and quarterly series.9

In Table I, we report the summary statistics for the five strategies defined above. For every

strategy, we report the annualized mean, standard-deviation, skewness and kurtosis of the distri-

bution of returns, the annualized Sharpe Ratio (SR) and modified Sharpe Ratio (MV aR), the ratio

of the mean return to the mean drawdown (MDD) and finally the ratio of the mean return to

the maximum drawdown (MMaxDD). The definition of these ratios can be found in Appendix B.

We also report the non annualized version of these ratios and their standard errors obtained by

bootstrapping the statistics.10

The carry trade strategy works: all the strategies produce a large mean return. They also

exhibit significant risk and crash risk as measured by the skewness of the distribution of returns.

Turning to the risk-return measures, we see that all the strategies but Carryall exhibit significant

Sharpe Ratios. The largest ones are obtained with the Carrysize and, in particular, Carrymax

versions of the carry trade. The other performance ratios are also highest for these two strategies.

8We share 16 currencies with DB GCH which does not include Russia, Thailand and the European countries

which have adopted the EUR.9Data for 1-week maturity contracts were not available on a daily basis for certain currencies in the early 2000. As

a consequence, the sample of weekly data covers only the period from October 2002 to May 2013. Also as mentioned

in Darvas (2009), there are errors in the data such as infrequent revision of the forward prices leading to jumps in

the series. These jumps have been removed. They mainly concern the USD/BRL and USD/TRL.10We estimate the distribution of the statistics by generating 10000 block bootstrap samples of the carry trade

returns. We presents the results with non-annualized figures as to follow Lo (2002) who shows that annualization

is correct only under very special circumstances. Our aim is to test whether the statistics are significantly different

from zero. This is different to Villanueva (2007) and Darvas (2009) who test whether the returns to the carry trade

and UIP conforming returns are significantly different from one another. Hence, for our test, we do not impose UIP

to hold in the bootstrap Data Generating Process.

9

Behind these, Carryn and Carryba generate performance ratios which are broadly similar. Al-

though Carryba, which invests in a time-varying set of currencies, provides somewhat better ratios

than Carryn, some of its performance measures are not statistically significant. Investors should

stay away from the Carryall version of the strategy because it underperforms all alternatives, and

often provides insignificant performance.

We conclude that investors should favor Carrymax and Carrysize. This may be surprising

since it calls for the construction of non-equally weighted portfolios (Carrysize) or non diversified

portfolios (Carrymax) which are rarely seen in the business side and only rarely used in academic

work. However our findings echo the point developed in Ang et al. (2010): creating equally

weighted portfolios seems to destroy the useful information conveyed by the signal.

Table I about here

2.3 DISCUSSION

In Table II, we report the summary statistics for the returns to the carry trade observed at time

t+1 for currencies sorted on their one-month forward discount (f−s) observed in t. At each point

in time, the currency with the smallest interest rate is located in C1 and the currency with the

highest interest rate is in C2111. For every currency C1 to C21, we report the annualized mean,

standard-deviation, skewness and kurtosis of the distribution of returns, the annualized Sharpe Ra-

tio (SR) and modified Sharpe Ratio (MV aR), the ratio of the mean return to the mean drawdown

(MDD) and finally the ratio of the mean return to the maximum drawdown (MMaxDD)12. Beside

these statistics, we plot in Figure 2, the mean return of the currencies sorted on their forward

discount. In this graph, the black points are for the unconditional ex-ante profitable currencies.

We define the unconditional ex-ante profitable currencies as the currencies which are statistically

not significantly different from being always ex-ante profitable. Those currencies are the ones

which have turned to be ex-ante profitable in, at least, 95% of the dataset, i.e. currencies for

which f bt /sat > 1 or fat /s

bt < 1 have been verified in, at least, 95% of the dataset. Empty points

are for ex-ante non profitable currencies. We report the frequency, at which each ranked currency

11In our sample, many currencies are alternatively funding or investment currencies but the CHF, the JPY and

the SGD are never investment currencies while the AUD, the BRL, the MXN, the NZD, the TRY and the ZAR are

never funding currencies.12The non annualized version of these ratios and their standard errors obtained by bootstrapping the statistics are

available upon request

10

is ex-ante profitable, in our sample, in the last column of Table II. Sorting currencies according

to their forward discount reveals several interesting features of the cross section of the currencies

which help to justify the above conclusions:

1. Higher mean returns tend to be associated with higher forward discounts and lower mean

returns with lower forward discounts. This is why all the versions of the carry trade strategy,

which all consist of buying high yielding currencies while selling low yielding currencies, offer

positive average returns.

2. However, this association suffers from obvious exceptions: the relationship is not monotonic.

The association is very poor for middle range yielding currencies. The currencies which

significantly break the monotonicity of the relationship somewhat impair the statistics of the

carry trade strategy: they might not be desirable from the investor’s point of view. As a

result, the Carryall version of the strategy which mixes all the currencies of the sample into

one equally weighted portfolio generates significantly lower risk-return ratios.

3. On the contrary, the black points in Figure 2, i.e. the average returns of ex-ante profitable

currencies, increase almost monotonically from the funding currencies C1 to the investment

currencies. Limiting the investment set for these currencies enables one to steer clear of those

significantly breaking the monotonicity of the relationship, especially the middle range ones.

Consequently, Carryba exhibits better risk-return ratios than Carryall. Similarly, Carryn

produces better risk-return ratios than Carryall because in this version of the carry trade the

set of investable currencies is de facto limited to the extreme ones. However, as the number of

bets is pre-fixed, at any time, some non-desirable currencies might remain in the investment

set. For instance, if the investor set n = 3, he (she) would sell currency C3 which exhibits

the third smaller forward discount but a positive average return. This investor would also

miss buying the currency C1813. As a result, Carryba exhibits somewhat better statistics

than Carryn.

4. When only the ex-ante profitable currencies are examined, the relationship between the

forward discounts and the average mean returns is more monotonic: the larger the forward

discount the larger the mean return. As a result, weighting the currencies in the portfolio

according to their relative forward discount, as in Carrysize, further improves the risk-return

ratios of the strategy. On the other hand, equally weighting the currencies, as in Carryba

13The number of investable currencies varies according to their observed transaction costs. It fluctuates from a

minimum of 2 currencies to a maximum of 13 currencies for the high yielding ones and from 1 currency to 8 currencies

for the funding ones.

11

and Carryn is like minimizing the information we hold about the dispersion of the future

returns14.

To get further intuition on this point, we report that the R2 of the regression of the mean

returns of the currencies on their mean forward discount improves significantly once we

exclude non ex-ante profitable currencies. For instance, this R2 is 45% in the full sample.

When we exclude from the calculation and for each currency, the observations for which the

latter was not ex-ante profitable, the R2 jumps to 54%. Then, if we exclude the currencies

which are ex-ante profitable in only a limited period of time, the R2 improves further. It

equals 68% if the currencies which are ex-ante profitable in only 40% of the sample are

excluded and it equals 86% if we raise the threshold to 95%. The slope of the relationship

remains almost unchanged at around 0.60 in the different samples.

5. The highest yielding currency, C21, exhibits the best set of risk-return ratios which tends

to indicate that building portfolios does not diversify away the risk, at least not sufficiently

to compensate for the change in the mean returns. This is the case because the currencies

with a comparable forward discount tend to vary together (see Table III)15. Brunnermeier

et al. (2009) find similar results especially concerning the crash risk as measured by the

skewness of the distribution of returns. We have also known for a long time that correlations

tend to be particularly high in periods of poor performance, precisely when we would like

the risk to be diversified away (e.g. Erb et al., 1994; Longin and Solnik, 2001). Instead of

diversifying the portfolio, this observation calls for concentrating it in the extreme currencies,

as in Carrymax.

Table II and III about here

In the next section we perform several robustness tests to confirm our findings. We look espe-

cially at the sensitivity of the results in relation to the size of the sample in the cross-section and

the time series. Beyond, we provide Monte Carlo simulations in section 4.

14Weighting the currencies according to their rank as in Asness et al. (2013) is another sort of minimization of

the information we hold. But, even though the investor uses the forward discount order as a weighting scheme, she

omits to use the true dispersion of the signal.15As a result minimum variance portfolios do not improve the results.

12

2.4 ROBUSTNESS TESTS

As a first robustness test, in Table IV, we reduce the sample to G10 countries only. Results are

largely similar: in particular, we note that the strategy Carrysize still offers a better set of risk-

return statistics than the conventional Carryn strategy. This is because reducing the sample to G10

countries does not significantly alter the main features of the cross-section of the currencies. For

instance, in Figure 3, we plot the mean return of the currencies sorted on their forward discount for

various cuts of the data. Again, the black points are for the currencies which are ex-ante profitable

(95% threshold as above) and the empty points for ex-ante non-profitable currencies. For the

period from December 1984 to May 2013 for G10 countries (top left graph), we find the same

linear pattern between the forward discount and mean returns, especially for the ex ante profitable

currencies, which again explains why Carrysize tends to outperform the alternative strategies.

However, in this sample, Carryall also exhibits good statistics. This is because in this reduced

cross section sample the returns of the middle range currencies are also ordered somewhat in line

with their forward discount. The monotonicity of the relationship is, however, less strong in the

post 1999 sample (top right graph).

Table IV about here

To check whether only a limited number of currencies are behind our results, in Table V, we

randomly split our large sample of developed and emerging countries into two sub-samples. To do

so, we sorted countries alphabetically and consider two groups: the first contains the ten countries

from Australia to Korea and the second contains the eleven countries from Mexico to the UK. The

return-discount plots are given in the bottom row of figure 3. We build carry trade portfolios as

defined in section 2.1 for both groups and find results in line with the preceding ones: Carrymax

and Carrysize offer better statistics although Carrysize exhibit lower figures in the sample using

the ten countries from Australia to Korea.

Table V about here

We now turn to the conditional return of the carry trade strategy. For every strategy, we

calculate time-varying risk-return ratios over rolling windows of 36 months from 1999 to 2013

in the large sample of 21 countries. In Figure 4, we report rolling Sharpe Ratios for strategies

13

Carryn, Carrymax and Carrysize. The graphs show that Carrymax and Carrysize perform better

than Carryn. The results hold whether we look at a quiet period (mid 2000 for instance) or

turbulent periods (2008-2009).

Figure 4 about here

Finally, in Table VI, we test whether the results are altered by a change in the frequency of

observation of the data. To do so, we run all tests using quarterly and weekly data. Quarterly data

are observations of spot and 3-month forward contracts at the end of each quarter while weekly

data are observations of spot and 1-week forward contracts on every Friday. Again, the results

show that considering a pre-fixed number of currencies, Carryn, yields poor results. This is inde-

pendent of the frequency of observation of the data. Considering a time-varying set of investable

currencies significantly improves the statistics. This is particularly visible in the weekly dataset:

Carryn and Carryall exhibit negative returns and risk-return ratios while Carryba and Carrysize

generate significantly positive ratios.

Table VI about here

This result might be surprising since weekly, monthly and quarterly statistics are extracted from

similar samples of daily observations. However, returning to the rationale for currency speculation

helps to justify these results. The rationale, notably, sets the binding minimal horizon of investment

in the currency. Using the definition of any forward exchange rate, we can rewrite the necessary

but insufficient condition for currency speculation to be profitable as follow:

(ra

f

t −rbUS

t )n/b

(1+rbUS

t n/b)>

Sat

Sbt− 1 if the investor buys the FCU

(rbf

t −raUS

t )n/b

(1+raUS

t n/b)<

SbtSat− 1 if the investor sells the FCU

(5)

raf

t is the foreign short-term interest rate at time t, rbUS

t is the US short-term interest rate at time

t, n is the maturity of the forward contract and b is the number of reference period in one year. For

instance, for a 3-month forward contract, n = 3 and b = 12. Of course, as the costs of transaction

are very similar whether one buys a 1-week contract or a 3-month contract, the maturity, n, of the

14

contract stands as the only key parameter defining the ex-ante profitability of the currency.16 For

instance, the average bid-ask spread is around 0.14% in our sample, including episodes of crises in

emerging markets. This means that, on average, a currency is ex-ante profitable if the annualized

differential of interest rates with the US is around 7.3% for weekly contracts, 1.7% for monthly

contracts and 0.6% for 3-month contracts.17

Finally, rearranging these two inequalities we find:

if the investor buys the FCUn

b>

Sat

Sbt

−1

(raf

t −rbUS

t )−(Sat

Sbt

−1)rbUSt

if the investor sells the FCUn

b<

Sbt

Sat−1

(rbf

t −raUS

t )−( Sbt

Sat−1)raUS

t

(6)

These two equations set the binding horizon for investment in one currency to be ex-ante

profitable. They show that any rise in transaction costs commands a revision of the investment

horizon, all else being equal. The non revision of the investment horizon turns some bets into

ex-ante non profitable ones. The same is true for a decrease in the interest rate differential. This

is an important result because it shows that rebalancing the portfolio of carry trade positions too

often, as in a strategy based on weekly contracts, might cause an increase in costs not covered by

the proceeds of the strategy.

3 Risk Factors in the Currency Market.

We turn now to the sensitivity of conclusions of academic research into the carry trade to the

design of portfolios. The good performance of the carry trade (Figure 1) is puzzling to academics

because, according to the uncovered interest rate parity (UIP), arbitrage should eliminate the

gains arising from the interest rate differential between currencies. Lustig, Roussanov and Verdel-

han (2011) offer a partial resolution to the puzzle by testing the relevance of non-traditional risk

factors derived directly from currency returns. Notably, they test a factor called HML-FX which

is the return to a portfolio combining long positions in high yielding currencies and short positions

in low yielding ones (our Carryn). Similarly, Menkhoff et al. (2012) and Burnside (2012) test a

16This is exactly the case if the differentials of interest rate between the two currencies are similar over the possible

horizon, i.e. weekly, monthly and quarterly.17Of course, these numbers are only rough estimates since the bid ask spreads are time-varying and heterogenous

across currencies.

15

global volatility factor extracted from the currency market while Della Corte et al. (2013) and

Della Corte et al. (2015) find respectively a relationship with global imbalances and sovereign risk.

Together, Mancini et al. (2013) explore a link with liquidity, Dupuy (2015) a link with VaR-based

constraints and Dobrynskaya (2014) with crash risk18.

The papers in this literature typically follow the same methodology: to reduce errors in parame-

ter estimation, they form equally weighted portfolios of currencies sorted on their forward discount.

Then, they look for a significant spread across these portfolios in the covariance, β, between their

return and the SDF. Looking at the conclusions of these papers, we see that the concurrent SDF

are numerous but that none of them enjoy a clear consensus. For example, Burnside (2012) rejects

the global carry trade factor of Lustig et al. (2011) and the volatility factor of Menkhoff et al.

(2012), which in turn rejects a skewness indicator proposed in Burnside (2012). Similarly, Dupuy

(2015) rejects the skewness factor and finds weak results for the volatility factor. In this section,

we show that the way one designs the test portfolios can produce significantly different results, and

in particular the rejection of models that might be accepted otherwise. Notably, we show that for

any given factor and sample, many portfolio designs do not reduce the statistical noise by enough

to allow researchers to accept the risk premia story. Together, we show that relying, again, on the

ex-ante profitable currencies to define the test portfolios enables one to focus on a set of assets

which is particularly interesting from an economic standpoint.

3.1 THE RATIONALE OF TEST ASSET PORTFOLIO CONSTRUCTION.

Following Fama and MacBeth (1973) among others, many researchers package assets into portfolios

to shrink the dispersion of their returns by offsetting their idiosyncratic components. However,

grouping assets is like considering that there is no interesting information outside the portfolio (i.e.

Cochrane, 2005, p224). Indeed, grouping may also shrink the total dispersion of their βs on the

risk factor, leading to poorer estimates of the cross-sectional risk premia and potentially to the

rejection of the model for non significant parameters (e.g., Ang et al., 2010): different designs of

18An alternative solution to the puzzle is the peso story: risk averse agents assign small but non-zero probabilities

to rare events with larger negative payoff than can be observed in sample. This rare event solution has also received

renewed attention in the literature (see Barro and Ursua, 2011 and Gourio, 2008). For instance, Jurek (2010), Farhi

et al. (2009) and Burnside et al. (2010) use hedged versions of the carry trade to test the possibility that rare

events outside the sample may explain returns. The results seem to indicate that losses associated with rare events

are relatively small supporting the alternative view that the salient feature of a peso state is a large value of the

SDF. Cen and Marsh (2014) address the related issue of the in-sample underrepresentation of extreme events by

extending the sample analysed to include data from 1921-1936, a period characterised by a very large number of

extreme returns.

16

the test assets might produce significantly different conclusion to the tests. This result has been

also reported by Kandel and Stambaugh (1995) for equities.

Which portfolio design should a researcher choose? Absent any rationale for currency portfo-

lio construction, researchers have the freedom to choose from many alternatives, including those

discussed earlier in this paper. And, as already mentioned, there is a tendency to favor equally

weighted portfolios. However, this leaves two dimensions, the number of portfolios p to form and

the number of currencies they receive each, especially when n/p is not an integer. Even this lim-

ited freedom encourages debate between researchers. For instance, in their seminal paper, Lustig,

Roussanov and Verdelhan (2011) accept their risk premia story on the back of tests based on six

portfolios while Burnside (2012) disputes their findings on the back of results obtained from tests

based on five portfolios. Similarly, Menkhoff et al. (2012) reject the skewness factor proposed in

Burnside (2012) and Rafferty (2012) but they provide only the results obtained from a 5-portfolio

test19.

Yet, there exists∑n

p=3Cn−1p−1 =

∑np=3

(n−1)!(p−1)!(n−p)! ways to combine n ranked elements in p sets

for p ≥ 3. This makes 1,048,555 possible combinations in our sample, each of them being an

opportunity to accept or reject the risk premia story. In this section, we show that the conclusions

to asset pricing tests can be sensitive to the choice of p and to the way one combines the assets in

the p portfolios, especially when n/p is small and not an integer. As a consequence, we recommend

that the rejection of a factor should be justified by results obtained from a large set of possible

p-portfolio tests. Of course, most of these repackagings are not economically interesting and we

further argue that the rationale for currency speculation, as described in Equation (2), offers a

promising and economically interesting way to handle the cross section of currencies which should

be tested above all.

From section 2, we know that Equation (2) exactly defines the set of ex-ante profitable curren-

cies in which rational carry traders may invest. Hence, in theory, these currencies should exhibit

the largest covariances, in absolute terms, with the candidate SDF. On the other hand, in the-

ory, changes in ex-ante non profitable currencies, in which rational carry traders may not invest,

should be orthogonal to the global carry trade risk factor. As a consequence, they should be the

ones conveying the largest errors. We propose to filter out this noise by mixing these ex-ante

19To the best of our knowledge, none of these papers explain precisely the justification for the number of portfolios

p and their construction, especially when n/p is not an integer. At best, Lustig and Verdelhan (2007) justifies

eight portfolios, and not fewer, to isolate high inflation countries (equivalent to currencies with the largest forward

discounts) in one portfolio located in the top right of their universe.

17

non profitable currencies in one equally weighted portfolio. Together, we rely on all the ex-ante

profitable currencies as specific test assets to take full profit of their dispersion to estimate the

parameters. The logic is as follows. First, we group in one portfolio all the currencies which, in

theory, are orthogonal to the candidate SDF to minimize the pricing errors. Second, we rely on the

full dispersion of the currencies which are, in theory, correlated to the candidate SDF to improve

the estimation of the risk premia. These assets are the ones which are economically interesting.

In our sample, this is equivalent to setting-up a test based on 6 portfolios receiving, for five of

them, one of the five ex-ante profitable currencies (see Figure 2) and for one of them the remaining

ex-ante non profitable currencies.

For our data, we confirm that different designs of the test assets produce significantly different

conclusions. Indeed, conditional on the value of p and on the distribution of the currencies in the

portfolios, we can alternatively accept or reject the risk premia story of Lustig, Roussanov and

Verdelhan (2011). Especially, we may reject it on the back of their 6-portfolio test but we may

well accept it using precisely the number of portfolios (p=5) which leads Burnside (2012) to reject

it. Together, our non-diversified 6-portfolio test is favorable to the story.

Clearly, these observations weaken the conclusions, especially the unfavorable ones, drawn from

asset pricing tests using one and only one portfolio design. This comment applies, for instance,

to the rejection of a factor mimicking skewness in Menkhoff et al. (2012), to the weak results

obtained by Burnside (2012) and Dupuy (2015) for a factor mimicking currency volatility and to

the rejection of the model of Lustig, Roussanov and Verdelhan (2011) by Burnside (2012).

Also, we report that, in our sample, we find, for every factor, at least, one conventional portfo-

lio construction favorable to the risk premia story. This observation seems to indicate that these

currency-based factors are all informative about investors’ SDF. However, we stress that it is not

our intention to validate or dismiss risk factors. Rather, the rest of this section seeks to illustrate

our main point: that conclusions from cross-sectional factor tests in the currency market can be

sensitive to the way the test portfolios are designed. We present evidence using the seminal work

of Lustig, Roussanov and Verdelhan (2011). First, however, we introduce the Stochastic Discount

Factor (SDF) procedure as presented by Cochrane (2005).

3.2 ASSET PRICING TESTS

One way to test whether there is a Stochastic Discount Factor that prices the returns to the carry

trade is to test if the returns to the currencies, sorted on their forward discount, covary with some

18

risk factors in the times series and then in the cross-section. Positive answers to these questions

support a risk based explanation of the returns to the carry trade. This is the common two-step

procedure inspired by Fama and Mc Beth (1973).

Therefore, in this paper, first we look whether a linear combination of factors can significantly

justify the returns to the carry trades, in the time series, for each currency or portfolio of currencies

i:

Rit+1 = αi + f ′t+1βi + εit+1 (7)

Then, we test whether the betas of Equation (7) combined with estimates of risk premia (λ)

might justify the returns to the carry trade in the cross section. To do so, in the traditional Fama

and McBeth (1973) procedure, one runs a cross-sectional regression of average excess returns on

betas. Instead, following Cochrane (2005), Burnside (2012), Lustig, Roussanov and Verdelhan

(2011) and Menkhoff and al. (2012) among others, we co-estimate the vector of SDF parameters

and their moments using the Generalized Method of Moments of Hansen (1982).

We use the iterated GMM estimator20 starting from the identity matrix as weighting matrix

WT = I. In this case, GMM treats all assets symmetrically. However, when we package them, we

impose a structure on the primitive assets which forces the GMM to pay less attention to some of

them, especially when n/p is not an integer. In our case, we choose to downweight the assets which,

in theory, should be orthogonal to the SDF (i.e. the non ex-ante profitable currencies mixed in a

single portfolio). We could deemphasized further the assets with the largest variance by starting,

for instance, from the optimal matrix of Hansen (1982) instead of the identity matrix. However,

as mentioned by cochrane (2005), with the iterated version of the GMM, the estimates should not

much depend on the initial weighting matrix21. The detail of computation of GMM, especially the

moment conditions and the J-test, can be found in Appendix C.

In this paper, we run and compare the results from several tests: i) the p-portfolio tests with

p ranging from three to seven, ii) the test using the entire universe of currencies, and iii) the test

using the five ex-ante profitable currencies as defined by Equation (2) plus the portfolio of ex-ante

20We follow Burnside (2012) because the iterated estimator has much greater power to reject mispecified models.

However, using the iterated GMM or the two-step GMM does not change the main conclusions to this paper.21This point can be extended to the second-moment matrix of Hansen and Jagannathan (1997). However, as

reported by Cochrane (2005), the second-moment matrix is often nearly singular providing an unreliable weighting

matrix when inverted. This is precisely what we find in our sample and probably the reason why none of the papers

studying the carry trade in an APT framework use it, with the noticeable exception of Menkhoff et al. (2012).

Furthermore, the second-moment matrix gives an objective which is invariant to the initial choice of portfolios only

if the packaging does not throw away information.

19

non profitable ones. We test four risk factors similar to those proposed in Lustig, Roussanov and

Verdelhan (2011) and Menkhoff et al. (2012): RX, a dollar risk factor which is the average excess

return of the p portfolios of Carryn; HML − FX, the difference between the returns of the two

extreme portfolios defined in each strategy; V OLEQTY , a proxy for the volatility of global equity

market returns; and V OLFX , a proxy for the volatility of global currency market returns. We

report the results for the largest cross section of currencies we have in hand, i.e. 21 currencies

observed from 1999 to 2013, hence the largest number of possible combinations of currencies in the

portfolios. The results based on a smaller cross section, i.e. G10 currencies from 1984 to 2013, do

not change the main conclusions of this paper.

3.3 EMPIRICAL RESULTS

In Table VII, we report the results for the first step of the asset pricing tests, i.e. the time-series

regression of the portfolios excess returns on the two risk factors RX and HML − FX. As we

find that alternative p-portfolios generate similar conclusions, we only report results for a sample

of extreme and central ones (i.e. 3, 5, 7-portfolios), and the set of ex-ante profitable currencies22.

The formation of the p-portfolios is in line with the literature as we purposely attempt to avoid

non economically interesting packaging by first forming p portfolios containing each the integer

portion of n/p currencies and second by alternatively allocating the decimal part in each portfolio.

It is important to note that, if the portfolios based on alternative allocations of the decimal part,

produce different results, they illustrate our point well. Especially, the portfolios exhibit all an

increasing pattern in return from P1 to Pi which makes them economically interesting. For our

example, the allocation we report has been chosen at random23.

As in Lustig, Roussanov and Verdelhan (2011), in the time series, the betas of the RX factor

are all close to one in value and statistically significant for all designs of the test portfolios. Also,

the betas of the HML − FX factor are significant, with the expected signs and the difference

between the betas of the extreme portfolios adds up to almost one. Turning to the precision of

the estimations, we see that the standard-deviation of the parameters remain largely unchanged

whether we mix currencies in portfolios or not. This is due to the fact that in the currency market,

22In contrast with the usual practice, most of the test portfolios built upon the rationale for currency portfolio

construction are non diversified. They are, also, not equally weighted. Hence, we take account of the differences in

leverage by limiting the amount invested in each of them to 1 USD.23For instance, for a 5-portfolio test, one can build 4 equally weighted portfolios containing 4 currencies each and

one portfolio containing 5 currencies. Whether this portfolio is P1, P2...,P5 is chosen randomly. As 21 is a multiple

of 3 and 7, P3 and P7 are unique in this case.

20

as mentioned in section 1, forming portfolios does not diversify away the risk because adjacent

currencies covary to a high degree.

Table VII about here

Now, in Table VIII we summarize the results of estimating candidate SDF for currency factor

models. The results illustrate perfectly the trade-off that researchers face: grouping more curren-

cies into fewer portfolios mechanically improves the ability of the estimator to shrink the pricing

errors, but this also reduces the significance of the risk premia λ.

Looking at the results in Table VIII, we see that the 3-portfolio test does not produce significant

values for the parameter b or for the risk premia λ for HML−FX, but neither does it produce too

large pricing errors according to the J-test. Conversely, the 7-portfolio test produces significant

parameters, at least at the 10% threshold, but it is rejected for over-large pricing errors. Moving

from the 3-portfolio test to the 7-portfolio test, we see an improvement in the estimation of λ, as

measured by the ratio of the point estimate to the standard deviation, while the tests of the pricing

errors deteriorates.24 Enlarging the set of portfolios to a maximum of 10 portfolios or running the

tests on individual currencies confirms these results (not reported). In our example, among the set

of conventional p-portfolio tests, only the 5-portfolio one is not rejected. This illustrates our point

that if p is set randomly there is a large probability of rejecting the model. Of course, this casts

also doubts on the conclusions drawn on the back of a unique p-portfolio test. We return to this

point in our simulations in Section 4.

Table VIII about here

Turning to tests based on the ex-ante profitable currencies, we see in Table VIII that the model

is not rejected and that the parameters are more precisely estimated.25 We find a point estimate

24In line with these results, the 6-portfolio test of Lustig, Roussanov and Verdelhan (2011) is rejected for too large

pricing errors and the 4-portfolio test is rejected for non significant premia. Of course, one cannot compare the value

of the J statistics as the final weighting matrices differ across the tests.25For this test, we work with the unconditional ex-ante profitable currencies, the currencies which have turned to

21

of 1.44 for the parameter λ with a standard-deviation of 0.51 using the set of ex-ante profitable

currencies compared with 0.69 and 0.28 respectively from the 5-portfolio test. This improvement

is perhaps expected since we have been able to add one asset to the cross-sectional sample.26 Also

as expected, the statistics relating to the errors deteriorate somewhat, especially the R2. However,

if we consider the sampling errors as emphasized by Lewellen et al. (2010) any deterioration is

insignificant.

Looking at alternative risk factors, we reach the same conclusions: different designs of the port-

folios produce significant different results. Concerning the packaging based on ex-ante profitable

currencies, we have to mention that while it might offer, in certain cases, no reason to reject the

model, for some others, the reduction of the statistical noise might not be enough. For instance,

in Table IX we test the global equity market volatility factor of Lustig et al. (2011) (the construc-

tion of the equity volatility factor is detailed in Appendix D). Using any of the randomly chosen

p-portfolio tests we would reject this model (we report results for p=5 in the upper panel of Table

IX). However, using the set of ex-ante profitable currencies as test assets we see that the model

produces significant parameters and is not rejected on the basis of the J − test or the R2.

Conversely, in the lower panel of Table IX, we report the results of estimating a model using the

Menkhoff et al. (2012) currency market volatility factor. In this case, the test based on the ex-ante

currencies rejects the model for insignificant parameters. Even if the rationale for currency basket

construction is a promising way to study the cross-section of currencies, one should not make con-

clusions without also studying the full set of possible p-portfolios.

Table IX about here

We again stress that our intention in this paper is to point out the importance of the con-

struction of the test assets but not to validate or dismiss the factors proposed in the literature.

Rather, we show that the conclusions reached based on the usual test results are sensitive to the

be ex-ante profitable according to Equation (2) in, at least, 95% of the dataset. The results are similar if we use

dynamically defined portfolios: for each time t we build three portfolios, P1 which receives the conditional ex-ante

profitable short positions, P2 which receives the conditional ex-ante non profitable currencies and P3 which receives

the conditional ex-ante profitable long positions.26In the special case of HML− FX, the precision of the estimates can only come marginally from changes in the

spread of the βs since the difference between the two extreme βs is always approximately one, irrespective of the

way one designs the test assets (see Table VII).

22

way one builds the test assets, suggesting that any evaluation of a candidate risk factor should be

fully justified by a range of approaches, including the use of the set of ex-ante profitable currencies.

Especially, in our data, we find, at least, one conventional, economically interesting construction

that is favourable to the risk premia story for each factor we tested27. However, we did not find

a single economically interesting construction favourable jointly to all the factors. Also, when a

construction enables one to accept several factors, the test statistics tend to favor one in particular.

As a result, the conclusion drawn from the traditional horse races between the factors might be

largely conditioned by the initial test assets construction even when the factors are jointly signifi-

cant. We think this explains why researchers can obtain contradictory results when testing similar

risk factors using one and only one portfolio design.

3.4 DISCUSSION

Ideally, for the cross-sectional estimations, researchers would like the test assets: i) to be as nu-

merous as possible ii) with βs widely spread from their mean on the X-axis and iii) lying on the

factor line (e.g., Burnside, 2010; Kan and Zhang, 1999). In that case, the market risk premia

would be precisely estimated and the pricing errors would be minimized. In Figure 5, we sort

currencies on their β to the pricing factor HML − FX. The solid points are associated with the

currencies labeled C1 and C18 to C21 which are ex-ante profitable according to Equation (2) in,

at least, 95% of the dataset. Empty points are associated with ex-ante non profitable currencies.

These currencies can be combined into a single portfolio labeled P1. Tests based on ex-ante prof-

itable currencies are run on the set of solid points. Figure 5 reveals a key feature of the cross

section of currencies which helps to explain some of our previous observations. Ranking the full

set of currencies by their betas to the risk factor gives non-monotonic mean returns. Conversely,

mean returns increase almost monotonously from C1 to C21 (the main exception being C19) if

we consider only the ex-ante profitable currencies and the portfolio P1 of ex-ante non profitable

currencies. Furthermore, the average distance of the βs from the mean is larger if we consider

only the ex-ante profitable currencies rather than the entire universe. Considering the ex-ante non

profitable currencies (the empty points) makes clear the noise they add to the relationship. Their

27We find that all the factors tested are significant for several portfolio formation approaches, indicating that

these currency-based indicators are all informative about investors’ SDF. The HML − FX factor of Lustig et al.

(2011) shows that currency excess returns can be understood as a compensation for time-varying risk. Menkhoff

et al. (2012) volatility factor measures this time-varying risk but one has to take account of the skewness of the

distribution as proposed by Rafferty (2011) to price the cross-section during extreme events. Volatility and Skewness

are combined in the encompassing tail risk factor proposed by Dupuy (2015). Dobrynskaya (2014) downside factor

can be understood as a higher tail risk factor applied to equity returns.

23

distribution appears essentially orthogonal to the factor line: they exhibit significant variation in

mean returns but have largely similar βs which, in most cases, are not significantly different from

zero. This is what we expect since rational carry trader would not invest in ex-ante non profitable

currencies. As a result, these currencies should be orthogonal to the global carry trade factor.

Hence, the candidate SDF does not help to price this subset of currencies. When the models

are estimated on the complete universe of currencies there are larger chances to reject them for

over-large pricing errors.

The alternative approach of grouping many assets in each of the portfolios may give monoton-

ically increasing returns but may remove the risk premia leading to the rejection of the model for

insignificant parameters. As an example, the crosses in Figure 5 are the test assets for the full

universe of currencies allocated to three portfolios. Clearly, in that configuration, assets exhibit

significant variation in βs but not in the mean return: the candidate SDF is not priced in this

subset. We reach similar conclusions when we look at alternative risk factors. Our results echo

those of Ang et al. (2010) who find that using portfolios, instead of stocks, often produces in-

significant and sometimes negative points estimates of the market premium. Related to this, Kan

(1998) notes that the explanatory power of an asset pricing model at the individual firm level can

be nullified at the portfolio level.

Finally, to better understand the relationship between the ranked currencies and the factor,

we lower the threshold defining the set of ex ante profitable currencies from 95% to 70%. This

enables us to include up to three more currencies in the cross-section for the tests (C17, C16 and

C15). Lowering further the threshold would define too small samples for the number of assets in

the cross-section. When we add C17 only, the model is still accepted but with a very small R2 of

4%. Both tests including C17 and C16 only and C17, C16 and C15 are rejected for negative R2.

However, interestingly, if we run the test on the subset of observations for which these currencies

are ex-ante profitable, hence on respectively 89%, 81% and 72% of the sample (see Table II), we

obtain more favourable results. For instance, the R2 of the test including C17 and C16 jumps to

34% indicating that these currencies tend to covary much more with the SDF in the subsample

in which they are ex-ante profitable. Also, in these subsets, the results of our benchmark test

including only C21, C20, C19 and C18 as investment currencies are better, especially when looking

at the R2: on average, the larger the number of currencies which are ex-ante profitable, the more

the extreme ones covary with the SDF.

24

4 Monte Carlo Simulations

So far, the estimation of the risk premia is based on the estimated vector of betas, B because the

true one, B, is not observed. Hence, all the conclusions we have been able to draw are tied to the

sample we have to hand. In a bid to generalize our results, we conduct Monte Carlo experiments

with a large number of simulated samples based on an assumed Data Generating Process (DGP).

As we control the properties of the DGP, we are able to test our conclusions in samples based on

alternative assumptions.

First we estimate the vector of N unconditional betas and their covariance matrix from the

actual data using OLS (N=21). We base our analysis on the Lustig et al. two factor model (RX

and HML − FX of section 3.2). In the second step we simulate 1000 vectors of 21 betas which

enables us to generate 1000 samples of expected currency returns. To produce random multivariate

observations of the betas and control for their cross-sectional correlation, we rely on the Cholesky

decomposition of their covariance matrix estimated from actual data. The simulated betas are also

adjusted such that their means match the observed betas.28 In our benchmark simulation, only

the betas to HML − FX are simulated while the intercepts and the betas to RX are extracted

from the actual data. Simulated returns data are obtained by adding generated iid errors of mean

zero and variances matching those of the idiosyncratic errors in the actual data. Each sample has

the size of the observed sample in the cross section and time series (i.e. 21x172 observations).

Under these assumptions, the simulated data should exhibit means and variances approximately

equivalent to the observed ones. Indeed, we find that the average annualized risk premia in this

set of simulated data is 3.45% with a standard-deviation of 6.30% which match the annualized risk

premia in the actual data (i.e. 3.60% and 6.76%).

Starting from this benchmark simulation, we first investigate the effects of changes in assump-

tions on betas and error distributions on the return to the carry trade. In the second step, also

using the benchmark simulation, we illustrate the consequences of badly constructed test assets on

asset pricing tests in the currency market.

4.1 RETURNS FROM CURRENCY SPECULATION

Table X reports the descriptive statistics of the different versions of the carry trade strategy as de-

scribed in section 2 when the data are simulated 1000 times under the benchmark assumptions. We

28We also consider cross-sectional correlations among both factors and asset returns when calibrating the simulated

data but altering these does not produce significantly different results (e.g. Ahn and Gadarowski, 2004).

25

report the average value and the standard error of each statistic. As expected, with the benchmark

simulation, they are in line with the ones obtained in the actual data: Carrymax and Carrysize

exhibit the largest mean return accross the 1000 trials even corrected for risk. More importantly,

the results of the Monte Carlo simulations show that the performance statitics are all significantly

different from zero. Finally, the ranking of the strategies is similar to the one obtained in the

actual data even though the spread of each strategy to the others is smaller.

Table X about here

In a second step we investigate the effects of the distribution of the betas on the return to the

strategy. In particular, we generate data under the assumption that the betas to HML − FX

increase linearly from the low yielding currency to the high yielding currency, everything else be-

ing equal. In doing so, we minimize the noise generated by the actual betas which are loosely

ordered, in particular for the middle range currencies (see Figure 5). This assumption does not

really impact the descriptive statitics of Carrymax and Carrysize. This is not surprising since, in

the actual data, the betas of the extreme currencies are already well ordered. But the reduction

of the noise in the distribution of the middle range currencies boosts the performance statistics

of portfolios that include them. This is because the positive effect of diversification is no longer

erased by badly distributed returns. As a result, under this assumption, the strategy Carryall

exhibits significantly better statistics than in the actual data. However, even with these linearly

increasing betas, Carryall exhibits lower statistics than both Carrymax and Carrysize.

To further explore the carry trade strategy, we generate data under even more restrictive as-

sumptions: if the returns to the currencies C1 to C21 were mutually uncorrelated and had common

variance the construction of the factors RX and HML−FX would imply a beta of 1 for RX and

betas of -0.5, 0, 0,..., 0, 0.5 for HML − FX. Under these assumptions, the middle range curren-

cies have similar mean returns equal to the mean return of RX and are uncorrelated with each

other or with the extreme currencies. As a result, diversification works particularly well. However,

again, the observed improvement in the statistics of Carryall is not enough to exceed the ones of

Carrymax and Carrysize (not reported).

These simulations show that Carrymax and Carrysize work particularly well because the asso-

ciation between the discount forward and the returns for the extreme currencies, and particularly

26

the ex-ante profitable ones, is strong. In this case diversifying with middle range currencies impairs

somewhat the performance statistics of the carry trade. Conversely, when we impose a high level

of association to the middle range currencies (linearly increasing or zero betas), diversification

becomes attractive. However, the impact of diversification is not strong enough for the statistcs of

Carryall to exceed the ones of the competing strategies. We test many other departures from the

benchmark assumptions, particularly on the covariance matrix of the betas (near constant betas

for instance) and on the distribution of the residual errors, and we repeat all tests using only the

set of G10 currencies. We do not find any significant differences in the conclusions of these tests,

confirming that Carrymax and Carrysize exhibit the best set of risk statistics and that diversifica-

tion works only if one can find currencies with uncorrelated betas, especially with respect to the

extreme currencies.

4.2 RISK FACTORS IN THE CURRENCY MARKET.

Table XI reports the results of running asset pricing tests on the 1000 samples of simulated data

under the benchmark assumptions. As mentioned earlier, with 21 ranked assets in the cross sec-

tion, there are 1,048,555 combinations to test per simulated sample hence more than a billion

combinations in total. This is beyond the scope of this paper to analyse the full range of possible

p-portfolios across the simulated data especially because many of them might not be economically

interesting. Also the computational burden limits our ability to be systematic. Rather, in Table

XI, we challenge our point on the impact of portfolio construction on the conclusion for asset

pricing tests by looking at the results obtained from the randomly chosen number of portfolios (see

section 3.3) in the set of simulated samples. Especially, we would like to know whether this impact

is visible in a large number of samples.

When we choose a 6-portfolio test randomly (n/p is not an integer) similar to the one used

in Lustig et al. (2011), we find that their model is rejected for about 88% of the trials. This

number jumps to 94% for a 7-portfolio test and decreases to 55% for a 3-portfolio test: indeed,

different designs produce significantly different rejection rates and conclusion to the tests for each

trial. The same conclusions can be reached when the p-portfolios are set on alternative allocation

of the decimal part for each of them (alternative 6-portfolio tests for instance). Interestingly, using

the randomly chosen portfolios, only the results to one trial among the 1000 look independent to

the way we set the portfolios. Inversely, for most of the trials, there are many constructions that

enable one to reject the risk premia story.

We find that the tests based on undiversified portfolios are more often rejected for over-large

27

errors than for insignificant risk premia λ. For instance, in the case of the 7-portfolio tests we

report, all the rejections are motivated by over-large errors. Conversely, the tests based on very

diversified portfolios are often rejected for non significant premia: for the 3-portfolio tests we re-

port, 95% of the rejections are motivated by a non significant λ29.

The rejection rate for the tests based on the ex-ante profitable currencies is rather high at 83%.

However, this design of the test assets offers a significant number of new opportunities to accept

the risk premia story. Indeed, when we look in more detail at the results, we see that for every

1000 trials, there is only a limited number of test asset constructions that enables one to accept the

risk premia story. Even, in our simulation, for 21% of these trials, there is only one construction

favourable to the model. This unique construction is the one based on ex-ante profitable currencies

in 7% of the cases. This number is not very different for 5-portfolio test (7%) and 3-portfolio tests

(6%) but significantly lower for 6-portfolio tests (1%) and nil for the other p-portfolio tests.

The rejection rate we obtain might look very high to the reader giving the impression that the

asset pricing tests are extremely fragile. However, one has to keep in mind that we only report 7

combinations among the 1,048,555 possible (0.0007% of the total). While these combinations have

been chosen randomly they enable one to accept the risk premia story in more than 50% of the

simulated samples. Of course, spanning the set of conventional test asset constructions, especially

when n/p is not an integer, increases mechanically the overall rate of acceptance. However, as

already mentioned, it is out of the scope of this paper to be systematic. Also, we recall that the

iterated GMM places a higher hurdle than the two-step GMM or the Fama-MacBeth procedure

often used in the literature. Finally, we do not filter out the data as reported in Lustig et al. (2011)

or Menkhoff and al. (2012).

All together, our simulations are rather supportive to the risk premia story. They show that

for most of the cases if the number of portfolios p is chosen randomly, there is a large probability

of rejecting the model which, again, casts doubt on the conclusions drawn using one and only

one portfolio design. For instance, in our simulated samples, a researcher focusing only on our

randomly chosen 6-portfolio tests would reject the model in 88% of the trials while, in the limited

set of portfolio constructions we consider, for 38% of them there exists at least one alternative

p-portfolio favourable to the risk premia story. Also, considering other alternative 6-portfolio tests

(n/p is not an integer) may increase further the acceptance rate. Finally, we mention that the

29We also run the asset pricing tests under alternative assumptions such as the ones described earlier. However,

results change little, although we do note that with a diagonal covariance matrix (uncorrelated betas) the tests

exhibit more similar rejection rates which are higher than obtained under benchmark assumptions.

28

impact of the construction of the portfolios on the result of the tests is visible for all the factors

we tested. This makes the probability to get two factors jointly significant for a given design, very

low and may explain why researchers tend to reject the concurrent factors to their own when they

use one and only one portfolio construction.

Looking at the errors of the cross-sectional tests helps to understand why alternative samples

call for alternative portfolio construction. We know that for the risk premia to be precisely esti-

mated we need numerous assets widely spread and lying on the factor line. When the errors are

low and homogeneous across the assets many portfolio constructions can work. For instance, on

average, the trials for which most of the test asset work (at least 4) exhibit lower squared errors

(∑e2 = 0.000155). The ones for which only one construction or none of the possible designs work

exhibit on average significantly higher errors: respectively∑e2 = 0.000206 and

∑e2 = 0.000244.

Finally, when the mass of errors is concentrated on the ex-ante non profitable currencies, the test

based on the differentiation between ex-ante profitable and non profitable currencies turns to be

favourable more often than the alternative designs.

Table XI about here

5 Conclusion

The aim of this paper is to show that the way one designs carry trade portfolios produces sig-

nificantly different returns for practitioners and statistical results for academics. Together, we

suggest that both academics and practitioners may want to consider non-diversified and non-

equally-weighted portfolios built upon the rationale of portfolio construction that we study. Non-

diversified and non-equally-weighted portfolios might outperform the conventional diversified and

equally weighted ones because of a special feature of the variance-covariance matrix in the currency

market: adjacent forward-ranked currencies exhibit large covariances. This point echoes the work

of Lustig et al. (2011) who show that there is a large systematic risk in the currency market30. As

a result, grouping currencies in portfolios does not really diversify the risk away while erasing the

information conveyed by the dispersion of the signals. Grouping currencies in portfolios might then

produce lower returns for practitioners and sometimes misleading results for academics. Especially,

30This is in opposition to the results of Bakshi and Panayotov (2013) but they mainly look at risk measures

(volatility and skewness) instead of risk-return ratios.

29

for academics, some designs might lead to the rejection of cross sectional tests of asset pricing in

the currency market that could be accepted otherwise. Especially, when we package the asset

on the back of the rationale we study, we highlights more precisely the economic phenomenon of

interest, which might offer new chances of obtaining favorable results. This packaging makes sense

since, for the tests, we filter out currencies which should be, in theory, orthogonal to the pricing

factor, while relying on the full dispersion of the currencies correlated, in theory, to the candidate

SDF in order to estimate the risk premia. This alternative way to build the test assets may help

academics to reach a consensus in the debate on the common risk factors in the currency markets.

Also, our work shows that one should not reject a factor without mining the full set of possible and

economically interesting p-portfolio tests. For example, in our sample, while we find at least one

construction favorable to the risk premia story for each factor we tested, we did not find a single

one favorable jointly to all the factors. This might be one of the reasons why researchers obtain

sometimes contradictory results when testing the risk premia story. Finally, the large number of

concurrent SDF proposed recently in the literature calls for a comparative analysis which should

benefit from our findings. For practitioners our non-diversified and non-equally-weighted portfolios

look attractive since they seem to offer higher average returns, even corrected for risk and tail risk.

This idea of non diversification is not new. For instance, many studies on the US equity market

have concluded that focused managers, who take big bets in a few equities, tend to outperform

their counterparts (e.g., Huij and Derval, 2011). The novelty of this paper is that our results

extend this conclusion to the currency market.

30

Appendix A: Portfolio design.

Markowitz’ approach has well known drawbacks. In particular it is very sensitive to estimation

errors in the parameters. To overcome this problem, investors often rely on simple assumptions in

the parameters and their structure.

Carrysize The no-arbitrage condition implies that the cross-section of assets’ risk should match

expected returns. When the investors believe that currencies have different expected returns hence

level of risk, they might consider that the relative weights of the currencies in the portfolio are

somewhat in line with their relative expected returns and/or risks. Then the portfolios of ex-ante

profitable currencies (Equation 2) are:

xt =

Buy wi=fbt/sat − 1 if f bt /s

at > 1,

Sell wi=fat /sbt − 1 if fat /s

bt < 1,

0 otherwise

(8)

Equation (1) of this Appendix specifies portfolios in which the currencies are weighted accord-

ing to the size of the signals (f − s). We call this design Carrysize.

Carryba If traders think that the ex-ante profitable currencies have similar expected returns,

variances and correlations, they might choose to form portfolios of equally weighted ex-ante prof-

itable currencies because nothing helps to discriminate among them:

xt =

Buy wi = 1 if f bt /sat > 1,

Sell wi = −1 if fat /sbt < 1,

0 otherwise

(9)

We call this design Carryba because the time-varying transaction costs - the bid-ask spreads -

define the set of investable currencies for every period.

Carrymax. when investors believe that the currencies share the same level of risk (same

31

variances and correlations equal to one) but have different expected returns, there is no benefit to

diversification, hence the traders’ problem become:

xt =

Buy wi = 1 if f bt /sat = max(f bt /s

at ),

Sell wi = −1 if fat /sbt = min(fat /s

bt),

0 otherwise

(10)

We call this design Carrymax because only the currencies with the maximum (minimum) in-

terest rates enter the portfolio. In this case, the portfolios of long and short positions contain one

currency each and the carry trade strategy consists of a long-short equally weighted position on

the two portfolios.

Carryall When agents consider that transaction costs are insignificant, they do not discriminate

between ex-ante profitable and non profitable currencies. In this case, when they believe that the

assets share similar parameters, they might form equally weighted portfolios containing all the

currencies together. Investors’ allocation rule becomes:

xt =

Buy wi = 1 if ft/st > 1,

Sell wi = −1 if ft/st < 1,

(11)

We call this design Carryall.

Carryn However, even in the case of near-zero transaction costs, casual evidence shows that,

limiting the investment to the extreme currencies produces higher return for lower risk. We call

this design Carryn. Most of the ”carry trade” products sold by banks are based on this evidence.

For currency ranked by their forward discount from 1 (lowest) to N (highest):

xt =

Buy wi = 1 if N − n+ 1 < Rankft/st < N,

Sell wi = −1 if 1 < Rankft/st < n,

(12)

32

Appendix B: Risk statistics

As with most financial returns, carry trades returns are not normally distributed. They exhibit

large negative skewness and substantial kurtosis (see Table I in Appendix E). Therefore, the two

first moments of the distributions are not sufficient to describe them. In this Appendix, we intro-

duce several popular risk ratios that have been developed by academics and the financial industry

to account for non-normal distributions and complement the traditional Sharpe ratio. These ra-

tios relate the mean return of the distribution to different risk statistics: the standard-deviation

for the Sharpe ratio, a quantile-based statistics such as VaR for the modified Sharpe ratios and

some measures of drawdown for several ratios which are alternative definitions of the Sterling ratio.

We study the generic risk ratio (RR):

RR =mean(xt)

risk(xt)(13)

with risk(xt) ∈ {sd(x);V aR(x);Dmean;DMax}

The standard-deviation, sd(x), is a sufficient measure of risk only if the distribution is symmet-

rical. But since most distributions are not, one needs to focus more precisely on negative outcomes

to estimate the risk. Especially, agents might fear infrequent but profound outcomes that might

cause their ruin. In this case, the risk is measured by estimating the extreme quantiles of the

distribution of returns. The likely loss associated to these quantiles is then called the Value at

Risk in practitioners’ words. A quantile is defined as a number µ that set

Prob (x ≤ µ) = p (14)

As it is traditional, in this paper, we estimate the quantile for µ = 95% such as: Prob (x ≤ µ) =

95%. Now, the question is: how to estimate the quantile? Arguably, the simplest way is to use the

sample quantile extracted from the historical data. When these estimations are conditional, they

might be very sensitive, especially in short samples, to the inclusion or not of some extreme values.

One way to mitigate this bias is to extrapolate the VaR with a parametric method. For instance,

initially, RiskMetrics (1996) estimated the VaR from the variance of the distribution of the returns.

But this estimation is valid only if returns are normally distributed otherwise the statistic might be

only a rough approximation of the true one. Hence we follow Longerstaey and Zangari (1996) and

correct for the non-normality of the distributions, especially for excess skewness and kurtosis using

another parametric method based on the Cornish-Fisher expansion. These estimations will give

33

a larger loss estimate than traditional quantile calculations when outcomes are negatively skewed

or highly kurtotic. Conversely it will give smaller loss magnitude when historical outcomes are

positively skewed or leptokurtic. This approximation is based on a Taylor series expansion using

higher moments of the distribution. The approximation of the Qα quantile to the fourth moment

can be written as31:

Qα ∼= µ−(Zα +

1

6

(Z2α − 1

)SK +

1

24

(Z3α − 3Zα

)K

)σ (15)

with Zα the critical value associated to the standard normal distribution qth quantile, µ, σ,

SK and K the in-sample mean, standard-deviation, skewness and excess kurtosis of the returns.

To take account of tail risk, Favre and Galeano (2002) propose modifying the traditional Sharpe

ratio by substituting the VaR to the standard-deviation. In this paper the modified Sharpe ratios

is reported under the name MV aR.

Beside this measure of tail risk, we look at the risk of severe outcomes. According to Thaler

and Johnson (1990) economic agents might also exhibit preferences in term of the distributions

of cumulative returns especially negative ones, i.e. drawdowns. Long lasting episodes of negative

outcomes might be particularly painful. We measure this risk by the maximum drawdown and the

average drawdown observed in the series of returns. A drawdown is defined as the difference in

value of the series of cumulated returns between any local maximum and the next local minimum.

Obviously, less pronounced drawdowns are preferred. With X a vector of time series observations

of length T, we define any drawdown Dt in X as:

Dt = xmaxt − xt (16)

with

xmaxt = max {xi |i [0, t]} (17)

then

∀xt xt = xmaxt ⇔ Dt = 0

otherwise

Dt ≥ 0

The mean drawdown in X is then defined as:

31Higher moments expansion can be found in Stuart and al. (1999).

34

Dmean =1

n

n∑i=1

Dt (18)

n being the number of drawdowns in X.

and the maximum drawdown is defined as:

DMax = max (Dt) (19)

In this paper the ratio of the mean return to the mean drawdown is called MDD while the ratio of

the mean return to the maximum drawdown is called MMaxDD.

35

Appendix C: Asset Pricing.

The carry trade is a zero-cost investment (see section 2), hence its excess returns in level, Rt+1,

satisfy the basic Euler equation:

Et(Mt+1.Rt+1) = 0 (20)

with Mt+1 a linear SDF of the form:

Mt+1 = ξ[1− (ft+1 − µ)′b)

](21)

b being the vector of SDF parameters, f the vector of risk factors and µ the vector of the sample

mean of the risk factors. ξ is a scalar we set at one (see Cochrane, 2005).

Equation (21) implies a beta representation of the model in which expected excess returns

depend on factor risk premia λ and risk loadings β:

E(Rt+1) = cov(Rt+1, ft+1)b (22)

or

E(Rt+1) = βλt+1 (23)

β is the vector of coefficients of the regression of Rt+1 on ft+1, in the sample, while λ is a vector

of risk premia.

To estimate this relationship, the literature follows the common two-step procedure inspired by

Fama and Mc Beth (1973). First we look whether a linear combination of factors can significantly

justify the returns to the carry trades, in the time series, for each currency or portfolio of currencies

i:

Rit+1 = αi + ft+1′βi + εit+1 (24)

Then, following Cochrane (2005) among others, we estimate the parameters of Equation (21)

using the Generalized Method of Moments of Hansen (1982). To remove estimation uncertainty, as

recommended by Burnside (2010), factor means, µ and the variance covariance matrix of factors

Σf are co-estimated. The vector of moment conditions is:

36

g(Rt+1, ft+1, θ) =

[1− (ft+1 − µ)′b)]Rt+1

ft+1 − µ

Σ((ft+1 − µ)(ft+1 − µ)′)− Σft+1

(25)

where θ contains the parameters (b, µ, Σft+1).

Then, estimates of λ are obtained from b and Σf as λ = bΣf . Standard errors of λ come from the

estimation of the variance of the function bΣf .

We use the iterated GMM estimator starting from the identity matrix as weighting matrix

WT = I. The iterated GMM estimator has much greater power to reject mispecified models than

the two-step version (e.g. Burnside, 2010). Reported standard errors are estimated by the Newey-

West procedure, with the number of lags determined according to Andrews (1991). Predicted mean

returns are cov(R, f)b and pricing errors are α = µR − cov(R, f)b. As a mean to test the validity

of the model, we run a test of the pricing errors J = T α′V −1T α where VT is a consistent estimate of

the asymptotic covariance matrix of√T α. The test statistic is asymptotically distributed as a χ2

with n− k degrees of freedom, n being the number of test assets and k the number of risk factors

f . Also we estimate the cross-sectional fit of the model as R2 = Σα2/Σ(Rit− Rt)2.

37

Appendix D: global volatility mimicking portfolio.

To build our global equity volatility factor, using daily observations, we compute the standard

deviation over one month of daily changes in the MSCI World index. Our risk factor corresponds

to volatility innovations, obtained as log differences of our global volatility series. We call it

V OLEQTY . Similarly we build a global volatility factor extracted from the currency market. We

call it V OLFX . In line with Menkhoff and al. (2012), this indicator is measured as the innovations

to the average sample standard deviation of the daily returns of the currencies in the sample. We

expect innovations to global volatility to be negatively correlated with the return to the carry trade.

As both factors are not traded assets, we convert them into returns as to be able to evaluate their

economic significance. Following Breeden et al. (1989) and the prescription of Lewellen, Nagel and

Shanken (2010), we build factor-mimicking portfolios of VOL innovations. To obtain the factor

mimicking portfolios, we regress VOL innovations on respectively the p carry trade portfolio excess

returns of the p-portfolio test and as a matter of comparison on the ex-ante profitable currencies

C1, C18 to C21 plus the residual currencies mixed in one unique portfolio P1. The results in Table

IX are reported for p = 5.

δFMt+1 = α+ β′Rpt+1 + µt+1 (26)

where Rpt+1 is the vector of excess returns of the test assets and δFMt+1 is alternatively

V OLEQTY t+1 and V OLFXt+1 . Then the factor-mimicking portfolio’s excess return is given by

RFMt+1 = β′Rpt+1.

38

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43

Appendix E: the return to the carry trade

Table I: Portfolios’ descriptive statistics

The table reports the mean returns, standard deviation, skewness and kurtosis of currency portfolios invested in the carry trade.Carrysize is a strategy in which the set of investable currencies is determined dynamically by the time varying transactioncosts while the relative weights of the currencies depend on their time-varying relative expected returns, Carryba is the equallyweighted version of Carrysize, Carryn is also an equally version of the strategy but over a fixed and predefined number, n,of currencies, Carryall is similar to Carryn except that all the currencies are considered investable and finally, Carrymax isan equally weighted portfolio of the two extreme currencies (largest and smallest interest rate). The statistics are reported forannualized log excess returns accounting for transaction costs (bid-ask spreads). The data are monthly. We also report theannualized Sharpe Ratios (SR), modified Sharpe Ratio MV aR, and the ratios of the mean return to the mean drawdown MDD

and to the maximum drawdown MMaxDD as defined in Appendix B of this paper. Monthly ratios and their standard-errors arealso reported (e.g. Lo, 2002). They are obtained by bootstrapping the sample 10000 times. The figures are in basis points notpercentages. The panel uses all countries with data from January 1999 to May 2013.

mean sd SK KT SR MV aR MDD MMaxDD

Carrysize annualized 0.0893 0.1222 -0.6083 7.2268 0.7307 0.5365 0.1750 0.0424monthly 0.2109 0.1548 0.1750 0.0424sd-dev (0.0843) (0.0943) (0.0906) (0.0187)

Carryba annualized 0.0394 0.0684 -0.6101 6.0700 0.5760 0.3892 0.1363 0.0185monthly 0.1662 0.1123 0.1363 0.0185sd-dev (0.0826) (0.0640) (0.0838) (0.0201)

Carryn annualized 0.0242 0.0467 -0.3976 4.1057 0.5196 0.3515 0.1288 0.0238monthly 0.1500 0.1014 0.1288 0.0238sd-dev (0.0791) (0.0664) (0.0811) (0.0140)

Carryall annualized 0.0223 0.0512 -0.7560 5.6684 0.4359 0.2748 0.0940 0.0140monthly 0.1258 0.0793 0.0940 0.0140sd-dev (0.0819) (0.0630) (0.0695) (0.0158)

Carrymax annualized 0.0808 0.0858 0.2104 5.3331 0.9416 0.7559 0.2349 0.0419monthly 0.2718 0.2182 0.2349 0.0419sd-dev (0.0773) (0.0795) (0.0984) (0.0238)

44

Table II: Currencies’ descriptive statistics

The table reports the mean returns, standard deviation, skewness and kurtosis for currencies sorted on their forwarddiscount f − s. C1 is the lowest yielding currency while C21 is the highest yielding currency. We also report the meanforward discount for each currency (f − s). The statistics are reported for annualized log excess returns accounting fortransaction costs (bid-ask spreads). The data are monthly. We also report the annualized Sharpe Ratios (SR), modifiedSharpe Ratio MV aR, and the ratios of the mean return to the mean drawdown MDD and to the maximum drawdownMMaxDD as defined in Appendix B of this paper. The last column reports the frequency at which each ranked currencywas ex-ante profitable. The figures are in basis points not percentages. The panel uses all countries with data fromJanuary 1999 to May 2013.

f-s mean sd SK KT SR MV aR MDD MMaxDD Freq

C1 -0.0027 -0.0030 0.1214 -1.2945 11.8697 -0.0251 -0.0165 -0.0046 -0.0012 1.00C2 -0.0015 -0.0039 0.0840 0.1938 3.6402 -0.0465 -0.0299 -0.0085 -0.0026 0.73C3 -0.0011 0.0086 0.0867 -0.3340 4.9061 0.0996 0.0573 0.0225 0.0048 0.54C4 -0.0008 -0.0252 0.1149 -0.4427 5.2630 -0.2191 -0.1252 -0.0382 -0.0064 0.43C5 -0.0006 -0.0131 0.0919 -0.2846 5.7095 -0.1427 -0.0814 -0.0287 -0.0049 0.35C6 -0.0005 0.0092 0.0996 -2.1338 15.1546 0.0927 0.0674 0.0207 0.0023 0.30C7 -0.0004 0.0607 0.1132 0.0322 6.4764 0.5363 0.4015 0.1451 0.0301 0.23C8 -0.0002 0.0205 0.1144 -0.4910 6.2439 0.1794 0.1292 0.0371 0.0058 0.17C9 0.0000 0.0523 0.1050 -0.0495 3.5613 0.4981 0.3402 0.1123 0.0341 0.08C10 0.0002 0.0361 0.1131 -0.1063 4.3952 0.3195 0.2247 0.0608 0.0148 0.30C11 0.0003 0.0265 0.0934 -0.1952 3.4465 0.2833 0.1723 0.0514 0.0127 0.34C12 0.0005 0.0211 0.1034 -0.1965 3.9938 0.2045 0.1207 0.0485 0.0075 0.39C13 0.0007 -0.0048 0.1142 -0.8046 6.1240 -0.0420 -0.0256 -0.0088 -0.0012 0.47C14 0.0011 -0.0329 0.1243 -0.8578 5.5035 -0.2647 -0.1248 -0.0442 -0.0069 0.56C15 0.0016 0.0462 0.1158 -0.3886 6.4500 0.3988 0.2898 0.0958 0.0153 0.72C16 0.0022 0.0379 0.1217 -0.6822 4.6988 0.3115 0.1769 0.0614 0.0108 0.81C17 0.0030 -0.0164 0.1333 -1.5933 8.8547 -0.1227 -0.0764 -0.0237 -0.0045 0.89C18 0.0041 0.0148 0.1289 -0.8460 5.9012 0.1144 0.0685 0.0240 0.0030 0.98C19 0.0061 0.0022 0.1405 -1.1856 6.5273 0.0159 0.0110 0.0032 0.0005 0.99C20 0.0095 0.0665 0.1859 -1.8157 16.5783 0.3579 0.2737 0.0857 0.0151 1.00C21 0.0185 0.1489 0.1616 -0.7329 5.2441 0.9218 0.6806 0.2111 0.0311 1.00

45

Table III: Variance-CoVariance Matrix

The table reports the variance-covariance matrix for currencies sorted on their forward discount f − s. C1 is the lowest yieldingcurrency while C21 is the highest yielding currency. Only the statistics for extreme currencies are reported. The statistics arereported for monthly log excess returns accounting for transaction costs (bid-ask spreads). The data are monthly. The figuresare in basis points not percentages. The panel uses all countries with data from January 1999 to May 2013.

C1 C2 C3 C4 C5 . . . C16 C17 C18 C19 C20 C21

C1 0.0011 0.0002 0.0005 0.0005 0.0003 . . . 0.0006 0.0004 0.0005 0.0004 0.0002 0.0004C2 0.0002 0.0006 0.0004 0.0004 0.0003 . . . 0.0004 0.0004 0.0003 0.0003 0.0001 0.0002...

......

......

.... . .

......

......

......

C18 0.0004 0.0004 0.0005 0.0005 0.0004 . . . 0.0007 0.0015 0.0006 0.0007 0.0006 0.0008C19 0.0005 0.0003 0.0006 0.0006 0.0005 . . . 0.0008 0.0006 0.0014 0.0008 0.0006 0.0008C20 0.0004 0.0003 0.0005 0.0003 0.0005 . . . 0.0007 0.0007 0.0008 0.0016 0.0009 0.0010C21 0.0002 0.0001 0.0004 0.0002 0.0003 . . . 0.0006 0.0006 0.0006 0.0009 0.0029 0.0010C22 0.0004 0.0002 0.0004 0.0003 0.0004 . . . 0.0007 0.0008 0.0008 0.0010 0.0010 0.0021

46

Appendix F: Robustness tests

Table IV: The return to the carry trade in G10 currencies.

The table reports the mean returns, standard deviation, skewness and kurtosis of currency portfoliosinvested in the carry trade. Carrysize is a strategy in which the set of investable currencies is determineddynamically by the time varying transaction costs while the relative weights of the currencies depend ontheir time-varying relative expected returns, Carryba is the equally weighted version of Carrysize, Carrynis also an equally version of the strategy but over a fixed and predefined number, n, of currencies, Carryallis similar to Carryn except that all the currencies are considered investable and finally, Carrymax is anequally weighted portfolio of the two extreme currencies (largest and smallest interest rate). The statisticsare reported for annualized log excess returns accounting for transaction costs (bid-ask spreads). We alsoreport the annualized Sharpe Ratios (SR), modified Sharpe Ratio MV aR, and the ratios of the meanreturn to the mean drawdown MDD and to the maximum drawdown MMaxDD as defined in Appendix Bof this paper. The data are monthly. The figures are in basis points not percentages. The panel A usesG10 countries from January 1999 to May 2013.The panel B uses G10 countries from December 1984 toMay 2013.

Panel A: G10 99-13

mean sd SK KT SR MV aR MDD MMaxDD

Carrysize 0.0699 0.1014 -0.0469 5.9169 0.6893 0.5411 0.1955 0.0495Carryba 0.0504 0.0796 -0.1795 5.9958 0.6336 0.4859 0.1518 0.0285Carryn 0.0259 0.0442 -0.3806 4.1166 0.5858 0.3332 0.1071 0.0199Carryall 0.0354 0.0497 -0.4433 4.0781 0.7120 0.4517 0.1596 0.0339Carrymax 0.0729 0.1211 -0.4352 4.5317 0.6021 0.4263 0.1403 0.0181

Panel B: G10 84-13

mean vol SK KT SR MSR meanDD MaxDD

Carrysize 0.0659 0.0946 -0.4673 6.2259 0.6973 0.4498 0.1703 0.0270Carryba 0.0414 0.0697 -0.3712 6.3016 0.5948 0.4174 0.1247 0.0234Carryn 0.0236 0.0472 -0.7682 4.5867 0.5002 0.2799 0.0948 0.0182Carryall 0.0331 0.0507 -0.8183 5.1138 0.6524 0.4033 0.1382 0.0207Carrymax 0.0703 0.1277 -0.6546 5.0869 0.5505 0.3219 0.1244 0.0175

47

Table V: The return to the carry trade in random subsets.

The table reports the mean returns, standard deviation, skewness and kurtosis of currency portfoliosinvested in the carry trade. Carrysize is a strategy in which the set of investable currencies is determineddynamically by the time varying transaction costs while the relative weights of the currencies depend ontheir time-varying relative expected returns, Carryba is the equally weighted version of Carrysize, Carrynis also an equally version of the strategy but over a fixed and predefined number, n, of currencies, Carryallis similar to Carryn except that all the currencies are considered investable and finally, Carrymax is anequally weighted portfolio of the two extreme currencies (largest and smallest interest rate). The statisticsare reported for annualized log excess returns accounting for transaction costs (bid-ask spreads). We alsoreport the annualized Sharpe Ratios (SR), modified Sharpe Ratio MV aR, and the ratios of the meanreturn to the mean drawdown MDD and to the maximum drawdown MMaxDD as defined in AppendixB of this paper. The data are monthly. The figures are in basis points not percentages. Panel A uses allcountries from A to L with data from January 1999 to May 2013. Panel B uses all countries from M toZ with data from January 1999 to May 2013.

Sample: All countries A to L 99-13

mean sd SK KT SR MV aR MDD MMaxDD

Carrysize 0.0678 0.1863 -3.4638 31.9020 0.3640 0.3425 0.1069 0.0102Carryba 0.0464 0.0710 -0.3443 5.8389 0.6534 0.4526 0.1484 0.0357Carryn 0.0050 0.0251 -0.1119 4.6627 0.1994 0.1095 0.0416 0.0068Carryall 0.0259 0.0558 -0.8863 6.3613 0.4649 0.3215 0.0969 0.0182Carrymax 0.0721 0.0891 0.2764 6.4706 0.8100 0.5954 0.2030 0.0365

Panel B: All countries M to Z 99-13

mean vol SK KT SR MSR meanDD MaxDD

Carrysize 0.0871 0.1746 1.0607 24.8980 0.4988 0.4210 0.1648 0.0218Carryba 0.0301 0.0757 -1.0801 8.1980 0.3980 0.2780 0.0936 0.0114Carryn 0.0020 0.0444 -0.8873 4.9769 0.0451 0.0265 0.0093 0.0018Carryall 0.0212 0.0537 -0.7163 5.6426 0.3944 0.2824 0.0812 0.0082Carrymax 0.0468 0.0840 -0.5739 5.0454 0.5573 0.3763 0.1118 0.0243

48

Table VI: The return to the carry trade. Weekly and quarterly observations.

The table reports the mean returns, standard deviation, skewness and kurtosis of currency portfolios in-vested in the carry trade. Carrysize is a strategy in which the set of investable currencies is determineddynamically by the time varying transaction costs while the relative weights of the currencies depend ontheir time-varying relative expected returns, Carryba is the equally weighted version of Carrysize, Carrynis also an equally version of the strategy but over a fixed and predefined number, n, of currencies, Carryallis similar to Carryn except that all the currencies are considered investable and finally, Carrymax is anequally weighted portfolio of the two extreme currencies (largest and smallest interest rate). The statisticsare reported for annualized log excess returns accounting for transaction costs (bid-ask spreads). We alsoreport the annualized Sharpe Ratios (SR), modified Sharpe Ratio MV aR, and the ratios of the mean returnto the mean drawdown MDD and to the maximum drawdown MMaxDD as defined in Appendix B of thispaper. The figures are in basis points not percentages. The panel A uses all countries with weekly datafrom October 2002 to May 2013. The panel B uses all countries with quarterly data from January 1999 toMay 2013.

Panel A: Weekly data all countries 02-13

mean sd SK KT SR MV aR MDD MMaxDD

Carrysize 0.1517 0.2846 -1.1622 76.7866 0.5330 0.7659 0.1132 0.0057Carryba 0.0422 0.0853 -0.2943 8.2086 0.4943 0.3509 0.0571 0.0063Carryn -0.0233 0.0435 -0.9198 6.8637 -0.5345 -0.3103 -0.0442 -0.0034Carryall -0.0439 0.0459 -0.8210 8.5390 -0.9571 -0.5671 -0.0714 -0.0055Carrymax 0.0198 0.0901 -0.5346 5.4684 0.2201 0.1340 0.0216 0.0022

Panel B: Quarterly data all countries 99-13

mean sd SK KT SR MV aR MDD MMaxDD

Carrysize 0.0780 0.1264 -0.1241 4.7886 0.6171 0.5874 0.2518 0.0779Carryba 0.0266 0.0578 -0.5150 3.7844 0.4607 0.3006 0.1581 0.0504Carryn 0.0145 0.0546 -1.1359 4.3622 0.2652 0.1360 0.1064 0.0369Carryall 0.0181 0.0510 -0.8573 4.4144 0.3558 0.2245 0.1254 0.0341Carrymax 0.0334 0.1040 -0.8640 6.2811 0.3211 0.2262 0.1072 0.0290

49

Appendix G: Asset pricing.

Table VII: Statistics of the time series regression of portfolios’ excess returns on risk factors.

Monthly returns from January 1999 to May 2013. Test assets are currency portfolios sorted on their forward discount. Excess returns used as tests assets and risk factorstake into accounts bid-ask spreads. The RX factor is the average excess return to all the currencies included in the sample. The HML-FX portfolio is the excess returnof a strategy buying the currencies with the largest forward discount and selling those with the smallest discount. Ex-ante profitable currencies C1 and C18 to C21 aredefined according to equation (2). For this test, P1 contains the 16 currencies which are ex-ante non-profitable (95% threshold). The table reports OLS estimates of theβ of equation (7) as well as heteroskedasticity consistent standard errors and R2. Significant loadings at the 5% level are indicated with two*.

Panel: All countries 99-13

3 EQW portfolios 5 EQW portfolios 7 EQW portfolios ex-ante profitable ccyPtf RX HML-FX R2 Ptf RX HML-FX R2 Ptf RX HML-FX R2 Ptf RX HML-FX R2

P1 0.96** -0.39** 0.95 P1 0.88** -0.33** 0.89 P1 0.77** -0.31** 0.82 C1 0.76** -0.36** 0.60(0.02) (0.02) (0.02) (0.02) (0.03) (0.02) (0.06) (0.03)

P2 1.07** -0.16** 0.90 P2 1.00** -0.09** 0.89 P2 1.03** -0.11** 0.75 P1 1.01** -0.04** 0.96(0.03) (0.02) (0.03) (0.02) (0.05) (0.02) (0.02) (0.01)

P3 0.95** 0.58** 0.98 P3 1.06** -0.15** 0.84 P3 1.06** -0.10** 0.78 C18 1.18** 0.05 0.54(0.02) (0.02) (0.03) (0.02) (0.04) (0.02) (0.08) (0.03)

P4 1.19** 0.02 0.83 P4 1.03** -0.13** 0.79 C19 1.12** 0.12** 0.43(0.04) (0.03) (0.04) (0.02) (0.10) (0.04)

P5 0.87** 0.63** 0.95 P5 1.12** -0.03 0.81 C20 0.90** 0.25** 0.33(0.03) (0.02) (0.04) (0.02) (0.12) (0.05)

P6 1.20** 0.03 0.80 C21 0.87** 0.61** 0.80(0.04) (0.03) (0.07) (0.03)

P7 0.79** 0.66** 0.93(0.03) (0.02)

50

Table VIII: Iterated GMM estimates of linear factor models for sorted currency portfolios.

Monthly returns from January 1999 to May 2013 for all countries. Test assets are currency portfolios sorted ontheir forward discount. Excess returns used as tests assets and risk factors take into accounts bid-ask spreads.The RX factor is the average excess return to all the currencies included in the sample. The HML-FX portfoliois the excess return of a strategy buying the currencies with the largest forward discount and selling those withthe smallest discount. Ex-ante profitable currencies C1 and C18 to C21 are defined according to equation (2).P1 contains the 16 currencies which are ex-ante non-profitable (95% threshold). The table reports iteratedGMM estimates of the SDF parameter, b, and the factor risk premia λ reported in monthly percentages. Crosssectional R2 as well as test statistics J for the overidentifying restrictions are reported. Standard errors forstatistics are in brackets. The p-value of the J test is also in brackets.

b λ R2 J b λ R2 J

3 Equally Weighted portfolios 5 Equally Weighted portfolios

RX 1.68 0.18 0.99 0.01 RX 2.53 0.23 0.39 4.22(4.36) (0.21) (0.91) (4.68) (0.21) (0.23)

HML-FX 7,38 0.36 HML-FX 7.81** 0.69**(4.77) (0.22) (3.26) (0.28)

7 Equally Weighted portfolios Ex-ante profitable currencies

RX 4.11 0.24 0.18 13.16 RX 7.34 0.31 0.30 4.86(4.52) (0.18) (0.03) (4.49) (0.17) (0.30)

HML-FX 4.53* 0.63* HML-FX 5.50** 1.44**(2.86) (0.33) (2.07) (0.51)

51

Table IX: Asset Pricing - Global volatility factor.

Monthly returns from January 1999 to May 2013 for all countries. Test assets are currency portfolios sorted on theirforward discount. Excess returns used as tests assets and risk factors take into accounts bid-ask spreads. The RXfactor is the average excess return to all the currencies included in the sample. The HML-FX portfolio is the excessreturn of a strategy buying the currencies with the largest forward discount and selling those with the smallest discount.V OLEQTY is a global volatility factor estimated on the equity market. V OLFX is a global volatility factor estimatedon the currency market. Ex-ante profitable currencies are C1, C18, C19, C20 and C21. The test using ex-ante profitablecurrencies includes also an equally weighted portfolio of non ex-ante profitable currencies as defined by equation (2) atthe 95% threshold. The table reports iterated GMM estimates of the SDF parameter, b, and the factor risk premia λreported in monthly percentages. Cross sectional R2 as well as test statistics J for the overidentifying restrictions arereported. Standard errors for statistics are in brackets. The p-value of the J test is also in brackets.

Panel: all countries 99-13

b λ R2 J b λ R2 J

5 Equally Weighted portfolios Ex-ante profitable currencies

RX 6.46 0.33 -0.58 7.12 RX -4.27 0.33 0.33 6.88(6.17) (0.27) (0.06) (6.55) (0.27) (0.14)

VOLEQTY -1.37 -6.00 VOLEQTY -2.10** -5.80**(0.61) (3.18) (0.72) (2.81)

RX -5.48 0.21 0.42 4.62 RX 8.97 0.29 -0.10 13.36(5.69) (0.29) (0.20) (4.59) (0.27) (0.01)

VOLFX -0.81** -8.63 VOLFX 0.24 -1.38(0.32) (5.75) (0.22) (4.74)

52

Table X: Simulated Portfolios’ descriptive statistics

The table reports the mean returns, standard deviation, skewness and kurtosis of simulated currency portfolios invested in the carrytrade. Carrysize is a strategy in which the set of investable currencies is determined dynamically by the time varying transactioncosts while the relative weights of the currencies depend on their time-varying relative expected returns, Carryall is a strategy inwhich all the currencies are considered investable and finally, Carrymax is an equally weighted portfolio of the two extreme currencies(largest and smallest interest rate). The statistics are reported for annualized log excess returns accounting for transaction costs(bid-ask spreads). The data are monthly. We also report the annualized Sharpe Ratios (SR), modified Sharpe Ratio MV aR, and theratios of the mean return to the mean drawdown MDD and to the maximum drawdown MMaxDD as defined in Appendix I of thispaper. We also report the standard deviation of these statistics over 1000 trials. The figures are in basis points not percentages. Thepanel uses all countries with data from January 1999 to May 2013.

mean sd SK KT SR MV aR MDD MMaxDD

Benchmark simulationsCarrysize annualized 0.0740 0.1056 -0.5703 5.5640 0.7004 0.5439 0.1517 0.0171

sd-dev (0.0256) (0.0305) (0.023) (0.7033) (0.192) (0.237) (0.062) (0.006)Carryall annualized 0.0312 0.0466 -0.8313 6.3781 0.6678 0.4569 0.1785 0.0214

sd-dev (0.0201) (0.0228) (0.079) (1.285) (0.311) (0.195) (0.092) (0.0117)Carrymax annualized 0.1542 0.1703 -0.7650 7.0741 0.9053 0.8191 0.2428 0.0228

sd-dev (0.0623) (0.0421) (0.027) (1.54) (0.364) (0.241) (0.129) (0.009)

Linearly increasing betasCarrysize annualized 0.0819 0.1197 -0.6533 4.8532 0.6844 0.4903 0.1352 0.0237

sd-dev (0.0322) (0.0295) (0.026) (0.807) (0.246) (0.242) (0.068) (0.011)Carryall annualized 0.0483 0.0589 -0.9807 5.9623 0.8191 0.5571 0.1787 0.0165

sd-dev (0.0195) (0.0212) (0.156) (0.995) (0.279) (0.191) (0.090) (0.0079)Carrymax annualized 0.1397 0.1411 -0.9144 6.7625 0.9902 0.8618 0.2281 0.0242

sd-dev (0.0639) (0.0623) (0.101) (1.23) (0.396) (0.314) (0.106) (0.007)

53

Table XI: Simulation results

The table reports the rejection rate of simulated asset pricing tests using alternative designs for thetest asset. We simulate 1000 samples on which we test Lustig, Roussanov and Verdelhan (2011) model.The tests are based alternatively on 3, 4, 5, 6, 7 equally weighted portfolios, on 6 portfolios definedaccording to equation (2) and finally on all the currencies. The second column of the table says that therejection rate of the tests based on 3 portfolios is 55%. The table reports also the number of trials forwhich 0 to 7 designs where favourable to the risk premia story. The second column says that for 217trials there is only one design favourable. The percentage of trials for which we report no favourabledesign is almost 50%. Monthly returns from January 1999 to May 2013 for all countries. Test assetsare currency portfolios sorted on their forward discount. The tests are similar to the tests run in section 3.

Sample: G10 99-13 - Benchmark simulations

Type of ptf 3-p 4-p 5-p 6-p 7-p Ex-ante P All ccyRejection rate 0.55 0.73 0.68 0.88 0.94 0.83 1

Nbr of designs favourable 1 2 3 4 5 6 7 0Nbr of trials 217 144 103 27 13 1 0 495

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Figure 1: The return to the carry trade

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Figure 2: Mean returns for currencies sorted on their forward discount f − s: full sample.

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Sample January 1999 to May 2013 for all countries. Monthly observations. The graph shows the monthly meanreturns for currencies sorted on their forward discount. The forward discounts are in monthly percentage in X-axiswhile the monthly mean returns are reported along the Y-axis. Black points are for currencies which are ex-anteprofitable in the sample according to Equation (2) using the 95% threshold. Empty points are for ex-ante nonprofitable currencies. The vertical line separates the set of currencies which have, on average, a negative forwarddiscount (funding currencies) from those with a positive forward discount (investment currencies). The horizontalline discriminates between the currencies with a positive average return (above the line) and a negative averagereturn (below the line). The graph shows that the relationship between the forward discount and the mean returnis almost monotonic for ex-ante profitable currencies but not for ex-ante non profitable currencies. This observationsuggests that high yielding currencies offer a premium to investors. This premium looks almost linearly distributed:the size of the premium increases as much as the forward premium. As a consequence, carry trade strategies thatuse all the information conveyed by the dispersion of the signal produce better statistics than strategies in whichbets are normalized.

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Figure 3: Mean returns for currencies sorted on their forward discount f − s: subsamples.

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Upper left graph: sample December 1984 to May 2013 for G10 countries. Upper right graph: sample January 1999to May 2013 for G10 countries. Lower left graph: sample January 1999 to May 2013 for alphabetically orderedcountries from A to L. Lower right graph: sample January 1999 to May 2013 for alphabetically ordered countriesfrom M to Z. Monthly observations. The graphs show the monthly mean returns for currencies sorted on theirforward discount. The forward discounts are in monthly percentage in X-axis while the monthly mean returnsare reported along the Y-axis. Black points are for currencies which are ex-ante profitable in the sample (95%threshold). Empty points are for ex-ante non profitable currencies. The vertical line separates the set of currencieswhich have, on average, a negative forward discount (funding currencies) from those with a positive forward discount(investment currencies). The horizontal line discriminates between the currencies with a positive average return(above the line) and a negative average return (below the line). The graphs show that the relationship between theforward discount and the mean return is almost monotonic for ex-ante profitable currencies but not for ex-ante nonprofitable currencies. This observation suggests that high yielding currencies offer a premium to investors. Thispremium looks almost linearly distributed: the size of the premium increases as much as the forward premium. Asa consequence, carry trade strategies that use all the information conveyed by the dispersion of the signal producebetter statistics than strategies in which bets are normalized.

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Figure 4: 36-month rolling Sharpe Ratios for Carrymax, Carryn and Carrysize

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Sample January 1999 to May 2013 for all countries. Monthly observations. The graph shows the annualized SharpeRatio calculated over a rolling window of 36 months. Carrysize is a strategy in which the set of investable currenciesis determined dynamically by the time varying transaction costs while the relative weights of the currencies dependon their time-varying relative expected returns, Carryn is also an equally version of the strategy but over a fixedand predefined number, n, of currencies we set, here at 4, and Carrymax is an equally weighted portfolio of thetwo extreme currencies (largest and smallest interest rate). The graph shows that the strategies Carrysize andCarrymax tend to overperform Carryn in most of the sample.

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Figure 5: Mean returns for currencies sorted on their β to HML-FX

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Sample January 1999 to May 2013 for all countries. Monthly observations. The graph shows the monthly meanreturns for currencies sorted on their beta to the risk factor HML-FX. The betas are on the X-axis while the monthlymean returns are reported along the Y-axis. Black points are for currencies which are ex-ante profitable in thesample according to Equation (2) using the 95% threshold. They are labeled C1 and C18 to C21. Empty pointsare for ex-ante non profitable currencies while P1 is the portfolio mixing them. The crosses are for the assets ofthe 3-portfolio test. The vertical line separates the set of currencies which exhibit a negative beta from those witha positive beta. The horizontal line discriminates between the currencies with a positive average return (above theline) and a negative average return (below the line). The graph shows that the relationship between the betas andthe mean returns is almost monotonic for ex-ante profitable currencies but not for ex-ante non profitable currencies.This observation suggests that non-profitable currencies convey noise which might be filtered out by grouping themin a unique portfolio. However, looking at the 3-portfolio test, we see that grouping too many assets in a smallnumber of portfolios erases the phenomenon of interest.

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