Working Paper Series Faculty of Finance. No. 15 Currency Portfolios, Returns and Asset Pricing Tests. Philippe Dupuy, Jessica James, Ian W.

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1 Working Paper Series Faculty of Finance No. 15 Currency Portfolios, Returns and Asset Pricing Tests Philippe Dupuy, Jessica James, Ian W. Marsh

2 Currency portfolios, returns and asset pricing tests Philippe Dupuy Jessica James Ian W. Marsh Grenoble EM Cass Business School Cass Business School Risk premia may justify the return to the FX carry trade but the identity of the risk factors is still open to debate. We show that one may obtain very different results in terms of risk adjusted returns and the results of asset pricing tests depending upon the design of the carry trade portfolios. In particular, both investors and academics may want to consider non-equally weighted and/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 We also provide simulation-based evidence confirming our conclusions. JEL Classification numbers: F31, G12 Keywords: Exchange Rate, Carry trade, Portfolio, Risk premium 1

3 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 characteristics 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 coefficients 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 1 Baroso and Santa Clara (2013) and Della Corte et al. (2008) use optimized, hence non equally weighted, portfolios. 2 The 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). 3 The 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

4 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 attractive 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 portfolios 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 currencies 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 conclusions of some recent studies relating the cross-section of the currencies to currency-based risk 3

5 factors (e.g., Lustig, Roussanov and Verdelhan, 2011; Menkhoff et al., 2012; Burnside, 2012; Dobrynskaya, 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 monotonic 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 portfolios. 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 important 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

6 can produce significantly different results for alternative designs of the carry trade portfolios. Finally, 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 nonequally 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: R c t+1 = f c t s c t+1 with Rc t+1 the log periodic excess return of currency c observed in t+1, f c t the log forward rate of currency c observed at time t and s c t+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 4 Sorting 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

7 trade is profitable if: 5 R long t+1 = f t b s a t+1 > 0 if the investor buys the FCU Rt+1 short = ft a + s b t+1 > 0 if the investor sells the FCU (1) Portfolio design. If agents think 1/s b t+1 and 1/sa t+1 are martingales6 then the decision rule is the following: Buy the FCU if E(f b t /s a t+1 ) = f b t /s a t > 1 (2) Sell the FCU if E(f a t /s b t+1 ) = f a t /s b t < 1 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: x t = Max n i=1 w i(ft b /s a t ) i j w iw j V ij if ft b /s a t > 1 Min n i=1 w i(ft a /s b t) + i j w iw j V ij if ft a /s b t < 1 (3) with n the number of currencies, w i the portfolio weights and V ij, the variance-covariance ma- 5 We 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 + r US ) from the strategy that borrows funds in a low-interest-rate currency and lends them in a high-interest-rate currency. 6 Martingale: E t(1/s b t+1) = 1/s b t and E t(1/s a t+1) = 1/s a t 6

8 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 standarddeviation (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 Carry size. 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 Carry ba 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 Carry max. 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

9 and non-profitable currencies and they might invest in equally weighted portfolios containing all the currencies. We call this strategy Carry all. 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 Carry n. The return to currency speculation. For all these designs, the returns to the portfolio of long positions and the portfolio of short positions are: R long t+1 = 1 nl (nl+ns) i=1 w i rt+1 i Rt+1 short 1 = ns (nl+ns) j=1 w j r j t+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 w i > 0 and w j < 0, the return of the carry trade strategy for the 5 types of portfolio designs is the sum R long t for Carry max and Carry n, nl = ns. and Rt short. Of course, But for Carry all, Carry ba and Carry size, 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 positions 7 ). In the next section, we examine the risk-return statistics of the five designs of the portfolio of carry trade. For Carry n, 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 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 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 7 Alternatively, 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

10 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, Thailand, Turkey and the UK 8. 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 distribution of returns, the annualized Sharpe Ratio (SR) and modified Sharpe Ratio (M V ar ), the ratio of the mean return to the mean drawdown (M DD ) and finally the ratio of the mean return to the maximum drawdown (M MaxDD ). 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 Carry all exhibit significant Sharpe Ratios. The largest ones are obtained with the Carry size and, in particular, Carry max versions of the carry trade. The other performance ratios are also highest for these two strategies. 8 We share 16 currencies with DB GCH which does not include Russia, Thailand and the European countries which have adopted the EUR. 9 Data for 1-week maturity contracts were not available on a daily basis for certain currencies in the early As a consequence, the sample of weekly data covers only the period from October 2002 to May 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. 10 We estimate the distribution of the statistics by generating 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

11 Behind these, Carry n and Carry ba generate performance ratios which are broadly similar. Although Carry ba, which invests in a time-varying set of currencies, provides somewhat better ratios than Carry n, some of its performance measures are not statistically significant. Investors should stay away from the Carry all version of the strategy because it underperforms all alternatives, and often provides insignificant performance. We conclude that investors should favor Carry max and Carry size. This may be surprising since it calls for the construction of non-equally weighted portfolios (Carry size ) or non diversified portfolios (Carry max ) 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 C For every currency C1 to C21, we report the annualized mean, standard-deviation, skewness and kurtosis of the distribution of returns, the annualized Sharpe Ratio (SR) and modified Sharpe Ratio (M V ar ), the ratio of the mean return to the mean drawdown (M DD ) and finally the ratio of the mean return to the maximum drawdown (M MaxDD ) 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. which have turned to be ex-ante profitable in, at least, 95% of the dataset, i.e. Those currencies are the ones currencies for which f b t /s a t > 1 or f a t /s b t < 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 11 In 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. 12 The non annualized version of these ratios and their standard errors obtained by bootstrapping the statistics are available upon request 10

12 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 Carry all 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, Carry ba exhibits better risk-return ratios than Carry all. Similarly, Carry n produces better risk-return ratios than Carry all 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 C As a result, Carry ba exhibits somewhat better statistics than Carry n. 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 Carry size, further improves the risk-return ratios of the strategy. On the other hand, equally weighting the currencies, as in Carry ba 13 The 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

13 and Carry n is like minimizing the information we hold about the dispersion of the future returns 14. To get further intuition on this point, we report that the R 2 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 R 2 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 R 2 jumps to 54%. Then, if we exclude the currencies which are ex-ante profitable in only a limited period of time, the R 2 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 Carry max. Table II and III about here In the next section we perform several robustness tests to confirm our findings. We look especially 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 Weighting 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. 15 As a result minimum variance portfolios do not improve the results. 12

14 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 Carry size still offers a better set of riskreturn statistics than the conventional Carry n 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 Carry size tends to outperform the alternative strategies. However, in this sample, Carry all 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: Carry max and Carry size offer better statistics although Carry size 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

15 Carry n, Carry max and Carry size. The graphs show that Carry max and Carry size perform better than Carry n. The results hold whether we look at a quiet period (mid 2000 for instance) or turbulent periods ( ). 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, Carry n, yields poor results. This is independent 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: Carry n and Carry all exhibit negative returns and risk-return ratios while Carry ba and Carry size 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: (r af t rt bus )n/b n/b) (1+r bus t > Sa t S b t 1 if the investor buys the FCU (5) (r bf t rt aus )n/b n/b) (1+r aus t < Sb t S a t 1 if the investor sells the FCU r af t is the foreign short-term interest rate at time t, r bus 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

16 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 FCU n b > S (r af t a t S t b 1 rt bus ) ( Sa t S t b 1)rt bus if the investor sells the FCU n S b t b < S t a 1 (r bf t rt aus ) ( Sb t S t a 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 Verdelhan (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 Carry n ). Similarly, Menkhoff et al. (2012) and Burnside (2012) test a 16 This 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. 17 Of course, these numbers are only rough estimates since the bid ask spreads are time-varying and heterogenous across currencies. 15

17 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 risk 18. The papers in this literature typically follow the same methodology: to reduce errors in parameter 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 18 An 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 , a period characterised by a very large number of extreme returns. 16

18 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 portfolio 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 limited 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 test 19. Yet, there exists n p=3 Cn 1 p 1 = n (n 1)! p=3 (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 currencies 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 theory, 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 19 To 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

19 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 portfolio 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

20 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: R it+1 = α i + f t+1β i + ɛ it+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 estimator 20 starting from the identity matrix as weighting matrix W T = 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 matrix 21. 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 20 We 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. 21 This 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