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. Marsh
Grenoble EM
Cass Business School
Cass Business School
[email protected]
[email protected]
[email protected]
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 1984-2013. We also provide simulation-based evidence confirming our conclusions.
JEL Classification numbers: F31, G12 Keywords: Exchange Rate, Carry trade, Portfolio, Risk premium
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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.
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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
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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 SDFGMM methodology implemented in the seminal work of Lustig, Roussanov and Verdelhan (2011)
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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 c c the log periodic excess return of currency c observed in t + 1, ftc the as: Rt+1 = ftc − sct+1 with Rt+1
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 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)
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trade is profitable if:5 Rlong = ftb − sat+1 > 0 t+1
if the investor buys the FCU (1)
Rshort = −f a + sb > 0 if the investor sells the FCU t t+1 t+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(ftb /sat+1 ) = ftb /sat > 1 Sell the FCU if
(2)
E(fta /sbt+1 ) = fta /sbt < 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:
xt =
P P P M ax ni=1 wi (ftb /sat ) − i j wi wj Vij
if ftb /sat > 1
M in Pn w (f a /sb ) + P P w w V t i j i j ij i=1 i t
if fta /sbt < 1
(3)
with n the number of currencies, wi the portfolio weights and Vij , the variance-covariance ma5
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 + rU S ) from the strategy that borrows funds in a low-interest-rate currency and lends them in a high-interest-rate currency. 6 Martingale: Et (1/sbt+1 ) = 1/sbt and Et (1/sat+1 ) = 1/sat
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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 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
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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: Rlong = t+1 Rshort = t+1
1 (nl+ns)
Pnl
i=1 wi
i ∗ rt+1
(4) 1 (nl+ns)
Pns
j=1 wj
j ∗ rt+1
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 Rtlong and Rtshort . 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 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.
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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 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 distribution 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 (MM axDD ). 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. 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 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. 10 We 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.
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Behind these, Carryn and Carryba generate performance ratios which are broadly similar. Although 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 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 (MM axDD )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 ftb /sat > 1 or fta /sbt < 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
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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 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.
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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 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 4.
14
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.
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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 riskreturn 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 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: 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: f
US
(rta −rtb
f US (rtb −rta )n/b US (1+rta n/b)
)n/b
US (1+rtb n/b)
>
Sta Stb
− 1 if the investor buys the FCU (5)
b
Sta −1 Stb Sa f U S US (rta −rtb )−( tb −1)rtb St
n < b
Stb −1 Sta Sb f US US (rtb −rta )−( S at −1)rta 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 Carryn ). 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
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 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 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 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 test19 .
Yet, there exists
Pn
n−1 p=3 Cp−1
=
(n−1)! p=3 (p−1)!(n−p)!
Pn
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
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
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: 0 Rit+1 = αi + ft+1 βi + 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 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 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
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 ; HM L − F X, the difference between the returns of the two extreme portfolios defined in each strategy; V OLEQT Y , a proxy for the volatility of global equity market returns; and V OLF X , 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 HM L − F X. 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 P 1 to P i 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 HM L − F X 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, 22
In 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. 23 For 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 currencies 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 HM L − F X, 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 24
In 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. 25 For 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 portfolios 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 construction 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 conclusions 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 construction 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. 26 In the special case of HM L − F X, 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 significant. 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 numerous 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 HM L − F X. 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 profitable 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 27
We 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 HM L − F X 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 monotonically 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 insignificant 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
ˆ because the So far, the estimation of the risk premia is based on the estimated vector of betas, B 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 HM L − F X 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 HM L − F X 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 assumptions 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 described in section 2 when the data are simulated 1000 times under the benchmark assumptions. We 28
We 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 HM L − F X increase linearly from the low yielding currency to the high yielding currency, everything else being 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 assumptions: if the returns to the currencies C1 to C21 were mutually uncorrelated and had common variance the construction of the factors RX and HM L − F X would imply a beta of 1 for RX and betas of -0.5, 0, 0,..., 0, 0.5 for HM L − F X. Under these assumptions, the middle range currencies 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 association 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 diversification 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 section, 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 report, 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 29
We 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 estimated 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 P ( e2 = 0.000155). The ones for which only one construction or none of the possible designs work P 2 P 2 exhibit on average significantly higher errors: respectively e = 0.000206 and e = 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 significantly different returns for practitioners and statistical results for academics. Together, we suggest that both academics and practitioners may want to consider non-diversified and nonequally-weighted portfolios built upon the rationale of portfolio construction that we study. Nondiversified 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, 30
This 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: Buy wi =fbt /sat − 1 if ftb /sat > 1, xt = Sell wi =fat /sbt − 1 if fta /sbt < 1, 0 otherwise
(8)
Equation (1) of this Appendix specifies portfolios in which the currencies are weighted according 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 profitable currencies because nothing helps to discriminate among them: Buy wi = 1 if ftb /sat > 1, xt = Sell wi = −1 if fta /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: Buy wi = 1 if ftb /sat = max(ftb /sat ), xt = Sell wi = −1 if fta /sbt = min(fta /sbt ), 0 otherwise
(10)
We call this design Carrymax because only the currencies with the maximum (minimum) interest 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 Sell
if ft /st > 1, (11)
wi = −1 if ft /st < 1,
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 Sell
if N − n + 1 < Rankft /st < N, (12)
wi = −1 if 1 < Rankft /st < n,
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 introduce 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 ratios 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 ; DM ax }
The standard-deviation, sd(x), is a sufficient measure of risk only if the distribution is symmetrical. 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 P rob (x ≤ µ) = p
(14)
As it is traditional, in this paper, we estimate the quantile for µ = 95% such as: P rob (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 : 1 2 1 3 ∼ Qα = µ − Z α + Z − 1 SK + Z − 3Zα K σ 6 α 24 α
(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 = xmax − xt t with xmax = max {xi |i [0, t]} (17) t then ∀xt
xt = xmax ⇔ Dt = 0 t
otherwise Dt ≥ 0 The mean drawdown in X is then defined as: 31
Higher moments expansion can be found in Stuart and al. (1999).
34
(16)
n
Dmean
1X = Dt n
(18)
i=1
n being the number of drawdowns in X.
and the maximum drawdown is defined as: DM ax = 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 MM axDD .
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)
Mt+1 = ξ 1 − (ft+1 − µ)0 b)
(21)
with Mt+1 a linear SDF of the form:
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)
E(Rt+1 ) = βλt+1
(23)
or
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 0 βi + it+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
[1 − (ft+1 − µ)0 b)] Rt+1
g(Rt+1 , ft+1 , θ) = ft+1 − µ Σ((ft+1 − µ)(ft+1 − µ)0 ) − Σft+1
(25)
where θ contains the parameters (b, µ, Σft+1 ).
ˆ as λ ˆ = ˆbΣf ˆ . Standard errors of λ come from the Then, estimates of λ are obtained from ˆb and Σf ˆ . estimation of the variance of the function ˆbΣf
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 NeweyWest procedure, with the number of lags determined according to Andrews (1991). Predicted mean returns are cov(R, f )ˆb and pricing errors are α ˆ = µR − cov(R, f )ˆb. As a mean to test the validity ˆ where VT is a consistent estimate of of the model, we run a test of the pricing errors J = T α ˆ 0 VT−1 α √ the asymptotic covariance matrix of T α ˆ . 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 = Σ¯ α2 /Σ(R¯i t − R¯t )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 OLEQT Y . Similarly we build a global volatility factor extracted from the currency market. We call it V OLF X . 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. 0
p + µt+1 δF Mt+1 = α + β Rt+1
(26)
p where Rt+1 is the vector of excess returns of the test assets and δF Mt+1 is alternatively
V OLEQT Y t+1 and V OLF X t+1 . Then the factor-mimicking portfolio’s excess return is given by F M = βˆ0 Rp . Rt+1 t+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 transaction costs while the relative weights of the currencies depend on their time-varying relative expected returns, Carryba is the equally weighted 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 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 the ratios of the mean return to the mean drawdown MDD and to the maximum drawdown MM axDD as defined in Appendix B of this paper. Monthly ratios and their standard-errors are also reported (e.g. Lo, 2002). They are obtained by bootstrapping the sample 10000 times. The figures are in basis points not percentages. The panel uses all countries with data from January 1999 to May 2013.
44
Carrysize annualized monthly sd-dev Carryba annualized monthly sd-dev Carryn annualized monthly sd-dev Carryall annualized monthly sd-dev Carrymax annualized monthly sd-dev
mean 0.0893
sd 0.1222
SK -0.6083
KT 7.2268
0.0394
0.0684
-0.6101
6.0700
0.0242
0.0467
-0.3976
4.1057
0.0223
0.0512
-0.7560
5.6684
0.0808
0.0858
0.2104
5.3331
SR 0.7307 0.2109 (0.0843) 0.5760 0.1662 (0.0826) 0.5196 0.1500 (0.0791) 0.4359 0.1258 (0.0819) 0.9416 0.2718 (0.0773)
MV aR 0.5365 0.1548 (0.0943) 0.3892 0.1123 (0.0640) 0.3515 0.1014 (0.0664) 0.2748 0.0793 (0.0630) 0.7559 0.2182 (0.0795)
MDD 0.1750 0.1750 (0.0906) 0.1363 0.1363 (0.0838) 0.1288 0.1288 (0.0811) 0.0940 0.0940 (0.0695) 0.2349 0.2349 (0.0984)
MM axDD 0.0424 0.0424 (0.0187) 0.0185 0.0185 (0.0201) 0.0238 0.0238 (0.0140) 0.0140 0.0140 (0.0158) 0.0419 0.0419 (0.0238)
Table II: Currencies’ descriptive statistics
The table reports the mean returns, standard deviation, skewness and kurtosis for currencies sorted on their forward discount f − s. C1 is the lowest yielding currency while C21 is the highest yielding currency. We also report the mean forward discount for each currency (f − s). 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 the ratios of the mean return to the mean drawdown MDD and to the maximum drawdown MM axDD as defined in Appendix B of this paper. The last column reports the frequency at which each ranked currency was ex-ante profitable. The figures are in basis points not percentages. The panel uses all countries with data from January 1999 to May 2013.
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21
f-s -0.0027 -0.0015 -0.0011 -0.0008 -0.0006 -0.0005 -0.0004 -0.0002 0.0000 0.0002 0.0003 0.0005 0.0007 0.0011 0.0016 0.0022 0.0030 0.0041 0.0061 0.0095 0.0185
mean -0.0030 -0.0039 0.0086 -0.0252 -0.0131 0.0092 0.0607 0.0205 0.0523 0.0361 0.0265 0.0211 -0.0048 -0.0329 0.0462 0.0379 -0.0164 0.0148 0.0022 0.0665 0.1489
sd 0.1214 0.0840 0.0867 0.1149 0.0919 0.0996 0.1132 0.1144 0.1050 0.1131 0.0934 0.1034 0.1142 0.1243 0.1158 0.1217 0.1333 0.1289 0.1405 0.1859 0.1616
SK -1.2945 0.1938 -0.3340 -0.4427 -0.2846 -2.1338 0.0322 -0.4910 -0.0495 -0.1063 -0.1952 -0.1965 -0.8046 -0.8578 -0.3886 -0.6822 -1.5933 -0.8460 -1.1856 -1.8157 -0.7329
KT 11.8697 3.6402 4.9061 5.2630 5.7095 15.1546 6.4764 6.2439 3.5613 4.3952 3.4465 3.9938 6.1240 5.5035 6.4500 4.6988 8.8547 5.9012 6.5273 16.5783 5.2441
45
SR -0.0251 -0.0465 0.0996 -0.2191 -0.1427 0.0927 0.5363 0.1794 0.4981 0.3195 0.2833 0.2045 -0.0420 -0.2647 0.3988 0.3115 -0.1227 0.1144 0.0159 0.3579 0.9218
MV aR -0.0165 -0.0299 0.0573 -0.1252 -0.0814 0.0674 0.4015 0.1292 0.3402 0.2247 0.1723 0.1207 -0.0256 -0.1248 0.2898 0.1769 -0.0764 0.0685 0.0110 0.2737 0.6806
MDD -0.0046 -0.0085 0.0225 -0.0382 -0.0287 0.0207 0.1451 0.0371 0.1123 0.0608 0.0514 0.0485 -0.0088 -0.0442 0.0958 0.0614 -0.0237 0.0240 0.0032 0.0857 0.2111
MM axDD -0.0012 -0.0026 0.0048 -0.0064 -0.0049 0.0023 0.0301 0.0058 0.0341 0.0148 0.0127 0.0075 -0.0012 -0.0069 0.0153 0.0108 -0.0045 0.0030 0.0005 0.0151 0.0311
Freq 1.00 0.73 0.54 0.43 0.35 0.30 0.23 0.17 0.08 0.30 0.34 0.39 0.47 0.56 0.72 0.81 0.89 0.98 0.99 1.00 1.00
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 yielding currency while C21 is the highest yielding currency. Only the statistics for extreme currencies are reported. The statistics are reported for monthly log excess returns accounting for transaction costs (bid-ask spreads). The data are monthly. The figures are in basis points not percentages. The panel uses all countries with data from January 1999 to May 2013.
C1 C2 .. .
C1 0.0011 0.0002 .. .
C2 0.0002 0.0006 .. .
C3 0.0005 0.0004 .. .
C4 0.0005 0.0004 .. .
C5 0.0003 0.0003 .. .
... ... ... .. .
C16 0.0006 0.0004 .. .
C17 0.0004 0.0004 .. .
C18 0.0005 0.0003 .. .
C19 0.0004 0.0003 .. .
C20 0.0002 0.0001 .. .
C21 0.0004 0.0002 .. .
C18 C19 C20 C21 C22
0.0004 0.0005 0.0004 0.0002 0.0004
0.0004 0.0003 0.0003 0.0001 0.0002
0.0005 0.0006 0.0005 0.0004 0.0004
0.0005 0.0006 0.0003 0.0002 0.0003
0.0004 0.0005 0.0005 0.0003 0.0004
... ... ... ... ...
0.0007 0.0008 0.0007 0.0006 0.0007
0.0015 0.0006 0.0007 0.0006 0.0008
0.0006 0.0014 0.0008 0.0006 0.0008
0.0007 0.0008 0.0016 0.0009 0.0010
0.0006 0.0006 0.0009 0.0029 0.0010
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 portfolios invested in the carry trade. Carrysize is a strategy in which the set of investable currencies is determined dynamically by the time varying transaction costs while the relative weights of the currencies depend on their time-varying relative expected returns, Carryba is the equally weighted 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 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). We also report the annualized 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 MM axDD as defined in Appendix B of this paper. The data are monthly. The figures are in basis points not percentages. The panel A uses G10 countries from January 1999 to May 2013.The panel B uses G10 countries from December 1984 to May 2013.
Carrysize Carryba Carryn Carryall Carrymax
mean 0.0699 0.0504 0.0259 0.0354 0.0729
sd 0.1014 0.0796 0.0442 0.0497 0.1211
Carrysize Carryba Carryn Carryall Carrymax
mean 0.0659 0.0414 0.0236 0.0331 0.0703
vol 0.0946 0.0697 0.0472 0.0507 0.1277
Panel A: G10 SK KT -0.0469 5.9169 -0.1795 5.9958 -0.3806 4.1166 -0.4433 4.0781 -0.4352 4.5317 Panel B: G10 SK KT -0.4673 6.2259 -0.3712 6.3016 -0.7682 4.5867 -0.8183 5.1138 -0.6546 5.0869
47
99-13 SR 0.6893 0.6336 0.5858 0.7120 0.6021 84-13 SR 0.6973 0.5948 0.5002 0.6524 0.5505
MV aR 0.5411 0.4859 0.3332 0.4517 0.4263
MDD 0.1955 0.1518 0.1071 0.1596 0.1403
MM axDD 0.0495 0.0285 0.0199 0.0339 0.0181
MSR 0.4498 0.4174 0.2799 0.4033 0.3219
meanDD 0.1703 0.1247 0.0948 0.1382 0.1244
MaxDD 0.0270 0.0234 0.0182 0.0207 0.0175
Table V: The return to the carry trade in random subsets. 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 transaction costs while the relative weights of the currencies depend on their time-varying relative expected returns, Carryba is the equally weighted 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 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). We also report the annualized 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 MM axDD as defined in Appendix B of this paper. The data are monthly. The figures are in basis points not percentages. Panel A uses all countries from A to L with data from January 1999 to May 2013. Panel B uses all countries from M to Z with data from January 1999 to May 2013.
Carrysize Carryba Carryn Carryall Carrymax
mean 0.0678 0.0464 0.0050 0.0259 0.0721
Carrysize Carryba Carryn Carryall Carrymax
mean 0.0871 0.0301 0.0020 0.0212 0.0468
Sample: All sd SK 0.1863 -3.4638 0.0710 -0.3443 0.0251 -0.1119 0.0558 -0.8863 0.0891 0.2764 Panel B: All vol SK 0.1746 1.0607 0.0757 -1.0801 0.0444 -0.8873 0.0537 -0.7163 0.0840 -0.5739
countries KT 31.9020 5.8389 4.6627 6.3613 6.4706 countries KT 24.8980 8.1980 4.9769 5.6426 5.0454
48
A to L 99-13 SR MV aR 0.3640 0.3425 0.6534 0.4526 0.1994 0.1095 0.4649 0.3215 0.8100 0.5954 M to Z 99-13 SR MSR 0.4988 0.4210 0.3980 0.2780 0.0451 0.0265 0.3944 0.2824 0.5573 0.3763
MDD 0.1069 0.1484 0.0416 0.0969 0.2030
MM axDD 0.0102 0.0357 0.0068 0.0182 0.0365
meanDD 0.1648 0.0936 0.0093 0.0812 0.1118
MaxDD 0.0218 0.0114 0.0018 0.0082 0.0243
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 invested in the carry trade. Carrysize is a strategy in which the set of investable currencies is determined dynamically by the time varying transaction costs while the relative weights of the currencies depend on their time-varying relative expected returns, Carryba is the equally weighted 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 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). We also report the annualized 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 MM axDD as defined in Appendix B of this paper. The figures are in basis points not percentages. The panel A uses all countries with weekly data from October 2002 to May 2013. The panel B uses all countries with quarterly data from January 1999 to May 2013.
Carrysize Carryba Carryn Carryall Carrymax
mean 0.1517 0.0422 -0.0233 -0.0439 0.0198
Carrysize Carryba Carryn Carryall Carrymax
mean 0.0780 0.0266 0.0145 0.0181 0.0334
Panel A: Weekly data all countries 02-13 sd SK KT SR MV aR 0.2846 -1.1622 76.7866 0.5330 0.7659 0.0853 -0.2943 8.2086 0.4943 0.3509 0.0435 -0.9198 6.8637 -0.5345 -0.3103 0.0459 -0.8210 8.5390 -0.9571 -0.5671 0.0901 -0.5346 5.4684 0.2201 0.1340 Panel B: Quarterly data all countries 99-13 sd SK KT SR MV aR 0.1264 -0.1241 4.7886 0.6171 0.5874 0.0578 -0.5150 3.7844 0.4607 0.3006 0.0546 -1.1359 4.3622 0.2652 0.1360 0.0510 -0.8573 4.4144 0.3558 0.2245 0.1040 -0.8640 6.2811 0.3211 0.2262
49
MDD 0.1132 0.0571 -0.0442 -0.0714 0.0216
MM axDD 0.0057 0.0063 -0.0034 -0.0055 0.0022
MDD 0.2518 0.1581 0.1064 0.1254 0.1072
MM axDD 0.0779 0.0504 0.0369 0.0341 0.0290
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 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 portfolio is the excess return of 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 are defined 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 the β of equation (7) as well as heteroskedasticity consistent standard errors and R2 . Significant loadings at the 5% level are indicated with two*.
Ptf P1 P2 50
P3
3 EQW portfolios RX HML-FX R2 0.96** -0.39** 0.95 (0.02) (0.02) 1.07** -0.16** 0.90 (0.03) (0.02) 0.95** 0.58** 0.98 (0.02) (0.02)
Ptf P1 P2 P3 P4 P5
Panel: All countries 99-13 5 EQW portfolios 7 EQW portfolios 2 RX HML-FX R Ptf RX HML-FX 0.88** -0.33** 0.89 P1 0.77** -0.31** (0.02) (0.02) (0.03) (0.02) 1.00** -0.09** 0.89 P2 1.03** -0.11** (0.03) (0.02) (0.05) (0.02) 1.06** -0.15** 0.84 P3 1.06** -0.10** (0.03) (0.02) (0.04) (0.02) 1.19** 0.02 0.83 P4 1.03** -0.13** (0.04) (0.03) (0.04) (0.02) 0.87** 0.63** 0.95 P5 1.12** -0.03 (0.03) (0.02) (0.04) (0.02) P6 1.20** 0.03 (0.04) (0.03) P7 0.79** 0.66** (0.03) (0.02)
R2 0.82
Ptf C1
0.75
P1
0.78
C18
0.79
C19
0.81
C20
0.80
C21
0.93
ex-ante profitable ccy RX HML-FX 0.76** -0.36** (0.06) (0.03) 1.01** -0.04** (0.02) (0.01) 1.18** 0.05 (0.08) (0.03) 1.12** 0.12** (0.10) (0.04) 0.90** 0.25** (0.12) (0.05) 0.87** 0.61** (0.07) (0.03)
R2 0.60 0.96 0.54 0.43 0.33 0.80
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 on their 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 portfolio is the excess return of 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 are defined according to equation (2). P1 contains the 16 currencies which are ex-ante non-profitable (95% threshold). The table reports iterated GMM estimates of the SDF parameter, b, and the factor risk premia λ reported in monthly percentages. Cross sectional R2 as well as test statistics J for the overidentifying restrictions are reported. Standard errors for statistics are in brackets. The p-value of the J test is also in brackets.
b
λ
R2
J
3 Equally Weighted portfolios RX
1.68 (4.36)
0.18 (0.21)
HML-FX
7,38 (4.77)
0.36 (0.22)
0.99
4.11 (4.52)
0.24 (0.18)
HML-FX
4.53* (2.86)
0.63* (0.33)
0.18
R2
J
0.39
4.22 (0.23)
0.30
4.86 (0.30)
λ
5 Equally Weighted portfolios 0.01 (0.91)
7 Equally Weighted portfolios RX
b
RX
2.53 (4.68)
0.23 (0.21)
HML-FX
7.81** (3.26)
0.69** (0.28)
Ex-ante profitable currencies 13.16 (0.03)
RX
7.34 (4.49)
0.31 (0.17)
HML-FX
5.50** (2.07)
1.44** (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 their 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 portfolio is the excess return of a strategy buying the currencies with the largest forward discount and selling those with the smallest discount. V OLEQT Y is a global volatility factor estimated on the equity market. V OLF X is a global volatility factor estimated on the currency market. Ex-ante profitable currencies are C1, C18, C19, C20 and C21. The test using ex-ante profitable currencies includes also an equally weighted portfolio of non ex-ante profitable currencies as defined by equation (2) at the 95% threshold. The table reports iterated GMM estimates of the SDF parameter, b, and the factor risk premia λ reported in monthly percentages. Cross sectional R2 as well as test statistics J for the overidentifying restrictions are reported. 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
5 Equally Weighted portfolios RX
6.46 (6.17)
0.33 (0.27)
VOLEQT Y
-1.37 (0.61)
-6.00 (3.18)
RX
-5.48 (5.69)
0.21 (0.29)
VOLF X
-0.81** (0.32)
-8.63 (5.75)
λ
R2
J
0.33
6.88 (0.14)
-0.10
13.36 (0.01)
Ex-ante profitable currencies -0.58
0.42
7.12 (0.06)
4.62 (0.20)
52
RX
-4.27 (6.55)
0.33 (0.27)
VOLEQT Y
-2.10** (0.72)
-5.80** (2.81)
RX
8.97 (4.59)
0.29 (0.27)
VOLF X
0.24 (0.22)
-1.38 (4.74)
Table X: Simulated Portfolios’ descriptive statistics The table reports the mean returns, standard deviation, skewness and kurtosis of simulated 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 transaction costs while the relative weights of the currencies depend on their time-varying relative expected returns, Carryall is a strategy in which 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 the ratios of the mean return to the mean drawdown MDD and to the maximum drawdown MM axDD as defined in Appendix I of this paper. We also report the standard deviation of these statistics over 1000 trials. The figures are in basis points not percentages. The panel uses all countries with data from January 1999 to May 2013.
53
Benchmark simulations Carrysize annualized sd-dev Carryall annualized sd-dev Carrymax annualized sd-dev Linearly increasing betas Carrysize annualized sd-dev Carryall annualized sd-dev Carrymax annualized sd-dev
mean
sd
SK
KT
SR
MV aR
MDD
MM axDD
0.0740 (0.0256) 0.0312 (0.0201) 0.1542 (0.0623)
0.1056 (0.0305) 0.0466 (0.0228) 0.1703 (0.0421)
-0.5703 (0.023) -0.8313 (0.079) -0.7650 (0.027)
5.5640 (0.7033) 6.3781 (1.285) 7.0741 (1.54)
0.7004 (0.192) 0.6678 (0.311) 0.9053 (0.364)
0.5439 (0.237) 0.4569 (0.195) 0.8191 (0.241)
0.1517 (0.062) 0.1785 (0.092) 0.2428 (0.129)
0.0171 (0.006) 0.0214 (0.0117) 0.0228 (0.009)
0.0819 (0.0322) 0.0483 (0.0195) 0.1397 (0.0639)
0.1197 (0.0295) 0.0589 (0.0212) 0.1411 (0.0623)
-0.6533 (0.026) -0.9807 (0.156) -0.9144 (0.101)
4.8532 (0.807) 5.9623 (0.995) 6.7625 (1.23)
0.6844 (0.246) 0.8191 (0.279) 0.9902 (0.396)
0.4903 (0.242) 0.5571 (0.191) 0.8618 (0.314)
0.1352 (0.068) 0.1787 (0.090) 0.2281 (0.106)
0.0237 (0.011) 0.0165 (0.0079) 0.0242 (0.007)
Table XI: Simulation results
54
The table reports the rejection rate of simulated asset pricing tests using alternative designs for the test 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 defined according to equation (2) and finally on all the currencies. The second column of the table says that the rejection rate of the tests based on 3 portfolios is 55%. The table reports also the number of trials for which 0 to 7 designs where favourable to the risk premia story. The second column says that for 217 trials there is only one design favourable. The percentage of trials for which we report no favourable design is almost 50%. Monthly returns from January 1999 to May 2013 for all countries. Test assets are 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 Rejection rate 0.55 0.73 0.68 0.88 0.94 0.83 Nbr of designs favourable 1 2 3 4 5 6 Nbr of trials 217 144 103 27 13 1
All ccy 1 7 0
0 495
0.6 0.4 0.0
0.2
Cumulated return
0.8
1.0
1.2
Figure 1: The return to the carry trade
12/1984
01/1992
02/1999
03/2006
04/2013
Time
Sample December 1984 to May 2013 for G10 countries currencies. Monthly observations. The figure plots the cumulated return of investing one dollar in a portfolio of long positions in high interest rate currencies and short positions in currencies with low interest rate. The strategy is rebalanced every month and the proceeds are not reinvested. The portfolio is equally weighted and contains 3 currencies of each type. The strategy has offered a large positive mean return over the sample period with few episodes of crises.
55
Figure 2: Mean returns for currencies sorted on their forward discount f − s: full sample.
●
0.5
Monthly mean returns (%)
1.0
●
● ● ● ●
●
●
● ● ●
0.0
● ● ● ●
●
● ●
●
● ●
0.0
0.5
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1.5
Average Discount (%)
Sample January 1999 to May 2013 for all countries. Monthly observations. The graph shows the monthly mean returns for currencies sorted on their forward discount. The forward discounts are in monthly percentage in X-axis while the monthly mean returns are reported along the Y-axis. Black points are for currencies which are ex-ante profitable in the sample according to Equation (2) using the 95% threshold. Empty points are for ex-ante non profitable currencies. The vertical line separates the set of currencies which 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 graph shows that the relationship between the forward discount and the mean return is almost monotonic for ex-ante profitable currencies but not for ex-ante non profitable currencies. This observation suggests 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 that use all the information conveyed by the dispersion of the signal produce better statistics than strategies in which bets are normalized.
56
0.5
Figure 3: Mean returns for currencies sorted on their forward discount f − s: subsamples.
●
●
●
0.4
0.4
● ●
0.2
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0.0
0.3 0.2
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0.1
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●
0.0
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−0.1
−0.2
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−0.2
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0.2
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0.0
0.2
0.4
0.6
Average Discount (%)
1.0
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●
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0.4 0.2
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0.6 0.4
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0.6
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0.0
0.2
0.4
0.6
0.8
●
0.0
Average Discount (%)
0.5
1.0
1.5
Average Discount (%)
Upper left graph: sample December 1984 to May 2013 for G10 countries. Upper right graph: sample January 1999 to May 2013 for G10 countries. Lower left graph: sample January 1999 to May 2013 for alphabetically ordered countries from A to L. Lower right graph: sample January 1999 to May 2013 for alphabetically ordered countries from M to Z. Monthly observations. The graphs show the monthly mean returns for currencies sorted on their forward discount. The forward discounts are in monthly percentage in X-axis while the monthly mean returns are 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 currencies which 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 the forward discount and the mean return is almost monotonic for ex-ante profitable currencies but not for ex-ante non profitable currencies. This observation suggests 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 that use all the information conveyed by the dispersion of the signal produce better statistics than strategies in which bets are normalized.
57
Figure 4: 36-month rolling Sharpe Ratios for Carrymax , Carryn and Carrysize Rolling Sharpe Ratios
Rolling Sharpe Ratios size n=3
1.5 1.0
Sharpe Ratio
1.0
−0.5
−0.5
0.0
0.0
0.5
0.5
Sharpe Ratio
1.5
2.0
2.0
2.5
max n=3
01/2002
10/2004
08/2007
06/2010
04/2013
01/2002
Time
10/2004
08/2007
06/2010
04/2013
Time
Sample January 1999 to May 2013 for all countries. Monthly observations. The graph shows the annualized Sharpe Ratio calculated over a rolling window of 36 months. Carrysize is a strategy in which the set of investable currencies is determined dynamically by the time varying transaction costs while the relative weights of the currencies depend on their time-varying relative expected returns, Carryn is also an equally version of the strategy but over a fixed and predefined number, n, of currencies we set, here at 4, and Carrymax is an equally weighted portfolio of the two extreme currencies (largest and smallest interest rate). The graph shows that the strategies Carrysize and Carrymax 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
C21
C20 ●
0.5
Monthly mean returns (%)
1.0
●
● ● ● ●
● ● ●
C18 ●
P1 0.0
● ●
●
C19 ●
●
●
C1
● ●
●
● ●
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
Beta
Sample January 1999 to May 2013 for all countries. Monthly observations. The graph shows the monthly mean returns for currencies sorted on their beta to the risk factor HML-FX. The betas are on the X-axis while the monthly mean returns are reported along the Y-axis. Black points are for currencies which are ex-ante profitable in the sample according to Equation (2) using the 95% threshold. They are labeled C1 and C18 to C21. Empty points are for ex-ante non profitable currencies while P1 is the portfolio mixing them. The crosses are for the assets of the 3-portfolio test. The vertical line separates the set of currencies which exhibit a negative beta from those with a positive beta. The horizontal line discriminates between the currencies with a positive average return (above the line) and a negative average return (below the line). The graph shows that the relationship between the betas and the 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 them in a unique portfolio. However, looking at the 3-portfolio test, we see that grouping too many assets in a small number of portfolios erases the phenomenon of interest.
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