Do Peso Problems Explain the Returns

8/2/2019 Do Peso Problems Explain the Returns

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Do Peso Problems Explain the Returns

to the Carry Trade?

Craig BurnsideDuke University and NBER

Martin Eichenbaum

Northwestern University, NBER, and Federal Reserve Bank o Chicago

Isaac Kleshchelski

Washington University in Saint Louis

Sergio RebeloNorthwestern University, NBER, and CEPR

We study the properties o the carry trade, a currency speculation strategy in which an in-

vestor borrows low-interest-rate currencies and lends high-interest-rate currencies. This

strategy generates payos that are on average large and uncorrelated with traditional

risk actors. We argue that these payos reect a peso problem. The underlying peso

event eatures high values o the stochastic discount actor rather than very large negative

payos. (JEL F31)

The orward exchange rate is a biased orecaster o the uture spot exchange

rate. This act is oten reerred to as the orward-premium puzzle. We study

the properties o a widely used currency speculation strategy, known as the

carry trade, which exploits this puzzle. This strategy involves selling curren-

cies orward that are at a orward premium and buying currencies orward

that are at a orward discount. Transaction costs aside, this strategy is equiva-

lent to borrowing low-interest-rate currencies in order to lend high-interest-rate

currencies, without hedging the associated currency risk. Consistent with re-

sults in the literature, we fnd that the carry trade strategy applied to portolios

o currencies yields high average payos, as well as Sharpe ratios that are

substantially higher than those associated with the U.S. stock market.

This article is a substantially revised version o NBER Working Paper 12489, titled The Returns to CurrencySpeculation. We thank the editor, Geert Bekaert, two anonymous reerees, John Cochrane, John Heaton, JakubJurek, and Ravi Jagannathan or very useul comments. We also thank the Chicago Mercantile Exchange orproviding us with the currency-options data used in this article. Burnside is grateul to the National ScienceFoundation or fnancial support (SES-0516697). Send correspondence to Sergio Rebelo, Kellogg Graduate

School o Management, Northwestern University, Leverone Hall, Evanston, IL 60208, USA; telephone: 1-847-467-2329. E-mail: [email protected].

c The Author 2010. Published by Oxord University Press on behal o The Society or Financial Studies.All rights reserved. For Permissions, please e-mail: [email protected]:10.1093/rs/hhq138 Advance Access publication December 22, 2010

atVrijeUniversiteit,Libraryon

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The Review o Financial Studies / v 24 n 3 2011

The most natural explanation or the high average payo to the carry trade

is that it compensates agents or bearing risk. However, we show that linear

stochastic discount actors built rom conventional measures o risk, such as

consumption growth, the returns to the stock market, and the Fama-French

(1993) actors, ail to explain the payos to the carry trade. This ailure re-ects the absence o a statistically signifcant unconditional correlation be-

tween the payos to the carry trade and traditional risk actors. Our results are

consistent with previous work documenting that one can reject consumption-

based asset-pricing models using data on orward exchange rates (e.g., Bekaert

and Hodrick 1992 and Backus, Foresi, and Telmer 2001). More generally, it

has been difcult to use structural asset-pricing models to rationalize the risk-

premium movements required to account or the time-series properties o the

orward premium (see Bekaert 1996).

The most natural alternative explanation or the high average payo to thecarry trade is that it reects the presence o a peso problem. We use the term

peso problem to reer to the eects on inerence caused by low-probability

events that do not occur in the sample.1 Peso problems can in principle ex-

plain the positive average payo to the carry trade. To understand the basic

argument, suppose that a oreign currency is at a orward premium so that a

carry trade investor sells this currency orward. Assume that a substantial ap-

preciation o the oreign currency occurs with small probability. The investor

must be compensated or the negative payo to the carry trade in this state o

the world. So, the average risk-adjusted payo to the carry trade is positive innon-peso states.

In this article, we address the question o whether the large average payo

to the unhedged carry trade is compensation or peso-event risk. Our approach

to answering this question relies on analyzing the payos to a version o the

carry trade strategy that does not yield high negative payos in a peso state.

This strategy works as ollows. When an investor sells the oreign currency

orward, he simultaneously buys a call option on that currency. I the oreign

currency appreciates beyond the strike price, the investor can buy the oreign

currency at the strike price and deliver the currency in ulfllment o the or-ward contract. Similarly, when an investor buys the oreign currency orward,

he can hedge the downside risk by buying a put option on the oreign currency.

By construction, this hedged carry trade is immune to large losses such as

those potentially associated with a peso event.

We use data on currency options to estimate the average risk-adjusted pay-

o to the hedged carry trade. We fnd that this payo is smaller than the cor-

responding payo to the unhedged carry trade. This fnding is consistent with

1 In related work, Farhi and Gabaix (2008) and Brunnermeier, Nagel, and Pedersen (2008) emphasize the impor-tance o rare in-sample events. There are some moderately large negative payos to the unhedged carry tradein our sample. While these payos aect the proftability o the strategy, the average risk-adjusted payo to thecarry trade is positive.

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Do Peso Problems Explain the Returns to the Carry Trade?

the view that the average payo to the unhedged carry trade reects a peso

problem. Given this result, an obvious question is: What is the nature o the

peso event or which agents are being compensated? It is useul to distinguish

between two extreme possibilities. The frst possibility is that the salient ea-

ture o a peso state is large carry trade losses.

2

The second possibility is thatthe salient eature o a peso state is a large value o the stochastic discount ac-

tor (SDF). A key contribution o this article is to assess the relative importance

o these two possibilities. We fnd that a peso event reects high values o the

SDF in the peso state rather than very large negative payos to the unhedged

carry trade in that state.

The intuition or why the losses to the unhedged carry trade are small in

the peso state is as ollows. Any risk-adjusted payos associated with the

carry trade in the non-peso states must on average be compensated, on a risk-

adjusted basis, or losses in the peso state. According to our estimates, theaverage risk-adjusted payos o the hedged and unhedged carry trade in the

non-peso states are not very dierent. Consequently, the risk-adjusted losses

to these two strategies in the peso state cannot be very dierent. Since the value

o the SDF in the peso state is the same or both strategies, the actual losses o

the two strategies in the peso state must be similar. By construction, there is an

upper bound to the losses o the hedged carry trade. This upper bound tells us

how much an agent loses in the peso state i he is pursuing the hedged carry

trade. Since these losses turn out to be small, the losses to the unhedged carry

trade in the peso state must also be small.The rationale or why the SDF is much larger in the peso state than in the

non-peso states is as ollows. We just argued that the unhedged carry trade

makes relatively small losses in the peso state. At the same time, the average

risk-adjusted payo to the unhedged carry trade in the non-peso states is large.

The only way to rationalize these observations is or the SDF to be very high

in the peso state. So, even though the losses o the unhedged carry trade in the

peso state are moderate, the investor attaches great importance to those losses.

A possible shortcoming o our methodology is that we can always produce

values o the SDF and the carry trade payo in the peso state that rational-ize the observed average payos to the carry trade. The skeptical reader may

conclude that we have documented an interesting puzzle without providing a

credible resolution o that puzzle. So, we bring additional data to bear on the

plausibility o our estimates. We consider two versions o an equity strategy

that involve borrowing one dollar at the Treasury-bill rate and investing it in

the stock market. In the frst version, the agent does not hedge against adverse

movements in the stock market. In the second version, the agent buys at-the-

money put options, which compensate him or a all in the stock market. We

2 See Bates (1996a) or a related argument that exchange rate jumps can explain the volatility smile present incurrency options.

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fnd that, in sharp contrast to the carry trade, the hedged stock market strategy

yields large, negative average payos. Using the average risk-adjusted payo

to the two stock market strategies, we generate an independent estimate o the

value o the SDF in the peso state. Remarkably, this estimate is similar to the

estimate o the peso state SDF that rationalizes the average risk-adjusted pay-os to the carry trade. We interpret these fndings as supportive o the view that

the positive average payo to the unhedged carry trade reects the presence o

a peso problem.

Our article is organized as ollows. In Section 1, we describe the carry trade

strategy and discuss our method or estimating carry trade losses and the value

o the SDF in the peso state. We describe our data in Section 2. In Section 3, we

examine the covariance between the payos to the carry trade and traditional

risk actors, using both time-series and panel data. In Section 4, we examine

the properties o the hedged carry trade. In Section 5, we report our estimateso the payos to the carry trade and stock market strategies in the peso state,

and our estimates o the SDF in the peso state. We also generalize the analysis

to multiple peso states. Section 6 concludes.

1. Peso Problems and the Carry Trade

The ailure o uncovered interest rate parity (UIP) motivates a variety o specu-

lative strategies. We ocus on the carry trade, the strategy most widely used by

practitioners (Galati and Melvin 2004). In this section, we describe a procedure

or analyzing peso-event explanations or carry trade payos.

The carry trade consists o borrowing a low-interest-rate currency and lend-

ing a high-interest-rate currency. Abstracting rom transaction costs, the payo

to the carry trade, denominated in dollars, is

yt

1 + rt

St+1St

(1 + rt)

. (1)

The variable St denotes the spot exchange rate expressed as dollars per oreign

currency unit (FCU). The variables rt and rt represent the domestic and or-eign interest rate, respectively. We normalize the amount o dollars bet on this

strategy to one. The amount o dollars borrowed, yt, is given by

yt =

+1 irt < rt ,1 irt rt.

(2)

Consider the case in which St is a martingale:

EtSt+1 = St, (3)where Et denotes the time-t conditional expectations operator. This martin-

gale property is not an implication o market efciency, but it is a reasonable

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description o the data. Equation (3) implies that the expected payo to the

carry trade is positive and equal to the dierence between the higher and the

lower interest rate:

yt rt rt > 0.The carry trade strategy can also be implemented by selling the oreign cur-

rency orward when it is at a orward premium ( Ft St) and buying theoreign currency orward when it is at a orward discount ( Ft < St). The value

owt, the number o FCUs sold orward, is given by

wt =

+1/Ft i Ft St,1/Ft i Ft < St.

(4)

This value owt is equivalent to buying/selling one dollar orward. The dollar-denominated payo to this strategy at t + 1, denoted zt+1, is

zt+1 = wt (Ft St+1) . (5)

Covered interest-rate parity implies that

(1 + rt) = FtSt 1 + r

t . (6)

When Equation (6) holds, the payos to the strategies defned by Equations ( 4)

and (2) are proportional to each other. In this sense, the strategies are equiva-

lent. We ocus our analysis on strategy (4) because o data considerations.

1.1 The impact o peso problems

Since the carry trade is a zero-net-investment strategy, the payo, zt, must

satisy:

Et (Mt+1zt+1) = 0. (7)Here, Mt+1 denotes the SDF that prices payos denominated in dollars. Equa-tion (7) implies that

E(zt+1) = cov (Mt+1,zt+1)

E(Mt+1). (8)

In light o Equation (8), a natural explanation or the positive average payos

to the carry trade is that these payos compensate agents or the risk result-ing rom negative covariance between M and z. In our empirical work (see

Section 3), we document that the covariance between the payos to the carry

trade and a host o traditional risk actors is not statistically dierent rom

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zero.3 This fnding implies that traditional risk-based explanations are not a

plausible rationale or the positive average payos to the carry trade.

An alternative explanation relies on the existence o peso events. To pursue

this explanation, we use the ollowing notation. Let t denote the state

o the world at timet, let

z(t) denote the payo to the carry trade in state t,and let M(t) denote the value o the SDF in state t. We partition , the set

o possible states, t, into two sets. The frst set, N, consists o those values

ot corresponding to non-peso events. The second set, P , consists o those

values ot corresponding to a peso event. For simplicity, we assume that or

all t P , z(t) = z < 0 and M(t) = M.We denote by FN(t+1) the unconditional distribution ot+1 in non-peso

states. For uture reerence, we defne FN(t+1|t) as the conditional distri-bution ot+1 given t, where both t+1 and t are in N. To simpliy, we

assume that the conditional and unconditional probability o the peso state isp. The unconditional version o Equation (7) is

(1 p)EN (M z) + p Mz = 0, (9)where EN () denotes the expectation over non-peso states,

EN (M z) =

NM(t+1)z(t+1)dFN(t+1).

Since z is negative, Equation (9) implies that the average risk-adjusted pay-o over non-peso states, EN(M z), is positive. This observation captures the

conventional view that a peso problem can rationalize positive average payosto the carry trade.

We ocus on whether the existence o peso events provides a plausible ra-

tionale or our estimate o the average payo to the carry trade in non-peso

states. To study this question, we develop a version o the carry trade strategy

that does not yield large losses when a peso event occurs. We call this strategy

the hedged carry trade.

1.2 The hedged carry trade

We begin by defning notation or options contracts. A call option gives anagent the right, but not the obligation, to buy oreign currency at a strike price

o Kt dollars per FCU. We denote the dollar price o this option by Ct. The

payo to the call option in dollars, net o the option price, is

zCt+1 = max (0, St+1 Kt) Ct (1 + rt) . (10)

3 See Villanueva (2007) or additional evidence on this point. Lustig and Verdelhan (2007) argue that aconsumption-based SDF explains the cross-sectional variation in the excess returns to going long on currencyportolios that are sorted by their interest rate dierential with respect to the U.S. Burnside (2007) challenges

their results based on two fndings. First, the time-series covariance between the excess returns to the Lustig-Verdelhan portolios and standard risk actors, including consumption-based SDFs, is not signifcantly dierentrom zero. Second, imposing the constraint that a zero beta asset has a zero excess return leads to a substan-tial deterioration in the ability o their model to explain the cross-sectional variation in excess returns to theportolios.

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A put option gives an agent the right, but not the obligation, to sell oreign

currency at a strike price o Kt dollars per FCU. We denote the dollar price

o this option by Pt. The payo to the put option in dollars, net o the option

price, is

zPt+1 = max (0, Kt St+1) Pt (1 + rt) . (11)

Suppose that an agent sells one FCU orward. Then, the worst-case scenario

in the standard carry trade arises when there is a large appreciation o the or-

eign currency. In this state o the world, the agent realizes large losses because

he has to buy oreign currency at a high value o St+1 to deliver on the orwardcontract. Suppose that the agent buys, at time t, a call option on the oreign cur-

rency with a strike price Kt. So, whenever St+1 > Kt, the agent buys FCUs atthe price Kt. In this case, the minimum payo to the hedged carry trade is

(Ft St+1) + (St+1 Kt) Ct (1 + rt) = Ft Kt Ct (1 + rt) . (12)Similarly, suppose that an agent buys one FCU orward. Then, the worst-

case scenario in the standard carry trade arises when there is a large depre-

ciation o the oreign currency. In this state o the world, the agent sells the

oreign currency that he receives rom the orward contract at a low value o

St+1. Suppose that the agent buys, at time t, a put option on the oreign cur-rency with a strike price Kt. So, whenever St+1 < Kt, the agent sells FCUs ata price Kt. In this case, the minimum payo to the hedged carry trade is

(St+1 Ft) + (Kt St+1) Pt (1 + rt) = Kt Ft Pt (1 + rt) . (13)In order to normalize the size o the bet to one dollar, we set the amount o

FCUs traded equal to 1/Ft. We defne the hedged carry trade strategy as

I Ft St, sell 1/Ft FCUs orward and buy 1/Ft call options,I Ft < St, buy 1/Ft FCUs orward and buy 1/Ft put options.

The dollar payo to this strategy is

zHt+1 =

zt+1 + zCt+1/Ft i Ft St,zt+1 + z Pt+1/Ft i Ft < St,

(14)

where zt+1, zCt+1, and zPt+1 are the carry trade payos defned in Equations (5),

(10), and (11), respectively.

An alternative way to implement the hedged carry trade is to use only op-

tions, instead o using a combination o orwards and options. Under this alter-

native implementation, we buy 1/Ft call options on the oreign currency when

it is at a orward discount and 1/Ft put options on the oreign currency whenit is at a orward premium. Using the put-call-orward parity condition,

(Ct Pt) (1 + rt) = Ft Kt, (15)

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it is easy to show that this strategy or hedging the carry trade is equivalent to

the one described above as long as the strike price o the options is the same in

the two strategies.

The minimum payo to the hedged carry trade, ht+1, is negative. To see this

property, use the put-call-orward parity condition, (15), and Equations (12)and (13) to write the minimum payos as ollows:

ht+1 =Pt(1 + rt)/Ft i Ft St,

Ct (1 + rt) /Ft i Ft < St.(16)

Since option prices are positive, ht+1 is negative.We summarize the realized payos to the hedged carry trade as ollows:

zHt+1 = ht+1 i the option is in the money,zt+1 ct(1 + rt)/Ft i the option is out o the money.

The variable ct denotes the cost o the put or call option. Note that the option

is in the money in the peso state as well as in some non-peso states.

Using options to assess the eect o peso problems Since the hedged carry

trade is a zero-net-investment strategy, the payo, zH (t+1), must satisy:

(1 p) N

M(t+1)zH (t+1) dFN(t+1|t) + p Mh(t) = 0.

Taking expectations over all non-peso states, we obtain

(1 p)EN(M zH) + p MEN(h) = 0. (17)

Using Equation (17) to solve or p M and replacing this term in Equation (9),we obtain

z = EN

(h)

EN (M z)

EN(M zH) . (18)

We can estimate the variables on the right-hand side o Equation (18) and com-

pute an estimate o z. In estimating z, we do not have to take a stand on thevalues o p or M. Given our estimate o z and a value o p, we can useEquation (9) to estimate M:

M = (1 p)EN(M z)

p (z) . (19)

There are two possible outcomes o these calculations. The frst possible

outcome is that our estimate oz is a large negative value, consistent with theconventional view about the payos to the carry trade in the peso state. The

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second possible outcome is that our estimate o z is a small negative value. Inthis case, a peso event can still explain the average payo to the carry trade,

but only i M is large. So, in this case, the carry trade makes relatively smalllosses in the peso event, but traders value those losses very highly.

A natural question is whether the implied value oM

is empirically plausi-ble. We consider an equity strategy whose payo is also potentially aected bythe peso event to answer this question. Using hedged and unhedged versions o

this strategy, we obtain an alternative estimate o M. We then assess whetherthis estimate o M is consistent with the one implied by Equation (19). Theequity strategy involves borrowing one dollar at the Treasury-bill rate, rt, and

investing it in the S&P 100 index.4 We denote the ex-dividend price o the in-

dex and the associated dividend yield by Vt and dt, respectively. The payo to

this strategy is given by

xt+1 = Vt+1/ Vt + dt (1 + rt).Now consider the ollowing hedged version o the equity strategy: borrow

at the Treasury-bill rate to invest in the S&P 100 index and buy at-the-money

put options on the S&P 100 index. These put options compensate an investor

or a all in the S&P 100. It ollows that, anytime the S&P 100 index alls, the

payo to the hedged equity strategy is the dividend yield o the index minus

the dollar interest rate and the price o the option, cxt (1 + rt). By assumption,the stock index alls in the peso state, as well as in some non-peso states. In

these states, the payo to the hedged stock strategy is dt rt cxt (1 + rt).

In summary, the payo to the hedged equity strategy net o the options cost is

given by

xHt+1 =

xt+1 cxt (1 + rt) iVt+1 Vt,dt rt cxt (1 + rt) iVt+1 < Vt.

The payo to the unhedged equity strategy in the peso state is x . The pay-os to the unhedged and hedged equity strategies must satisy

(1 p)EN

(M x ) + p Mx = 0, (20)(1 p)EN(M xH) + p MEN d r cx (1 + r)= 0. (21)

Combining these two equations, we obtain

x = EN d r cx (1 + r) EN(M x)EN

M xH

, (22)M

=(1 p)EN (M x)

p(x ). (23)

4 The choice o this index is driven by data considerations.

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Given estimates o EN(M x ) and ENM xH

, we can use Equations (22)

and (23) to estimate M and x . Given a value o p, we then estimate x andM. A test o the second interpretation o the peso event is whether the value oM that emerges rom this procedure is consistent with the value o M implied

by Equation (19).The procedure just described assumes that a peso-based explanation o the

positive average payo to the carry trade is required. Ater describing the data,

we motivate this assumption in Sections 3 and 4, at least in the context o linear

asset-pricing models. In Section 5, we report the results o implementing the

procedure described in this section.

2. Data

In this section, we describe our data sources or spot and orward exchange

rates and interest rates. We also describe the options data that we use to analyze

the importance o the peso problem.

2.1 Spot and orward exchange rates

Our dataset on spot and orward exchange rates, obtained rom Datastream,

covers the euro (EUR) and the currencies o 20 countries: Australia, Austria,

Belgium, Canada, Denmark, France, Germany, Ireland, Italy, Japan, the

Netherlands, New Zealand, Norway, Portugal, South Arica, Spain, Sweden,

Switzerland, the United Kingdom, and the United States.

The data consist o daily observations or bid and ask spot exchange rates

and one-month-orward exchange rates. We convert daily data into non-

overlapping monthly observations (see Appendix A or details).

Our data span the period rom January 1976 to July 2009. However, the

sample period varies by currency (see Appendix A or details). Exchange rate

quotes (bid, ask, and mid, defned as the average o bid and ask) against the

British pound (GBP) are available beginning as early as 1976. Bid and ask ex-

change rate quotes against the U.S. dollar (USD) are available only rom Jan-

uary 1997 to July 2009. We obtain mid-quotes over the longer sample against

the dollar by multiplying GBP/FCU quotes by USD/GBP quotes.

2.2 Interbank interest rates and covered interest parity

We also collected data on interest rates in the London interbank market rom

Datastream. These data are available or 17 countries/currencies: Australia,

Belgium, Canada, Denmark, France, Germany, Italy, Japan, the Netherlands,

New Zealand, Norway, South Arica, Sweden, Switzerland, the U.K., the U.S.,

and the euro.

The data consist o daily observations or bid and ask eurocurrency interestrates. We convert daily data into non-overlapping monthly observations. Our

data span the period rom January 1976 to July 2009, with the exact sample

period varying by currency.

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To assess the quality o our dataset, we investigate whether covered-interest

parity (CIP) holds taking bid-ask spreads into account. We fnd that deviations

rom CIP are small and rare. We provide details o our interest-rate data and

analysis o CIP in the online appendix (www.duke.edu/acb8/bekr app.pd).

2.3 Option prices

We use two options datasets. Our frst dataset is rom the Chicago Mercan-

tile Exchange (CME). These data consist o daily observations or the period

January 1987 to April 2009 on the prices o put and call options against USD

utures or the Australian dollar (AUD), the Canadian dollar (CAD), EUR, the

Japanese yen (JPY), the Swiss ranc (CHF), and the GBP. Appendix B speci-

fes the exact period o availability or each currency. When a utures contract

and an option contract expire on the same date, an option on currency u-

tures is equivalent to an option on the spot exchange rate. This equivalenceresults rom the act that the price o a utures contract coincides with the spot

exchange rate at maturity. In practice, the expiration dates o the two con-

tracts do not generally coincide in our dataset. In Appendix B, we describe

how to modiy our hedging strategy to take this act into account. This mod-

ifcation involves adjusting both the number o options purchased and their

strike price. Our modifed hedging strategy is exactly equivalent to the hedg-

ing strategy described in Section 1 whenever interest rates are constant over the

period between the expiration o the option and the expiration o the utures

contract.Since we compute carry trade payos at the monthly requency, we use data

on options that are one month rom maturity (see Appendix B or details).

We work exclusively with options expiring near the beginning o each month

(two Fridays prior to the third Wednesday). We measure option prices using

settlement prices or transactions that take place exactly 30 days prior to the

options expiration date. We measure the time-t orward, spot, and option strike

and settlement prices on the same day, and measure the time t+1 utures priceon the option expiration date. We multiply option prices by the gross 30-day

eurodollar interest rate obtained rom the Federal Reserve Board to computenet payos. This 30-day interest rate is matched to the maturity o the options

in our dataset.

Our second options dataset is rom J. P. Morgan. These data consist o daily

observations o one-month at-the-money-orward implied volatility quotes and

orward and spot exchange rates or the ollowing currencies: the AUD, CAD,

Danish krone (DKK), EUR, JPY, CHF, GBP, New Zealand dollar

(NZD), Norwegian krone (NOK), and Swedish krone (SEK). Our sample pe-

riod is February 1995 to July 2009. We convert the implied volatility quotes

to option prices using the Black-Scholes ormula. Given the structure o thisdataset, we select monthly trades that mature on a date near the end o the

month and are initiated 30 days earlier. We select the last maturity date that is

a business day and or which the initiation date is a business day.

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3. Payos to the Carry Trade

In this section, we examine the properties o the payos to the carry trade.

First, we compute the mean and variance o the payo to the carry trade with

and without transaction costs. Second, we investigate whether the payo dis-

tribution has at tails. Third, we study the covariance between the payos tothe carry trade and various risk actors using both time-series and panel data.

We consider two versions o the carry trade. In the carry trade without

transaction costs, we assume that agents can buy and sell currency at the av-

erage o the bid and ask rates. We compute St as the average o the bid ( Sbt )

and the ask (Sat ) spot exchange rates,

St =

Sat + Sbt

/2,

and Ft as the average o the bid ( Fbt ) and the ask (Fat ) orward exchange rates,

Ft =

Fat + Fbt

/2.

The ask (bid) exchange rate is the rate at which a participant in the interdealer

market can buy (sell) oreign currency rom (to) a currency dealer.

In the carry trade with transaction costs, we take bid-ask spreads into ac-

count when deciding whether to buy or sell oreign currency orward and in

calculating payos. In this case, the number o FCUs sold orward, wt, is

given by

wt =

+1/Fbt i Fbt /Sat > 1,1/Fat i Fat /Sbt < 1,0 otherwise.

(24)

The payo to this strategy is

zt+1 =

wt Fbt Sat+1iwt > 0,

wt

Fat Sbt+1

iwt < 0,

0 iwt = 0.(25)

3.1 Characteristics o carry trade payos

We consider the carry trade strategy or individual currencies, as well as or

portolios o currencies. For now, we ocus attention on the payos to an equally

weighted portolio o carry trade strategies.5 This portolio is constructed by

betting 1/nt USD in each individual currency carry trade. Here, nt denotes

5 We report results or individual currencies in the online appendix.

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Table 1

Annualized payos o the equally weighted carry trade strategy

No Transactions Costs With Transactions Costs

Standard Sharpe Standard SharpeMean Deviation Ratio Mean Deviation Ratio

GBP is the Base CurrencyFebruary 1976 to July 2009 0.0319 0.043 0.748 0.0288 0.057 0.507

(0.0080) (0.003) (0.194) (0.0111) (0.004) (0.203)

USD is the Base CurrencyJanuary 1997 to July 2009 0.0440 0.051 0.865 0.0431 0.062 0.694

(0.0171) (0.005) (0.358) (0.0213) (0.007) (0.356)

Payos are measured either in GBP, per GBP bet, or in USD, per USD bet. The carry trade portolio is ormedas the equally weighted average o up to 20 individual currency carry trades against either the GBP or theUSD. The 20 currencies are indicated in Appendix A. Heteroskedasticity-consistent GMM standard errors arein parentheses.

the number o currencies in our sample at time t. In the remainder o the ar-

ticle, unless otherwise noted, we use the term carry trade strategy to reer

to the equally weighted carry trade. Also, we report all statistics on an annu-

alized basis. Table 1 reports the mean, standard deviation, and Sharpe ratio

o the monthly payos to the carry trade, with and without transaction costs.

We consider two alternative home currencies, the GBP and the USD. Using

the GBP as the home currency allows us to assess the importance o bid-ask

spreads using a much longer time series than would be the case i we used only

the USD as the home currency.Consider the results when the GBP is the home currency. Ignoring trans-

action costs, the Sharpe ratio o the equally weighted carry trade portolio is

0.748. Taking bid-ask spreads into account reduces the Sharpe ratio to 0.507.

But the Sharpe ratio is statistically dierent rom zero with and without trans-

action costs. Next, consider the results when the dollar is the home currency.

Ignoring transaction costs, the Sharpe ratio o the equally weighted carry trade

portolio is 0.865. Taking bid-ask spreads into account reduces the Sharpe ratio

to 0.694. The impact o transaction costs is smaller when the dollar is the base

currency because bid-ask spreads are lower or the dollar than or the pound.Taken together, our results indicate that, while transaction costs are quanti-

tatively important, they do not explain the proftability o the carry trade. For

the remainder o this article, we abstract rom transaction costs and work with

spot and orward rates that are the average o bid and ask rates.6 Given this de-

cision, we can work with the longer dataset (rom January 1976 to July 2009)

using the USD as the home currency.

Table 2 reports statistics or the payos to the equally weighted carry trade

and summary statistics or the individual-currency carry trades. We compute

the latter by taking the average o the statistics or the carry trade applied to

6 In an earlier version o this article (Burnside et al. 2006), we present a more comprehensive set o results or thecarry trade payos taking bid-ask spreads into account.

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Table 2

Annualized payos o investment strategies: February 1976 to July 2009

Standard Sharpe Excess Jarque-BeraMean Deviation Ratio Skewness Kurtosis Statistic

U.S. stock market 0.0582 0.156 0.373

0.808 2.53 150.9

(0.0281) (0.010) (0.192) (0.288) (1.17) (0.000)Equally-weighted carry trade 0.0482 0.053 0.911 0.648 5.81 592.6

(0.0101) (0.005) (0.222) (0.520) (2.02) (0.000)

Average o individual-currency 0.0492 0.114 0.442 0.229 1.57 67.4carry trades

USD is the base currency. Payos are measured in USD, per USD bet. The payo at time t to the U.S. stockmarket is the value-weighted excess return on all U.S. stocks reported in Kenneth Frenchs database, divided by1 + rt1 (this normalizes the excess stock returns to the same size o bet as the carry trade payos). The carrytrade portolio is ormed as the equally weighted average o up to 20 individual currency carry trades againstthe USD. The individual currencies are indicated in the appendix. Heteroskedasticity-consistent GMM standarderrors are in parentheses, except or the Jarque-Bera statistic or which the p-value is reported in parentheses.The mean payo o the equally weighted carry trade is not equal to the average mean payo o the individual

currency carry trades because the sample periods or which the currencies are available varies (see Appendix A).

each o the 20 currencies in our sample. To put our results into perspective, we

also report statistics or excess returns to the value-weighted U.S. stock market.

Two results emerge rom this table. First, there are large gains to diversifca-

tion. The average Sharpe ratio across currencies is 0.442, while the Sharpe

ratio or an equally weighted portolio o currencies is 0.911. This large di-

erence between the Sharpe ratios is due to the act that the standard deviation

o the payos is much lower or the equally weighted portolio. Second, theSharpe ratio o the carry trade is substantially larger than that o the U.S. stock

market (0.911 vs. 0.373). While the average excess return to the U.S. stock

market is larger than the payo to the carry trade (0.058 vs. 0.048), the returns

to the U.S. stock market are much more volatile than the payos to the carry

trade (0.156 vs. 0.053).

So ar we have emphasized the mean and variance o the payos to the

carry trade. These statistics are sufcient to characterize the distribution o the

payos only i this distribution is normal. Table 2 reports skewness and excess

kurtosis statistics, as well as the results o the Jarque-Bera normality tests. Itis evident that the distributions o both payos are leptokurtic, exhibiting at

tails.

3.2 The impact o the fnancial crisis on carry trade payos

Panel A o Figure 1 shows the cumulative payos to investing one dollar in

January 1976 in three dierent strategies. The frst strategy involves investing

in one-month Treasury bills. The second strategy involves investing in a value-

weighted index o the universe o U.S. stocks in the CRSP database. In bothstrategies the monthly proceeds are reinvested. The third strategy is the carry

trade. Since this strategy involves zero net investment, we compute the cumu-

lative payos as ollows. We initially deposit one USD in a bank account that

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Figure 1

Cumulative returns o investment strategies

Panel A: Twenty Currency Carry Trade and U.S. Stocks, January 1976 to July 2009Panel B: Six Currency Carry Trade (Hedged and Unhedged) and U.S. Stocks, January 1987 to April 2009The fgure plots the cumulative returns o a trader who begins with 1 dollar in January 1976 (Panel A) or January1987 (Panel B) and invests his accumulated earnings exclusively in one o our strategies. For T-bills and U.S.stocks, we use the risk-ree rate and value-weighted market return reported in Kenneth Frenchs database. Forthe carry trade strategies, we assume that the trader invests the initial dollar in T-bills and bets the uture nominalvalue o those T-bills in the carry trade. In each period, all proceeds are deposited in the T-bill account, and theuture value o the T-bill account is bet on the carry trade. Details o the strategies are provided in the text.

yields the same rate o return as the Treasury-bill rate. In the beginning o ev-

ery period, we bet the balance o the bank account on the carry trade strategy.

At the end o the period, payos to the carry trade are deposited into the bank

account.

Three eatures o Figure 1 are worth noting. First, the cumulative payos to

the carry trade and stock market strategies are very similar. Second, the payos

to the carry trade are much less volatile than those o the U.S. stock market.

These two eatures account or the Sharpe ratio o the carry trade strategy beingroughly 2.5 times higher than that o the U.S. stock market. Third, in the recent

fnancial crisis, the carry trade strategy loses money, but these losses are much

smaller than those o the U.S. stock market. The U.S. stock market cumulative

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return peaked at $44.32 in October 2007 and ell to a trough o $21.47 in

February 2009, a decline o 51.6%. The carry trade portolio cumulative return

peaked at $31.22 in July 2008 and ell to a trough o $27.87 in February 2009,

a decline o 10.7%. Both the cumulative payos to the carry trade and the U.S.

stock market strategies have partially recovered rom their trough values.The worst monthly payos (i.e., the largest drawdowns) to the carry trade

rom February 1976 to July 2009 are 8.9% (March 1991), 5.8% (October1992), and 5.1% (June 1993). The three worst monthly payos to the carrytrade rom July 2008 to July 2009 are 4.2% (September 2008), 3.9% (Au-gust 2008), and 3.7% (January 2009).7 The three worst monthly payos tothe stock market strategy in our sample are 23.0% (October 1987), 18.5%(October 2008), and 16.1% (August 1998). Note that the largest drawdownso the carry trade strategy did not occur during the recent fnancial crisis. In

contrast, one o the three worst payos to the stock market strategy did occurduring the recent fnancial crisis. It is worth emphasizing that, while there are

some reasonably large negative payos to the carry trade in the sample, the

average payo is still positive.

During the recent fnancial crisis (July 2008January 2009), an alternative

carry trade index, the Deutsche Bank G10 Currency Future Harvest (DBCFH),

lost 30.9% o its value. In contrast, our equally weighted carry trade portolio

declined by only 10.4%. The DBCFH index is based on a bet o two dol-

lars, while our index is based on a bet o one dollar. This dierence explains

60% o the disparity in the perormance o the two indexes during the recentcrisis. The remaining 40% is explained by dierences in the way the porto-

lios are ormed. There are two key dierences. First, the DBCFH index takes

positions in up to six currencies rom a list o ten. The index is ormed by

taking equally weighted long positions vis-a-vis the USD in the three curren-

cies with the highest interest rates, and symmetric short positions vis-a-vis the

USD in the three currencies with the lowest interest rates.8 In contrast, we use

a larger set o currencies and take positions in all o them based on the sign

o the orward premium versus the USD. Second, the currency composition

o the DBCFH portolio is rebalanced quarterly, while the composition o ourportolio is rebalanced monthly.

Over the period or which data on the DBCFH index are available (April

1993 to July 2009), our index outperormed the DBCFH index once the dier-

ence in the size o the bet is taken into account. Our equally weighted index

earned a mean annualized excess return o 4.1%, compared to 3.2% or the

DBCFH index (Deutsche Bank reports an average excess return o about 5.7%

in this period, during which the average size o the bet was approximately 1 .80

7 Because o data limitations we cannot compute the drawdowns on the carry trade strategy at a daily requency.

8 When the dollar is one o the three high-interest-rate currencies or one o the three low-interest-rate currencies,the amount bet by the index is 1.67 instead o2. During the fnancial crisis, the dollar was one o the low-interest-rate currencies.

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USD instead o one USD). The Sharpe ratios o the two indices are 0.80 and

0.62, respectively.

3.3 Risk actor analysis o carry trade payos

We now turn to the question o whether the payos are correlated with tradi-tional risk actors. In this subsection, we show that the covariances between

the payos to the carry trade and traditional risk actors are not statistically

dierent rom zero. We do so using both time-series and panel-data analysis.

We consider data at both the monthly and quarterly requencies. When data

on the risk actors are available at the monthly requency, we defne a 26 1vector Rt containing the time-t nominal payo to the carry trade strategy and

the nominal excess returns o the 25 Fama-French (1993) portolios o equities

sorted by frm size and the ratio o book value to market value. When data on

the risk actors are available at the quarterly requency, we defne a 26 1vector Rt containing the time-t real quarterly payo to the carry trade strategyand the 25 Fama-French (1993) portolios.9 These payos or excess returns

must satisy:

Et (Rt+1mt+1) = 0, (26)where, when the data are monthly, mt+1 is the SDF that prices nominal USD-denominated excess returns and, when the data are quarterly, mt+1 is the SDFthat prices real USD-denominated excess returns. We consider linear SDFs o

the ormm t =

1 (t ) b

. (27)

Here, is a scalar, t is a vector o risk actors, = E(t), and b is a con-ormable vector. To simpliy our analysis, we abstract rom non-negativity con-

straints on mt (see Li, Xu, and Zhang 2010 or a discussion o the

potential importance o this issue).

It ollows rom Equation (26) and the law o iterated expectations that

E(Rtm t)

=0. (28)

Equations (28) and (27) imply that

E(Rt) = ,where

= cov(Rt, t)V1 , (29) = Vb.

Here, V is the covariance matrix o the actors, is a measure o the sys-

tematic risk o the payos, and is a vector o risk premia. Note that is

9 In Appendix C, we show how we convert monthly payos to real quarterly excess returns.

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the population value o the regression coefcient o Rt on t. Our time-series

analysis ocuses on estimating the betas o the carry trade payos or dierent

candidate risk actors. Our panel analysis provides complementary evidence

on the importance o dierent risk actors by estimating alternative SDF mod-

els using the moment condition (28). One o these models is the Fama-French(1993) model that we later use to estimate M.

3.3.1 Time-series risk-actor analysis. We consider the ollowing risk ac-

tors: the excess returns to the value-weighted U.S. stock market, the Fama-

French (1993) actors (the excess return to the value-weighted U.S. stock

market, the size premium (SMB), and the value premium (HML)), real U.S.

per-capita consumption growth (nondurables and services), the actors pro-

posed by Yogo (2006) (the growth rate o per-capita consumption o non-

durables and services, the growth rate o the per-capita service ow romthe stock o consumer durables, and the return to the value-weighted U.S.

stock market), luxury sales growth (obtained rom At-Sahalia, Parker, and

Yogo 2004), GDP growth, industrial production growth, the Fed Funds Rate,

the term premium (th

Do Peso Problems Explain the Returns

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