The Term Structure of Currency Risk Premia...bringing information from the term structure of...

The Term Structure of Currency Risk Premia

Hanno Lustig

UCLA and NBER

Andreas Stathopoulos

USC

Adrien Verdelhan

MIT and NBER

June 2013∗

Abstract

The returns to the currency carry trade are much smaller at longer maturities. Countries

with a high local term premium have an offsetting low or negative currency risk premium.

The local term premium in bond markets compensates investors for the risk associated with

temporary innovations to the pricing kernel. In the limiting case in which the permanent

shocks are fully shared across countries and exchange rates are only driven by temporary

innovations, the currency exposure completely hedges the interest rate exposure of the foreign

bond portfolio, and the term premium in dollars is identical across countries. The empirical

evidence suggests that there is more cross-country sharing of permanent shocks to the pricing

kernel than of temporary shocks. That accounts for the downward sloping term structure of

currency risk premia.

∗Please do not quote. Incomplete Lustig: UCLA Anderson School of Management, 110 Westwood Plaza,Suite C4.21, Los Angeles, CA 90095 ([email protected]). Stathopoulos: USC Marshall School of Busi-ness, 3670 Trousdale Parkway, Hoffman Hall 711, Los Angeles, CA 90089 ([email protected]). Verdelhan:MIT Sloan School of Management, 77 Massachusetts Avenue E62-621, Cambridge, MA 02139 ([email protected]).

1 Introduction

There is no point in chasing high long term bond yields around the world, at least not without

hedging currency risk. The returns to the currency carry trade are much smaller at longer

maturities. Figure 1 plots the cumulative log returns on an investment strategy that goes long

in T-bills of high interest rate currencies and short in T-bills of low interest rate currencies

against the returns on 10-year government bond portfolios for the same currencies. The returns

on the bond investment strategy are much smaller.

Between 1950 and 2012, the average spread in dollar returns between the high and low

interest rate portfolios for T-bills is 4.36%, but only a 40 basis points for the 10-year bond

portfolios. Countries with a high local term premium have an offsetting low or negative currency

risk premium. The portfolio of high interest rate currencies yields a currency risk premium of

2.88% and a local term premium of -51 bps per annum, while the portfolio of low interest rate

currencies yields a currency risk premium of -148 bps and a term premium of 198 bps. The

dollar term premium, which is the sum of the local currency term premium and the currency

risk premium, only differs by 40 bps.

No arbitrage implies that foreign currency risk premia are high when there is less risk in

those foreign countries’ pricing kernels than at home. If most of this variation in risk concerns

temporary shocks to the pricing kernel, then these countries would also have lower term premia

in foreign bond markets, because the local term premium in bond markets compensates investors

for the risk associated with temporary innovations to the pricing kernel (Alvarez and Jermann

(2005) and Hansen and Scheinkman (2009a)).

In the limiting case in which the permanent shocks are fully shared across countries and ex-

change rates are only driven by temporary innovations, the currency exposure completely hedges

the interest rate exposure of the foreign bond portfolio, because the exchange rate completely

offsets the effect of ‘unshared’ temporary foreign shocks on the foreign bond portfolio. We refer

to this as uncovered bond return parity. In this case, the term premium in dollars is identical

across countries.

Long-run uncovered bond parity is a better fit in the cross-section than in the time series.

1

Figure 1: The Carry Trade Premium and Term Premium

1943 1957 1971 1984 1998 2012 2026−1

−0.5

0

0.5

1

1.5

2

2.5

3

HML on Bonds

HML on Currencies

Cumulative log returns on high-minus-low in Currencies (sorting on monthly T-bill returns at t into 5 portfolios)and high-minus-low in 10-year Bonds (same sorting). Monthly data. 1950.1-2012.12.

While we reject long-run uncovered bond return parity in the time series, we do find a secular

increase in the sensitivity of foreign long-term bond returns to U.S. bond returns over time, our

measure of risk sharing of permanent shocks in international financial markets. After 1991, a 100

basis points increase in U.S. long-term bond returns increases foreign bond returns in dollars by

an average of 56 basis points. The exchange rate exposure accounts for a 1/3 to a quarter of this

effect: The dollar appreciates on average against a basket of foreign currencies when the U.S.

bond returns are lower than average, and vice-versa, except during flight-to-liquidity episodes.

While permanent innovations are not completely shared across countries, the empirical ev-

idence suggests that there is substantially more cross-country sharing of permanent shocks to

the pricing kernel than of temporary shocks. That accounts for the downward sloping term

structure of currency risk premia.

An important question in international finance is the extent to which countries leave op-

portunities for risk sharing unexploited. As pointed out by Brandt, Cochrane, and Santa-Clara

(2006), the combination of relatively smooth exchange rates (10% per annum) and much more

2

volatile stochastic discount factors (50% per annum) implies that state prices are highly corre-

lated across countries (at least 0.98). 1

This paper sheds some light on the nature of international risk sharing by decomposing the

pricing kernel of each country into a permanent component and a transitory component. Alvarez

and Jermann (2005), Hansen and Scheinkman (2009a) and Hansen, Heaton, and Li (2008) have

explored the implications of that decomposition for asset prices. From the relative size of the

equity premium (large) and the term premium (small), Alvarez and Jermann (2005) infer that

almost all the variation in stochastic discount factors arises from permanent fluctuations. By

bringing information from the term structure of currency risk premia to bear, we learn that the

shocks driving exchange rates and currency risk premia are much less persistent.

The bulk of the persistent shocks to the pricing kernel may have been effectively traded

away in international financial markets. This result is relevant to economists. The welfare gains

from removing all aggregate consumption uncertainty are large, but almost exclusively because

of the low frequency component in consumption, not the business cycle component Alvarez and

Jermann (2004)). While international risk sharing gains may not have been fully exploited, they

may be smaller than commonly assumed.

Our paper makes contact with the the vast literature on UIP (Uncovered Interest Rate

Parity) and the currency carry trade. We derive general conditions under which long-run UIP

follows from no-arbitrage: if all permanent shocks to the pricing kernel are common, then foreign

and domestic yield spreads in dollars on long maturity bonds will be equalized, regardless of the

properties of the pricing kernel. Chinn and Meredith (2004) have documented some time-series

evidence that supports UIP at longer holding periods.

In closely related work, Koijen, Moskowitz, Pedersen, and Vrugt (2012) and Wu (2012)

examine the currency-hedged returns on ‘carry’ portfolios of international bonds, sorted by

a proxy for the carry on long-term bonds, but they do not examine the interaction between

currency and term risk premia, the topic of our paper. We focus on portfolios sorted by interest

1Colacito and Croce (2011) argue that only the persistent component of consumption growth is highly corre-lated across countries. Our finding provide model-free evidence in support of the view that the bulk of permanentshocks are shared across countries

3

rates, as well as yield spreads. Ang and Chen (2010) show that yield curve variables also forecast

currency excess returns, but they do not examine the returns on foreign bond portfolios. Finally,

Dahlquist and Hasseltoft (2013) study international bond risk premia in an affine asset pricing

model and find evidence for local and global risk factors. Jotikasthira, Le, and Lundblad (2012)

report similar findings.

Asymmetric exposure to global or common innovations to the pricing kernel are key to

understanding the global currency carry trade premium (Lustig, Roussanov, and Verdelhan

(2011)). They identify innovations in the volatility of global equity markets as candidate shocks,

while Menkhoff, Sarno, Schmeling, and Schrimpf (2012) propose the volatility in global currency

markets instead. If these global shocks are temporary, and the permanent shocks are completely

shared between countries, then there should be no carry trade premium for portfolios of bonds

with long maturities. The downward sloping term structure of currency risk premia lend support

to the view that the shocks driving exchange rates and currency returns are much less persistent

than the bulk of the innovations to the pricing kernel.

The rest of the paper is organized as follows. Section 2 derives the no-arbitrage restrictions

imposed on currency and term risk premia. Section 3 describes the data and section 4 explores

the correlation and volatility of foreign bond returns at various maturities. Section 5 documents

a strong negative relation between currency risk premia and local currency term risk premia in

the data. Section 6 directly tests uncovered bond return parity in the time-series and in the

cross-section.

2 The Term Premium and the Currency Risk Premium

We use Λt to denote the nominal pricing kernel, or the marginal value of a dollar delivered at t

in some state of the world $; the nominal stochastic discount factor (SDF) is the growth rate of

the pricing kernel (Mt+1 = Λt+1/Λt). The price of a zero-coupon bond with maturity k periods

into the future is given by

Vt [1t+k] = Et

(Λt+k

Λt

)(2.1)

4

We define the one-period return on a zero-coupon bond with maturity k as:

Rt+1,1 [1t+k] =Vt+1 [1t+k]

Vt [1t+k](2.2)

We use rxkt+1 to denote the log excess returns logRt+1,1[1t+k]Rt+1,1[1t+1] . We define the term premium as:

ht[k] = Et

[log

Rt+1,1[1t+k]

Rt+1,1[1t+1]

]

Let us define the yield spread at long maturities:

yt[k] = log

(Vt [1t+1]

Vt [1t+k]1/k

)(2.3)

If the limits of yt[k] and ht[k] are well defined and the unconditional expectations of holding

returns are independent of calendar time, then, according to results derived by Alvarez and

Jermann (2005), the average term premium equals the average yield difference:

limk→∞

E(ht[k]) = limk→∞

E(yt[k]).

Hence, the average term premium is the average yield spread at longer maturities.

2.1 Pricing Kernel without Permanent Innovations

We start by considering a pricing kernel that is not subject to permanent innovations.

Definition 1 The pricing kernel has no permanent innovations if and only if (Alvarez and

Jermann (2005)):

limk→∞

Et logEt+1[Λt+k]

Et[Λt+k]= 0.

Under regularity conditions, this condition is equivalent to limk→∞Et+1[Λt+k]Et[Λt+k] = 1.

5

Term Premium on Domestic Bonds If there are no permanent innovations to the pric-

ing kernel, then the return on the bond with longest maturity equals the inverse of the SDF:

limk→∞Rt+1,1[1t+k] = Λt/Λt+1. High marginal utility growth translates into higher yields on

long maturity bonds and low long bond returns, and vice-versa. As a result, this long bond

commands the largest possible risk premium in an economy without permanent innovations to

the pricing kernel.

Proposition 2.1 If the pricing kernel has no permanent innovations, then the term premium

is the largest risk premium in the economy (Alvarez and Jermann (2005)).

ht[∞] = limk→∞

Et

[log

Rt+1,1[1t+k]

Rt+1,1[1t+1]

]≥ Et

[log

Rt+1

Rt+1,1[1t+1]

]

for any Rt+1.

Example 2.1 Consider a log-normal model of a pricing kernel:

log Λt =

∞∑i=0

αiεt−i + (t) log β (2.4)

with ε ∼ N(0, σ2), α0 = 1.

It is easy to check that the condition outlined in definition 1 is satisfied provided that limk→∞ α2k =

0.

Proposition 2.2 In example 2.1, the term premium equals one half of the variance (Alvarez

and Jermann (2005)).

limk→∞

Etrxkt+1 = lim

k→∞Et

[log

Rt+1,1[1t+k]

Rt+1,1[1t+1]

]= (1/2)σ2.

Term Premium on Foreign Bonds What is the term premium for foreign bonds? We use

St to denote the spot exchange rate in pounds per dollar. Similarly, we use Ft to denote the

6

one-period forward exchange rate. The log local currency return on a foreign bond position in

excess of the domestic risk-free rate can be restated as the sum of the log excess return in local

currency plus the return on a long position in forward currency:

log

[StSt+1

R∗t+1,1[1t+k]

Rt+1,1[1t+1]

]= rxk,∗t+1 + (ft − st)−∆st+1,

where we have used covered interest rate parity:

Ft

St=R∗t+1,1[1t+1]

Rt+1,1[1t+1]

The last two components represent the log excess return on a long position in foreign currency,

given by the forward discount minus the rate of depreciation. By taking expectations, we see

that the total term premium in dollars consists of a foreign bond risk premium Et[rxk,∗t+1] plus a

currency risk premium (ft − st)− Et∆st+1.

Proposition 2.3 In example 2.1, the foreign term premium in dollars is identical to the do-

mestic term premium.

h∗t [∞] + (ft − st)− Et[∆st+1] = (1/2)σ2 = ht[∞]

In a log-normal model, the currency risk premium equals (1/2)(σ∗,2 − σ2.

)(see Bansal

(1997), Bekaert (1996) and Backus, Foresi, and Telmer (2001)). Currencies with a high local

currency term premium (high σ2) also have an offsetting negative currency risk premium, while

those with a small term premium have a larger currency risk premium. Hence, U.S. investors

get the same dollar premium on foreign as on domestic bonds. There is no point in chasing high

term premia around the world, at least not in economies with only temporary innovations to the

pricing kernel. Currencies with the highest local term premium also have the lowest (i.e. most

negative) currency risk premium.

This result is equivalent to uncovered interest rate parity for very long holding periods.

7

Corollary 2.1 In example 2.1, the average foreign yield spread in dollars is identical to the

domestic yield spread (Long-run UIP):

y∗t [∞] + (ft − st)− E[ limk→∞

(1/k)k∑

j=1

∆st+j ] = (1/2)σ2 = yt[∞]

Non-normalities This result does not require log-normality. Gavazzoni, Sambalaibat, and

Telmer (2012) convincingly argue that higher moments are critical for understanding currency

returns. 2 We obtain the same result.

Proposition 2.4 If the pricing kernels do not have permanent innovations, the foreign term

premium in dollars equal the domestic term premium.

h∗t [∞] + (ft − st)− Et[∆st+1] = ht[∞].

The proof is straightforward. In general the foreign currency risk premium is equal to the

difference in entropy (see Backus, Foresi, and Telmer (2001)):

(ft − st)− Et[∆st+1] = Lt

(Λt+1

Λt

)− Lt

(Λ∗t+1

Λ∗t

)(2.5)

Also in the absence of permanent innovations, the term premium is equal to the entropy of the

pricing kernel. The result follows.

Actually, it turns out that we can prove a much stronger result. Not only are the risk premia

identical. The returns on the foreign bond position are the same to those on the domestic bond

position, because the foreign bond position automatically hedges the currency risk exposure.

2More recently, Brunnermeier, Nagel, and Pedersen (2008) show that risk reversals increase with interest rates.Jurek (2008) provides a comprehensive empirical investigation of hedged carry trade strategies. Farhi, Fraiberger,Gabaix, Ranciere, and Verdelhan (2009) estimate a no-arbitrage model with disaster risk using a cross-section ofcurrency options. Chernov, Graveline, and Zviadadze (2011) study jump risk at high frequencies.

8

We consider the multiplicative one-period excess return of the foreign k-maturity bond over the

domestic k-maturity bond, with both returns in domestic currency:

StSt+1

R∗t+1,1[1t+k]

Rt+1,1[1t+k]

Proposition 2.5 If the domestic and foreign pricing kernels have no permanent innovations,

then the one-period returns on the foreign longest maturity bonds in domestic currency are

identical to the domestic ones Long-run Uncovered Bond Return Parity):

limk→∞

StSt+1

R∗t+1,1[1t+k]

Rt+1,1[1t+k]= 1,

in all states.

In this class of economies, the returns on long-term bonds expressed in domestic currency

are equalized. We refer to this uncovered long-run bond return parity. For large maturity k, the

spread in log bond returns expressed in domestic currency equals the rate of depreciation:

limk→∞

(rx∗,kt+1 − rxkt+1) ≈ − ((ft − st)−∆st+1) .

In countries which experience higher marginal utility growth, the domestic currency appreciates

but that is exactly offset by the capital loss on the bond. The foreign bond position automatically

hedges the currency exposure.

2.2 Pricing Kernel with Permanent Innovations

Following Alvarez and Jermann (2005), Hansen, Heaton, and Li (2008), and Hansen and Scheinkman

(2009b), we decompose the pricing kernel into a transitory and a permanent component:

Λt = ΛPt ΛT

t . (2.6)

9

Definition 2 The transitory component, ΛTt , is defined as

ΛTt = lim

k→∞

δt+k

Vt[1t+k], (2.7)

where the constant δ is chosen to satisfy the following regularity condition:

0 < limk→∞

Vt[1t+k]

δk<∞. (2.8)

The permanent component, ΛPt is a martingale. To see why, note that

ΛPt = lim

k→∞

Vt[1t+k]

δt+kΛt, (2.9)

This expression is a martingale. That follows directly from the Euler equation for the zero

coupon bond with maturity k. The one-period growth rate of transitory SDF components is

given by

ΛTt+1

ΛTt

= limk→∞

δVt[1t+k]

Vt+1[1t+k]

The infinite maturity bond return is given by

Rt+1,1[1t+∞] = limk→∞

Rt,1[1t+k] =ΛTt

ΛTt+1

We can decompose exchange rate changes into a permanent component and a transitory

component, defined below:

St+1

St=

(Λ∗Pt+1

Λ∗Pt

ΛPt

ΛPt+1

)(Λ∗Tt+1

Λ∗Tt

ΛTt

ΛTt+1

)=SPt+1

SPt

STt+1

STt

Therefore, we can think of exchange rate changes as capturing differences in both the transitory

and the permanent component of the two countries’ stochastic discount factors. We can use

returns on long bonds to extract the permanent component of exchange rates.

Example 2.2 We consider a log-normal model of the pricing kernel( Alvarez and Jermann

10

(2005)):

log ΛPt+1 = −1

2σ2P + log ΛP

t + εPt+1,

log ΛTt+1 = log βt+1 +

∞∑i=0

αiεTt+1−i,

where α is a square summable sequence, and εP and εT are i.i.d. normal variables with mean

zero and covariance σTP . A similar decomposition applies to the foreign stochastic discount

factor, where a ? denotes a foreign variable:

log Λ?Pt+1 = −1

2σ?2P + log Λ?P

t + ε?Pt+1,

log Λ?Tt+1 = log β?t+1 +

∞∑i=0

α?i ε

?Tt+1−i.

Proposition 2.6 In example 2.2, the term premium is given by the following expression:

ht[∞] = (1/2)σ2T + σTP

.

Corollary 2.2 In example 2.2, the foreign term premium in dollars is identical to the domestic

term premium.

h∗t [∞] + (ft − st)− Et[∆st+1] = (1/2)(σ2 − σ2,∗P ).

Provided that σ2,∗P = σ2

P , the foreign term premium in dollars equals the domestic term premium:

h∗t [∞] + (ft − st)− Et[∆st+1] = (1/2)σ2T + σTP

High local currency term premia coincide with low currency risk premia and vice-versa. In

the symmetric case, dollar term premia are identical across currencies.

11

Proposition 2.7 In general, the foreign term premium in dollars equal the domestic term pre-

mium plus the difference in the entropy of the permanent component of the pricing kernel.

h∗t [∞] + (ft − st)− Et[∆st+1]− ht[∞] = Lt

(ΛPt+1

ΛPt

)− Lt

(ΛP,∗t+1

ΛP,∗t

).

In order to deliver a currency risk premium at longer maturities, entropy differences in the

permanent component of the pricing kernel are required. At short maturities, the currency

risk premium is determined by the entropy difference of the entire pricing kernel (see equation

2.5). Since carry trade returns are base-currency-invariant, heterogeneity in the exposure of

the pricing kernel to a global component of the pricing kernel is required to explain the carry

trade premium (Lustig, Roussanov, and Verdelhan (2011)). To deliver a carry trade premium

at longer maturities, we would need heterogeneous exposure to a global permanent component.

Certainty, the permanent component of the pricing kernel is important.

Proposition 2.8 There is a lower bound on the volatility of the permanent component of the

pricing kernel (Alvarez and Jermann (2005)):

Lt(ΛPt+1

ΛPt

) ≥ Et (logRt+1)− Et (logRt+1,1[1t+∞]) .

Given the size of the equity premium relative to the term premium, Alvarez and Jermann (2005)

conclude that the permanent component of the pricing kernel is large and accounts for most of

the risk. Lots of persistence is needed to deliver a low term premium and a high equity premium.

2.3 Measure of Risk Sharing

The valuation of long-maturity bonds in bond markets encodes information about the nature

of shocks that drive changes in exchange rates in currency markets. Using the prices of long-

maturity bonds in two countries, we can decompose the changes in the bilateral spot exchange

12

rate into two parts: a part that captures cross-country differences in the transitory components

of the pricing kernel and a part that encodes differences in the permanent components of the

the pricing kernel.

We assume that the transitory components of the domestic and foreign stochastic discount

factors are bounded from below and above:

0 <ΛTt+1

ΛTt

<∞ and 0 <Λ∗Tt+1

Λ∗Tt<∞

We consider the one-period multiplicative excess return of the foreign k-maturity bond over the

domestic k-maturity bond, where both returns are expressed in domestic currency terms:

RXt+1,1[1k] ≡ St+1

St

R∗t+1,1[1k]

Rt+1,1[1k]

As maturity k approaches infinity, we can apply the boundedness condition above and show that

differences in the returns of infinite maturity bonds allow us to trace how well countries share

risk that arises from permanent innovations in their marginal utility:

Proposition 2.9 In two economies with complete markets, the multiplicative excess return on

the longest maturity foreign bonds in domestic currency measures the permanent component of

exchange rates.

RXt+1,1[1∞] ≡ limk→∞

RXt+1,1[1k] =Λ∗Pt+1

Λ∗Pt

ΛPt

ΛPt+1

=SPt+1

SPt

Corollary 2.3 If the domestic and foreign pricing kernels have common permanent innovations,

ΛPt+1

ΛPt

=Λ∗Pt+1

Λ∗Pt

for all states, then the one-period returns on the foreign longest maturity bonds in domestic

currency are identical to the domestic ones: RXt+1,1[1∞] = 1 for all states.

We recover uncovered long-bond return parity. In this polar case, most of the innovations to

13

the pricing kernel are highly persistent, but the shocks that drive exchange rates are not, simply

because the persistent shocks are shared more efficiently across countries.

Brandt, Cochrane, and Santa-Clara (2006) show that the combination of relatively smooth

exchange rates and much more volatile stochastic discount factors implies that state prices are

very highly correlated across countries. A 10% volatility in exchange rate changes and a volatility

of marginal utility growth rates of 50% implies a correlation of at least 0.98. We can derive a

tighter bound on the covariance of the permanent component across different countries.

Proposition 2.10 The cross-country covariance of the SDF permanent components is bounded

below by:

covt

(log

ΛP?t+1

ΛP?t

, logΛPt+1

ΛPt

)=

1

2[V art

(log

ΛP?t+1

ΛP?t

)+ V art

(log

ΛPt+1

ΛPt

)− V art (logRXt,1[1∞])]

≥ Et

(logR?

t+1

)− Et

(logR?

t+1,1[1t+∞])

+ Et (logRt+1)− Et (logRt+1,1[1t+∞])

− 1

2V art (logRXt+1,1[1∞]) .

3 Data

We use two different panels of panels: a smaller panel of countries consisting of zero coupon

prices for the whole yield curve and a larger panel consisting of bond returns for a 10-year bond

index.

Small Panel First, we construct a small panel of countries with zero coupon bond return

data. We use end-of-the-month data for the riskless zero-coupon yield curves, as proxied by the

government debt zero-coupon yield curve, of 10 currencies: the US dollar, the German mark (the

euro from 1999 onwards), the UK pound, the Japanese yen, the Canadian dollar, the Australian

dollar, the Swiss franc, the New Zealand dollar, the Swedish krona and the Norwegian krone.

14

The sample starts in November 1971 and ends in September 2012, but we have full data only

for the US dollar; for the rest of the currencies, the sample period is given in Table 1. From

November 1971 to May 2009, we use the data in Wright (2011). From June 2009 to September

2011, we source the data from the Bank of International Settlements (for the US, German,

Canadian, Swiss and Swedish sovereign debt yield curves) and the Bank of England (for the UK

sovereign debt yield curve). For each currency, continuously-compounded yields are available at

maturities from 3 months to 120 months (10 years), in 3-month increments.

We also collect end-of-the-month data on spot exchange rates against the US dollar from

MSCI (available through Datastream) for the same set of countries.

Large Panel We also construct a larger panel. We collect data from Global Financial Data

for a much larger panel of developed countries and a larger panel that includes all countries.

The dataset includes a 10-year Government Bond Total Return Index for each of these countries

in dollars and local currency and a T-bill Total Return index. We will use the 10-year bond

returns as a proxy for the bonds with the longest maturity. We will check the robustness of our

results.

The entire sample of countries includes Australia, Austria, Belgium, Canada, Denmark,

Finland, France, Germany, Greece, Ireland, Israel, Italy, Japan, Malaysia, Mexico, Netherlands,

New Zealand, Norway, Pakistan, Philippines, Poland, Portugal, Singapore, South Africa, Spain,

Sweden, Switzerland, Taiwan, Thailand, United Kingdom and the United States. The sample

of developed countries includes Australia Austria, Belgium, Canada, Denmark, Finland, France,

Germany, Greece, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Spain,

Sweden, Switzerland, United Kingdom, and the United States.

4 International Bond Return Correlation and Volatility

We use bond return data from the small panel of 10 developed countries to show that long-

maturity bond returns expressed in US dollar terms are much more correlated across countries

than the dollar returns of short-maturity bonds. We then formally test the long bond par-

15

ity condition and show that unshared permanent innovations do contribute to exchange rate

variation.

4.1 Correlation and Maturity of Foreign Bond Returns

If international risk sharing is mostly due to countries sharing their permanent pricing kernel

fluctuations, we would expect to see that holding period returns on zero-coupon bonds, once

converted to a common currency (the US dollar, in particular), become increasingly similar as

bond maturities approach infinity. To determine whether this hypothesis has merit, we calcu-

late the correlation coefficient between one-period nominal USD returns on foreign bonds and

corresponding returns on US bonds for bonds of maturity ranging from 1 year to 10 years. To

determine whether the patterns in correlation coefficients arise as a result of exchange rate prop-

erties, we also calculate the correlation coefficient between one-period nominal local currency

returns on foreign bonds and corresponding returns on US bonds. The results for both USD and

local currency returns, regarding overlapping 3-month holding periods, are presented in Figure

2.

As we can see, all 9 foreign currency yield curves exhibit the conjectured pattern: correlation

coefficients for USD returns start from negative or zero values and increase monotonically with

bond maturity, tending towards 1 for long-term bonds. Interestingly, this striking monotonicity

is not observed if we focus on local currency returns. The local currency 3-month return cor-

relations do not exhibit any discernible pattern with maturity, implying that the convergence

of USD return correlations towards 1 results from exchange rate changes which partially offset

differences in local currency bond returns.

4.2 Volatility and Maturity of Foreign Bond Returns

To further explore our intuition that USD bond returns of different countries become increasingly

similar as bond maturity increases, we also calculate the ratio of foreign to domestic USD bond

16

Figure 2: The Maturity structure of Bond Return Correlations

2 4 6 8 10

0

0.5

1NOK

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t

Local

USD

2 4 6 8 10

0

0.5

1GBP

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t

2 4 6 8 10

0

0.5

1JPY

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t

2 4 6 8 10

0

0.5

1CAD

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t

2 4 6 8 10

0

0.5

1AUD

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t

2 4 6 8 10

0

0.5

1CHF

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t2 4 6 8 10

0

0.5

1NZD

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t

2 4 6 8 10

0

0.5

1SEK

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t

2 4 6 8 10

0

0.5

1NOK

Maturity (in years)

Co

rre

latio

n c

oe

ffic

ien

t

Panel B: Developed Countries

Notes: Correlation with U.S. bond returns. Sample: country-dependent. Monthly data. Holding period is3-months.

returns; the k-year maturity volatility ratio is given by

V olRk =σ(r∗t+1,k + ∆st+1

)σ (rt+1,k)

For comparison, we also calculate the corresponding volatility ratio for local currency returns,

given by

V olRLk =

σ(r∗t+1,k

)σ (rt+1,k)

and report both V olRk and V olRLk for k = 1, 2, ...10 years in Figure 3 for 3-month returns.

The pattern is unambiguous: the unconditional volatility of the USD 3-month foreign returns is

much higher than that of the corresponding volatility of US bond returns for small maturities,

but the volatility ratio falls sharply for higher maturities and is close to 1 for 10-year bonds.

17

In contrast with the observed pattern for V olRk, the local currency volatility ratio V olRLk is

virtually flat with maturity, implying that the convergence in USD return bond volatility is

due to the properties of the nominal exchange rate. Of course, even if exchange rates followed a

random walk and exchange rate innovations are uncorrelated with returns, we could still observe

this pattern, simply the exchange rates account for a smaller share of overall return volatility at

longer maturities. However, we will show that exchange rates actually hedge interest rate risk.

Figure 3: The Maturity structure of Bond Return Volatility

2 4 6 8 100

5

10NOK

Maturity (in years)

Vo

latil

ity r

atio

Local

USD

2 4 6 8 100

5

10GBP

Maturity (in years)

Vo

latil

ity r

atio

2 4 6 8 100

5

10JPY

Maturity (in years)

Vo

latil

ity r

atio

2 4 6 8 100

5

10CAD

Maturity (in years)

Vo

latil

ity r

atio

2 4 6 8 100

5

10AUD

Maturity (in years)

Vo

latil

ity r

atio

2 4 6 8 100

5

10CHF

Maturity (in years)

Vo

latil

ity r

atio

2 4 6 8 100

5

10NZD

Maturity (in years)

Vo

latil

ity r

atio

2 4 6 8 100

5

10SEK

Maturity (in years)

Vo

latil

ity r

atio

2 4 6 8 100

5

10NOK

Maturity (in years)

Vo

latil

ity r

atio

Notes: Volatility of Foreign and U.S. bond returns. Sample: country-dependent. Monthly data. Holding periodis 3-months.

Our results are robust to an increase of the holding period. Specifically, in unreported

results, 6-month and 12-month returns produce the same patterns: for both holding periods

and for virtually all currencies, there is an almost monotonic relationship between correlation

coefficients of USD returns and bond maturity. Furthermore, 6-month and 12-month local

currency return correlations are not sensitive to maturity, the USD return volatility ratio is

very high for short maturities, but quickly converges towards 1, and the local currency return

18

volatility ratio is flat with maturity.

In sum, the behavior of USD bond returns and local currency bond returns differs markedly

as bond maturity changes. While USD bond returns become more correlated and roughly equally

volatile across countries as the maturity increases, the behavior of local currency returns do not

appear to change when bond maturity changes.

5 The Maturity Structure of Currency Carry Trade Returns

To test the predictions of the theory, we sort currencies into portfolios based on variables that

predicts bond and currency returns. Interest rates predict currency returns. Hence, from the

perspective of the theory, sorting by interest rates is equivalent to sorting by the total entropy

of the foreign pricing kernel relative to the U.S.

5.1 Sorting Currencies by Interest Rates

Following the work of Lustig and Verdelhan (2005), we start by sorting currencies into 5 portfolios

based on the interest rate differences. We use the one-month return on the GFD Treasury-bills

Total Return index that is realized at t − 1 to sort currencies into portfolios at time t.3 Then,

we compute monthly returns between t and t+ 1. The portfolios are rebalanced each month. In

the large sample of countries, we have 5 to 6 currencies in each portfolio.

Table 1 reports the annualized moments of log returns. The top panel uses the entire sample.

The first three rows report moments of currency excess returns rxfx = (f − s) − ∆s. As we

expected (see Lustig and Verdelhan (2005)) for detailed analysis, average excess returns increase

from the first portfolio to the last portfolio. The average excess return on the first portfolio is

-73 basis points per annum. The average excess return on the last portfolio is 269 basis points.

The spread between the first and the last portfolio is 341 basis points per annum. The volatility

of these returns increases only slightly from the first to the last portfolio. As a result, the Sharpe

ratio (annualized) increases from -0.12 to 0.34 on the last portfolio. The Sharpe ratio on a long

3We are being conservative by sorting on the T-bill return at t− 1. The Tbill return at t is largely known att− 1.

19

position in the last portfolio and a short position in the first portfolio is 0.54 per annum. The

results for the post-Bretton-Woods sample in the bottom panel are very similar. Hence, the

currency carry trade is profitable at the short end of the maturity spectrum.

Table 1: Interest-Rate Sorted Portfolios: All Countries

1 2 3 4 5 5-1

Panel A: 1950-2012rxfx Mean -0.73 0.60 1.85 1.85 2.69 3.41

Std 6.24 7.07 6.98 7.19 7.89 6.28SR -0.12 0.09 0.27 0.26 0.34 0.54

rx? Mean 2.49 1.45 1.78 1.76 0.45Std 3.43 4.71 3.80 3.94 5.34SR 0.73 0.31 0.47 0.45 0.08

rx$ Mean 1.76 2.06 3.63 3.61 3.14 1.38Std 7.52 8.94 8.22 8.24 9.02 8.03SR 0.23 0.23 0.44 0.44 0.35 0.17

∆s Mean 0.79 0.43 0.61 -0.87 -2.95f − s Mean -1.52 0.18 1.24 2.72 5.64r? − rUS Mean -0.54 0.05 1.44 2.89 4.58

Panel B: 1971-2012rxfx Mean -0.50 0.85 2.37 2.02 3.49 3.99

Std 7.53 8.64 8.33 8.75 8.89 6.66SR -0.07 0.10 0.28 0.23 0.39 0.60

rx? Mean 3.14 2.22 2.06 2.55 0.78Std 4.06 5.63 4.49 4.48 5.26SR 0.77 0.39 0.46 0.57 0.15

rx$ Mean 2.63 3.07 4.43 4.57 4.27 1.64Std 9.04 10.85 9.79 9.88 10.73 9.44SR 0.29 0.28 0.45 0.46 0.40 0.17

∆s Mean 1.30 0.61 0.91 -1.23 -3.86f − s Mean -1.80 0.24 1.46 3.25 7.35r? − rUS Mean -1.23 -0.11 0.95 3.23 5.57

Portfolios of currencies sorted at t− 1 by returns on T-bills realized at end of t− 1. Annualized monthly returnsrealized at t on 10-year Bond Index and T-bills.

As we know from Bekaert (1996), Bansal (1997) and Backus, Foresi, and Telmer (2001), The

currency risk premia reflect differences in the entropy of the domestic and the foreign pricing

kernels:


(Λt+1

Λt

)− Lt

(Λ∗t+1

Λ∗t

).

20

Table 2: Interest-Rate Sorted Portfolios: Developed Countries

1 2 3 4 5 5-1

Panel A: 1950-2012rfx Mean -0.06 0.97 1.64 2.35 2.94 3.01

Std 7.22 7.75 7.71 7.65 8.18 5.88SR -0.01 0.12 0.21 0.31 0.36 0.51

rx Mean 2.04 1.80 1.61 0.96 0.39Std 3.75 5.68 4.62 4.35 5.30SR 0.55 0.32 0.35 0.22 0.07

rx$ Mean 1.98 2.76 3.24 3.30 3.34 1.36Std 8.75 9.91 9.39 8.97 9.36 8.01SR 0.23 0.28 0.35 0.37 0.36 0.17

∆s Mean 1.22 0.56 0.31 -0.21 -1.93f − s Mean -1.28 0.41 1.33 2.56 4.88r − rUS Mean -0.75 0.69 1.43 2.00 3.76

Panel B: 1971-2012rfx Mean 0.28 1.32 1.90 2.84 3.52 3.24

Std 8.80 9.47 9.41 8.97 9.74 6.82SR 0.03 0.14 0.20 0.32 0.36 0.48

rx Mean 2.58 2.65 1.89 1.40 1.18Std 4.44 6.82 5.39 4.88 6.10SR 0.58 0.39 0.35 0.29 0.19

rx$ Mean 2.86 3.97 3.80 4.25 4.70 1.84Std 10.60 12.03 11.34 10.39 11.02 9.27SR 0.27 0.33 0.33 0.41 0.43 0.20

∆s Mean 1.75 0.77 0.38 -0.03 -2.17f − s Mean -1.47 0.55 1.52 2.88 5.70r − rUS Mean -1.45 0.64 0.85 1.72 4.31

Portfolios of currencies sorted at t− 1 by returns on T-bills realized at end of t− 1. Annualized monthly returnsrealized at t on 10-year Bond Index and T-bills.

21

High interest rate currencies have low entropy and low interest rate currencies have high entropy.

This follows directly from no-arbitrage. In a log-normal world, entropy is just one half of the

variance. In that case, high interest rate currencies have low variance of the pricing kernel, while

low interest rate currencies have high variance of the pricing kernel. Hence, sorting by interest

rates (from low to high) seems equivalent to sorting by pricing kernel entropy (from high to

low).

The next three rows report the excess return rx? on 10-year bond positions in each of these

currencies. To be clear, these returns are reported in local currency. There is a strong decreasing

pattern in local currency bond risk premia. The average excess return on the first portfolio is

249 basis points per annum. These excess returns decrease monotonically to 45 basis points on

the last portfolio. Hence, there is a 294 basis points spread per annum between the first and

the last portfolio. The Sharpe ratio on the first portfolio is 0.73. Hence, there is a very strong

negative correlation between local currency bond risk premia and currency risk premia. Low

interest rate currencies tend to produce high local currency bond risk premia, while high interest

rate currencies tend to produce low local currency bond risk premia.

In the absence of arbitrage, we know that the foreign term premium in local currency is

given by:

h?t (∞) = limk→∞

Etrxk,?t+1 = Lt(m

?t+1)− Lt(m

?,Pt+1). (5.1)

Hence, the decreasing term premia are consistent with the decreasing entropy Lt(m?t+1) from

the low interest rate portfolio 1 to the high interest rate portfolio 5 that we had inferred from

the foreign currency risk premia. These are apparently not offset by equivalent increases in the

entropy of the entropy of the permanent component of the foreign pricing kernel. In a log-normal

world, the term premium is determined by (1/2) of the variance of the temporary component

of the pricing kernel plus a covariance term. Hence, if the currency risk premia are driven (to

seem extent) by the variance of the temporary component, that would explain why term premia

are high for low interest rate currencies, with high currency risk premia.

The monotonically decreasing pattern in term risk premia is direct evidence in favor of a

risk-based explanation of foreign currency returns. Bond markets agree with currency markets

22

that there is more temporary risk in the pricing kernel of low interest rate currencies. Hence,

temporary shocks to the pricing kernel play a major role as drivers of currency risk premia. If

all of the shocks driving currency risk premia were permanent, then there would be no relation

between currency risk premia and term premia.

A natural question is whether U.S. investors can ‘combine’ the currency risk premium and

the bond risk premium. To compute the dollar bond excess returns rx$, we simply add the

currency excess returns rxfx = (f − s) − ∆s and the local currency bond returns rx?. The

results are reported in the next three rows. The decline in the local currency bond risk premia

partly offsets the increase in currency risk premia. As a result, the average excess return on the

last portfolio is only 138 basis points per annum higher than the returns on the first portfolio.

The SR on a long-short position in bonds of the last and the first portfolio is only 0.17. U.S.

investors cannot simply combine the currency carry trade with a yield carry trade, because

these risk premia roughly offset each other. Interest rates are great predictors of currency excess

returns and local currency bond excess returns, but not dollar excess returns. To get long-term

carry trade returns, we need differences in the quantity of permanent risk, as can be verified

from:

h∗t [∞] + (ft − st)− Et[∆st+1]− ht[∞] = Lt

(ΛPt+1

ΛPt

)− Lt

(ΛP,∗t+1

ΛP,∗t

).

The data do not seem to lend support to these differences in permanent risk. These results

are essentially unchanged in the post-Bretton-Woods sample. The Sharpe ratio on the currency

carry trade is 0.60, achieved by going long int he last portfolio and short in the first portfolio.

However, there is a strong decreasing pattern in local currency bond risk premia, from 314 basis

points per annum in the first portfolio to 78 basis points in the last portfolio. As a result, there

is essentially no discernible pattern in dollar bond risk premia.

Table 2 excludes non-developed countries and performs the same spring exercise. These

moments look similar. The Sharpe ratio on the carry trade strategy is lower (0.51). We see a

monotonically increasing pattern in local currency bond risk premia. The spread in returns is

243 basis points, but this spread shrinks to 136 basis points in dollars, because of the offsetting

effects of the currency risk premia.

23

Sorting by Contemporaneous T-bill returns Since the construction of the total return

index by Global Financial Data assumes the Tbill-price does not change, we could also use the

return realized at t to sort currencies into portfolios of currencies at t − 1, because the return

at t would be known at t− 1. The results are reported in Table 3.

The annualized spread in currency risk premia between the first and the last portfolio is even

larger: 436 bps per annum. The returns increase from minus 148 bps on the first portfolio to

288 bps per annum on the last portfolio. The SR on the long-short strategy is 0.68. The spread

between the first and the last portfolio in local currency term premia is 397 bps per annum.

Hence, the term spread is almost the same order of magnitude as the currency risk premium.

As a result, the dollar term risk premium spread is only 40 bps per annum. Hence, the results

on these portfolios sorted by returns at t are even starker. The term premia almost completely

offset the currency risk premia.

We replicate the same portfolio-building exercise on the subsample of developed countries,

partly to guard against the possibility of credit risk contaminating our findings. These are

reported in 4.

Figure 6 and 7 depict the currency risk premia and the local currency bond risk premia. In

both samples, there is a strong negative relation between these risk premia. Low interest rate

currencies tend to have high entropy of the pricing kernel. This also leads to higher bond risk

premia, as one would expect.

5.2 Sorting Currencies by Slope of the Yield Curve

We also sorted currencies into portfolio by the slope of the yield curve in each country. Recently,

Ang and Chen (2010) have documented that the slope of the yield curve adds additional foresting

power for currency excess returns. We use the yield on the 10-year government bonds at t − 1

minus the T-bill rate at t−1 to sort currencies into portfolios at t−1. Then we compute returns

at t. Table 5 reports the annualized moments of log returns on these portfolios.

The slope of the yield curve, a measure of the term premium, is determined largely by the

entropy of the temporary component of the pricing kernel (see equation 5.1). As this increases,

24

Table 3: Interest-Rate Sorted Portfolios: All Countries

1 2 3 4 5 5-1

Panel A: 1950-2012rfx Mean -1.48 0.56 1.85 2.45 2.88 4.36

Std 6.89 7.06 7.09 7.02 7.37 6.40SR -0.21 0.08 0.26 0.35 0.39 0.68

rx Mean 3.46 1.90 1.46 1.33 -0.51Std 4.37 4.46 4.06 3.86 4.59SR 0.79 0.43 0.36 0.34 -0.11

rx$ Mean 1.98 2.46 3.31 3.78 2.38 0.40Std 7.50 8.84 8.41 8.13 9.13 8.19SR 0.26 0.28 0.39 0.47 0.26 0.05

∆s Mean 0.47 0.41 0.63 -0.25 -3.27f − s Mean -1.95 0.15 1.22 2.70 6.15r? − rUS Mean 0.00 0.55 1.17 2.52 4.14

Panel B: 1971-2012rfx Mean -1.05 0.76 2.54 2.70 3.28 4.33

Std 7.59 8.64 8.47 8.56 8.83 6.77SR -0.14 0.09 0.30 0.32 0.37 0.64

r Mean 3.85 2.76 1.83 2.04 -0.05Std 4.04 5.33 4.70 4.50 5.29SR 0.95 0.52 0.39 0.45 -0.01

r$ Mean 2.80 3.52 4.36 4.74 3.22 0.43Std 9.01 10.74 9.98 9.80 10.83 9.62SR 0.31 0.33 0.44 0.48 0.30 0.04

∆s Mean 0.86 0.55 1.10 -0.54 -4.29f − s Mean -1.90 0.21 1.44 3.24 7.57r? − rUS Mean -0.62 0.41 0.70 2.71 4.96

Portfolios of currencies sorted at t − 1 by monthly returns on T-bills realized at end of t. Annualized monthlyreturns realized at t on 10-year Bond Index and T-bills.

25

Table 4: Interest-Rate Sorted Portfolios: Developed Countries

1 2 3 4 5 5-1

Panel A: 1950-2012rxfx Mean -0.41 1.03 1.77 2.47 2.92 3.33

Std 7.17 7.79 7.71 7.66 8.20 5.83SR -0.06 0.13 0.23 0.32 0.36 0.57

rx? Mean 2.95 2.29 0.80 0.97 -0.77Std 3.89 5.34 4.78 4.43 5.39SR 0.76 0.43 0.17 0.22 -0.14

rx$ Mean 2.54 3.32 2.57 3.44 2.15 -0.39Std 8.76 9.87 9.39 9.03 9.60 8.16SR 0.29 0.34 0.27 0.38 0.22 -0.05

∆s Mean 0.92 0.65 0.45 -0.10 -2.06f − s Mean -1.33 0.38 1.32 2.58 4.98r? − rUS Mean 0.11 1.16 0.61 2.03 2.70

Panel B: 1971-2012rxfx Mean -0.20 1.38 2.11 3.01 3.54 3.74

Std 8.74 9.53 9.38 9.04 9.72 6.74SR -0.02 0.14 0.23 0.33 0.36 0.55

rx? Mean 3.68 3.49 0.68 1.33 -0.13Std 4.62 6.31 5.68 4.98 6.16SR 0.80 0.55 0.12 0.27 -0.02

rx$ Mean 3.48 4.87 2.79 4.34 3.42 -0.06Std 10.63 11.93 11.36 10.53 11.26 9.44SR 0.33 0.41 0.25 0.41 0.30 -0.01

∆s Mean 1.33 0.86 0.60 0.11 -2.29f − s Mean -1.53 0.51 1.51 2.91 5.83r? − rUS Mean -0.42 1.44 -0.37 1.67 3.14

Portfolios of currencies sorted at t − 1 by monthly returns on T-bills realized at end of t. Annualized monthlyreturns realized at t on 10-year Bond Index and T-bills.

26

the local term premium increases as well. However, the dollar term premium only compensates

investors for the relative entropy of the permanent component of the U.S. and the foreign pricing

kernel. In the extreme case in which all permanent shocks are common, the dollar term premium

equals the U.S. term premium.

The first three rows repots the moments of the currency excess returns. These decline from an

average of 232 bps per annum on the first portfolio to -96 bps per annum on the fifth portfolio.

A long-short position delivers an excess return of -327 bps per annum and a Sharpe ratio of

0.45. This confirms the findings of Ang and Chen (2010). The slope of the yield curve predicts

currency excess returns. These findings confirm that the entropy of the temporary component

plays a large role in currency risk premia.

The next three rows report the local currency bond returns. As expected, the highest slope

portfolios produce large bond excess returns of 5.11 percent per annum, compared to -179 basis

points per annum on the first portfolio. Hence, a long-short position produces a spread of 690

basis points per annnum.

The next three rows report dollar returns. In dollars, this 690 spread is reduced to 363 basis

points, because of the partly offsetting pattern in currency risk premia. What is driving these

results? The high slope currencies tend to be low interest rate currencies, while the low slope

currencies tend to be the high interest rate currencies, as is apparent from the last four rows

in each the top panel. The first portfolio has an average slope of -86 bps and an interest rate

difference of 386 bps relative to the U.S., while the last portfolio has a slope of 379 bps, and

a negative interest rate difference of -54 bps per annum. These findings confirm that currency

risk premia are driven to a large extent by temporary shocks to the pricing kernel.

We also sorted currencies into portfolio at t− 1 based on the yield at t− 1 minus the T-bill

returns that is realized at t. These results are reported in Table 7 and Table 8. We observe

the same negative correlation between currency and term risk premia. The spread in local term

premia is 953 basis points when we include all countries. Of course, a large portion of this spread

is due to credit risk, because we’re sorting by the slope of the yield curve, provided that the

term structure of credit risk premia is upward sloping. This spread gets reduced by 338 bps per

27

Table 5: Slope-sorted Portfolios: Developed Countries

1 2 3 4 5 5-1

Panel A: 1950-2012rxfx Mean 2.32 1.51 1.82 0.50 -0.96 -3.27

Std 7.03 7.04 7.28 7.53 7.89 7.20SR 0.33 0.22 0.25 0.07 -0.12 -0.45

rx? Mean -1.79 1.15 1.86 2.66 5.11Std 3.76 3.64 4.27 4.84 6.95SR -0.48 0.32 0.44 0.55 0.74

rx$ Mean 0.52 2.66 3.67 3.16 4.16 3.63Std 8.18 7.86 8.74 9.18 8.49 7.61SR 0.06 0.34 0.42 0.34 0.49 0.48

y10 − y1 Mean -0.86 0.74 1.36 1.97 3.79∆s Mean -1.54 -0.21 0.70 0.08 -0.41f − s Mean 3.86 1.73 1.11 0.42 -0.54r − rUS Mean 0.56 1.36 1.46 1.58 3.06

Panel B: 1971-2012rxfx Mean 2.57 1.89 2.38 0.74 -0.29 -2.86

Std 8.55 8.57 8.82 8.91 7.23 6.00SR 0.30 0.22 0.27 0.08 -0.04 -0.48

rx? Mean -1.79 2.12 2.31 3.41 5.28Std 4.38 4.23 5.04 5.67 5.56SR -0.41 0.50 0.46 0.60 0.95

rx$ Mean 0.78 4.01 4.68 4.15 4.99 4.21Std 9.86 9.46 10.52 10.84 10.06 8.92SR 0.08 0.42 0.45 0.38 0.50 0.47

y10 − y1 Mean -1.19 0.64 1.31 1.91 3.31∆s Mean -2.06 -0.15 1.00 0.17 -0.50f − s Mean 4.63 2.04 1.37 0.57 0.21r − rUS Mean 0.28 1.59 1.12 1.41 2.92

Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1. Monthly returns at t on 10-year BondIndex and T-bills.

28


1 2 3 4 5 5-1

Panel A: 1950-2012rxfx Mean 2.68 2.37 1.04 0.41 0.28 -2.39

Std 7.48 7.35 7.87 7.76 7.00 4.93SR 0.36 0.32 0.13 0.05 0.04 -0.49

rx? Mean -1.08 1.51 1.69 2.26 2.96Std 4.21 4.15 4.56 5.33 7.31SR -0.26 0.36 0.37 0.42 0.40

rx$ Mean 1.59 3.88 2.73 2.67 3.24 1.65Std 8.52 8.43 9.21 9.92 11.06 9.51SR 0.19 0.46 0.30 0.27 0.29 0.17

y10 − y1 Mean -0.58 0.78 1.31 1.90 3.14∆s Mean -0.79 0.71 -0.04 -0.22 0.14f − s Mean 3.47 1.65 1.08 0.63 0.14r − rUS Mean 0.87 1.65 1.26 1.38 1.59

Panel B: 1971-2012rxfx Mean 3.09 3.07 1.32 0.61 0.29 -2.80

Std 9.08 8.95 9.46 9.22 8.53 5.91SR 0.34 0.34 0.14 0.07 0.03 -0.47

rx? Mean -0.74 2.64 2.29 2.82 3.15Std 4.87 4.81 5.33 6.31 8.69SR -0.15 0.55 0.43 0.45 0.36

rx$ Mean 2.35 5.71 3.61 3.43 3.44 1.09Std 10.22 10.12 10.97 11.79 13.33 11.29SR 0.23 0.56 0.33 0.29 0.26 0.10

y10 − y1 Mean -0.79 0.72 1.27 1.86 3.18∆s Mean -0.88 1.21 0.08 -0.24 0.06f − s Mean 3.97 1.86 1.24 0.85 0.23r − rUS Mean 0.66 1.94 0.97 1.11 0.81

Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1. Monthly returns at t on 10-year BondIndex and T-bills.

29

annum when we convert the returns into dollars. When we look at developed countries only,

the spread in local term premia of 727 bps is reduced to 413 basis points, because the spread

in currency risk premia is 388 bps per annum. This spread in currency risk premia increases to

388 bps per annum in the post-Bretton-Woods sample.

Table 7: Slope-sorted Portfolios: All Countries

1 2 3 4 5 5-1

Panel A: 1950-2012rxfx Mean 2.39 1.95 1.39 0.57 -0.99 -3.38

Stdev 6.97 6.98 7.18 7.53 8.00 7.31SR 0.34 0.28 0.19 0.08 -0.12 -0.46

rx? Mean -2.95 0.32 1.59 3.62 6.58Stdev 3.75 3.83 4.14 4.72 6.97

SR -0.79 0.08 0.38 0.77 0.94

rx$ Mean -0.55 2.27 2.98 4.18 5.59 6.14Stdev 8.14 8.06 8.40 9.22 8.60 8.06

SR -0.07 0.28 0.35 0.45 0.65 0.76

y10 − y1 Mean -0.96 0.72 1.38 2.00 3.89∆s Mean -1.57 0.20 0.31 0.17 -0.36f − s Mean 3.97 1.75 1.08 0.40 -0.63r − rUS Mean -0.49 0.56 1.16 2.50 4.44

Panel B: 1971-2012rxfx Mean 2.70 2.45 1.78 1.01 -0.48 -3.18

Stdev 8.45 8.52 8.68 8.92 7.44 6.16SR 0.32 0.29 0.21 0.11 -0.07 -0.52

rx? Mean -3.11 0.89 1.79 4.59 7.22Stdev 4.40 4.39 4.90 5.54 5.63

SR -0.71 0.20 0.37 0.83 1.28

rx$ Mean -0.42 3.33 3.58 5.59 6.74 7.16Stdev 9.79 9.70 10.10 10.89 10.23 9.46

SR -0.04 0.34 0.35 0.51 0.66 0.76

y10 − y1 Mean -1.33 0.63 1.32 1.94 3.43∆s Mean -2.08 0.38 0.45 0.46 -0.59f − s Mean 4.77 2.07 1.33 0.55 0.10r − rUS Mean -0.90 0.39 0.56 2.57 4.76

Portfolios of currencies sorted at t− 1 by slope of yield curve at t− 1 (defined as yield at t− 1 minus return onT-bill at t). Monthly returns at t on 10-year Bond Index and T-bills.

30


1 2 3 4 5 5-1

Panel A: 1950-2012rxfx Mean 2.97 2.41 0.93 0.59 -0.17 -3.14

Std 7.46 7.38 7.84 7.69 7.08 5.02SR 0.40 0.33 0.12 0.08 -0.02 -0.63

rx? Mean -2.39 0.64 1.14 3.22 4.88Std 4.16 4.28 4.57 5.24 7.28SR -0.58 0.15 0.25 0.61 0.67

rx$ Mean 0.57 3.04 2.07 3.81 4.71 4.13Std 8.52 8.56 9.10 9.83 11.10 9.64SR 0.07 0.36 0.23 0.39 0.42 0.43

y10 − y1 Mean -0.68 0.77 1.33 1.93 3.22∆s Mean -0.59 0.74 -0.18 0.00 -0.21f − s Mean 3.56 1.67 1.11 0.59 0.04r − rUS Mean -0.35 0.79 0.74 2.30 3.41

Panel B: 1971-2012rxfx Mean 3.54 2.97 1.13 0.99 -0.34 -3.88

Std 9.02 9.02 9.40 9.15 8.62 5.99SR 0.39 0.33 0.12 0.11 -0.04 -0.65

rx? Mean -2.28 1.54 1.15 4.25 5.68Std 4.86 4.81 5.43 6.07 8.74SR -0.47 0.32 0.21 0.70 0.65

rx$ Mean 1.26 4.51 2.28 5.25 5.33 4.07Std 10.19 10.24 10.88 11.63 13.45 11.54SR 0.12 0.44 0.21 0.45 0.40 0.35

y10 − y1 Mean -0.91 0.71 1.29 1.89 3.29∆s Mean -0.54 1.08 -0.16 0.20 -0.42f − s Mean 4.08 1.89 1.30 0.80 0.08r − rUS Mean -0.76 0.87 -0.12 2.49 3.19

Portfolios of currencies sorted at t− 1 by slope of yield curve at t− 1 (defined as yield at t− 1 minus return onT-bill at t). Monthly returns at t on 10-year Bond Index and T-bills.

31

6 Testing Uncovered Bond Return Parity

This section directly tests the Uncovered Bond Return Parity Condition. Uncovered bond return

parity should hold for long bonds provided that countries share the permanent component. If

the permanent component of the pricing kernel is common, then exchange rate exactly hedge

the foreign interest rate risks in long foreign bond position, because exchange rates respond only

to temporary innovations to the pricing kernels. These are the innovations driving long-term

bond prices and yields.

6.1 Testing Uncovered Bond Return Parity in the Cross-section

The last three rows in Table 3 decompose the results for the portfolios sorted by returns at t−1.

The currency excess return equals the interest rate difference minus the rate of depreciation

(f−s)−∆s. The rate at which the high interest rate currencies depreciate (327 bps per annum)

is not high enough to offset the interest rate difference 615 bps. Similarly, the rate at which the

low interest rate currencies appreciate (47 bps per annum) is not high enough to offset the low

interest rates (minus 195 bps). UIP fails in the cross-section.

However, the bond return differences (in local currency) are closer to being offset by the rate

of depreciation. The bond return spread is 414 bps per annum for the last portfolio, compared to

an annual depreciation rate of 327 bps, while the spread on the first portfolio is 0 bps, compared

to depreciation of 47 bps. Figure 8 plots the rate of depreciation against the interest rate (bond

return) differences with the U.S. The vertical distance from the 45-degree line is an indication

of how far we are from UIP or long-run UBRP. Especially for portfolio 1 and portfolio 5, UBRP

is a much better fit for the data. The currency exposure hedges the interest rate exposure in

the bond position. High returns are off-set by higher depreciations. As a result, foreign bond

portfolios are almost hedged against foreign interest rate risk.

6.2 Testing Uncovered Bond Return Parity in the Time-Series

Alternatively, we could check whether bond return parity holds in the time series. To the ex-

tent the 10-year bond is a reasonable proxy for the infinite-maturity bond, uncovered long-bond

32

parity implies that the unconditional USD 10-year bond returns are not statistically different.

To determine whether exchange rate changes completely eliminate differences in countries’ per-

manent SDF component, we test the long-bond return parity condition by regressing nominal

USD holding period returns on 10-year foreign bonds on corresponding USD returns on 10-year

US bonds:

r$t+1,10 + ∆st+1 = α+ βrUS

t+1,10 + εt+1,

where small letters denote the log of their capital letter counterpart. Uncovered long-bond parity

implies α = 0 and β = 1. We run the same regression for the local currency bond returns (in

logs) and the change in the exchange rates r$, r? and ∆s on the U.S. bond return rUS . The sum

of the local currency and the FX beta equal the total dollar bond return beta.

Large Panel of Countries Table 9 reports the results for the entire sample in Panel A. Panel

B and C report the results for the post-Bretton-Woods sample and for 1991-2012. We report

the regression coefficients for the log local currency returns, the log exchange rate changes and

for the log dollar returns on foreign bonds. The sum of the local currency coefficient (first two

rows) and the exchange rate coefficient produces the dollar return coefficient in the last two

rows.

First, the average sloe coefficient for dollar returns is increasing over time for most of the

countries in the sample. For the whole sample, the average is 0.38. This number increases to

0.43 in the post-Bretton-Woods sample and to 0.56 in the sample that starts in 1991. More

than 50% of the permanent shocks are shared with the U.S. For some countries, the number is

closer to 75%.

The exchange rates account for up to 1/3 of this coefficient. When dollar returns are higher

than average, the dollar tends to depreciate relative to other currencies. When dollar returns

are lower than average, the dollar tends to appreciate relative to other currencies. Hence,

exchange rates actively enforce long-run uncovered bond return parity. Interestingly, the AUD,

the NZD, the NOK and to some extent the CAD are the main exceptions. We find negative

slope coefficients in these currencies. These are positive carry currencies (with on average high

33

interest rates) of countries that are commodity exporters.

To learn more about the time-variation in those coefficients, we use an equal-weighted port-

folio of all currencies. We run a regression of average returns on U.S. bond returns. Figure

10 plots the 60-month rolling window of the regression coefficients for the basket of developed

currencies. There are large increases in the dollar beta after the demise of the Bretton-Woods

regime, mostly driven increases in the exchange rate betas, as well around the early 90s. In

addition, there is a secular increase in the local return betas over the entire sample. There is

clear evidence that the currency exposure hedges the interest rate exposure of the foreign bond

position.

When U.S. bond returns are higher than usual, the dollar depreciates on average, relative to

all foreign currencies. There are two main exceptions: the LTCM crisis in 1998 and the recent

financial crisis. During these episodes, the dollar appreciated even though U.S. bond returns

were higher than usual.

Small Panel of Countries Using the zero coupon bond returns, we test the long-bond return

parity condition by regressing nominal USD holding period returns on 10-year foreign bonds on

corresponding USD returns on 10-year US bonds:

r∗t+1,10 + ∆st+1 = α+ βrt+1,10 + εt+1,

where small letters denote the log of their capital letter counterpart. Uncovered long-bond parity

implies α = 0 and β = 1. We test the two hypotheses separately, as well as jointly, and present

the results for 3-month, 6-month and 12-month holding period returns in Table 2. We report

both Newey-West standard errors (with 12 lags) and bootstrap standard errors. A mixed pattern

emerges: we can mostly reject the null hypothesis of long-bond return parity of US and foreign

bonds for Germany, the UK, Canada, Australia and Switzerland, while we are unable to reject

the parity condition between US and Japapese, New Zealand, Swedish and Norwegian bonds.

Overall, permanent exchange rate components appear to non-trivially contribute to nominal

exchange rate variation.

34

Tab

le9:

Bon

dR

etu

rnP

arit

y:

Lar

geP

anel

AU

DA

TS

BE

LC

AD

DK

KF

IMF

FD

EK

IRP

ITL

JP

YN

LG

NZ

DN

OK

PT

EE

SP

SE

KC

HF

GB

P

Pan

elA

:19

50-2

012

rloca

l0.3

30.1

60.

270.6

30.

280.

280.

340.

300.

190.

200.

400.

140.

11

0.1

60.1

50.

14

0.1

7s.

e.(0

.03)

(0.0

3)(0

.02)

(0.0

2)(0

.03)

(0.0

3)(0

.02)

(0.0

4)(0

.03)

(0.0

3)(0

.02)

(0.0

4)(0

.02)

(0.0

3)(0

.02)

(0.0

1)

(0.0

2)

−∆s

-0.0

30.1

90.

190.0

10.

180.

190.

210.

170.

150.

230.

21-0

.06

0.0

70.

17

0.1

00.

23

0.1

0s.

e.(0

.05)

(0.0

5)(0

.05)

(0.0

3)(0

.04)

(0.0

5)(0

.06)

(0.0

4)(0

.04)

(0.0

5)(0

.05)

(0.0

5)(0

.04)

(0.0

5)(0

.05)

(0.0

5)

(0.0

4)

r$0.3

00.3

60.

470.6

50.

470.

470.

560.

470.

340.

440.

610.

080.

18

0.3

20.2

50.

37

0.2

7(0

.05)

(0.0

5)

(0.0

5)(0

.04)

(0.0

5)

(0.0

5)(0

.06)

(0.0

7)(0

.06)

(0.0

6)(0

.05)

(0.0

6)

(0.0

5)(0

.05)

(0.0

5)(0

.05)

(0.0

5)

Pan

elB

:19

71-2

012

rloca

l0.3

70.1

90.

290.6

80.

340.

260.

320.

390.

360.

220.

240.

450.

16

0.1

10.

17

0.1

60.

15

0.1

8s.

e.(0

.04)

(0.0

3)(0

.03)

(0.0

3)(0

.04)

(0.0

4)(0

.03)

(0.0

3)(0

.05)

(0.0

4)(0

.04)

(0.0

3)(0

.05)

(0.0

3)(0

.04)

(0.0

3)(0

.02)

(0.0

3)

−∆s

-0.0

40.2

30.

220.0

10.

210.

150.

210.

230.

190.

180.

270.

24-0

.08

0.08

0.1

90.

12

0.2

70.1

1s.

e.(0

.06)

(0.0

6)(0

.06)

(0.0

4)(0

.06)

(0.0

6)(0

.06)

(0.0

6)(0

.06)

(0.0

6)(0

.06)

(0.0

6)(0

.07)

(0.0

6)(0

.06)

(0.0

6)(0

.07)

(0.0

6)

r$0.3

30.4

20.

520.6

90.

550.

410.

530.

610.

550.

400.

510.

690.

070.

19

0.3

70.

28

0.4

30.

29

s.e.

(0.0

7)(0

.06)

(0.0

7)

0.05

0.0

70.

070.

060.

070.

080.

070.

080.

070.

08

0.0

60.0

70.

06

0.0

70.

06

Pan

elC

:19

91-2

012

rloca

l0.7

10.4

20.

410.5

90.

490.

460.

480.

470.

520.

380.

230.

520.

45

0.2

00.

20

0.2

40.

31

0.2

50.

31

s.e.

(0.0

4)(0

.04)

(0.0

4)

(0.0

3)(0

.04)

(0.0

5)(0

.04)

(0.0

4)(0

.07)

(0.0

6)(0

.04)

(0.0

3)(0

.05)

(0.0

3)(0

.22)

(0.0

4)(0

.03)

(0.0

3)

(0.0

2)

−∆s

-0.0

60.2

30.

23-0

.09

0.2

20.

170.

220.

230.

200.

180.

340.

23-0

.01

0.04

0.2

30.

26

0.1

30.

27

0.0

4s.

e.(0

.10)

(0.0

9)(0

.09)

(0.0

7)(0

.09)

(0.1

0)(0

.09)

(0.0

9)(0

.09)

(0.0

9)(0

.09)

(0.0

9)(0

.10)

(0.0

9)(0

.09)

(0.0

9)(0

.10)

(0.0

9)

(0.0

8)

r$0.6

60.6

50.

640.5

00.

710.

640.

710.

710.

720.

560.

570.

750.

440.

24

0.4

30.

50

0.4

40.

52

0.3

5s.

e.(0

.10)

(0.0

9)(0

.09)

(0.0

7)(0

.08)

(0.1

0)(0

.09)

(0.0

9)(0

.11)

(0.1

2)(0

.10)

(0.0

8)(0

.10)

(0.0

9)(0

.23)

(0.1

0)(0

.10)

(0.0

9)

(0.0

8)

Month

lyR

eturn

s.R

egre

ssio

nof

log

retu

rnon

bonds

inlo

cal

curr

ency

rlocal ,

log

change

inth

eex

change

rate

and

the

log

retu

rnin

dollars

on

the

log

retu

rnon

U.S

.b

onds

indollars

.O

LS

standard

erro

rs.

35

r∗t+1,10 = α+ βrt+1,10 + εt+1,

Currency Portfolio Betas Finally, we computed the same regression coefficients for each

interest-rate sorted portfolio. These results are reported in Table 11. The top panel looks

at developed currencies. There are interesting differences in the slope coefficient across these

portfolios. The dollar slope coefficient declines from 46 (51) to 29 (32)% over the entire (Post-

Bretton-Woods) sample. This due to a decline in the local currency betas from 33% (36%) to

20%(22%) and a decline in the exchange rate betas from 13% (15%) to 9% (9%). In the bottom

panel, we see ben larger differences between portfolios. The dollar slope coefficient declines from

32 (34) to 13 (13)% over the entire (Post-Bretton-Woods) sample. This due to a decline in the

local currency betas from 24% (25%) to 13%(13%) and a decline in the exchange rate betas

from 8% (9%) to 0% (0%). As a result, it does look like there is more sharing of permanent

innovations between the U.S. and lower interest rate countries than with higher interest rate

countries.

7 Conclusion

The term structure of currency risk premia is downward sloping. That implies that the shocks not

shared in international financial markets are much less persistent than the overall shocks driving

pricing kernels. This model-free evidence supports the mechanism proposed by Colacito and

Croce (2011) to explain the Brandt-Cochrane-SantaClara puzzle of low exchange rate volatility,

high Sharpe ratios and low correlation of consumption growth.

36

Table 10: Bond Return Parity: Small Panel

DEM GBP JPY CAD AUD CHF NZD SEK NOK

3-month holding period USD returns

α 0.01∗∗ 0.01∗ 0.01 0.01∗∗ 0.02∗∗ 0.00 0.01 0.01∗ 0.01NW s.e. (0.01) (0.01) (0.01) (0.00) (0.01) (0.01) (0.01) (0.01) (0.01)BS s.e. (0.00) (0.01) (0.01) (0.00) (0.01) (0.00) (0.01) (0.01) (0.01)β 0.63∗∗∗ 0.58∗∗∗ 0.69∗∗ 0.71∗∗∗ 0.57∗∗∗ 0.61∗∗∗ 0.84 0.60∗∗∗ 0.48∗∗∗

NW s.e. (0.08) (0.08) (0.12) (0.11) (0.13) (0.10) (0.11) (0.10) (0.17)BS s.e. (0.07) (0.09) (0.14) (0.09) (0.13) (0.08) (0.10) (0.11) (0.13)

Wald 25.17∗∗∗ 27.13∗∗∗ 6.11∗∗ 8.87∗∗ 11.60∗∗∗ 16.65∗∗∗ 2.45 17.42∗∗∗ 11.29∗∗∗


α 0.02∗∗ 0.02∗ 0.01 0.02∗∗ 0.04∗∗ 0.01 0.01 0.02 0.01NW s.e. (0.01) (0.01) (0.02) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01)BS s.e. (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01)β 0.66∗∗∗ 0.58∗∗∗ 0.76 0.65∗∗∗ 0.48∗∗∗ 0.61∗∗∗ 0.80 0.68∗∗ 0.65∗

NW s.e. (0.12) (0.10) (0.19) (0.13) (0.16) (0.13) (0.14) (0.15) (0.21)BS s.e. (0.09) (0.10) (0.14) (0.09) (0.14) (0.10) (0.12) (0.14) (0.18)Wald 8.67∗∗ 16.22∗∗∗ 1.71 8.08∗∗ 12.04∗∗∗ 9.52∗∗∗ 3.07 5.08∗ 4.08


α 0.04∗ 0.05∗∗ 0.03 0.05∗∗ 0.06∗∗ 0.02 0.03 0.03 0.01NW s.e. (0.02) (0.02) (0.04) (0.02) (0.03) (0.03) (0.03) (0.03) (0.03)BS s.e. (0.01) (0.02) (0.02) (0.01) (0.02) (0.02) (0.02) (0.02) (0.02)β 0.69∗ 0.49∗∗∗ 0.73 0.63∗∗ 0.59∗∗ 0.54∗∗ 0.83 0.80 0.91

NW s.e. (0.16) (0.13) (0.29) (0.15) (0.21) (0.21) (0.20) (0.22) (0.34)BS s.e. (0.11) (0.10) (0.18) (0.09) (0.15) (0.15) (0.15) (0.16) (0.26)

Wald 4.05 15.64∗∗∗ 0.89 6.96∗∗ 5.47∗ 5.62∗ 1.50 1.52 0.21

We regress the USD holding period return of a foreign 10-year bond on the corresponding holding period returnof the US bond and report the constant and the slope coefficient. Standard errors are reported in the parentheses;we first report the Newey-West standard error (with 12 lags) and then the bootstrap standard error. For thelatter, we apply block bootstrapping using 1,000 bootstrap samples, with block length equal to 3. We test thenull hypotheses of constant equal to zero and slope coefficient equal to one both individually and jointly foreach currency; Wald is the Wald test statistic for the joint hypothesis test. One, two and three asterisks denoterejection of the null hypothesis at the 10, 5 and 1 percent level of significance, respectively.

37

Table 11: Bond Return Parity: Currency Portfolios

1 2 3 4 5 1 2 3 4 5

1950-2012 1971-2012Panel A: Developed

rlocal 0.33 0.30 0.28 0.24 0.20 0.36 0.32 0.30 0.26 0.22(0.02) (0.03) (0.02) (0.02) (0.03) (0.02) (0.03) (0.03) (0.03) (0.03)

rfx 0.13 0.16 0.17 0.11 0.09 0.15 0.19 0.19 0.13 0.09(0.04) (0.04) (0.04) (0.04) (0.04) (0.05) (0.05) (0.05) (0.05) (0.05)

r$ 0.46 0.47 0.44 0.36 0.29 0.51 0.51 0.49 0.40 0.32(0.04) (0.05) (0.04) (0.04) (0.05) (0.05) (0.06) (0.06) (0.05) (0.06)

Panel B: Allrlocal 0.24 0.28 0.28 0.23 0.13 0.25 0.31 0.31 0.24 0.13

(0.01) (0.02) (0.02) (0.02) (0.02) (0.02) (0.03) (0.02) (0.02) (0.03)

rfx 0.08 0.15 0.16 0.07 0.00 0.09 0.17 0.17 0.09 0.00(0.03) (0.03) (0.03) (0.03) (0.04) (0.04) (0.05) (0.05) (0.05) (0.05)

r$ 0.32 0.43 0.44 0.31 0.13 0.34 0.49 0.48 0.33 0.13(0.04) (0.04) (0.04) (0.04) (0.04) (0.05) (0.05) (0.05) (0.05) (0.06)

Monthly Returns. Regression of log return on bonds in local currency rlocal, log change in the exchange rate andthe log return in dollars on the log return on U.S. bonds in dollars. OLS standard errors. Portfolios of currenciessorted at t− 1 by monthly returns on T-bills realized at end of t.

38

Figure 4: Sorts by Interest Rates: Whole Sample

Panel A: All Countries

1 2 3 4 5−2

0

2

4Currency Risk Premium

1 2 3 4 5−2

0

2

4Local Term Risk Premium

1 2 3 4 50

1

2

3

4Dollar Term Risk Premium


1 2 3 4 5−1

0

1

2


1 2 3 4 5−1

0

1

2


1 2 3 4 50

1

2

3


Notes: Annualized Monthly Returns. Sample: 1950-2012. Portfolios of Currencies sorted by T-bill Interest Rates.The top panel shows all countries. The bottom panel shows developed countries.

39

Figure 5: Sorts by Interest Rates: Post-Bretton-Woods


1 2 3 4 5−2

0

2

4


1 2 3 4 50

2

4


1 2 3 4 50

2

4

6



1 2 3 4 5−2

0

2


1 2 3 4 5−2

0

2


1 2 3 4 50

2

4



40

Figure 6: Sorts by Slope: Whole Sample


1 2 3 4 5−1

0

1

2


1 2 3 4 5−2

0

2

4


1 2 3 4 50

2

4



1 2 3 4 50

1

2


1 2 3 4 5−2

0

2


1 2 3 4 50

1

2

3



41

Figure 7: Sorts by Slope: Post-Bretton-Woods


1 2 3 4 5−1

0

1

2


1 2 3 4 5−2

0

2

4


1 2 3 4 50

2

4



1 2 3 4 50

1

2

3


1 2 3 4 5−2

0

2


1 2 3 4 50

2

4



42

Figure 8: UIP and Uncovered Bond Return Parity: All Countries

−2 −1 0 1 2 3 4 5 6 7 8−2

−1

0

1

2

3

4

5

6

7

8

Spread with U.S.

Ra

te o

f D

ep

reci

atio

n

Bond Return Spread

Interest Rate Spread

45−degree

Notes: Annualized Monthly Returns. Sample: 1950-2012. Portfolios of Currencies sorted by T-bill returns at t.

43

Figure 9: UIP and Uncovered Bond Return Parity: All Countries

−2 −1 0 1 2 3 4 5 6 7 8−2

−1

0

1

2

3

4

5

6

7

8

Spread with U.S.

Ra

te o

f D

ep

reci

atio

n

Bond Return Spread

Interest Rate Spread

45−degree

Notes: Annualized Monthly Returns. Sample: 1971-2012. Portfolios of Currencies sorted by T-bill returns at t.

44

Figure 10: Foreign Bond Return Betas: Developed Countries

1957 1971 1984 1998 2012−0.2

0

0.2

0.4

0.6

0.8

Average Betas

Dollar

Local

FX

Notes: Sample: 1950-2012. 60-month rolling window estimation of beta with respect to US bond returns for theequal-weighted average of log bond returns in local currency, the log change in the exchange rate and the logdollar bond returns. Developed Countries.

45

Figure 11: Foreign Bond Return Betas: All Countries

1957 1971 1984 1998 2012−0.2

0

0.2

0.4

0.6

0.8

Average Betas

Dollar

Local

FX

Notes: Sample: 1950-2012. 60-month rolling window estimation of beta with respect to US bond returns for theequal-weighted average of log bond returns in local currency, the log change in the exchange rate and the logdollar bond returns. All countries.

46

References

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Working Paper Columbia University.

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counting for the Forward Premium Anomaly,” Journal of Finance, 56, 279–304.

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Review of Financial Studies, 10, 369–403.

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A General Equilibrium Perspective,” The Review of Financial Studies, 9(2), 427–470.

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is Better Than You Think, or Exchange Rates are Too Smooth,” Journal of Monetary Eco-

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Annualchap. Carry Trades and Currency Crashes. University of Chicago Press.

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turns,” Working Paper London School of Economics.

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International Economics, 90, 12–32.

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“Crash Risk in Currency Markets,” Working Paper MIT Sloan.

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Volatility,” Working Paper, Carnegie Mellon University.

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versity.

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49

A Proofs

• Proof of Proposition 2.2.

Proof. We use Mt+1 to denote the stochastic discount factor. To see why, consider theinvestor’s Euler equation:

0 = Et[logMt+1] + limk→∞

Et[logRt+1,1[1t+k]] + (1/2) limk→∞

V art[logMt+1 + logRt+1,1[1t+k]].

The last term drops out because we know that limk→∞Rt+1,1[1t+k] = Λt/Λt+1. Also, notethat

logEt[Mt+1] = Et[logMt+1] + (1/2)V art[logMt+1].

Hence, we obtain:

0 = logEt[Mt+1] + limk→∞

Et[logRt+1,1[1t+k]]− (1/2)V art[logMt+1].

It follows directly that the term premium equals one half of the variance:

limk→∞

Etrxkt+1 = lim

k→∞Et

[log

Rt+1,1[1t+k]

Rt+1,1[1t+1]

]= (1/2)σ2.


Proof. In the log-normal model, the foreign term premium –the bond risk premium onthe longest maturity bond limk→∞Et[rx

k,∗t+1] – equals (1/2)σ∗,2 . In a log-normal model,

the currency risk premium equals (1/2)(σ∗,2 − σ2.

)(Backus, Foresi, and Telmer (2001)).

To see why, consider the investor’s Euler equation:

0 = Et[logMt+1] + Et[logRt+1] + (1/2)V art[logMt+1 + (ft − st)−∆st+1].

We use the complete markets expression for the change in the spot rates:

∆st+1 = logMt+1 − logM∗t+1.

The result follows immediately.

• Proof of Corollary 2.1:

Proof. Using the equivalence between long-term yield spreads and expected excess re-turns, we know that:

y∗t [∞] + (ft − st)− E[∆st+1] = (1/2)σ2 = yt[∞]

Note that

E[ limk→∞

(1/k)

k∑j=1

∆st+j ] = limk→∞

(1/k)

k∑j=1

E[∆st+j ]

by the Lebesgue dominated convergence theorem. Also, note that E[∆st+j ] = E[∆st+1]

50

by assumption (because we assumed that expected holding period returns do not dependon calendar time). That delivers the result.


Proof. First, note that the foreign currency risk premium is equal to:


(Λt+1

Λt

)− Lt

(Λ∗t+1

Λ∗t

)See Backus, Foresi, and Telmer (2001). Also, note that:

Lt

(Λt+1

Λt

)= Lt

(ΛPt+1

ΛPt

)+ Et log

[Rt+1,1[1t+∞]

Rt+1,1[1t+1]

]This result is established in Alvarez and Jermann (2005). The final result follows imme-diately.

• Proof of Proposition 2.7:

Proof. First, note that the foreign currency risk premium is equal to:


(Λt+1

Λt

)− Lt

(Λ∗t+1

Λ∗t

)See Backus, Foresi, and Telmer (2001). Also, note that:

Lt

(Λt+1

Λt

)= Lt

(ΛPt+1

ΛPt

)+ Et log

[Rt+1,1[1t+∞]

Rt+1,1[1t+1]

]This result is established in Alvarez and Jermann (2005). The final result follows imme-diately.


Proof. This result follows directly from the definition of a pricing kernel without perma-nent innovations and from the no-arbitrage expression for the spot exchange rates whenmarkets are complete:

StSt+1

=M∗t+1/M

∗t

Λt+1/Λt

• Proof of Proposition 2.6: Proof. We use Mt+1 to denote the stochastic discount factor.To see why, consider the investor’s Euler equation:

0 = Et[logMt+1]+ limk→∞

Et[logRt+1,1[1t+k]]+(1/2) limk→∞

V art[logMPt+1+logMT

t+1+logRt+1,1[1t+k]].

The last term simplifies to V art[logMPt+1] because we know that limk→∞Rt+1,1[1t+k] =

ΛTt /Λ

Tt+1. Also, note that

logEt[Mt+1] = Et[logMt+1] + (1/2)V art[logMt+1].

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Hence, we obtain:

0 = logEt[Mt+1] + limk→∞

Et[logRt+1,1[1t+k]]− (1/2)V art[logMt+1] + (1/2)V art[logMPt+1].

It follows directly that the term premium equals one half of the variance:

limk→∞

Etrxkt+1 = lim

k→∞Et

[log

Rt+1,1[1t+k]

Rt+1,1[1t+1]

]= (1/2)σ2

T + σTP .

• Proof of Corollary 2.2:

Proof. The result follows immediately from Proposition 2.6.


Proof. The proof is immediate: the ratio of foreign to domestic holding period returns forinfinite maturity bonds (once converted to the same currency) measures the ratio of thedomestic and permanent components of the SDF. Theorem 2.10 refines the main result inBrandt, Cochrane, and Santa-Clara (2006): the permanent components of the SDFs mustbe highly correlated across countries.

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