The Term Structure of Currency Risk Premia

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The Term Structure of Currency Risk Premia
The Term Structure of Currency Risk Premia
Hanno Lustig
Andreas Stathopoulos
Adrien Verdelhan
UCLA and NBER
USC
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Please do not quote.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 Business, 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 exchange 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
3
HML on Bonds
HML on Currencies
2.5
2
1.5
1
0.5
0
−0.5
−1
1943
1957
1971
1984
1998
2012
2026
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 evidence 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 opportunities 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 correlated 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
1
Colacito and Croce (2011) argue that only the persistent component of consumption growth is highly correlated across countries. Our finding provide model-free evidence in support of the view that the bulk of permanent
shocks 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
4
Λt+k
Λt
(2.1)
We define the one-period return on a zero-coupon bond with maturity k as:
Rt+1,1 [1t+k ] =
We use rxkt+1 to denote the log excess returns log
ht [k] = Et
Vt+1 [1t+k ]
Vt [1t+k ]
(2.2)
Rt+1,1 [1t+k ]
Rt+1,1 [1t+1 ] .
Rt+1,1 [1t+k ]
log
Rt+1,1 [1t+1 ]
We define the term premium as:
Let us define the yield spread at long maturities:
yt [k] = log
Vt [1t+1 ]
!
(2.3)
Vt [1t+k ]1/k
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:
lim E(ht [k]) = lim E(yt [k]).
k→∞
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)):
lim Et log
k→∞
Et+1 [Λt+k ]
= 0.
Et [Λt+k ]
Under regularity conditions, this condition is equivalent to limk→∞
5
Et+1 [Λt+k ]
Et [Λt+k ]
= 1.
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)).
Rt+1,1 [1t+k ]
Rt+1
≥ Et log
ht [∞] = lim Et log
k→∞
Rt+1,1 [1t+1 ]
Rt+1,1 [1t+1 ]
for any Rt+1 .
Example 2.1 Consider a log-normal model of a pricing kernel:
log Λt =
∞
X
αi t−i + (t) log β
(2.4)
i=0
with ∼ N (0, σ 2 ), α0 = 1.
It is easy to check that the condition outlined in definition 1 is satisfied provided that limk→∞ αk2 =
0.
Proposition 2.2 In example 2.1, the term premium equals one half of the variance (Alvarez
and Jermann (2005)).
lim
k→∞
Et rxkt+1
= lim Et
k→∞
Term Premium on Foreign Bonds
Rt+1,1 [1t+k ]
log
= (1/2)σ 2 .
Rt+1,1 [1t+1 ]
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:
∗
[1t+k ]
St Rt+1,1
= rxk,∗
log
t+1 + (ft − st ) − ∆st+1 ,
St+1 Rt+1,1 [1t+1 ]
where we have used covered interest rate parity:
∗
Rt+1,1
[1t+1 ]
Ft
=
St
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 domestic term premium.
h∗t [∞] + (ft − st ) − Et [∆st+1 ] = (1/2)σ 2 = ht [∞]
In a log-normal model, the currency risk premium equals (1/2) σ ∗,2 − σ 2 . (see Bansal
(1997), Bekaert (1996) and Backus, Foresi, and Telmer (2001)). Currencies with a high local
currency term premium (high σ 2 ) 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):
yt∗ [∞]
Non-normalities
+ (ft − st ) − E[ lim (1/k)
k
X
k→∞
∆st+j ] = (1/2)σ 2 = yt [∞]
j=1
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.
2
More 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 of
currency 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:
∗
[1t+k ]
St Rt+1,1
St+1 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):
∗
[1t+k ]
St Rt+1,1
= 1,
k→∞ St+1 Rt+1,1 [1t+k ]
lim
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:
k
lim (rx∗,k
t+1 − rxt+1 ) ≈ − ((ft − st ) − ∆st+1 ) .
k→∞
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 ΛTt .
9
(2.6)
Definition 2 The transitory component, ΛTt , is defined as
δ t+k
,
k→∞ Vt [1t+k ]
ΛTt = lim
(2.7)
where the constant δ is chosen to satisfy the following regularity condition:
0 < lim
k→∞
Vt [1t+k ]
< ∞.
δk
(2.8)
The permanent component, ΛPt is a martingale. To see why, note that
ΛPt = lim
k→∞
Vt [1t+k ]
Λt ,
δ t+k
(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
Vt [1t+k ]
= lim δ
T
k→∞ Vt+1 [1t+k ]
Λt
The infinite maturity bond return is given by
Rt+1,1 [1t+∞ ] = lim Rt,1 [1t+k ] =
k→∞
ΛTt
ΛTt+1
We can decompose exchange rate changes into a permanent component and a transitory
component, defined below:
St+1
=
St
P
Λ∗P
t+1 Λt
Λ∗P
ΛPt+1
t
!
T
Λ∗T
t+1 Λt
T
Λ∗T
t Λt+1
!
=
P ST
St+1
t+1
StP 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)):
1
log ΛPt+1 = − σP2 + log ΛPt + εPt+1 ,
2
∞
X
log ΛTt+1 = log β t+1 +
αi εTt+1−i ,
i=0
where α is a square summable sequence, and εP and εT are i.i.d. normal variables with mean
zero and covariance σT P . A similar decomposition applies to the foreign stochastic discount
factor, where a
?
denotes a foreign variable:
1 ?2
?P
?P
log Λ?P
t+1 = − σP + log Λt + εt+1 ,
2
∞
X
?t+1
log Λ?T
=
log
β
+
αi? ε?T
t+1
t+1−i .
i=0
Proposition 2.6 In example 2.2, the term premium is given by the following expression:
ht [∞] = (1/2)σT2 + σT P
.
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)(σ 2 − σP2,∗ ).
Provided that σP2,∗ = σP2 , the foreign term premium in dollars equals the domestic term premium:
h∗t [∞] + (ft − st ) − Et [∆st+1 ] = (1/2)σT2 + σT P
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 premium 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
) ≥ Et (log Rt+1 ) − Et (log Rt+1,1 [1t+∞ ]) .
ΛPt
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 longmaturity 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<
Λ∗T
ΛTt+1
t+1
<
∞
and
0
<
<∞
T
Λt
Λ∗T
t
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 ] ≡
∗
[1k ]
St+1 Rt+1,1
St 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∞ ] ≡ lim RXt+1,1 [1k ] =
k→∞
P
P
St+1
Λ∗P
t+1 Λt
=
Λ∗P
ΛPt+1
StP
t
Corollary 2.3 If the domestic and foreign pricing kernels have common permanent innovations,
ΛPt+1
Λ∗P
t+1
=
ΛPt
Λ∗P
t
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
ΛPt+1
ΛP ?
,
log
log t+1
ΛPt ?
ΛPt
!
=
1
[V art
2
ΛP ?
log t+1
ΛPt ?
!
+ V art
ΛP
log t+1
ΛPt
!
− V art (log RXt,1 [1∞ ])]
?
?
≥ Et log Rt+1
− Et log Rt+1,1
[1t+∞ ]
+ Et (log Rt+1 ) − Et (log Rt+1,1 [1t+∞ ])
−
3
1
V art (log RXt+1,1 [1∞ ]) .
2
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 longmaturity 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 calculate 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 properties, 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 correlations 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
10
1
0.5
0
2
4
6
8
Maturity (in years)
NZD
10
1
0.5
0
2
4
6
8
Maturity (in years)
10
0
2
4
6
8
Maturity (in years)
AUD
10
1
0.5
0
2
4
6
8
Maturity (in years)
SEK
10
1
0.5
0
2
4
6
8
Maturity (in years)
10
Correlation coefficient
4
6
8
Maturity (in years)
CAD
0.5
Correlation coefficient
2
JPY
1
Correlation coefficient
0
Correlation coefficient
Local
USD
0.5
Correlation coefficient
GBP
Correlation coefficient
Correlation coefficient
Correlation coefficient
Correlation coefficient
NOK
1
1
0.5
0
2
4
6
8
Maturity (in years)
CHF
10
2
4
6
8
Maturity (in years)
NOK
10
2
4
6
8
Maturity (in years)
10
1
0.5
0
1
0.5
0
Panel B: Developed Countries
Notes: Correlation with U.S. bond returns. Sample: country-dependent. Monthly data. Holding period is
3-months.
returns; the k-year maturity volatility ratio is given by
V olRk =
∗
+ ∆st+1
σ rt+1,k
σ (rt+1,k )
For comparison, we also calculate the corresponding volatility ratio for local currency returns,
given by
V olRkL =
∗
σ rt+1,k
σ (rt+1,k )
and report both V olRk and V olRkL 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 olRkL 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
NOK
GBP
5
2
4
6
8
Maturity (in years)
CAD
2
4
6
8
Maturity (in years)
NZD
2
4
6
8
Maturity (in years)
SEK
4
6
8
Maturity (in years)
10
10
2
4
6
8
Maturity (in years)
NOK
10
2
4
6
8
Maturity (in years)
10
10
5
0
4
6
8
Maturity (in years)
CHF
5
0
10
Volatility ratio
Volatility ratio
2
2
10
10
5
5
0
10
5
0
10
10
Volatility ratio
4
6
8
Maturity (in years)
AUD
Volatility ratio
5
0
2
10
Volatility ratio
Volatility ratio
10
0
5
0
10
10
Volatility ratio
Local
USD
0
JPY
10
Volatility ratio
Volatility ratio
10
2
4
6
8
Maturity (in years)
10
5
0
Notes: Volatility of Foreign and U.S. bond returns. Sample: country-dependent. Monthly data. Holding period
is 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 rxf x = (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
3
We are being conservative by sorting on the T-bill return at t − 1. The Tbill return at t is largely known at
t − 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
Panel A: 1950-2012
1.85
1.85
6.98
7.19
0.27
0.26
5
5-1
2.69
7.89
0.34
3.41
6.28
0.54
rxf x
Mean
Std
SR
-0.73
6.24
-0.12
0.60
7.07
0.09
rx?
Mean
Std
SR
2.49
3.43
0.73
1.45
4.71
0.31
1.78
3.80
0.47
1.76
3.94
0.45
0.45
5.34
0.08
rx$
Mean
Std
SR
1.76
7.52
0.23
2.06
8.94
0.23
3.63
8.22
0.44
3.61
8.24
0.44
3.14
9.02
0.35
∆s
f −s
r? − rU S
Mean
Mean
Mean
0.79
-1.52
-0.54
0.43
0.18
0.05
rxf x
Mean
Std
SR
-0.50
7.53
-0.07
0.85
8.64
0.10
rx?
Mean
Std
SR
3.14
4.06
0.77
2.22
5.63
0.39
2.06
4.49
0.46
2.55
4.48
0.57
0.78
5.26
0.15
rx$
Mean
Std
SR
2.63
9.04
0.29
3.07
10.85
0.28
4.43
9.79
0.45
4.57
9.88
0.46
4.27
10.73
0.40
∆s
f −s
r? − rU S
Mean
Mean
Mean
1.30
-1.80
-1.23
0.61
0.24
-0.11
0.91
1.46
0.95
-1.23
3.25
3.23
-3.86
7.35
5.57
0.61
-0.87
1.24
2.72
1.44
2.89
Panel B: 1971-2012
2.37
2.02
8.33
8.75
0.28
0.23
1.38
8.03
0.17
-2.95
5.64
4.58
3.49
8.89
0.39
3.99
6.66
0.60
1.64
9.44
0.17
Portfolios of currencies sorted at t − 1 by returns on T-bills realized at end of t − 1. Annualized monthly returns
realized 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:
(ft − st ) − Et [∆st+1 ] = Lt
20
Λt+1
Λt
− Lt
Λ∗t+1
Λ∗t
.
Table 2: Interest-Rate Sorted Portfolios: Developed Countries
1
2
3
4
Panel A: 1950-2012
1.64
2.35
7.71
7.65
0.21
0.31
5
5-1
2.94
8.18
0.36
3.01
5.88
0.51
rf x
Mean
Std
SR
-0.06
7.22
-0.01
0.97
7.75
0.12
rx
Mean
Std
SR
2.04
3.75
0.55
1.80
5.68
0.32
1.61
4.62
0.35
0.96
4.35
0.22
0.39
5.30
0.07
rx$
Mean
Std
SR
1.98
8.75
0.23
2.76
9.91
0.28
3.24
9.39
0.35
3.30
8.97
0.37
3.34
9.36
0.36
∆s
f −s
r − rU S
Mean
Mean
Mean
1.22
-1.28
-0.75
0.56
0.41
0.69
rf x
Mean
Std
SR
0.28
8.80
0.03
1.32
9.47
0.14
rx
Mean
Std
SR
2.58
4.44
0.58
2.65
6.82
0.39
1.89
5.39
0.35
1.40
4.88
0.29
1.18
6.10
0.19
rx$
Mean
Std
SR
2.86
10.60
0.27
3.97
12.03
0.33
3.80
11.34
0.33
4.25
10.39
0.41
4.70
11.02
0.43
∆s
f −s
r − rU S
Mean
Mean
Mean
1.75
-1.47
-1.45
0.77
0.55
0.64
0.38
1.52
0.85
-0.03
2.88
1.72
-2.17
5.70
4.31
0.31
-0.21
1.33
2.56
1.43
2.00
Panel B: 1971-2012
1.90
2.84
9.41
8.97
0.20
0.32
1.36
8.01
0.17
-1.93
4.88
3.76
3.52
9.74
0.36
3.24
6.82
0.48
1.84
9.27
0.20
Portfolios of currencies sorted at t − 1 by returns on T-bills realized at end of t − 1. Annualized monthly returns
realized 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:
?,P
?
h?t (∞) = lim Et rxk,?
t+1 = Lt (mt+1 ) − Lt (mt+1 ).
k→∞
(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 rxf x = (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
Panel A: 1950-2012
1.85
2.45
7.09
7.02
0.26
0.35
5
5-1
2.88
7.37
0.39
4.36
6.40
0.68
rf x
Mean
Std
SR
-1.48
6.89
-0.21
0.56
7.06
0.08
rx
Mean
Std
SR
3.46
4.37
0.79
1.90
4.46
0.43
1.46
4.06
0.36
1.33
3.86
0.34
-0.51
4.59
-0.11
rx$
Mean
Std
SR
1.98
7.50
0.26
2.46
8.84
0.28
3.31
8.41
0.39
3.78
8.13
0.47
2.38
9.13
0.26
∆s
f −s
r? − rU S
Mean
Mean
Mean
0.47
-1.95
0.00
0.41
0.15
0.55
rf x
Mean
Std
SR
-1.05
7.59
-0.14
0.76
8.64
0.09
r
Mean
Std
SR
3.85
4.04
0.95
2.76
5.33
0.52
1.83
4.70
0.39
2.04
4.50
0.45
-0.05
5.29
-0.01
r$
Mean
Std
SR
2.80
9.01
0.31
3.52
10.74
0.33
4.36
9.98
0.44
4.74
9.80
0.48
3.22
10.83
0.30
∆s
f −s
r? − rU S
Mean
Mean
Mean
0.86
-1.90
-0.62
0.55
0.21
0.41
1.10
1.44
0.70
-0.54
3.24
2.71
-4.29
7.57
4.96
0.63
-0.25
1.22
2.70
1.17
2.52
Panel B: 1971-2012
2.54
2.70
8.47
8.56
0.30
0.32
0.40
8.19
0.05
-3.27
6.15
4.14
3.28
8.83
0.37
4.33
6.77
0.64
0.43
9.62
0.04
Portfolios of currencies sorted at t − 1 by monthly returns on T-bills realized at end of t. Annualized monthly
returns realized at t on 10-year Bond Index and T-bills.
25
Table 4: Interest-Rate Sorted Portfolios: Developed Countries
1
2
3
4
Panel A: 1950-2012
1.77
2.47
7.71
7.66
0.23
0.32
5
5-1
2.92
8.20
0.36
3.33
5.83
0.57
rxf x
Mean
Std
SR
-0.41
7.17
-0.06
1.03
7.79
0.13
rx?
Mean
Std
SR
2.95
3.89
0.76
2.29
5.34
0.43
0.80
4.78
0.17
0.97
4.43
0.22
-0.77
5.39
-0.14
rx$
Mean
Std
SR
2.54
8.76
0.29
3.32
9.87
0.34
2.57
9.39
0.27
3.44
9.03
0.38
2.15
9.60
0.22
∆s
f −s
r? − rU S
Mean
Mean
Mean
0.92
-1.33
0.11
0.65
0.38
1.16
rxf x
Mean
Std
SR
-0.20
8.74
-0.02
1.38
9.53
0.14
rx?
Mean
Std
SR
3.68
4.62
0.80
3.49
6.31
0.55
0.68
5.68
0.12
1.33
4.98
0.27
-0.13
6.16
-0.02
rx$
Mean
Std
SR
3.48
10.63
0.33
4.87
11.93
0.41
2.79
11.36
0.25
4.34
10.53
0.41
3.42
11.26
0.30
∆s
f −s
r? − rU S
Mean
Mean
Mean
1.33
-1.53
-0.42
0.86
0.51
1.44
0.60
1.51
-0.37
0.11
2.91
1.67
-2.29
5.83
3.14
0.45
-0.10
1.32
2.58
0.61
2.03
Panel B: 1971-2012
2.11
3.01
9.38
9.04
0.23
0.33
-0.39
8.16
-0.05
-2.06
4.98
2.70
3.54
9.72
0.36
3.74
6.74
0.55
-0.06
9.44
-0.01
Portfolios of currencies sorted at t − 1 by monthly returns on T-bills realized at end of t. Annualized monthly
returns 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
Panel A: 1950-2012
1.82
0.50
7.28
7.53
0.25
0.07
5
5-1
-0.96
7.89
-0.12
-3.27
7.20
-0.45
rxf x
Mean
Std
SR
2.32
7.03
0.33
1.51
7.04
0.22
rx?
Mean
Std
SR
-1.79
3.76
-0.48
1.15
3.64
0.32
1.86
4.27
0.44
2.66
4.84
0.55
5.11
6.95
0.74
rx$
Mean
Std
SR
0.52
8.18
0.06
2.66
7.86
0.34
3.67
8.74
0.42
3.16
9.18
0.34
4.16
8.49
0.49
y 10 − y 1
∆s
f −s
r − rU S
Mean
Mean
Mean
Mean
-0.86
-1.54
3.86
0.56
0.74
-0.21
1.73
1.36
rxf x
Mean
Std
SR
2.57
8.55
0.30
1.89
8.57
0.22
rx?
Mean
Std
SR
-1.79
4.38
-0.41
2.12
4.23
0.50
2.31
5.04
0.46
3.41
5.67
0.60
5.28
5.56
0.95
rx$
Mean
Std
SR
0.78
9.86
0.08
4.01
9.46
0.42
4.68
10.52
0.45
4.15
10.84
0.38
4.99
10.06
0.50
y 10 − y 1
∆s
f −s
r − rU S
Mean
Mean
Mean
Mean
-1.19
-2.06
4.63
0.28
0.64
-0.15
2.04
1.59
1.31
1.00
1.37
1.12
1.91
0.17
0.57
1.41
3.31
-0.50
0.21
2.92
1.36
1.97
0.70
0.08
1.11
0.42
1.46
1.58
Panel B: 1971-2012
2.38
0.74
8.82
8.91
0.27
0.08
3.63
7.61
0.48
3.79
-0.41
-0.54
3.06
-0.29
7.23
-0.04
-2.86
6.00
-0.48
4.21
8.92
0.47
Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1. Monthly returns at t on 10-year Bond
Index and T-bills.
28
Table 6: Slope-sorted Portfolios: Developed Countries
1
2
3
4
Panel A: 1950-2012
1.04
0.41
7.87
7.76
0.13
0.05
5
5-1
0.28
7.00
0.04
-2.39
4.93
-0.49
rxf x
Mean
Std
SR
2.68
7.48
0.36
2.37
7.35
0.32
rx?
Mean
Std
SR
-1.08
4.21
-0.26
1.51
4.15
0.36
1.69
4.56
0.37
2.26
5.33
0.42
2.96
7.31
0.40
rx$
Mean
Std
SR
1.59
8.52
0.19
3.88
8.43
0.46
2.73
9.21
0.30
2.67
9.92
0.27
3.24
11.06
0.29
y 10 − y 1
∆s
f −s
r − rU S
Mean
Mean
Mean
Mean
-0.58
-0.79
3.47
0.87
0.78
0.71
1.65
1.65
rxf x
Mean
Std
SR
3.09
9.08
0.34
3.07
8.95
0.34
rx?
Mean
Std
SR
-0.74
4.87
-0.15
2.64
4.81
0.55
2.29
5.33
0.43
2.82
6.31
0.45
3.15
8.69
0.36
rx$
Mean
Std
SR
2.35
10.22
0.23
5.71
10.12
0.56
3.61
10.97
0.33
3.43
11.79
0.29
3.44
13.33
0.26
y 10 − y 1
∆s
f −s
r − rU S
Mean
Mean
Mean
Mean
-0.79
-0.88
3.97
0.66
0.72
1.21
1.86
1.94
1.27
0.08
1.24
0.97
1.86
-0.24
0.85
1.11
3.18
0.06
0.23
0.81
1.31
1.90
-0.04
-0.22
1.08
0.63
1.26
1.38
Panel B: 1971-2012
1.32
0.61
9.46
9.22
0.14
0.07
1.65
9.51
0.17
3.14
0.14
0.14
1.59
0.29
8.53
0.03
-2.80
5.91
-0.47
1.09
11.29
0.10
Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1. Monthly returns at t on 10-year Bond
Index 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
Panel A: 1950-2012
1.39
0.57
7.18
7.53
0.19
0.08
5
5-1
-0.99
8.00
-0.12
-3.38
7.31
-0.46
rxf x
Mean
Stdev
SR
2.39
6.97
0.34
1.95
6.98
0.28
rx?
Mean
Stdev
SR
-2.95
3.75
-0.79
0.32
3.83
0.08
1.59
4.14
0.38
3.62
4.72
0.77
6.58
6.97
0.94
rx$
Mean
Stdev
SR
-0.55
8.14
-0.07
2.27
8.06
0.28
2.98
8.40
0.35
4.18
9.22
0.45
5.59
8.60
0.65
y 10 − y 1
∆s
f −s
r − rU S
Mean
Mean
Mean
Mean
-0.96
-1.57
3.97
-0.49
0.72
0.20
1.75
0.56
1.38
0.31
1.08
1.16
2.00
0.17
0.40
2.50
3.89
-0.36
-0.63
4.44
Panel B: 1971-2012
1.78
1.01
8.68
8.92
0.21
0.11
rxf x
Mean
Stdev
SR
2.70
8.45
0.32
2.45
8.52
0.29
rx?
Mean
Stdev
SR
-3.11
4.40
-0.71
0.89
4.39
0.20
1.79
4.90
0.37
4.59
5.54
0.83
7.22
5.63
1.28
rx$
Mean
Stdev
SR
-0.42
9.79
-0.04
3.33
9.70
0.34
3.58
10.10
0.35
5.59
10.89
0.51
6.74
10.23
0.66
y 10 − y 1
∆s
f −s
r − rU S
Mean
Mean
Mean
Mean
-1.33
-2.08
4.77
-0.90
0.63
0.38
2.07
0.39
1.32
0.45
1.33
0.56
1.94
0.46
0.55
2.57
3.43
-0.59
0.10
4.76
-0.48
7.44
-0.07
6.14
8.06
0.76
-3.18
6.16
-0.52
7.16
9.46
0.76
Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1 (defined as yield at t − 1 minus return on
T-bill at t). Monthly returns at t on 10-year Bond Index and T-bills.
30
Table 8: Slope-sorted Portfolios: Developed Countries
1
2
3
4
Panel A: 1950-2012
0.93
0.59
7.84
7.69
0.12
0.08
5
5-1
-0.17
7.08
-0.02
-3.14
5.02
-0.63
rxf x
Mean
Std
SR
2.97
7.46
0.40
2.41
7.38
0.33
rx?
Mean
Std
SR
-2.39
4.16
-0.58
0.64
4.28
0.15
1.14
4.57
0.25
3.22
5.24
0.61
4.88
7.28
0.67
rx$
Mean
Std
SR
0.57
8.52
0.07
3.04
8.56
0.36
2.07
9.10
0.23
3.81
9.83
0.39
4.71
11.10
0.42
y 10 − y 1
∆s
f −s
r − rU S
Mean
Mean
Mean
Mean
-0.68
-0.59
3.56
-0.35
0.77
0.74
1.67
0.79
rxf x
Mean
Std
SR
3.54
9.02
0.39
2.97
9.02
0.33
rx?
Mean
Std
SR
-2.28
4.86
-0.47
1.54
4.81
0.32
1.15
5.43
0.21
4.25
6.07
0.70
5.68
8.74
0.65
rx$
Mean
Std
SR
1.26
10.19
0.12
4.51
10.24
0.44
2.28
10.88
0.21
5.25
11.63
0.45
5.33
13.45
0.40
y 10 − y 1
∆s
f −s
r − rU S
Mean
Mean
Mean
Mean
-0.91
-0.54
4.08
-0.76
0.71
1.08
1.89
0.87
1.29
-0.16
1.30
-0.12
1.89
0.20
0.80
2.49
3.29
-0.42
0.08
3.19
1.33
1.93
-0.18
0.00
1.11
0.59
0.74
2.30
Panel B: 1971-2012
1.13
0.99
9.40
9.15
0.12
0.11
4.13
9.64
0.43
3.22
-0.21
0.04
3.41
-0.34
8.62
-0.04
-3.88
5.99
-0.65
4.07
11.54
0.35
Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1 (defined as yield at t − 1 minus return on
T-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 extent 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’ permanent 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:
$
US
rt+1,10
+ ∆st+1 = α + βrt+1,10
+ t+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 rU S . 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 portfolio 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:
∗
rt+1,10
+ ∆st+1 = α + βrt+1,10 + t+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
35
0.30 0.36 0.47 0.65 0.47
(0.05) (0.05) (0.05) (0.04) (0.05)
0.19 0.21 0.17 0.15 0.23 0.21 -0.06 0.07
(0.05) (0.06) (0.04) (0.04) (0.05) (0.05) (0.05) (0.04)
0.69
0.05
0.55
0.07
0.41
0.07
0.53
0.06
0.61
0.07
0.19 0.12 0.27 0.11
(0.06) (0.06) (0.07) (0.06)
Monthly Returns. Regression of log return on bonds in local currency rlocal , log change in the exchange rate and the log return in dollars on the log return
on U.S. bonds in dollars. OLS standard errors.
r$ 0.66 0.65 0.64 0.50 0.71 0.64 0.71 0.71 0.72 0.56 0.57 0.75 0.44 0.24 0.43 0.50 0.44 0.52 0.35
s.e. (0.10) (0.09) (0.09) (0.07) (0.08) (0.10) (0.09) (0.09) (0.11) (0.12) (0.10) (0.08) (0.10) (0.09) (0.23) (0.10) (0.10) (0.09) (0.08)
−∆s -0.06 0.23 0.23 -0.09 0.22 0.17 0.22 0.23 0.20 0.18 0.34 0.23 -0.01 0.04 0.23 0.26 0.13 0.27 0.04
s.e. (0.10) (0.09) (0.09) (0.07) (0.09) (0.10) (0.09) (0.09) (0.09) (0.09) (0.09) (0.09) (0.10) (0.09) (0.09) (0.09) (0.10) (0.09) (0.08)
0.55 0.40 0.51 0.69 0.07 0.19
0.37 0.28 0.43 0.29
0.08 0.07 0.08 0.07 0.08 0.06
0.07 0.06 0.07 0.06
Panel C: 1991-2012
rlocal 0.71 0.42 0.41 0.59 0.49 0.46 0.48 0.47 0.52 0.38 0.23 0.52 0.45 0.20 0.20 0.24 0.31 0.25 0.31
s.e. (0.04) (0.04) (0.04) (0.03) (0.04) (0.05) (0.04) (0.04) (0.07) (0.06) (0.04) (0.03) (0.05) (0.03) (0.22) (0.04) (0.03) (0.03) (0.02)
r$ 0.33 0.42 0.52
s.e. (0.07) (0.06) (0.07)
−∆s -0.04 0.23 0.22 0.01 0.21 0.15 0.21 0.23 0.19 0.18 0.27 0.24 -0.08 0.08
s.e. (0.06) (0.06) (0.06) (0.04) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.07) (0.06)
0.17 0.16 0.15 0.18
(0.04) (0.03) (0.02) (0.03)
0.32 0.25 0.37 0.27
(0.05) (0.05) (0.05) (0.05)
0.17 0.10 0.23 0.10
(0.05) (0.05) (0.05) (0.04)
IRP ITL JPY NLG NZD NOK PTE ESP SEK CHF GBP
Panel A: 1950-2012
0.28 0.34 0.30 0.19 0.20 0.40 0.14 0.11
0.16 0.15 0.14 0.17
(0.03) (0.02) (0.04) (0.03) (0.03) (0.02) (0.04) (0.02)
(0.03) (0.02) (0.01) (0.02)
FF DEK
0.47 0.56 0.47 0.34 0.44 0.61 0.08 0.18
(0.05) (0.06) (0.07) (0.06) (0.06) (0.05) (0.06) (0.05)
Panel B: 1971-2012
rlocal 0.37 0.19 0.29 0.68 0.34 0.26 0.32 0.39 0.36 0.22 0.24 0.45 0.16 0.11
s.e. (0.04) (0.03) (0.03) (0.03) (0.04) (0.04) (0.03) (0.03) (0.05) (0.04) (0.04) (0.03) (0.05) (0.03)
r$
−∆s -0.03 0.19 0.19 0.01 0.18
s.e. (0.05) (0.05) (0.05) (0.03) (0.04)
rlocal 0.33 0.16 0.27 0.63 0.28
s.e. (0.03) (0.03) (0.02) (0.02) (0.03)
AUD ATS BEL CAD DKK FIM
Table 9: Bond Return Parity: Large Panel
∗
rt+1,10
= α + βrt+1,10 + t+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 (PostBretton-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
α
NW s.e.
BS s.e.
β
NW s.e.
BS s.e.
Wald
0.01∗∗
(0.01)
(0.00)
0.63∗∗∗
(0.08)
(0.07)
25.17∗∗∗
α
NW s.e.
BS s.e.
β
NW s.e.
BS s.e.
Wald
0.02∗∗
(0.01)
(0.01)
0.66∗∗∗
(0.12)
(0.09)
8.67∗∗
α
NW s.e.
BS s.e.
β
NW s.e.
BS s.e.
Wald
0.04∗
(0.02)
(0.01)
0.69∗
(0.16)
(0.11)
4.05
GBP
JPY
CAD
AUD
CHF
3-month holding period USD returns
0.01∗
0.01
0.01∗∗
0.02∗∗
0.00
(0.01)
(0.01) (0.00)
(0.01)
(0.01)
(0.01)
(0.01) (0.00)
(0.01)
(0.00)
0.58∗∗∗ 0.69∗∗ 0.71∗∗∗ 0.57∗∗∗
0.61∗∗∗
(0.08)
(0.12) (0.11)
(0.13)
(0.10)
(0.09)
(0.14) (0.09)
(0.13)
(0.08)
27.13∗∗∗ 6.11∗∗ 8.87∗∗ 11.60∗∗∗ 16.65∗∗∗
6-month holding period USD returns
0.02∗
0.01
0.02∗∗
0.04∗∗
0.01
(0.01)
(0.02) (0.01)
(0.01)
(0.01)
(0.01)
(0.01) (0.01)
(0.01)
(0.01)
∗∗∗
∗∗∗
∗∗∗
0.58
0.76
0.65
0.48
0.61∗∗∗
(0.10)
(0.19) (0.13)
(0.16)
(0.13)
(0.10)
(0.14) (0.09)
(0.14)
(0.10)
16.22∗∗∗
1.71
8.08∗∗ 12.04∗∗∗ 9.52∗∗∗
12-month holding period USD returns
∗∗
0.05
0.03
0.05∗∗
0.06∗∗
0.02
(0.02)
(0.04) (0.02)
(0.03)
(0.03)
(0.02)
(0.02) (0.01)
(0.02)
(0.02)
0.49∗∗∗
0.73
0.63∗∗
0.59∗∗
0.54∗∗
(0.13)
(0.29) (0.15)
(0.21)
(0.21)
(0.10)
(0.18) (0.09)
(0.15)
(0.15)
15.64∗∗∗
0.89
6.96∗∗
5.47∗
5.62∗
NZD
SEK
NOK
0.01
(0.01)
(0.01)
0.84
(0.11)
(0.10)
2.45
0.01∗
(0.01)
(0.01)
0.60∗∗∗
(0.10)
(0.11)
17.42∗∗∗
0.01
(0.01)
(0.01)
0.48∗∗∗
(0.17)
(0.13)
11.29∗∗∗
0.01
(0.01)
(0.01)
0.80
(0.14)
(0.12)
3.07
0.02
(0.01)
(0.01)
0.68∗∗
(0.15)
(0.14)
5.08∗
0.01
(0.01)
(0.01)
0.65∗
(0.21)
(0.18)
4.08
0.03
(0.03)
(0.02)
0.83
(0.20)
(0.15)
1.50
0.03
(0.03)
(0.02)
0.80
(0.22)
(0.16)
1.52
0.01
(0.03)
(0.02)
0.91
(0.34)
(0.26)
0.21
We regress the USD holding period return of a foreign 10-year bond on the corresponding holding period return
of 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 the
latter, we apply block bootstrapping using 1,000 bootstrap samples, with block length equal to 3. We test the
null hypotheses of constant equal to zero and slope coefficient equal to one both individually and jointly for
each currency; Wald is the Wald test statistic for the joint hypothesis test. One, two and three asterisks denote
rejection of the null hypothesis at the 10, 5 and 1 percent level of significance, respectively.
37
Table 11: Bond Return Parity: Currency Portfolios
1
rlocal
2
3
4
1950-2012
5
1
3
4
1971-2012
5
Panel A: Developed
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)
rf x
0.13
0.16
0.17
0.11
0.09
(0.04) (0.04) (0.04) (0.04) (0.04)
r$
0.46
0.47
0.44
0.36
0.29
0.51
(0.04) (0.05) (0.04) (0.04) (0.05)
(0.05)
Panel B: All
0.24
0.28
0.28
0.23
0.13
0.25
(0.01) (0.02) (0.02) (0.02) (0.02)
(0.02)
rlocal
2
0.15
0.19
0.19
0.13
0.09
(0.05) (0.05) (0.05) (0.05) (0.05)
0.51
0.49
0.40
0.32
(0.06) (0.06) (0.05) (0.06)
0.31
0.31
0.24
0.13
(0.03) (0.02) (0.02) (0.03)
rf x
0.08
0.15
0.16
0.07
0.00
(0.03) (0.03) (0.03) (0.03) (0.04)
0.09
0.17
0.17
0.09
0.00
(0.04) (0.05) (0.05) (0.05) (0.05)
r$
0.32
0.43
0.44
0.31
0.13
(0.04) (0.04) (0.04) (0.04) (0.04)
0.34
0.49
0.48
0.33
0.13
(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 and
the log return in dollars on the log return on U.S. bonds in dollars. OLS standard errors. Portfolios of currencies
sorted 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
Currency Risk Premium
4
2
0
−2
1
2
3
4
5
4
5
4
5
4
5
4
5
4
5
Local Term Risk Premium
4
2
0
−2
1
2
3
Dollar Term Risk Premium
4
3
2
1
0
1
2
3
Panel B: Developed Countries
Currency Risk Premium
3
2
1
0
−1
1
2
3
Local Term Risk Premium
3
2
1
0
−1
1
2
3
Dollar Term Risk Premium
4
3
2
1
0
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
Panel A: All Countries
Currency Risk Premium
6
4
2
0
−2
1
2
3
4
5
4
5
4
5
4
5
4
5
4
5
Local Term Risk Premium
6
4
2
0
1
2
3
Dollar Term Risk Premium
8
6
4
2
0
1
2
3
Panel B: Developed Countries
Currency Risk Premium
4
2
0
−2
1
2
3
Local Term Risk Premium
4
2
0
−2
1
2
3
Dollar Term Risk Premium
6
4
2
0
1
2
3
Notes: Annualized Monthly Returns. Sample: 1971-2012. Portfolios of Currencies sorted by T-bill Interest Rates.
The top panel shows all countries. The bottom panel shows developed countries.
40
Figure 6: Sorts by Slope: Whole Sample
Panel A: All Countries
Currency Risk Premium
3
2
1
0
−1
1
2
3
4
5
4
5
4
5
4
5
4
5
4
5
Local Term Risk Premium
6
4
2
0
−2
1
2
3
Dollar Term Risk Premium
6
4
2
0
1
2
3
Panel B: Developed Countries
Currency Risk Premium
3
2
1
0
1
2
3
Local Term Risk Premium
4
2
0
−2
1
2
3
Dollar Term Risk Premium
4
3
2
1
0
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.
41
Figure 7: Sorts by Slope: Post-Bretton-Woods
Panel A: All Countries
Currency Risk Premium
3
2
1
0
−1
1
2
3
4
5
4
5
4
5
4
5
4
5
4
5
Local Term Risk Premium
6
4
2
0
−2
1
2
3
Dollar Term Risk Premium
6
4
2
0
1
2
3
Panel B: Developed Countries
Currency Risk Premium
4
3
2
1
0
1
2
3
Local Term Risk Premium
4
2
0
−2
1
2
3
Dollar Term Risk Premium
6
4
2
0
1
2
3
Notes: Annualized Monthly Returns. Sample: 1971-2012. Portfolios of Currencies sorted by T-bill Interest Rates.
The top panel shows all countries. The bottom panel shows developed countries.
42
Figure 8: UIP and Uncovered Bond Return Parity: All Countries
8
Bond Return Spread
Interest Rate Spread
45−degree
7
6
Rate of Depreciation
5
4
3
2
1
0
−1
−2
−2
−1
0
1
2
3
4
Spread with U.S.
5
6
7
8
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
8
Bond Return Spread
Interest Rate Spread
45−degree
7
6
Rate of Depreciation
5
4
3
2
1
0
−1
−2
−2
−1
0
1
2
3
4
Spread with U.S.
5
6
7
8
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
Average Betas
Dollar
Local
FX
0.8
0.6
0.4
0.2
0
−0.2
1957
1971
1984
1998
2012
Notes: Sample: 1950-2012. 60-month rolling window estimation of beta with respect to US bond returns for the
equal-weighted average of log bond returns in local currency, the log change in the exchange rate and the log
dollar bond returns. Developed Countries.
45
Figure 11: Foreign Bond Return Betas: All Countries
Average Betas
Dollar
Local
FX
0.8
0.6
0.4
0.2
0
−0.2
1957
1971
1984
1998
2012
Notes: Sample: 1950-2012. 60-month rolling window estimation of beta with respect to US bond returns for the
equal-weighted average of log bond returns in local currency, the log change in the exchange rate and the log
dollar bond returns. All countries.
46
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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 the
investor’s Euler equation:
0 = Et [log Mt+1 ] + lim Et [log Rt+1,1 [1t+k ]] + (1/2) lim V art [log Mt+1 + log Rt+1,1 [1t+k ]].
k→∞
k→∞
The last term drops out because we know that limk→∞ Rt+1,1 [1t+k ] = Λt /Λt+1 . Also, note
that
log Et [Mt+1 ] = Et [log Mt+1 ] + (1/2)V art [log Mt+1 ].
Hence, we obtain:
0 = log Et [Mt+1 ] + lim Et [log Rt+1,1 [1t+k ]] − (1/2)V art [log Mt+1 ].
k→∞
It follows directly that the term premium equals one half of the variance:
Rt+1,1 [1t+k ]
= (1/2)σ 2 .
lim Et rxkt+1 = lim Et log
k→∞
k→∞
Rt+1,1 [1t+1 ]
• Proof of Proposition 2.3.
Proof. In the log-normal model, the foreign term premium –the bond risk premium on
∗,2
the longest maturity bond limk→∞ Et [rxk,∗
t+1 ] – equals
(1/2)σ . In a log-normal model,
∗,2
2
the currency risk premium equals (1/2) σ − σ . (Backus, Foresi, and Telmer (2001)).
To see why, consider the investor’s Euler equation:
0 = Et [log Mt+1 ] + Et [log Rt+1 ] + (1/2)V art [log Mt+1 + (ft − st ) − ∆st+1 ].
We use the complete markets expression for the change in the spot rates:
∗
∆st+1 = log Mt+1 − log Mt+1
.
The result follows immediately.
• Proof of Corollary 2.1:
Proof. Using the equivalence between long-term yield spreads and expected excess returns, we know that:
yt∗ [∞] + (ft − st ) − E[∆st+1 ] = (1/2)σ 2 = yt [∞]
Note that
E[ lim (1/k)
k→∞
k
X
∆st+j ] = lim (1/k)
k→∞
j=1
k
X
E[∆st+j ]
j=1
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 depend
on calendar time). That delivers the result.
• Proof of Proposition 2.4.
Proof. First, note that the foreign currency risk premium is equal to:
∗ Λt+1
Λt+1
(ft − st ) − Et [∆st+1 ] = Lt
− Lt
Λt
Λ∗t
See Backus, Foresi, and Telmer (2001). Also, note that:
!
ΛPt+1
Rt+1,1 [1t+∞ ]
Λt+1
= Lt
Lt
+
E
log
t
Λt
Rt+1,1 [1t+1 ]
ΛPt
This result is established in Alvarez and Jermann (2005). The final result follows immediately.
• Proof of Proposition 2.7:
Proof. First, note that the foreign currency risk premium is equal to:
∗ Λt+1
Λt+1
(ft − st ) − Et [∆st+1 ] = Lt
− Lt
Λt
Λ∗t
See Backus, Foresi, and Telmer (2001). Also, note that:
!
ΛPt+1
Rt+1,1 [1t+∞ ]
Λt+1
Lt
= Lt
+ Et log
Λt
Rt+1,1 [1t+1 ]
ΛPt
This result is established in Alvarez and Jermann (2005). The final result follows immediately.
• Proof of Proposition 2.5:
Proof. This result follows directly from the definition of a pricing kernel without permanent innovations and from the no-arbitrage expression for the spot exchange rates when
markets are complete:
M ∗ /M ∗
St
= t+1 t
St+1
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:
P
T
0 = Et [log Mt+1 ]+ lim Et [log Rt+1,1 [1t+k ]]+(1/2) lim V art [log Mt+1
+log Mt+1
+log Rt+1,1 [1t+k ]].
k→∞
k→∞
P ] because we know that lim
The last term simplifies to V art [log Mt+1
k→∞ Rt+1,1 [1t+k ] =
T
T
Λt /Λt+1 . Also, note that
log Et [Mt+1 ] = Et [log Mt+1 ] + (1/2)V art [log Mt+1 ].
51
Hence, we obtain:
P
0 = log Et [Mt+1 ] + lim Et [log Rt+1,1 [1t+k ]] − (1/2)V art [log Mt+1 ] + (1/2)V art [log Mt+1
].
k→∞
It follows directly that the term premium equals one half of the variance:
Rt+1,1 [1t+k ]
= (1/2)σT2 + σT P .
lim Et rxkt+1 = lim Et log
k→∞
k→∞
Rt+1,1 [1t+1 ]
• Proof of Corollary 2.2:
Proof. The result follows immediately from Proposition 2.6.
• Proof of Proposition 2.10:
Proof. The proof is immediate: the ratio of foreign to domestic holding period returns for
infinite maturity bonds (once converted to the same currency) measures the ratio of the
domestic and permanent components of the SDF. Theorem 2.10 refines the main result in
Brandt, Cochrane, and Santa-Clara (2006): the permanent components of the SDFs must
be highly correlated across countries.
52

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