fiscal cyclicality and currency risk premia
TRANSCRIPT
Fiscal Cyclicality and Currency Risk Premia
Zhengyang Jiang∗
November 12, 2018
Government surpluses load on a common factor, but to different degrees. In the cross-section,
countries whose government surpluses are more cyclical with respect to the common factor
tend to have higher nominal interest rates and higher currency returns. Their currency
returns are also more exposed to a common risk factor, leading to a correspondence between
the factor structure in government surpluses and the factor structure in currency returns.
In a frictionless model, I show these results are consistent with the idea that currencies are
priced as the claims to government surpluses.
∗ Department of Finance, Kellogg School of Management, Northwestern University. 2211 Campus Drive,Evanston, IL 60208. Email: [email protected]. This paper is a part of my PhDthesis. I acknowledge with deep gratitude the mentorship of John Cochrane and Hanno Lustig as my advisors,and the guidance of Adrien Auclert, Svetlana Bryzgalova, Sebastian Di Tella, and Darrell Duffie on mydissertation committee. For helpful comments, I thank Torben Andersen, Samuel Antill, Jonathan Berk, ShaiBernstein, YiLi Chien, Jesus Crespo Cuaresma (discussant), Peter DeMarzo, Ian Dew-Becker, Xiang Fang,Steven Grenadier, Benjamin Hebert, Robert Hodrick, Oleg Itskhoki, Patrick Kehoe, Peter Koudijs, ArvindKrishnamurthy, Ye Li, Edith X. Liu (discussant), Matteo Maggiori, Konstantin Milbradt, Sergio Rebelo,Rob Richmond, Dimitris Papanikolaou, Cheng Peng, Paul Pfleiderer, Jesse Schreger, Kenneth Singleton, IlyaStrebulaev, Viktor Todorov, Christopher Tonetti, Victoria Vanasco, Adrien Verdelhan, Jonathan Wallen, YiDavid Wang, Rui Xu, Mindy Xiaolan Zhang, and seminar participants at Northwestern Kellogg, UW Foster,NYU Stern, Imperial College Business School, LSE, New York Fed, USC Marshall, Chicago Booth, Wharton,WFA, Vienna Symposium on Foreign Exchange Markets, and Cubist Systematic Strategies. I thank DaojingZhai for excellent research assistance.
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I. Introduction
I find a high level of commonality in the changes in government surplus-to-debt ratios.
Across 11 developed countries, the first principal component explains 43% of their variations
from 1991 to 2017, and this fraction rises to 55% in the subsample starting from 2007. All
countries are exposed to this common factor, but to different degrees. A country has a
higher government surplus cyclicality if its government surplus-to-debt ratio is more exposed
to this factor. In this paper, I show how government surplus cyclicalities explain currency
risk premia in the cross-section.
To see this point, consider a model in which each country’s government only issues local
currency debt, and the debt is the claim to the government’s surpluses. Then, the real value
of the government debt reflects the present value of government surpluses, which fluctuate
across business cycles. On the other hand, since the notional payment of the government
debt is fixed in the unit of the local currency, the value of the local currency must adjust in
response to changes in government surpluses.
Therefore, currencies that are associated with more cyclical government surpluses tend to
depreciate more when the common factor in government surpluses declines. To compensate
investors for bearing this risk, these currencies have to offer higher risk premia. Notice,
however, their risk premia are not compensation for government default. In this model,
governments never default because they can always inflate away their local currency debt.
Currencies with higher risk premia can compensate investors by either raising nominal
interest rates or promising future appreciation. If each country’s monetary policy is set
so that its nominal exchange rate does not permanently drift upwards or downwards with
respect to other currencies, the country’s nominal interest rate must reflect its government
surplus cyclicality. So, a country with a higher government surplus cyclicality not only has
a higher currency return but also has a higher nominal interest rate.
Finally, the common variation in government surpluses also generates a factor structure
in currency returns. Lustig, Roussanov and Verdelhan (2011) define each currency’s carry
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beta as the exposure of its excess return with respect to the carry trade return, which is
the return differential between high interest rate currencies and low interest rate currencies.
They find currencies with higher carry betas tend to have higher returns. My model offers
an economic explanation for this factor structure: Currency returns load on a common risk
factor because their government surpluses are exposed to a common shock. The carry trade
bets on currencies whose government surpluses are more cyclical, and is therefore correlated
with the common factor in currency returns.
In summary, my model connects currency risk premia to the fiscal side of the economy. I
derive closed-form characterizations and test them in the sample of 11 developed economies.
Figure 1 summarizes the main result. A country with a higher government surplus cyclicality
tends to have a higher nominal interest rate, a higher currency expected return, and a higher
carry beta. Government surplus cyclicalities explain 62% of the cross-country variation in
average quarterly nominal interest rates, 78% of the variation in average quarterly currency
excess returns, and 52% of the variation in carry betas. This result is robust after I account
for the fact that government surplus cyclicalities are estimated.
1 2 3 4 5
−0.
6−
0.2
0.2
0.6
Government Surplus Cyclicality
Nom
inal
Inte
rest
Rat
e D
iffer
entia
l (%
)
Australia
CanadaDenmark
Germany
Japan
New ZealandNorway
Sweden
Switzerland
United Kingdom
United States
1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
Government Surplus Cyclicality
Cur
renc
y E
xces
s R
etur
n (%
)
Australia
Canada
Denmark
Germany
Japan
New Zealand
Norway
Sweden
Switzerland
United Kingdom
United States
1 2 3 4 5
−0.
6−
0.2
0.2
0.6
Government Surplus Cyclicality
Cur
renc
y R
etur
n’s
Car
ry B
eta
Australia
Canada
DenmarkGermany
Japan
New ZealandNorwaySweden
Switzerland
United Kingdom
United States
Fig. 1.—Government surplus cyclicality explains currency risk premia. I plot each country’s government
surplus cyclicality against its currency’s quarterly average nominal interest rate differential with respect to
the U.S. dollar, quarterly average excess return with respect to the U.S. dollar, and carry beta. Data are
quarterly, 1980Q2—2017Q4. I use the longest sample possible for each currency. The dashed line is the best
fitting straight line from ordinary least squares.
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Moreover, the government surplus-to-debt ratio can be decomposed into a GDP-to-debt
component, a tax-to-GDP component, and a spending-to-tax component. It is mainly the
spending-to-tax component that drives government fiscal shocks and determines government
surplus cyclicalities. This result suggests that currency risk premia are mainly influenced by
government fiscal policies rather than underlying economic conditions.
Finally, I construct a currency portfolio sorted by conditional government surplus cycli-
calities, which are estimated from rolling window regressions. The cross-country strategy
takes a long position in currencies whose conditional government surplus cyclicalities are
higher than the cross-country median, and a short position in other currencies. The return
of this strategy is strongly correlated with the carry trade return, and offers a Sharpe ratio
similar to that of the carry trade. Because conditional government surplus cyclicalities are
estimated from data available ex-ante, this approach is an out-of-sample evaluation of the
fiscal condition’s return predictability. This result confirms that the carry trade is profitable
because it loads on currencies whose government surpluses are cyclical.
This paper proceeds as follows. Section II formulates the model and derives its predictions.
Section III describes the data. Sections IV, V, and VI report the main empirical results. Sec-
tion VII concludes. The Appendix contains proof and data sources. The Internet Appendix,
available on my personal website, contains additional empirical results.
A. Literature review
This paper connects the currency literature with the fiscal literature. The currency liter-
ature has documented the carry trade anomaly (Brunnermeier, Nagel and Pedersen (2008);
Lustig and Verdelhan (2007); Lustig, Roussanov and Verdelhan (2011); Burnside, Eichen-
baum and Rebelo (2011); Engel (2014)) and found a factor structure in currency returns
(Lustig, Roussanov and Verdelhan (2014); Fourel et al. (2015); Verdelhan (2018)). Hassan
and Mano (2014) shows that this factor structure is related to the cross-sectional component
of currency risk premia. I offer a fiscal explanation for these patterns.
My model is closest to Gourio, Siemer and Verdelhan (2013); Colacito et al. (Forthcoming)
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that explore heterogeneous loadings on global shocks as the key determinant of currency risk
premia, to Engel and West (2005); Gourinchas and Rey (2007, 2014); Farhi and Gabaix
(2016) that derive exchange rates as present values, and to Maggiori and Gabaix (2015) that
model risk-averse international investors.
On the other hand, the fiscal literature studies how fiscal conditions affect domestic and in-
ternational prices. Burnside, Eichenbaum and Rebelo (2001, 2003); Corsetti and Mackowiak
(2001); Daniel (2001a) show how fiscal shocks affected exchange rates during currency crises.
The fiscal theory of the price level connects fiscal conditions to domestic price levels (Sargent
and Wallace (1984); Leeper (1991); Woodford (1994); Sims (1994); Cochrane (2001, 2005,
2018, 2017); Dupor (2000); Daniel (2001b)). My paper extends this link to study currency
risk premia.
Outside the currency crisis literature and the fiscal theory, Aguiar et al. (2013, 2015); Du,
Pflueger and Schreger (2016) study how fiscal commitment interacts with inflation and debt
crisis. Aguiar, Amador and Gopinath (2005); Farhi, Gopinath and Itskhoki (2013) study
fiscal policies and real allocations. Obstfeld (2011); Farhi, Gourinchas and Rey (2011);
Caballero and Farhi (2013) explore the fiscal production of safe assets. My paper provides
a novel mechanism for determining international asset returns through the risk exposures of
fiscal processes.
In addition, my fiscal explanation of currency risk premia takes each country’s government
surplus cyclicality as given. Deeper economic rationales are required to explain why some
countries have more cyclical government surpluses. These rationales are beyond the scope
of this paper, but they can be motivated by previous studies that find commodity-exporting
countries (Powers (2015); Ready, Roussanov and Ward (2016, 2017)), peripheral countries
in the international trade network (Richmond (2016)), and net debtor countries (Corte,
Riddiough and Sarno (2016); Wiriadinata (2018)) have higher currency risk premia.
Finally, my model assumes that governments do not default. Default is possible if gov-
ernments have real liabilities, in which case their fiscal conditions affect both their default
6
probabilities and their exchange rates (Chernov, Schmid and Schneider (2016); Bolton and
Huang (2017); Della Corte et al. (2016); Du and Schreger (2016); Augustin, Chernov and
Song (2018)). My model mutes this channel in order to focus on the connection between
fiscal conditions and currency risk premia.
II. Model
A. Environment
This section develops a simple model. There is only one type of consumption goods, which
is traded internationally at zero transportation cost and has a flexible price.
There are N countries, indexed by i ∈ {1, . . . , N}. Each country has a local currency. Let
Qit denote its value, which is the amount of consumption goods each unit of currency can
buy.
Each country also has a government, which collects tax and has government spending. Let
τ it denote the tax revenue, and let git denote the government spending. Both quantities are
in the unit of the consumption goods. Government surplus is defined as their difference:
sitdef= τ it − git.
The government issues debt denominated in the local currency. I make two simplifying
assumptions about the government debt. First, the government only issues one-period debt;
that is, each unit of the government debt pays 1 unit of the local currency in next period.
Let Bit denote the nominal quantity of the government debt issued in period t, which is the
amount of the local currency that the government pays in period t+ 1.
Second, the government does not default. Let Rf,it denote the nominal interest rate of the
government debt issued in period t. Then, this debt is worth (1 + Rf,it )−1Bi
t units of the
local currency. As this model does not distinguish between a country’s monetary authority
and its government, the nominal interest rate is set by the government.
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In sum, the government has the following intertemporal budget condition:
τ it + (1 +Rf,it )−1Bi
tQit = git +Bi
t−1Qit. (1)
On the left-hand side, the government collects tax revenue and receives the proceeds from
new debt issuance. On the right-hand side, it spends and pays out for expiring debt. Debt
quantities Bit and Bi
t−1 are converted to real values by a factor of Qit, the currency value.
B. International Investor
There is a representative international investor who has access to the complete market. In
period t, his consumption is ct and his utility is
u (ct) =c1−γt
1− γ.
He receives endowment yt. He pays tax τ it to country i’s government, and receives govern-
ment spending git from it. He also trades government debt. By market clearing conditions,
since he is the only non-government agent, he holds all government debt in equilibrium. His
budget constraint is
yt +∑i
git +∑i
Bit−1Q
it −∑i
(1 +Rf,it )−1Bi
tQit = ct +
∑i
τ it , (2)
where all Arrow-Debrew securities with zero supply are omitted.
Then, the Lagrangian for the international investor’s optimization problem is
∑t
e−δtu (ct) + Λt
(yt +
∑i
sit − ct +∑i
Bit−1Q
it −∑i
(1 +Rf,it )−1Bi
tQit
).
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The first-order condition implies
1 = Et[
Λt+1
Λt
(1 +Rf,it )
Qi+t+ 1
Qit
], (3)
where Λt+1/Λt is the international investor’s real pricing kernel:
Λt+1
Λt
= e−δu′ (ct+1)
u′ (ct)= exp(−δ − γ∆ log ct+1).
Next, assume the present value of future government surpluses grows slower than the
discount rate:
ASSUMPTION 1:
limT→∞
Et
[Λt+T+1
Λt
(∞∑k=0
Λt+T+1+k
Λt+T+1
sit+T+1+k
)]= 0.
Then I can iterate the intertemporal government budget condition Eq. (1) and obtain the
following present value relationship.
LEMMA 1: Currency value equals the present value of real government surpluses divided
by the nominal quantity of government debt:
Qit =
1
Bit−1
∞∑k=0
Et[
Λt+k
Λt
sit+k
]. (4)
Let ei,jt denote the nominal exchange rate between country i’s currency and country j’s cur-
rency. Because the law of one price holds for the consumption goods, the nominal exchange
rate between the two currencies equals the ratio between their purchasing powers:
ei,jt = Qit/Q
jt .
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C. Shocks and Processes
Now I specify the processes of the real pricing kernel, government surpluses, and interest
rate targets. All shocks are standard normal variables, mutually independent across countries
and across time.
The international investor’s endowment loads on a global consumption shock εct+1:
yt+1
yt= exp(σεct+1).
The government budget condition Eq. (1) and the international investor’s budget con-
straint Eq. (2) imply that his real pricing kernel is
Λt+1
Λt
= exp(−δ − γσεct+1).
The log growth rate of country i’s government surplus is exposed to the same global
consumption shock εct+1:
∆ log sit+1 =
(µ− 1
2(ϕi)2σ2 − 1
2ω2
)+ ϕiσεct+1 + ωεs,it+1,
where the exposure ϕi is defined as the country’s government surplus cyclicality. This value
is fixed for each country, but can vary across countries. In addition, the log growth rate
of government surplus is also exposed to a country-specific shock εs,it+1. The terms in the
bracket are set so that the expected growth rate of government surplus is the same for all
countries:
Et[sit+1/sit] = eµ.
Finally, the government in country i sets the nominal interest rate Rf,it according to the
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following rule:
log(1 +Rf,it ) = rf,i + ηεf,it , (5)
where rf,i is a country-specific constant and εf,it is an idiosyncratic shock.
With these assumptions, the currency value equals the government surplus-to-debt ratio
times a function of the government surplus cyclicality:
LEMMA 2: The currency value is
Qit =
sitBit−1
F (ϕi),
where the function F is defined as
F (ϕi) =∞∑k=0
Et[
Λt+k
Λt
sit+ksit
].
D. Characterizations
When the international investor invests one unit of consumption goods in country i’s
government debt, he earns (1 +Rf,it )Qi
t+1/Qit unit of consumption goods in the next period.
Define the log currency excess return of currency i against currency j as the difference in
their log returns from the international investor’s perspective:
ri,jt+1def= log
(Qit+1
Qit
(1 +Rf,it )
)− log
(Qjt+1
Qjt
(1 +Rf,jt )
).
Alternatively, we can take the perspective of a hypothetical investor in country j. The
log currency excess return can also be expressed as the log return of country i’s government
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debt in the unit of country j’s currency minus country j’s log nominal interest rate:
ri,jt+1 = log
(ei,jt+1
ei,t(1 +Rf,i
t )
)− log
(1 +Rf,j
t
).
PROPOSITION 1 (Currency Risk Premium): The expected log currency excess return of
currency i against currency j is
Et[ri,jt+1] = (γϕiσ2 − γϕjσ2)−(
1
2(ϕi)2σ2 − 1
2(ϕj)2σ2
).
For any country i, its expected log currency excess return has two components. The
first component γσ2ϕi reflects the currency risk premium, which is increasing in country
i’s government surplus cyclicality ϕi. The second component −(1/2)(ϕi)2σ2 is the Jensen’s
inequality term that comes from the expectation of a log-normal process. When risk aversion
γ is larger than government surplus cyclicality ϕi, the currency’s expected log excess return
is increasing in ϕi.
The next proposition shows that the risk premium term γϕiσ2 also manifests itself in the
currency’s nominal interest rate.
PROPOSITION 2 (Nominal Interest Rate): (a) The log nominal interest rate in country i
satisfies
log(1 +Rf,it ) = γϕiσ2 +
(δ − 1
2γ2σ2
)+ ∆ logBi
t − µ.
(b) The log expected exchange rate movement is
logEt
[ei,jt+1
ei,jt
]=
(γϕiσ2 − rf,i − ηεf,it
)−(γϕjσ2 − rf,j − ηεf,jt
).
If the log expected exchange rate movement has an unconditional mean of 0, the nominal
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interest rate target rf,i must reflect the currency risk premium:
rf,i = γϕiσ2.
Proposition 2(a) states that besides the risk premium term, the nominal interest rate
contains two additional terms. δ − (1/2)γ2σ2 is the log real interest rate implied from the
international investor’s pricing kernel:
Et[
Λt+1
Λt
· eδ−12γ2σ2
]= 1,
and ∆ logBit − µ is the inverse of the expected change in currency value:
logEt[Qit+1
Qit
]= logEt
[sit+1/s
it
Bit/B
it−1
]= −∆ logBi
t + µ;
since this expected change in currency value does not affect risk premium, the currency has
to offer a higher nominal interest rate if it is expected to depreciate.
Proposition 2(b) states that if exchange rates have no permanent drifts, currencies with
higher government surplus cyclicalities must also have higher nominal interest rates. Pro-
ceeding with this assumption, I set rf,i = γϕiσ2.
Lastly, I define the carry trade strategy as a long position in currencies whose nominal
interest rates are above the cross-country median plus a short position in currencies whose
nominal interest rates are below the median. The average log return of its holdings is
rcarryt+1def=
1
N
(∑i∈Ht
ri,jt+1 −∑i∈Lt
ri,jt+1
), (6)
Ht ={i : Rf,i
t ≥ median({Rf,kt }k)
},
Lt ={i : Rf,i
t < median({Rf,kt }k)
}.
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Fix a base currency j. I define a currency’s carry beta βicarry as the exposure of its excess
return against the base currency j with respect to the carry trade return (Lustig, Roussanov
and Verdelhan (2011)), obtained from the following regression:
ri,jt+1 = αi + βicarryrcarryt+1 + εit+1.
For tractability, I make the following assumption.
ASSUMPTION 2: There is a large number N of countries, and their government surplus
cyclicalities ϕi are normally distributed with a mean of ϕ and a standard deviation of ρ.
Then, the following proposition characterizes the carry beta.
PROPOSITION 3 (Factor Structure in Currency Returns): Currency returns loads on the
carry trade return to different degrees. Each currency’s carry beta βicarry equals its govern-
ment surplus cyclicality ϕi up to a linear transformation:
βicarry =1
C2
ϕi − ϕj
C2
,
where C2 = 2√2π
ρ2√1+(
η
γσ2ρ
)2is a positive constant.
This last proposition shows a correspondence between the factor structure in currency re-
turns and the factor structure in government surpluses. A currency with a higher government
surplus cyclicality also has a higher carry beta.
E. Discussions of Assumptions
This model makes the following assumptions to derive simple characterizations of nominal
currency returns and nominal interest rates.
No Government Default
In my model, governments never default. The local currency debt is like equity, and the
currency value is like its stock price. Whenever the present value of government surpluses
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declines in real terms, the local currency depreciates so that the government surplus is still
sufficient to honor the debt’s nominal payment.
However, if the government also issues real debt or foreign currency debt, the government
defaults whenever its surplus falls below the required debt payment in real terms or in
foreign currency terms. Governments can also default strategically (Arellano (2008)). In
these cases, a country’s fiscal condition affects both its default probability and its currency
value. Because there is one more degree of freedom, additional layers are required to pin
down the entire dynamics. These layers are abstracted away in my paper.
In the Internet Appendix, I document that government surplus cyclicality explains a large
fraction of variation in currency risk premia among developed countries, and a smaller frac-
tion of variation among developing countries. Sovereign CDS premium has the opposite
pattern: It explains no variation in currency risk premia among developed countries, and
a large fraction of variation among developing countries. Since government default mainly
happens in developing countries, it is reasonable to expect my fiscal explanation of currency
risk premia to work better for developed countries.
Constant Real Exchange Rate
In this frictionless model, a single type of consumption goods implies constant real exchange
rates. This assumption allows me to characterize currency returns in closed forms. In reality,
real exchange rates are not constant, and their movements are highly correlated with nominal
exchange rate movements (Mussa (1986)). In Jiang (2018), I develop a New Keynesian model
with differentiated goods and sticky prices. This model has two ex-ante symmetric countries,
and each country’s price level is determined by the present value of government surpluses
as in Eq. (1). Because prices are sticky, real and nominal exchange rates have to comove
in response to fiscal shocks, while inflation adjustment is sluggish and shaped by monetary
policy.
Local Currency Debt
This model assumes that governments only issue local currency debt. The Bank for In-
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ternational Settlements discloses the currency compositions of central government debt in
different countries. With the exception of Argentina, most government debt issued in recent
years is denominated in its local currency, with a large fraction of debt having fixed rates
(See Internet Appendix).
Negative Government Surplus
If the present value of all future government surpluses is negative, the currency should have
a negative value. However, current government deficits do not necessarily imply a negative
present value of future government surpluses, if government surpluses will turn positive in
the long run. In support of this view, Bohn (1998) documents that the U.S. government
tends to increase government surpluses when the debt-to-GDP ratio is high.
III. Data
A. Sample
I focus on 11 developed economies: Australia, Canada, Denmark, Germany, Japan, New
Zealand, Norway, Sweden, Switzerland, the United Kingdom, and the United States.
Data are quarterly, covering the period from 1980Q1 to 2017Q4. The sample is unbalanced,
but after each country’s time series starts, there is no missing observation. The Appendix
provides a detailed description of the data source.
B. Estimation of government surplus cyclicality
By the law of large numbers, the average log growth rate of the government surplus-to-debt
ratio in my model equals the global consumption shock εct+1 up to a linear transformation:
1
N
N∑i=1
(log
sit+1
Bit
− logsitBit−1
)=
1
N
N∑i=1
(−1
2(ϕi)2σ2 − 1
2ω2 + δ − 1
2γ2σ2
)+
(1
N
N∑i=1
ϕi
)σεct+1.
Therefore, each country’s government surplus cyclicality ϕi can be recovered by regressing
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the log growth rate of its government surplus-to-debt ratio on the average log growth rate:
logsit+1
Bit
− logsitBit−1
= ai + bi · 1
N
N∑k=1
(log
skt+1
Bkt
− logsktBkt−1
)+ εit+1, (7)
and the regression coefficient bi is a scaled version of the government surplus cyclicality:
bi =1
1N
∑Nk=1 ϕ
k· ϕi.
To estimate bi from this regression, I encounter two empirical issues. First, there are
seasonal effects in quarterly government surpluses and government debt. To control for
these seasonal effects, I use the four-quarter change in the government surplus-to-debt ratio
instead. Then Eq. (7) becomes:
logsit+1
Bit
− logsit−3
Bit−4
= ai + bi · 1
N
N∑j=1
(log
sjt+1
Bjt
− logsjt−3
Bjt−4
)+ εit+1. (8)
My model implies that bi from Eq. (8) is still equal to government surplus cyclicality ϕi
up to a linear transformation (See the Internet Appendix).
Second, government surpluses can be negative, in which case their logarithm log sit+1 is
undefined. This issue reflects my model’s trade-off between tractability and reality: While
the assumption of log-normal shocks allows me to derive closed-form results, it cannot match
observed government deficits. Since the focus of this paper is on the change in the government
surplus-to-debt ratio, I use the following approximation to derive testable implications:
logsit+1
Bit
− logsit−3
Bit−4
≈Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
, (9)
where Snom,it+1 is the nominal government surplus and P it+1 is the price level. The proof is in
17
the Appendix. Under this approximation, Eq. (8) becomes
Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
= ai + bi · ft+1 + εit+1, (10)
where ft+1 is the common surplus factor:
ft+1def=
1
N
N∑j=1
(Snom,jt+1
Bjt
P jt−3
P jt+1
−Snom,jt−3
Bjt−4
).
Since not all countries have government surplus data available at the start of the sample,
the common surplus factor is the equal-weighted average over countries that have data.
IV. Commonality in Government Fiscal Shocks
A. Exposures to the common surplus factor
Table 1 reports the estimation result of Eq. (10). All countries have positive and statisti-
cally significant loadings on the common surplus factor. Although the loadings vary across
countries, the common surplus factor explains large fractions of variations in all countries
except Japan and Germany. For example, United States’ coefficient is 0.35 while Norway’s
coefficient is 1.85, but the common surplus factor explains about the same fraction of the
variation in both countries.
Table 1
Loadings on the Common Surplus Factor
Japan US Switzerland Canada Germany Denmark UK Sweden Norway Australia New Zealand
bi 0.20 0.35 0.37 0.41 0.50 0.77 0.80 0.81 1.85 1.99 3.07
s.e. (0.05) (0.03) (0.05) (0.05) (0.18) (0.08) (0.08) (0.08) (0.18) (0.21) (0.30)R2 9.71 41.45 27.08 35.54 5.29 38.63 38.69 40.29 42.20 43.35 49.65
Note: I report the coefficient bi, its standard error, and the R2 of Eq. (10) in the time series of each country.
18
B. Principal component analysis
Alternatively, I use the principal component analysis to confirm the existence of a common
factor in the fiscal shocks across countries. I rescale all countries’ time series of the changes in
government surplus-to-debt ratios so that they all have unit variance. Under this approach,
a country with a volatile government surplus-to-debt ratio does not necessarily have a higher
loading on the common factor.
Figure 2 reports the result. The left panel shows the fraction of variance explained by each
principal component. The first principal component explains 43% of the variation in gov-
ernment surplus-to-debt ratios, while the remaining principal components explain much less
variation. In the subsample starting from 2007, this fraction rises to 55%. For comparison,
the first principal component of GDP growth rates explains 54% of the variation.
The right panel shows the loading of each country’s time series on the first principal
component. All countries have positive loadings on the first principal component. Since
2 4 6 8 10
010
2030
40
Principal Components
Prop
ortio
n of
Var
ianc
e
Ger
man
y
Japa
n
New
Zea
land
Aus
tral
ia
Switz
erla
nd
Nor
way
Den
mar
k
Swed
en
Uni
ted
Stat
es
Uni
ted
Kin
gdom
Can
ada
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Fig. 2.—Principal component analysis of the changes in government surplus-to-debt ratios. Each country’s
time series is rescaled so that it has a unit variance. The left panel shows the fraction of variance explained
by each principal component. The right panel shows each country’s loading on the first principal component.
Data are quarterly, from 1991Q1 to 2017Q4. This sample is shorter because some countries contain missing
data points in the 1980s.
19
each country’s time series is normalized by its volatility, the rank of countries by their
loadings is different from the rank in Table 1.
The first principal component is highly correlated with the common surplus factor ft+1
with a correlation of 0.95. Because the common surplus factor is easier to construct and has
a longer sample period, I focus on the common surplus factor in my remaining analysis.
C. What drives the common surplus factor
To examine what explains the variation in the common surplus factor, I regress the common
surplus factor ft+1 on concurrent variables xt+1:
ft+1 = α + βxt+1 + εt+1.
The explanatory variables xt+1 fall into three categories. First, government surpluses are
tax minus spending, and ultimately come from the production of goods. I include the average
log growth rates of GDP, tax revenue and government spending across all countries.
Second, Lustig, Roussanov and Verdelhan (2011, 2014) and Verdelhan (2018) show that
the carry factor and the dollar factor are systematic risk factors in currency returns. The
dollar factor is the holding return of the U.S. dollar against the equal-weighted portfolio of
foreign currencies in my sample. If government surplus-to-debt ratios affect currency risk
premia, the common surplus factor might comove with these currency factors.
Third, stock market performance also reflects economic fundamentals and government
policies. I use the MSCI world equity cum-dividend return in US dollar and the VIX index
to represent the fluctuations in the stock market.
Table 2 reports the regression results. The average log GDP growth rate is positively cor-
related with the common surplus factor, and explains 26% of the variation in the common
surplus factor. The common surplus factor is also positively correlated with the average
growth rate of tax revenue and negatively correlated with the average growth rate of gov-
ernment spending. Therefore, the common movements in tax revenue and in government
20
Table 2
Drivers of the Common Surplus Factor
(1) (2) (3) (4) (5) (6) (7) (8) (9)
GDP Growth 0.168∗∗
(0.070)
Tax Revenue Growth 0.231∗∗∗
(0.014)Govt Spending Growth −0.224∗∗∗
(0.025)Carry Factor 0.198∗ 0.199∗ 0.133∗∗∗
(0.103) (0.102) (0.051)
Dollar Factor −0.002 0.011 0.060∗∗
(0.051) (0.037) (0.029)
MSCI World Return 0.098∗∗∗ 0.127∗∗∗ 0.114∗∗∗
(0.029) (0.017) (0.019)VIX −0.047∗∗ −0.002 −0.002
(0.020) (0.011) (0.012)
Observations 147 147 133 133 133 147 112 112 112
R2 0.264 0.837 0.144 0.00004 0.145 0.271 0.139 0.350 0.403
Note: I regress the common surplus factor on fundamental variables, currency factors and stock marketperformance. Because the common surplus factor is the average 4-quarter change in government surplus-to-debt ratios, all explanatory variables except VIX are growth rates or cumulative returns over the same 4-quarter periods. The constant is not reported. The standard errors are heteroskedasticity and autocorrelationconsistent. ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01.
spending both contribute to the variation in the common surplus factor.
The common surplus factor also comoves with the carry trade factor. The common surplus
factor is not correlated with the dollar factor in the univariate regression, but it is positively
correlated with the dollar index once the MSCI world stock return index is controlled for.
Finally, the common surplus factor is higher when the stock market performs well. In
univariate regressions, the common surplus factor is positively correlated with the MSCI
world equity return and negatively correlated with the VIX index. The VIX index no longer
explains the common surplus factor once the MSCI world equity return is controlled for.
The last column in Table 2 regresses the common surplus factor on currency risk factors
and stock market performance. These financial variables explain 40% of the variation in the
common surplus factor.
21
V. Currency Risk Premia in the Cross-Section
A. Main results
Having established the existence of the common factor in government surplus-to-debt ra-
tios, I test my model’s key predictions. Proposition 1 and 2 predict that a currency with a
higher government surplus cyclicality has a higher average excess return and a higher average
nominal interest rate. As discussed in Section III, I use the regression coefficient bi from
Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
= ai + bi · ft+1 + εit+1
as a proxy for the government surplus cyclicality ϕi.
In Figure 1 in the introduction, I have shown that the regression coefficient bi is positively
associated with the average nominal interest rate differential and the average currency excess
return with respect to the U.S. dollar. I run 4 tests to quantify this relationship.
The first test is ordinary least squares (OLS). I regress each country’s average nominal
interest rate differential or average currency excess return with respect to the U.S. dollar on
its regression coefficient bi:
1
T
T∑t=1
(logRf,i
t − logRf,USt
)or
1
T
T∑t=1
ri,USt+1 = λ0 + λbi + ei. (11)
I regard each currency’s regression coefficient bi as a known constant, and run a linear
regression in the cross-section of countries. Table 3 reports the results. A one-standard
deviation increase in a country’s regression coefficient bi is associated with a 0.32% higher
nominal interest rate and a 0.24% higher currency excess return per quarter. The government
surplus cyclicality explains 62% of the cross-country variation in the average nominal interest
rate and 78% of the cross-country variation in the average currency excess return.
The next three tests recognize the fact that the regression coefficient bi is estimated from
a regression. The second test corrects for the estimation errors in bi using the generalized
22
Table 3
Currency Risk Premia in the Cross-Section
Dependent Variable Test #Quarters λ Std Error R2 (%) α Test p Value
Nominal Interest Rate OLS 152 35.40 (9.20) 62.19
Nominal Interest Rate GMM 108 47.35 (9.88) 7.48 0.68Nominal Interest Rate Shanken 108 32.53 (1.21) 1220.22 0.00
Nominal Interest Rate Fama-Macbeth 137 45.85 (5.40) 645.18 0.00
Currency Excess Return OLS 152 26.83 (4.80) 77.64Currency Excess Return GMM 108 38.21 (19.48) 3.40 0.97
Currency Excess Return Shanken 108 31.33 (18.12) 2.91 0.97
Currency Excess Return Fama-Macbeth 137 17.38 (21.11) 6.06 0.73
Note: I report the estimates of the risk premium parameter λ from the four tests. The estimates λ are scaledto express the change in the dependent variable in basis points for a unit increase in the government surpluscyclicality. #Quarters is the number of quarters used in each test. OLS and the Fama-Macbeth test allowsome countries to have missing observations. α Test is the test statistics against the null that all pricingerrors are jointly zero. Under the null, it follows a Chi-squared distribution, and I also report its p value.The standard errors from the GMM, the Shanken test, and the Fama-Macbeth test are heteroskedasticityand autocorrelation consistent.
method of moments (GMM). The moment conditions are
(Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
)− ai − bift+1 = 0,((
Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
)− ai − bift+1
)ft+1 = 0,(
logRf,it − logRf,US
t
)− λbi − λ0 = 0 or ri,USt+1 − λbi − λ0 = 0.
The first two moment conditions estimate the proxy bi for government surplus cyclicality.
The last moment condition estimates the relationship λ between government surplus cycli-
cality and currency risk premia. In order to estimate the covariance matrix of residuals, all
countries’ time series should have the same length. So, in this procedure I use the subsample
that contains no missing observation, which starts from 1991.
I report the first-stage GMM result in Table 3. The estimate λ is consistent with the
OLS results, suggesting a positive relationship between government surplus cyclicality and
currency risk premia.
23
The third test uses the Shanken (1992) correction. The sample is shorter because this pro-
cedure also requires that the sample contains no missing observation. When the dependent
variable is the currency excess return, the estimate λ and its standard error are similar to
those from the GMM.
The fourth test follows the Fama and MacBeth (1973) procedure. I estimate the regression
coefficient bi from the entire time series, and then estimate the coefficient λ from Eq. (11)
using the cross-section in each quarter. I only require that there are at least four countries
with non-missing observations to admit a quarter into my sample, and report the sample
average of the estimate λ in each quarter. When the dependent variable is the currency
excess return, the estimate λ is smaller than the estimates from the other tests.
B. The source of government surplus cyclicality
The government surplus-to-debt ratio can be decomposed into 3 components:
sit+1
Bit
≡GDP i
t+1
Bit
·τ it+1
GDP it+1
·τ it+1 − git+1
τ it+1
.
The GDP-to-debt ratio measures the quantity of domestic production per unit of govern-
ment debt, reflecting the country’s underlying economic condition. The tax-to-GDP ratio
measures the quantity of tax revenue per unit of domestic production, reflecting the govern-
ment’s tax policy. The surplus-to-tax ratio measures the quantity of government spending
per unit of tax revenue, reflecting the government’s spending policy.
Take the four-quarter log difference,
∆4 logsit+1
Bit
≡ ∆4 logGDP i
t+1
Bit
+ ∆4 logτ it+1
GDP it+1
+ ∆4 logτ it+1 − git+1
τ it+1
,
where ∆4 = (I − L4) takes the difference between the variable and its value 4 quarters ago.
24
On the left-hand side, I use the approximation formula Eq. (9):
∆4 logsit+1
Bit
≈Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
.
On the right-hand side, because the government surplus τ it − git can be negative, I use the
spending-to-tax ratio git/τit to represent log((τ it − git)/τ it ).
Then, I can examine which component explains the variation in the government surplus-
to-debt ratio. To do so, I regress the change in the government surplus-to-debt ratio on its
three components:
Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
= a+ c1∆4 logGDP i
t+1
Bit
+ c2∆4 logτ it+1
GDP it+1
+ c3∆4 git+1
τ it+1
+ εit+1.
Table 4 reports the regression results. All three components are correlated with the change
in the government surplus-to-debt ratio. However, the spending-to-tax ratio alone explains
69% of the variation, and drives out the explanatory power of the other two components.
This result suggests that the variation in the government surplus-to-debt ratio is mainly
driven by the government’s fiscal policy rather than the country’s economic condition.
Table 4
Decomposition of Government Fiscal Shock
(1) (2) (3) (4)
∆ log(GDP it+1/Bit) 0.034∗∗∗ −0.004
(0.010) (0.004)∆ log(τ it+1/GDP
it+1) 0.211∗∗∗ −0.0001
(0.019) (0.018)∆(git+1/τ
it+1) −0.245∗∗∗ −0.246∗∗∗
(0.009) (0.012)
Observations 1,578 1,578 1,578 1,578
R2 0.027 0.205 0.693 0.693
Note: I regress the change in government surplus-to-debt ratio on its three components. It is a panelregression across all countries and quarters. The constant is not reported. Standard errors are clustered byquarter. ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01.
25
Now that the spending-to-tax ratio drives the variation in the government surplus-to-debt
ratio, can its cyclicality explain currency risk premia? To answer this question, I repeat the
same tests in Table 3, but replace the government surplus-to-debt ratio with the spending-
to-tax ratio and other components.
For example, in the OLS test, I regress each of the three components on the common
surplus factor in each country’s time series:
∆ logGDP i
t+1
Bit
= ai1 + bi1 · ft+1 + εi1,t+1,
∆ logτ it+1
GDP it+1
= ai2 + bi2 · ft+1 + εi2,t+1,
∆gitτ it
= ai3 + bi3 · ft+1 + εi3,t+1,
and then regress each country’s average currency excess return with respect to the U.S. dollar
on one of the coefficients bi1, bi2, and bi3.
I also repeat the GMM, the Shanken test and the Fama-Macbeth for each of the components
of the government surplus-to-debt ratio. Table 5 reports the test results. The association
between the spending-to-tax ratio’s cyclicality and the currency excess return is the strongest:
It has the highest R2 and the highest t statistics, and the null that all pricing errors are jointly
zero is not rejected in its GMM test.
This result suggests that the cross-country variation in currency risk premia is mostly
due to the governments’ fiscal policies. In the Internet Appendix, I also report the results
using the nominal interest rate differential as the dependent variable. The cyclicality of
the spending-to-tax ratio also explains the cross-country variation in nominal interest rate
differentials.
VI. The Factor Structure of Currency Returns
Lustig, Roussanov and Verdelhan (2011) show that the carry trade factor explains the
26
Table 5
Decomposition of Government Surplus Cyclicality
Explanatory Variable Test #Qtrs λ Std Error R2 (%) α Test p Value
GDP-to-Debt Ratio OLS 152 6.59 (5.33) 14.53
GDP-to-Debt Ratio GMM 108 45.56 (47.61) 488.98 0.00GDP-to-Debt Ratio Shanken 108 7.26 (9.35) 11.04 0.27
GDP-to-Debt Ratio Fama-Macbeth 137 6.70 (9.15) 10.72 0.30
Tax-to-GDP Ratio OLS 152 8.78 (8.93) 9.71Tax-to-GDP Ratio GMM 108 48.93 (36.03) 44.11 0.00
Tax-to-GDP Ratio Shanken 108 16.56 (14.88) 4.39 0.88
Tax-to-GDP Ratio Fama-Macbeth 137 17.11 (13.88) 12.43 0.19Spending-to-Tax Ratio OLS 152 −9.98 (4.78) 32.60
Spending-to-Tax Ratio GMM 108 −26.80 (16.88) 4.32 0.93
Spending-to-Tax Ratio Shanken 108 −13.51 (9.11) 7.80 0.55Spending-to-Tax Ratio Fama-Macbeth 137 −17.48 (8.74) 14.94 0.09
Note: I report the estimates of the risk premium parameter λ from the four tests described in Table 3.The dependent variable is currency excess return. The estimates λ are scaled to express the change in thedependent variable in basis points for a unit increase in the explanatory variable. #Quarters is the numberof quarters used in each test. α Test is the test statistics against the null that all pricing errors are jointlyzero. Under the null, it follows a Chi-squared distribution, and I also report its p value. The standard errorsfrom the GMM, the Shanken test, and the Fama-Macbeth test are heteroskedasticity and autocorrelationconsistent.
cross-section of currency risk premia, and Fourel et al. (2015); Verdelhan (2018) show that it
also explains currency returns. Proposition 3 offers a fiscal explanation: Currency loadings
on the carry trade return have a factor structure because government surpluses are exposed
to the common surplus factor to different degrees. In this section, I examine the extent
to which the factor structure in currency returns corresponds to the factor structure in
government surpluses.
A. Government surplus cyclicality and currency return beta
As in Proposition 3, a currency’s carry beta βicarry is defined as the exposure of its excess
return in dollar with respect to the carry trade factor:
ri,USt+1 = αi + βicarryrcarryt+1 + εit+1. (12)
The last panel of Figure 1 in the introduction plots each country’s regression coefficient
27
bi against its carry beta. This figure confirms Proposition 3: A currency with a higher
government surplus cyclicality tends to be more exposed to the carry trade factor.
Table 6 provides a detailed analysis. Panel A reports each currency’s carry beta from
from Eq. (12). Countries with higher government surplus cyclicalities, such as Australia and
New Zealand, have positive carry betas, whereas countries with lower government surplus
cyclicalities, such as Japan and Switzerland, have negative carry betas. The R2 is higher for
countries whose carry betas are greater in absolute values. Compared with Table 1, the carry
trade factor explains a smaller fraction of variation in currency returns than the common
surplus factor does for the variation in government fiscal shocks.
By Proposition 3, since bi measures the government surplus cyclicality of country i, country
i’s carry beta has the following functional form:
βicarry = ζ0 + ζbi.
Panel B tests this relationship in two ways. The first test regards each currency’s carry beta
βicarry and government surplus cyclicality bi as known constants, and runs a linear regression
Table 6
Factor Structure in Currency Returns
Panel A: Carry Beta
Japan Switzerland Germany Denmark US UK Canada Sweden Norway New Zealand Australiaβicarry −0.65 −0.29 −0.17 −0.10 0.00 0.31 0.34 0.40 0.41 0.63 0.71
s.e. (0.15) (0.15) (0.14) (0.14) (0.00) (0.13) (0.08) (0.14) (0.14) (0.13) (0.13)
R2 12.80 2.88 1.09 0.38 - 4.25 11.31 5.70 6.45 15.02 18.14
Panel B: Carry Beta vs. Government Surplus Cyclicality
Test #Quarters ζ Std Error R2 (%)
OLS 152 0.34 (0.10) 54.95GMM 108 0.49 (0.21)
Note: Panel A reports the coefficient βicarry, its standard error, and the R2 of Eq. (12) for each country.
Panel B reports the test statistics. #Quarters is the number of quarters used in each test. The standarderrors from the GMM are heteroskedasticity and autocorrelation consistent.
28
in the cross-section of countries:
βicarry = ζ0 + ζbi + ei.
The OLS results suggest that a one-standard deviation increase in government surplus
cyclicality is associated with a 0.30 increase in slope beta, and government surplus cyclicality
explains 55% of the cross-country variation in the slope beta.
The second test corrects for the estimation errors using the generalized method of moments
(GMM). The moment conditions are
Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
− ai − bift+1 = 0,(Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
− ai − bift+1
)ft+1 = 0,
ri,USt+1 − (ζ0 + ζbi)rcarryt+1 − ci = 0,(ri,USt+1 − (ζ0 + ζbi)rcarryt+1 − ci
)rcarryt+1 = 0.
The first two moment conditions estimate the regression coefficient bi, which proxies for
the government surplus cyclicality. The last two moment conditions estimate the carry beta
βicarry, imposing the functional form (ζ0 + ζbi). Consistent with the OLS result, ζ is positive.
A higher government surplus cyclicality bi corresponds to a higher carry beta βicarry. A
country with a more cyclical fiscal condition also has riskier currency returns.
B. Currency portfolios sorted by conditional government surplus cyclicality
Now that a country’s government surplus cyclicality also reflects its currency’s risk expo-
sure, I can construct the carry trade from the fiscal data. First, I estimate the conditional
government surplus cyclicality of country i in quarter t by running the regression Eq. (10)
29
over a rolling window of T quarters:
Snom,ik+1
Bik
P ik−3
P ik+1
−Snom,ik−3
Bik−4
= ait + bitfk+1 + εi,tk+1, (13)
for k = {t− T, . . . , t− 1}.
In the earlier part of the sample, some countries’ government surpluses and debt quantities
are missing. I exclude a country/quarter observation (i, t) from panel the whenever there is
any missing variable in the entire rolling window from quarter t− T to quarter t− 1. I use
a look-back horizon of T = 4, 8, 20 or 40 quarters.
Then, I sort currencies into two quarterly-rebalanced portfolios based on their conditional
government surplus cyclicalities bit. Portfolio Low contains the currencies whose conditional
government surplus cyclicalities are below or equal to the cross-country median, and Portfolio
High contains those whose conditional government surplus cyclicalities are above the median.
The cross-country strategy invests a dollar in each currency in Portfolio High, and shorts a
dollar’s worth of each currency in Portfolio Low. The average log return of this strategy is
rxct+1def=
1
N
∑i∈Hxct
ri,USt+1 −∑i∈Lxct
ri,USt+1
,
where Hxct =
{i : bit > median({bjt}j)
}, Lxct =
{i : bit ≤ median({bjt}j)
}.
Table 7 reports the means, the Sharpe ratios, and the correlation matrix of the carry trade
return and the cross-country strategies’ returns. Regardless of the look-back horizon, the
Sharpe ratios of the cross-country strategies are slightly lower than that of the carry trade.
Surprisingly, a sample of four quarters is enough to estimate conditional government surplus
cyclicalities that predict currency returns in the cross-section.
Moreover, the cross-country strategies’ returns are positively correlated with the carry
trade return. I also report the alpha from regressing the carry trade return on the return
of each cross-country strategy. These strategies’ returns explain 24% to 52% of the average
30
Table 7
Portfolios Sorted By Conditional Government Surplus Cyclicality
Avg Return (%) SR Correlation Matrix Alpha of Carry Trade (%)
Carry Trade 0.25 0.16 1.00 0.50 0.53 0.51 0.53 0.00
(0.15) (0.10) (0.00) (0.08) (0.09) (0.10) (0.08) (0.00)Cross-Country, T=4 0.13 0.12 0.50 1.00 0.66 0.46 0.32 0.16
(0.10) (0.10) (0.08) (0.00) (0.07) (0.08) (0.11) (0.14)
Cross-Country, T=8 0.18 0.16 0.53 0.66 1.00 0.67 0.56 0.12(0.11) (0.10) (0.09) (0.07) (0.00) (0.08) (0.12) (0.14)
Cross-Country, T=20 0.16 0.13 0.51 0.46 0.67 1.00 0.77 0.14
(0.11) (0.11) (0.10) (0.08) (0.08) (0.00) (0.09) (0.14)Cross-Country, T=40 0.08 0.07 0.53 0.32 0.56 0.77 1.00 0.19
(0.11) (0.10) (0.08) (0.11) (0.12) (0.09) (0.00) (0.13)
Note: I estimate each currency’s conditional government surplus cyclicality using a rolling window regressionEq. (13), and sort currencies based on this estimate. Avg return is the quarterly average return, and SR isthe quarterly Sharpe ratio. The standard errors are obtained from 10,000 rounds of bootstrapping. In eachround, I resample the quarters with replacement.
excess return of the carry trade.
VII. Conclusion
In this paper, I show how government surplus cyclicalities explain the cross-country vari-
ation in currency risk premia and give rise to a factor structure in currency returns. These
results are consistent with the asset pricing view that an asset’s risk premium is driven by
the systematic risk exposure of its cash flows.
This framework has broader implications. In this model, I assume constant real exchange
rates in order to focus on currency risk premia. In Jiang (2018), I show that if prices are
sticky but exchange rates are flexible, government fiscal conditions drive both nominal and
real exchange rates.
In this model, investors hold government debt for its cash flows. In Jiang, Krishnamurthy
and Lustig (2018), we assume that investors also derive convenience benefits from holding
the US government debt, and show how this extension explains the dollar’s exchange rate.
31
Appendix
Appendix A: Proof
Proof of Lemma 1: Consider any country i. Combine the government budget condition Eq. (1) with Euler equation
Eq. (3),
sit + Et[
Λt+1
ΛtBitQ
it+1
]= Bit−1Q
it. (A1)
I iterate this equation forward, and obtain
Bit−1Qit = lim
T→∞
(T∑j=0
Et[
Λt+jΛt
sit+j
]+ Et
[Λt+T+1
ΛtBit+TQ
it+T+1
]). (A2)
If Assumption 1 holds, i.e.
limT→∞
Et
[Λt+T+1
Λt
(∞∑k=0
Λt+T+1+k
Λt+T+1sit+T+1+k
)]= 0, (A3)
then Eq. (4), reproduced below, is a solution to Eq. (A2):
Qit =
∞∑k=0
Et[
Λt+kΛt
sit+kBit−1
]. (A4)
Other solutions to Eq. (A2) create arbitrage opportunities: If the real value of the currency is
Qi∗t =
∞∑j=0
Et[
Λt+jΛt
sit+jBit−1
]+M i∗
t (A5)
for some positive M i∗t , then the international investor can short-sell one unit of this currency and trade Arrow-
Debreu securities to replicate the government’s budget from time t. This portfolio of Arrow-Debreu securities
requires the international investor to provide a stream of cash flows {sit+j}. This stream of cash flows costs∑∞j=0 Et[(Λt+js
it+j)/(ΛtB
it−1)] at time t. Therefore, the international investor makes a net profit of M i∗
t at time
t, which is an arbitrage opportunity. A similar argument also rules out the case of a negative M i∗t .
Proof of Lemma 2: Define
V T,itdef= Et
[ΛT s
iT
Λtsit
]
32
which implies a boundary condition V T,iT = 1 and an intertemporal relationship:
V T,it = Et[
Λt+1
Λt
sit+1
sitV T,it+1
]. (A6)
Conjecture
V T,it = exp(fT−t(ϕi)),
with the boundary condition f0(ϕi) = 0.
Then f can be solved by iterating Eq. (A6):
efT−t(ϕi) = eµ−δ+12γ2σ2−γϕiσ2+fT−t−1(ϕi),
which confirms the functional form of f . Then, the currency value can be expressed as
Qit =
∞∑τ=0
Et[
Λt+τΛt
sit+τBit−1
]=
sitBit−1
∞∑τ=0
exp(fτ (ϕi)),
where the function F is defined as
F (ϕi)def=
∞∑τ=0
exp(fτ ((νit)2, ϕi)).
Proof of Proposition 1 and 2: From the Euler equation
Et[
Λt+1
Λt
Qit+1
Qit(1 +Rf,it )
]= 1,
the nominal interest rate satisfies
1
1 +Rf,it= Et
[Λt+1
Λt
Qit+1
Qit
]=
Bit−1
Biteµ−δ+
12γ2σ2−γϕiσ2
,
which simplifies to the formula in the proposition.
33
Plug in the nominal interest rate rule,
logEt[Qit+1
Qit
]= logEt
[sit+1/s
it
Bit/Bit−1
]= −∆ logBit + µ
= γϕiσ2 +
(δ − 1
2γ2σ2
)− rf,i − ηεf,it .
The log currency excess return is
ri,jt+1def= log
(Qit+1
Qit(1 +Rf,it )
)− log
(Qjt+1
Qjt(1 +Rf,jt )
)
=
(γϕiσ2 − 1
2(ϕi)2σ2 + ϕiσεct+1 + ωεs,it+1
)−(γϕjσ2 − 1
2(ϕj)2σ2 + ϕjσεct+1 + ωεs,jt+1
).
So the expected log currency excess return is
Et[ri,jt+1] = (γϕiσ2 − γϕjσ2)−(
1
2(ϕi)2σ2 − 1
2(ϕj)2σ2
).
Proof of Proposition 3:
Plugging in the interest rate target rf,i = γϕiσ2, the log nominal interest rate is
log(1 +Rf,it ) = γσ2ϕi + ηεf,it .
Let φ denote the density function of the standard normal distribution. Then the distribution the log nominal
interest rate at time t is N (γσ2ϕ, (γσ2ρ)2 + η2). The median of this distribution is γσ2ϕ. By the Glivenko–Cantelli
theorem, the sample median converges to the population median almost surely. So, the carry trade return is
rcarryt+1 =
∫r≥γσ2ϕ
ri,jt+1 −∫r<γσ2ϕ
ri,jt+1
=
∫ϕi
∫εf,it
(2 · 1{γσ2ϕi+ηε
f,it ≥γσ2ϕ} − 1
)(γϕiσ2 − 1
2(ϕi)2σ2 + ϕiσεct+1 + ωεs,it+1
)φ
(ϕi − ϕρ
)φ(εct+1)dϕidεct+1
= C1 +
∫ϕi
∫εf,it
(2 · 1{γσ2ϕi+ηε
f,it ≥γσ2ϕ} − 1
)(ϕiσεct+1
)φ
(ϕi − ϕρ
)φ(εct+1)dϕidεct+1.
34
Then
rcarryt+1 = C1 + σεct+1
∫ϕi
(2Φ
(γσ2(ϕi − ϕ)
η
)− 1
)ϕiφ
(ϕi − ϕρ
)dϕi
= C1 + C2σεct+1,
where
C2 =
∫ϕi
(2Φ
(γσ2(ϕi − ϕ)
η
)− 1
)ϕiφ
(ϕi − ϕρ
)dϕi
=2√2π
ρ2√1 +
(η
γσ2ρ
)2> 0.
It then follows that currency i’s carry beta is
βicarry =cov(rcarryt+1 , ri,jt+1)
var(rcarryt+1 )
=ϕi − ϕj
C2.
Linear Approximation of the Change in Government Surplus-to-Debt Ratio:
The first step is to find a stationary time series. Let Snom,it denote the nominal government surplus, and let P it
denote the price level. Then, the government surplus-to-debt ratio can be written as
sit+1
Bit
def=
Snom,it+1 /Bit
P it+1
.
The numerator and the denominator of this fraction are not co-integrated: In the past 37 years, the numerator
Snom,it+1 /Bit fluctuates within a band, while the GDP deflator P it+1 has a strong trend. Figure A1 reports the time
series of their cross-country averages. The cross-country average of nominal surplus-to-debt ratios was −2.10% in
1980 and 0.00% in 2017; both values fall into the normal range of variation. In contrast, the cross-sectional average
of GDP deflators has increased from 0.51 in 1980 to 1.40 in 2017.
As a result, the government surplus-to-debt ratio sit+1/Bit has been declining. Economically, this pattern means
that the real government surplus backing each local currency unit of government debt has been decreasing, across all
countries.
On the other hand, the numerator Snom,it+1 /Bit is stationary. An augmented Dickey–Fuller test with 4 lags to
account for seasonal effects rejects the null hypothesis of a unit root at 5% level. Let si denote the average nominal
35
1980 1990 2000 2010 2020
−0.
020.
000.
02
date
Ave
rage
Nom
inal
Sur
plus
−to
−D
ebt R
atio
0.0
0.4
0.8
1.2
Ave
rage
GD
P D
efla
tor
Nominal Surplus−to−Debt RatioGDP Deflator
Fig. A1.—The nominal government surplus-to-debt ratio Snom,it /Bi
t−1 and the GDP deflator, averaged
across countries. The GDP deflator in each country is normalized so that its value in 2000Q1 is 1.
surplus-to-debt ratio in country i:
sidef= Snom,it+1 /Bit.
Assuming the average nominal surplus-to-debt ratio si is positive, I can linearize the change in government surplus-
to-debt ratio around si/P it−3:
logsit+1
Bit− log
sit−3
Bit−4
def= log
Snom,it+1 /Bit
P it+1
− logSnom,it−3 /Bit−4
P it−3
≈ 1
si/P it−3
(Snom,it+1 /Bit
P it+1
− si
P it−3
)− 1
si/P it−3
(Snom,it−3 /Bit−4
P it−3
− si
P it−3
)
=1
si
(Snom,it+1
Bit
P it−3
P it+1
−Snom,it−3
Bit−4
). (A7)
Intuitively, Eq. (A7) takes the nominal surplus-to-debt ratio at quarter t+ 1, adjusts it for the price level change
in the previous 4 quarters, and then compares it to the nominal surplus-to-debt ratio 4 quarters before. It accounts
for both the change in the nominal government surplus and the change in the price level.
Lastly, the average nominal surplus-to-debt ratio si may vary across countries, which affects the magnitude of Eq.
(A7). However, as the government surplus-to-debt ratio is highly persistent, its long-run average is very difficult to
estimate. For parsimony, I assume si is the same across all countries.
36
Appendix B: Data Source
Spot exchange rates and 3-month forward rates are closing rates at the end of each quarter, and they come from
three sources: WM/Reuters, Barclays Bank International and Thomson Reuters, all downloaded from Datastream.
For each currency and each quarter, I make sure the spot exchange rate and the 3-month forward rate come from
the same data source. Data from WM/Reuters take priority over data from Barclays Bank International, which take
priority over data from Thomson Reuters.
Following Du and Schreger (2016), I construct nominal interest rate differentials and currency returns based on
currency forward premia, which do not contain sovereign default risk. For robustness, I repeat my empirical analysis
in the Internet Appendix, using currency returns based on treasury yields.
The nominal government surplus, the nominal quantity of government debt, and the nominal GDP are downloaded
from Oxford Economics via Datastream. These nominal quantities are denominated in the unit of the local currency.
The GDP deflator is also downloaded from Oxford Economics via Datastream. Each country’s GDP deflator is
normalized so that its value in 2000Q1 is 1. Oxford Economics seasonally adjusts some, but not all, of these variables.
*
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