Foreign Exchange Intervention with UIP
and CIP Deviations: The Case of Small Safe
Haven Economies1
Philippe Bacchetta
University of Lausanne
Swiss Finance InstituteKenza Benhima
University of Lausanne
CEPR

CEPR
Brendan Berthold
University of LausanneAugust 31, 2023

1 We would like to thank Julien Bengui, Paolo Cavallino, Egemen Eren, Jiang Fang, Hanno

Lustig, Eric van Wincoop and participants at the CEPR-IMF and BdI-BdF-BoE International
Macroeconomics meetings in Rome, the 12th workshop on Exchange Rates at BIS, the Salento
Macro Meetings, and the EFA Congress in Amsterdam for comments. Philippe Bacchetta
gratefully acknowledges financial support from the Swiss National Science Foundation.

Abstract
We examine the welfare-based opportunity cost of foreign exchange (FX) intervention
when both CIP and UIP deviations are present. We consider a small open economy thatreceives international capital flows through constrained international financial intermediaries. Deviations from CIP come from limited arbitrage or through a convenience yield,
while UIP deviations are also affected by risk. We show that the sign of CIP and UIP
deviations may differ for safe haven countries. We find that there may be a benefit, rather
than a cost, of FX reserves if international intermediaries value the safe haven propertiesof a currency more than domestic households. We show that this has been the case for
the Swiss franc and the Japanese Yen. We examine the optimal policy of a constrained
central bank planner in this context.

1

Introduction

A vast literature examines the optimal level of central bank international reserves (seeBianchi and Lorenzoni (2022) for a recent survey). A recurrent feature is that the accumulation of reserves bears an opportunity cost arising from a return differential between
the liabilities and the assets of the central bank. In the recent literature on optimal Foreign Exchange (FX) interventions, some authors focus on Uncovered Interest Rate Parity(UIP) wedges (see Basu et al., 2020; Maggiori, 2021; Itskhoki and Mukhin, 2022). In
contrast, other researchers argue that what matters are deviations from Covered Interest
Rate Parity (CIP) (e.g., Amador et al., 2020; Fanelli and Straub, 2021). This distinction
between CIP and UIP appears particularly relevant for safe haven countries, since CIP
and UIP deviations may be of different signs. Figure 1 shows CIP and UIP deviationsfor Switzerland and Japan.1 They are computed from the perspective of international
investors and UIP deviations are estimated using survey expectation data. They showthat since 2008, both countries have experienced positive CIP deviations and negative
UIP deviations. The latter implies a negative excess return, which is typical of safe havencurrencies.
Figure 1: UIP and CIP DeviationsA) Deviations CHF/USD

B) Deviations JPY/USD
1.00%

10.0%

1.00%

5.0%

0.50%

5.0%

0.50%

0.0%

0.00%

0.0%

0.00%

−5.0%

−0.50%−5.0%−0.50%−10.0%−1.00%−10.0%−1.00%

10.0%

Deviations
CIP (rhs)

UIP (lhs)

2004 2006 2008 2010 2012 2014 2016 2018 20202004 2006 2008 2010 2012 2014 2016 2018 2020

Notes: This figure shows the UIP and CIP deviations in percentage points as defined in (5) and (4),taking the USD as the foreign currency and considering a 3-month horizon. Panel A) and B) consider the
CHF and the JPY as the domestic currencies. UIP deviations are computed using monthly data from
Datastream for the 3-month Libor rates and from Consensus Economics for the exchange rate forecasts
and the spot exchange rates. The CIP deviations are monthly averages of daily observations and arecomputed using 3-month Libor rates, spot exchange rates and forward rates with a 3-month maturity
from Datastream. All returns are annualised.

To clarify these issues, we develop a model where both CIP and UIP deviations arepresent. We consider a small open economy that receives international capital flows
through international financial intermediaries as in Gabaix and Maggiori (2015). The
1

See Appendix A for data description. Interestingly, Rime et al. (2022) show that CIP deviations for
the CHF and the JPY with respect to the USD have been the most profitable for financial institutions.

structure of the model is similar to that in recent papers examining the role of international
reserves (see Cavallino, 2019; Amador et al., 2020; Fanelli and Straub, 2021; Basu et al.,2020; Maggiori, 2021), but financial intermediaries are risk averse. Fang and Liu (2021)
propose a related framework in a two-country model, but they focus on the US dollar
and do not analyze FX interventions. The international financial intermediaries are themarginal investors and determine both UIP and CIP deviations through their unhedged
and hedged portfolio choices. These deviations typically do not coincide and may evenbe of different sign.
In this environment, we examine the opportunity costs of FX intervention in termsof welfare. We identify the conditions under which CIP or UIP deviations matter forthis cost. We find that there may be no opportunity cost, and that there may even be abenefit, of FX intervention in a safe haven country, even if that country faces a positive
CIP deviation. We examine the implications for optimal FX intervention in these cases.
The presence of systematic deviations from CIP in the wake of the Global Financial
Crisis is a major development in international finance (see Du and Schreger (2022) or
Cerutti et al. (2021) for recent surveys). The theoretical literature has provided explanations for CIP deviations, but has devoted limited attention to the link between CIP
and UIP deviation. Several papers analyzing interest rate differentials assume complete
markets so that either there are no UIP deviations or CIP deviations are equal to UIP
deviations. This is not consistent with the data.
The recent literature has followed two main approaches to explain interest rate differentials. First, there may be financial frictions that limit arbitrage, for example, byassuming constrained financial intermediaries. The other approach is to assume differences in convenience yields. The two approaches are present in our model and determine
deviations from CIP. However, we do not assume complete markets, so that UIP deviations
differ from CIP deviations. A basic result from this analysis is the following relationshipbetween UIP and CIP deviations:devU IPt = devCIPt −∗covt (m∗t+1 , Xt+1)∗
Et mt+1(1)∗
where m∗t+1 is the stochastic discount factor (SDF) of financial intermediaries and Xt+1
is the foreign currency excess return from the international intermediary perspective. If∗
the small open economy is a safe haven country, we have covt (m∗t+1 , Xt+1
) > 0, that is,
the safe haven currency yields a higher return in bad times. Therefore, it is possible tohave a positive CIP deviation with a negative UIP deviation.
We derive equation (1) in a simple two-period small economy model with two assumptions that differ from most of the literature. First, international financial intermediaries
face exchange rate risk. This risk could be hedged on the forward market, but it is not
optimal to fully hedge a safe haven currency. The other assumption is that the financial
constraint applies to the whole foreign exchange investment of financial intermediaries,2

whether it is hedged or not.2
We analyze the welfare cost of accumulating international reserves in this framework.
We show that if domestic households attribute less value to the safe haven properties of
their currency than international financial intermediaries (i.e., the domestic SDF is lesscorrelated to the excess return than for financial intermediaries), then FX reserves may
have a benefit, not a cost. We examine this issue empirically by estimating the SDF
of financial intermediaries following He et al. (2017). When considering the CHF and
JPY with respect to the USD, we find that the SDF of financial intermediaries is morecorrelated with excess returns than the SDF of domestic households.
We examine the implications of this analysis for optimal FX intervention, by modelingthe central bank as a constrained planner. We determine the various factors influencing
optimal policy decisions, focusing on various types of FX interventions. We show thatthe central bank incentives are similar for sterilized interventions and unsterilized interventions at the Zero Lower Bound (ZLB).
For a more specific analysis, we consider a linearized version of the model where the
distribution of shocks is such that the domestic currency is perceived as a safe haven byinternational investors. This allows us to derive precise expressions for the cost of FX
intervention and CIP and UIP deviations and examine the impact of various parameterson these variables and on optimal FX interventions. For example, an increase in global
risk leads to more beneficial FX interventions, larger positive CIP deviations and larger
negative UIP deviations.
This paper complements the literature on the opportunity cost of FX reserves. Thereis a long tradition of estimating the cost and benefits of accumulating FX reserves (e.g.,
Jeanne and Rancière, 2011). Adler and Mano (2021) estimate the quasi-fiscal cost of interventions for 73 countries using UIP deviations. Using survey expectations or assuming
a random walk for the nominal exchange rate, they find that the ex ante cost of intervention is negative for Japan and Switzerland in the period 2002-2013, while it is positive formost other countries.3 In this paper, we examine the cost of intervention from the welfare
point of view, and find that it is also negative for Japan and Switzerland, but that it is
not equal to UIP deviations in general.
By focusing on countries like Switzerland or Japan, this paper provides a different
perspective on safe haven economies. A growing literature has been analyzing the specialrole of the US dollar as a reserve currency. In particular, several papers have focusedon the role of convenience yields in generating currency movements and expected excess
returns (e.g., Jiang et al., 2021b,a; Valchev, 2020; Kekre and Lenel, 2021; Bianchi et al.,2

In contrast, in Itskhoki and Mukhin (2021), intermediation frictions generate UIP deviation without
CIP deviations. This is because the intermediation frictions originate in the intermediaries risk aversion.
3
In the case of developing or emerging economies, the opportunity cost may be based on the country’s
borrowing cost, which implies a credit risk (e.g., Edwards, 1985). However, Yeyati and Gómez (2022)
argue that when reserves are used for leaning-against-the-wind interventions, it is more appropriate touse UIP deviations.

3

2022). We show that convenience yields are not the sole determinant for exchange rate
movements and UIP deviations in safe haven economies.
The rest of the paper is organized as follows. Section 2 describes the model and thedecentralized equilibrium. Section 3 analyzes the opportunity cost of reserves in thiscontext. Section 4 discusses optimal FX intervention and Section 5 proposes a linearized
model of a safe haven country. Section 6 concludes.

2

The ModelThis section presents a two-period model of a small open economy facing international
financial intermediaries in the spirit of Gabaix and Maggiori (2015). These intermediaries
buy domestic bonds and are the marginal investors both in the spot and the forward market.4 They are risk averse so that there is a difference between their covered and uncovered
positions. After presenting the financial intermediaries, we describe the households, thegovernment and the central bank, as well as the equilibrium in the asset markets.
We call the foreign currency the dollar and assume that the foreign interest rate i∗t
is given. Purchasing power parity (PPP) is assumed to hold and the price of goods indollars is normalized to one. St is the spot price of dollars in terms of domestic currency
and Ft is the forward rate.

2.1

International Financial Intermediaries

International financial intermediaries value their expected profits with their stochastic
discount factor m∗t+1 , which will be further described below. They typically invest in
domestic bonds, but at the ZLB they may also hold domestic money.5 Denote bH∗
and
t
H∗
H∗
ht their net positions in domestic bonds and money, expressed in dollars, and at theirH∗
total position: aH∗= bH∗
tt +ht . Financial intermediaries have a zero net position and fundtheir investments in domestic assets in dollars. We also assume that they can use forward
contracts in quantities ft∗ and that they are the only players in the forward market.6
Moreover, financial intermediaries may value the liquidity of dollar assets. We assume
that investors have operating costs that are increasing in non-dollar assets holdings aH∗t
H∗
and that it is a linear function: χ · at , with χ ≥ 0. Their objective function is in dollars(and equivalently, in goods terms since the dollar price is constant):
4

Since the objective of the model is to highlight the consequences of the differences between CIP and
UIP deviations, we abstract from various interesting factors affecting the dynamics of spot and forward
rates that are considered in the recent literature. E.g., see Bacchetta et al. (2023) for a model and reviewof the literature.
5
For notational convenience, we assume that financial intermediaries only potentially hold money atH∗time t so that hH∗
t−1 = ht+1 = 0.
6These assumptions are similar to Fang and Liu (2021). They consider a two-country model withfinancial intermediaries in both countries, but only the Home country arbitrages CIP deviations.

4Vt∗ = Et







St
1
1
∗
H∗
∗
∗
mt+1 at(1 + it )− (1 + it ) − ft−
− χaH∗
t
St+1
St+1 Ft

represents the total funds invested in the country, covered or uncovered. ft∗ /(1 + it )StaH∗t− ft∗ /(1 + it )St is the uncovered amount. Here we takeis the covered amount, and aH∗t
as money is dominated by bonds.
= bH∗
into account the fact that when it > 0, aH∗tt
To capture the role of financial intermediaries, we assume, as Gabaix and Maggioriof the total invested funds, after the(2015), that intermediaries can divert a fraction ΓaH∗t
investment decisions are taken, but before shocks are realized. This yields a participation
constraint for investors:
2
Vt∗ ≥ Γ(aH∗t )Consider first the FOC with respsect to ft∗ :



1
1
∗
Et mt+1=0−
St+1 Ft(2)

The forward market is effectively frictionless, since it does not involve a transfer of funds
ex ante. This implies a relationship between CIP and UIP deviations:∗∗Et (m∗t+1 Zt+1
) = Et (m∗t+1 Xt+1
)

(3)

∗
is the excess return hedged by a forward contract or the CIP deviation:
where Zt+1∗
Zt+1
≡ (1 + it )St
− (1 + i∗t )
Ft

(4)

∗
is the domestic currency excess return, expressed in foreign currency:
and Xt+1∗
Xt+1≡ (1 + it )St
− (1 + i∗t )
St+1

(5)

Equation (3) can be rewritten as an equivalent of Equation (1):∗∗Et Xt+1= Zt+1−∗
covt (m∗t+1 , Xt+1)∗
Et mt+1(6)

Covered and uncovered carry trades yield the same returns in expectation, up to a covariance term, because intermediaries are risk-averse.
Using Equation (2), the participation constraint can be simplified as follows:∗
H∗2
Et m∗t+1 aH∗≥ Γ(aH∗t Xt+1 − χat
t )

(7)

If the participation constraint is binding, we have:∗Et m∗t+1 Xt+1
= ΓaH∗
t +χ

5

(8)

This, along with Equations (3) and (6), implies
∗
=
Zt+1

and
∗
Et Xt+1
=

ΓaH∗
t +χ
Et m∗t+1(9)∗
covt (m∗t+1 , Xt+1
)
ΓaH∗
t +χ
−
∗
∗
Et mt+1
Et mt+1(10)+ χ in Equations (9) and (10) shows the impact of limited arbitrage
The term ΓaH∗t
and of the convenience yield on CIP and UIP deviations. The intermediation frictions,which bear on both covered and uncovered intermediated funds, affect both CIP and UIP
deviations.7

2.2

Domestic Households

Households real consumption is ct and they receive a real endowment yt . They can holdmoney, HtH , domestic bonds BtH (both expressed in domestic currency), and foreign bonds
bFt (expressed in foreign currency). Domestic bonds and money are perfect substitutes at
the ZLB. We assume that households do not use the forward exchange market.
Since PPP holds, their budget constraint can be written as:
HF
ct = yt − hHt − bt − bt + tt
1
1 + it H
H
ct+1 = yt+1 +hH
b + (1 + i∗t )bFt + tt+1
t − ht+1 +1 + πt1 + πt t

(11)
(12)

H
H
H
where bHt = Bt /Pt and ht = Ht /Pt are the real levels of domestic bonds and money
holdings and tt = Tt /Pt and tt+1 = Tt+1 /Pt+1 are real transfers.
Potentially, households face a cash-in-advance constraint in t and t + 1:HhHt ≥ yt , ht+1 ≥ yt+1

(13)

They also face short-selling constraints :bHt ≥ 0,

bFt ≥ 0

(14)

Their utility function is:
U (ct ) + βEt U (ct+1 )(15)

Domestic households choose bonds and money holdings to maximize (15) subject toconstraints (11) to (14). Using the assumption of PPP (Pt = St ), the first-order conditions
associated with bond portfolio choices are:U ′ (ct ) − Et (βU ′ (ct+1 )(1 + i∗t ))



St
′
∗
Et βU (ct+1 ) (1 + it ) − (1 + it )St+1
7

−λF = 0(16)

+λF − λH = 0(17)

Fanelli and Straub (2021) discuss a similar setup with frictions in intermediation and frictionlessforward markets in an extension of their model. They find that, in this case, intermediation frictions
generate both UIP and CIP deviations.

6where λH and λF are the multipliers associated with the short-selling constraints (14).
Equation (16) shows that the borrowing constraints affect intertemporal allocations.
Equation (17) shows that they prevent households from reaching their optimal portfolio allocation between domestic and foreign currency bonds.

2.3

The Government

At time t the nominal government issues debt BtG (expressed in domestic currency) andtransfers the funds to households:(18)
BtG = Tt
At t + 1, the government receives the central bank profits, ΠCB
t+1 and repays its debt :Tt+1 = −(1 + it )BtG + ΠCB
t+1

(19)

G
We assume that the government is passive and that the level of real debt bG
t = Bt /Pt is
exogenous.8

2.4

The Central Bank

In period t, the central bank issues money Ht , and buys domestic and foreign bonds BtCB
and bCBF(expressed respectively in domestic and foreign currency). In period t + 1,t
the central bank issues new money Ht+1 − Ht and distributes its profits ΠCB
t+1 to thegovernment. The central bank’s budget constraint writes then as follows:
St bCBF+ BtCB = Ht
t

(20)

∗
CBF
ΠCB
+ (1 + it )BtCB + Ht+1 − Ht
t+1 = (1 + it )St+1 bt(21)

In period t, the central bank has the nominal interest rate it , the total money supply
and domestic bonds BtCB as instruments.
Ht and the choice of foreign reserves bCBFt
However, the interest rate cannot be negative, so the central bank loses the interest rate
instrument when it hits this zero lower bound (ZLB).
In period t + 1, we assume that the supply of money Ht+1 is exogenous: Ht+1 = H̄ehwhere h is an exogenous shock. It represents variations in the net money supply tohouseholds due for instance to liquidity trading, or money velocity shocks.
From the budget constraint (20), there are two ways to change the level of reserves
CBF
bt . First, through a sterilized intervention where an increase in bCBF
is compensatedt
CBby a decline in Bt . Second, through an unsterilized intervention where an increase inbCBF
is associated with an expansion in Ht . Another possibility would be to allow thetcentral bank to transfer funds to households in both periods. In this case, an increase inbCBF
could be implemented by introducing transfers from the central bank in period t.
t
We do not examine this fiscal foreign exchange intervention.
8

Alternatively, we could assume that it is the nominal debt level BtG that is exogenous. However,there would be an incentive for the central bank to alter the real debt level by moving the exchange rate.

7

2.5

Gross and Net Foreign Liabilities

For the rest of our analysis, it is convenient to focus on the Home country’s net andgross foreign liabilities. Moreover, since PPP holds, we replace Pt with St . Gross foreign
liabilities are domestic bonds and money not held domestically. They are given by
 


Ht
BtCB
H
CBF
H
H
G
(22)
− bH
− bt +− ht = bG
gf lt = bt −t − htt + bt
St
St
The first term of the first equality corresponds to the foreign holdings of domestic bonds.
The second term corresponds to the foreign holdings of domestic money. The second
equality is obtained by using the central bank’s budget constraint. It shows that FXinterventions directly affect the supply of domestic bonds to foreigners. The central bankcan increase its holding of foreign currency assets only through a balance-sheet expansion
and hence through an increase in its domestic currency liabilities, whether it is bonds ormoney.
Net foreign liabilities are given byHFH
nf lt = gf lt − (bFt + bCBF
) = bGtt − bt − bt − ht

(23)where bFt +bCBF
are the domestic holding of foreign assets. The second equality is obtainedt
CBF
by replacing bt
with (Ht − BtCB )/St , using the central bank budget constraint, andreplacing gf lt using (22).
It is useful to notice that FX intervention affects gf lt , but not nf lt : an increase in
CBF
bt
will increase gf l one-for-one, through an increase in Ht (unsterilized intervention)
or a decline in B CB (sterilized intervention), while in nf l the changes in bCBF
are offsettCBeither by changes in Bt or by changes in H.

2.6

Equilibrium in Asset Markets

The amount of domestic debt held by international intermediaries is equal to the net
CBHdomestic supply: bH∗= bGtt − Bt /St − bt . Similarly, foreign money holdings are equal to
the net domestic supply: hH∗= Ht /St − hHtt . In equilibrium, gross foreign liabilities areequal to the bonds and money held by foreigners: gf lt = aH∗
= bH∗
+ hH∗
t
t
t . From (10),this implies∗
covt (m∗t , Xt+1)∗
Γgf lt + χ = Et (Xt+1
)+
(24)
∗
Et mt+1
The net supply of domestic liabilities to foreigners, gf lt , determines the equilibrium ex∗pected domestic currency excess return Et (Xt+1), which is defined in (5). A higher gf lt
can only be absorbed by the intermediaries if it offers a higher excess return. On the∗contrary, an increase in covt (m∗t , Xt+1
) leads to a decline in the domestic currency excess
return. Intuitively, the increase in covariance makes the domestic bonds more attractive
to foreigners and generates an excess demand for domestic bonds. The decline in thedomestic currency excess return clears this excess demand.
8

How does the domestic currency excess return adjust in practice? Consider for instancean increase in the excess return. Since the foreign interest rate i∗t is exogenous, this implies
a higher domestic real interest rate (1+it )Et (St /St+1 ). At t+1, equilibrium on the moneyHmarket yields Ht+1
= St+1 yt+1 = Heh , which determines St+1 . Therefore, we can treat
St+1 as exogenous. As a result, a higher gf lt increases (1 + it )St . Whether it affects it or
St depends on whether the economy is at the ZLB or not, as discussed later.

3

On the Cost of Foreign Reserves

In this section, we derive the utility cost of FX reserves and assess its relation to UIP
and CIP deviations. We show that the key determinant of this cost is the covariance
between excess returns and the SDF of domestic households on the one hand and the
SDF of international financial intermediaries on the other. In the context of safe havencurrencies, it depends on whether international financial intermediaries value the safe
haven properties more than domestic investors. We examine this issue empirically andshow that this is the case for Switzerland and Japan.

3.1

Utility Cost of Reserves with UIP and CIP Deviations

After consolidating the household’s budget constraints using the equilibrium in the domestic asset markets, and substituting transfers in the household’s budget constraint, we
obtain the period resource constraints:
ct = yt + nf lt∗
t
gf lt + it SSt+1
ct+1 = yt+1 − (1 + i∗t )nf lt − Xt+1



Ht
− hH
t
St



(25)

The last term, which represents the economy’s seigniorage revenue from the foreign holdingt= hH
of domestic money, can be neglected since we will either have Ht (if it > 0) or it = 0.
St
The intertemporal resource constraint is:∗(1 + i∗t )ct + ct+1 = (1 + i∗t )yt + yt+1 − Xt+1
gf lt .

(26)

Everything else equal, FX interventions affect the economy’s intertemporal resources
through changes in gf lt .9 By holding more foreign reserve bCBF
, the central bank increasestthe economy’s gross foreign position by issuing more domestic bonds (decreasing BtCB ) ort
more money if the economy is at the ZLB (increasing H
− hH
t ).
St
The last term in (26) gives the monetary cost of holding reserves, since reserves affect∗gf lt . It depends on the sign of Xt+1
, which is the marginal cost of holding reserves,∗
evaluated in units of goods. In a safe haven case with Et Xt+1
< 0, there is an expected9

If the central bank could make transfers in t, then it could also affect the consumption profile throughthe economy’s net position nf lt . If the household is constrained, then the central bank could increase
the net borrowing of the economy and hence transfer consumption from t + 1 to t.

9

gain of holding reserves, i.e., the central bank can exploit the UIP deviation. However,
the increase in reserves also increases exchange rate risk, so the question is whether this
could increase households utility. We define the marginal utility cost of reserves as follows:
Definition 1 (The marginal utility cost of FX interventions) The marginal util∗
and theity cost of FX interventions is the expected product of the UIP deviation Xt+1
SDF of domestic households mt+1 , divided by the expected discount factor:
U CF Xt =∗Et (mt+1 Xt+1)
Et (mt+1 )(27)

where mt+1 = βU ′ (ct )/U ′ (ct+1 ).
∗The excess return on domestic bonds Xt+1
is valued using the utility-based stochastic
discount factor. It is normalized by the expected discount factor so that it coincides with∗the monetary cost Xt+1
in the absence of risk. In that sense, U CF Xt can be seen as thecertainty-equivalent cost of reserves in monetary terms.
The marginal utility cost of FX interventions can be rewritten as∗cov(mt+1 , Xt+1)∗
U CF Xt = Et Xt+1 +Et mt+1(28)

The utility cost is composed of the excess return on foreign bonds, minus the risk premium∗
associated with this excess return. Since Et Xt+1
< 0 for safe haven countries, there may
be a utility gain.
∗
Substituting Et Xt+1
using Equation (10), we can rewrite the utility cost of foreignexchange interventions:devCIPz }| {∗∗) cov(mt+1 , Xt+1)
Γgf lt + χ cov(m∗t+1 , Xt+1
U CF Xt =−+∗∗Et mt+1
Et mt+1
Et mt+1
|
{z
}

(29)

devU IP

Equation (29) shows how CIP and UIP deviations affect the utility cost. This can besummarized in the following proposition:
Proposition 1 Consider the SDF of domestic households, mt+1 , and of international∗
financial intermediaries m∗t+1 and the excess return in foreign currency, Xt+1
. The utility
cost (or benefit) of foreign exchange intervention depends on∗∗(i) CIP deviations when cov(mt+1 , Xt+1
) = cov(m∗t+1 , Xt+1).
∗
(ii) UIP deviations when cov(mt+1 , Xt+1
) = 0.

In fact, the intermediation friction generates two wedges that are relevant for welfare:
the CIP deviation, which is a riskless excess return; and the difference between the foreignand domestic risk premia. If the foreign and domestic agents have the same risk premium,10

only CIP deviations matter. This is the case in the absence of risk, as in Amador et al.
(2020), or when financial intermediaries have the same discount factor as households.
In contrast, in the limit case where the domestic agents have negligible risk aversion ascompared to financial intermediaries, then the sum of the two wedges is equal to the UIP
deviation and the cost of reserves would be equal to UIP deviations.10
However, in general, the sum of the two wedges does not coincide with either the
CIP or the UIP deviations. In particular, a safe haven currency may be more desirable
for foreign investors as a diversification hedge than for the domestic investors so that∗∗cov(m∗t+1 , Xt+1
) > cov(mt+1 , Xt+1). If the difference is large enough, there may be autility gain from accumulating reserves, instead of a cost.

3.2

Estimating the Utility Cost for Switzerland and JapanThe theoretical analysis has shown that the utility cost FX of interventions depends∗∗
crucially on the difference between cov(m∗t+1 , Xt+1)/Et m∗t+1 and cov(mt+1 , Xt+1)/Emt+1(Equation (29)).11 In this subsection, we provide estimates of these two terms for Switzerland and Japan. First, Appendix 2 confirms that both countries can be considered as safe
havens, in the sense that the excess return on their currencies is positively related to
global risk variables.
A key issue is the measurement of stochastic discount factors mt+1 and m∗t+1 . For
domestic households, we simply assume that mt+1 = β(ct+1 /ct )−γ , where 1/γ is the rateof intertemporal substitution. For international financial intermediaries, we follow theliterature on intermediary asset pricing (e.g., see He and Krishnamurthy (2011) or Brunnermeier and Sannikov (2014)), and assume that their SDF is proportional to their net
worth N Wt :12−γ
N Wt+1∗(30)
mt+1 = βN Wt
As in He et al. (2017), we assume that the financial intermediaries net worth is equal tothe aggregate wealth in the economy (denoted by Wt ) multiplied by the intermediaries
capital ratio (denoted by ηt ). This specification implies that the financial intermediaries
10

This is what Itskhoki and Mukhin (2021) implicitly assume. In their linear approximation, theytake the level of risk to zero but ensure that the risk premium of the financial intermediaries remains afirst order object by rescaling their risk aversion, but not that of the households. This implies that theintermediaries risk aversion is an order of magnitude higher than that of the households. As a result, it
is optimal to eliminate UIP deviations.
11
Although we need to compare these covariances numerically, they play a different role in the analy∗∗
sis: cov(m∗t+1 , Xt+1)/Et m∗t+1 determines asset pricing, while cov(mt+1 , Xt+1)/Emt+1 is used for welfare
evaluation.
12Here, we implicitly assume that, on top of the Gabaix-Maggiori constraint on their international
arbitrage, financial intermediaries face a borrowing constraint on their overall balance sheet, such that
their constraint depends on their net worth, e.g. as in Gertler and Kiyotaki (2010). This gives rise tointermediary asset pricing in the form of Equation (30).

11marginal utility of wealth rises when either the aggregate wealth in the economy or theequity capital ratio is low. The first term captures the asset pricing effect of weakerfundamentals, while the second captures the idea that the intermediaries risk bearing
capacity is impaired when the capital ratio is low. As a result, risk aversion increases
the marginal value of wealth. Using time-series and cross-sectional asset pricing tests,
He et al. (2017) show that this specification captures well the marginal utility of wealthof financial intermediaries, and find supporting evidence that financial intermediaries areindeed marginal investors for a wide class of assets.
In our empirical exercise, we consider two measures of the capital ratio (ηt ) and two
measures of aggregate wealth (Wt ), giving rise to four different possible specifications. For
the first capital ratio measure, we consider the equity capital ratio of financial intermediaries (Primary Dealer counterparties of the New York Federal Reserve) from He et al.
HKM. The second measure is from Adrian et al. (2014), and(2017), which we denote by ηt+1
is defined as the (inverse of) book leverage of security Brokers & Dealers.13 We denote
AEM. For total wealth, we consider a real measure using US GDP (WtGDP ) and ait as ηt+1
financial measure using the US MSCI Equity Index (WtM SCI ).14
As in He et al. (2017), our measure of net worth is obtained by interacting the capitalratio measure with the total wealth measure: N Wt = ηt ×Wt . To convert net worth into a
growth rate (as suggested by (30)), we adopt an approach similar to He et al. (2017). For
the capital ratio, we define the intermediary capital risk factor by dividing the residualfrom a regression of the capital ratio on its lag by the lagged capital ratio. For the financial
measure of wealth (WtM SCI ), we compute the excess returns on the equity index, using the
3-month US Libor as the risk-free rate. For the real measure of wealth (WtGDP ), we simply
compute the growth rate. NNWWt+1
is then defined by the interaction of the intermediarytcapital risk factor and the growth rate measure of total wealth. Appendix B provides
additional details about the sources of the data, as well as the construction of the excess
returns and the stochastic discount factors.
We consider excess returns using the CHF and the JPY as the domestic currency andthe USD as the foreign currency. Let us define the log excess returns of going long in thedomestic currency from the international investors perspective:x∗t+1 = it − i∗t + st − st+1

(31)∗
We use x∗t+1 as an approximation of Xt+1.
Table 1 displays an estimate of cov(m∗t+1 , x∗t+1 )/Et m∗t+1 and cov(mt+1 , x∗t+1 )/Emt+1
using either the CHF or the JPY as the domestic currency, keeping the USD as theforeign currency. We assume that β = 0.99 and γ = 10. For each currency, we consider two
subsamples (2000M1-2009M12 and 2010M1-2020M2) to highlight potential time-variation
13

It is obtained using balance sheet data reported in the Flow of Funds from the Federal Reserve Board.
It is computed as the ratio of total equity (total financial assets minus total financial liabilities) to total
financial assets.
14
As a robustness check, we also consider a ”world version” of these two variables.

12

Table 1:

Cov(x∗t+1 ,m∗t+1 )Cov(x∗t+1 ,mt+1 )and∗Et (mt+1 )Et (mt+1 )A) CHF domestic currency, USD foreign currency
Fin. Intermediaries

HH

N Wt+1 =

HKM
M SCI
ηt+1
× Wt+1AEM
M SCI
× Wt+1
ηt+1

HKM
GDP
ηt+1
× Wt+1

AEM
GDP
ηt+1
× Wt+1

CH
Ct+1

1999-20102010-2020

1.61
2.82∗∗

1.74
1.32

0.2
5.1∗

-1.17
2.13∗∗

0.25∗∗∗
0.01

B) JPY domestic currency, USD foreign currency
N Wt+1 =

HKM
M SCI
ηt+1
× Wt+1

AEM
M SCI
ηt+1
× Wt+1

HKM
GDP
ηt+1
× Wt+1

AEM
GDP
ηt+1
× Wt+1JP
Ct+1

1999-20102010-2020

1.85
6.39∗∗∗-2.9
3.31∗∗-3.57
7.93∗∗∗-2.56∗∗
2.63∗∗

0.7∗∗∗
0.33

Note:
∗
Cov(x∗
Cov(x∗
t+1 ,mt+1 )
t+1 ,mt+1 )This table estimatesandfrom equation (29) using different proxies ofEt (m∗)
Et (mt+1 )t+1
the SDF of (international) financial intermediaries and Swiss and Japanese households. Values are
expressed in percentage points. Appendix B provides details on their construction and the source ofthe data. Statistical significance is assessed by regressing excess returns on the different measures of
the SDF using Newey-West standard errors. ∗∗∗ : p < 0.01, ∗∗ : p < 0.05, ∗ : p < 0.1.

in these measures, as suggested by Figure 1. Columns 2 to 5 display the covariance terms
from the perspective of financial intermediaries using the capital ratio measure from He
et al. (2017) and Adrian et al. (2014) and the two measures of total wealth to compute theSDF. The last column displays the covariance term for the Swiss and Japanese households,
using real consumption growth to compute the SDF. Statistical significance is assessedby regressing the excess returns on the different measures of SDF and using Newey-Weststandard errors.
The results show that, since 2010, the covariance term for financial intermediaries isclearly positive and statistically significant for most of the specifications of the stochastic
discount factor, and quantitatively in line with the UIP deviations depicted in Figure 1,
reaching as high as 7.9% for Japan and 5.1% for Switzerland. Interestingly, the covarianceterm is generally an order of magnitude smaller (or negative) before 2010. Since 2010,
being long in CHF or JPY tends to provide higher returns when the marginal utility of
the wealth of financial intermediaries is high, which indicates that the CHF and the JPY
behave as a hedge for international intermediaries. On the other hand, the covarianceterm between excess returns and SDF based on real domestic consumption growth tends
to be much smaller and statistically not significant since 2010. These observations canhelp rationalise the large UIP deviations (and the low expected excess returns) observed
post 2010 in the data. For Switzerland and Japan, Proposition 1 implies that it is not CIP
but UIP deviations that should matter for FX interventions since cov(mt+1 , x∗t+1 )/Et mt+1
is not significantly different from zero.

13

4

The Central Bank as a Constrained Planner

To determine how the cost of reserves influences the policy trade-offs of the central bank,
we consider a central bank which maximizes households welfare. The households domestic
participation constraint provides an incentive for the central bank to distort the domestic
real interest rate. The cost of reserves may either conflict with this domestic objective,
or facilitate it. We first reframe the central bank problem as that of a constrained central
planner. We then show how the resulting optimal allocation can be decentralized using
foreign exchange interventions.
Before that, we relate the country’s consolidated financial liabilities to the household
participation constraints (14). Using the definition of gf lt in (22), we can show that thehouseholds constraint on domestic bond issuance translates into a constraint on grossforeign liabilities:
CBF
(32)
− hH
gf lt ≤ bGtt + btHowever, (32) is not an effective constraint since the central bank can change its holding.
of foreign bonds bCBFt
Similarly, the foreign currency no-borrowing constraint implies:
nf lt ≤ gf lt − bCBFt

(33)

This constraint cannot be relaxed by non-fiscal FX intervention since changes in gf lt areoffset by changes in bCBF
.15 This constraint is effective except if we allow the centraltbank to perform fiscal interventions, where changes in gf lt need not be offset by changes. Equations (32) and (33) are equivalent to the no-borrowing constraints (14).
in bCBF
t

4.1

The Constrained Planner Program

Based on the previous equations, we can examine the planner’s optimal choices.
Definition 2 (Constrained planner equilibrium) A constrained planner equilibrium
is an equilibrium where a planner maximizes objective (15) subject to the economy’s resource constraints (25); the asset pricing equation (8); the cash-in-advance constraintshhHt ≥ yt and H̄e = St+1 yt+1 ; the non-negativity of foreign domestic money holdings
H∗hH∗≥ 0; the equilibrium on the market for money Ht = St (hHtt + ht ); the consolidated bond and money market equilibrium aH∗= gf lt ; the zero lower bound it ≥ 0;t
and the foreign liability constraints (32) and (33). The planner’s choice variables are∗ ∗(it , St , St+1 , gf lt , nf lt , bCBF
, Ht , hHtt , ht , at ).
15

When capital controls are in place, however, Bacchetta et al. (2013) show that sterilized interventions
can affect the country’s intertemporal allocation.

14

The central bank’s program is:
max E U (ct ) + βU (ct+1 )+ηt (yth− ct + nf lt )

h
i
i

Ht
H
t
t
+ηt+1 yt+1 − ct+1 − (1 + i∗t )nf lt + (1 + i∗t ) − (1 + it ) SSt+1
gf lt + it SSt+1
−
h
t
St
+ξit

H
+∆H
t ht − yt 
+∆Ft
+Λ
+Λ̃

Ht
− hH
t
St

CBF
gf lt − bt
− nf lt

CBF
bG
− hH
t + btt − gf lt

+α0 Et



m∗t+1

h

t
(1 + i∗t ) − (1 + it ) SSt+1i+ Γgf lt + χand we treat St+1 as an exogenous variable since St+1 = Heh /yt+1 . Here, we substitutethe foreign demand for domestic assets aH∗
with gf lt and hH∗
with Ht /St − hHttt .
Consider the first order conditions for assets:/nf lt :/gf lt :/Ht :ηt − Et (ηt+1 (1 + i∗t ))



St
∗
Et ηt+1 (1 + it ) − (1 + it )
St+1



St
Et ηt+1 it
St+1

/bCBF
:
t

−Λ

=0

(34)

+Λ − Λ̃ + α0 Γ

=0

(35)

+∆Ft

=0

(36)

−Λ + Λ̃=0(37)Equation (37) implies that Λ̃ − Λ = 0. This means that the central bank equalizes themarginal benefit of relaxing the foreign-currency and domestic-currency debt constraints
by adjusting its assets and liabilities and going shorter in the asset whose shadow cost is
higher and longer in the asset whose shadow cost is lower. Also note that ηt = U ′ (ct ),ηt+1 = U ′ (ct+1 ), and that mt+1 = ηt+1 /ηt is the central bank’s discount factor, whichcoincides with the household’s (see Appendix C.1).

4.2

Optimal foreign exchange interventions

We can examine the impact of sterilized and unsterilized FX interventions by examining
Equation (35) with Λ − Λ̃ = 0.
Sterilized interventions Equation (35), with Λ − Λ̃ = 0, can be rewritten as follows:−U CF Xt

z

}|
{
∗
cov(m
,
X
)
α0
t+1
t+1
∗
−Et Xt+1−+
Γ=0
Et mt+1
ηt Et mt+1|{z}
M BF Xt

15(38)

The left-hand side, M BF Xt , corresponds to the marginal benefit of sterilized foreignexchange interventions, that is, of expanding the central bank’s balance sheet by going
long in foreign bonds and short in domestic bonds. This marginal benefit is composed ofthe marginal utility benefit of FX interventions (−U CF Xt ) and of the marginal benefit of
the resulting price distortions. If, in the absence of interventions, M BF Xt is positive, thenit would be optimal for the central bank to accumulate FX reserves. These interventions
can drive the marginal benefit to zero, achieving an optimal central bank balance-sheet,as we will see in more detail later.
Finally, we examine the last term in equation M BF Xt , which arises from the price(interest rate and exchange rate) distortions implied by the central bank’s interventions.
The central bank has an incentive to not fully shut down its risk-adjusted foreign currency
excess return in order to maximize its profit. Appendix C.2 shows that this term is equal
to:


St
E
m
t
t+1
St+1
α0


Γ = −Γgf lt(39)
ηt Et mt+1t
Et mt+1 Et m∗t+1 SSt+1
It is of the same sign as −gf lt , the gross external position in domestic currency. If thecountry is short in domestic currency (gf lt > 0), then this term is negative. Whenaccumulating foreign currency assets by issuing domestic currency liabilities, the planner
reduces the foreign currency excess return (by increasing it or depreciating the domestic
currency). The resulting opportunity cost is proportional to the economy’s gross external
position. This term also depends on Γ, which measures the impact of domestic currency
bond supply on the excess return (see Equation (24)). The higher Γ, the more difficult
it is for foreign intermediaries to absorb additional domestic currency assets, the higher
the impact of domestic currency bond supply on the excess return. This term reflects thecentral bank’s rent as a monopolistic issuer of domestic bonds.
To summarize, there could be a benefit of interventions for a safe haven currency if
its hedging property is more valued by international investors. However, the central bankalso has to consider how these interventions affect its monopoly rent.
Unsterilized interventions In our framework, unsterilized interventions are ineffective outside the ZLB and are equivalent to sterilized intervention at the ZLB. Equation(36) implies that ∆F > 0 if it > 0, meaning that Ht /St = hH
t outside the ZLB. Therefore, issuing more money outside the ZLB would be purely inflationary since domestic
households need a fixed real quantity of money and Ht /St = hH
t . This would not increase
the capacity of the central bank to buy foreign bonds. This arises from the absence of
nominal friction in our model and the resulting money neutrality.16
At the ZLB, money and bonds are perfect substitutes, so that sterilized and unsterilized interventions become equivalent. Then, M BF Xt is the marginal benefit of both
16

Note that open market operations would be equally ineffective because they also rely on changing
the money supply.

16

sterilized and unsterilized interventions, so that the above analysis applies. Whether foreign bonds are acquired by increasing Ht (unsterilized intervention) or by decreasing BtCB
(sterilized intervention) does not matter.
Since unsterilized interventions have no specific impact, in what follows we fix arbitrarily the quantity of money Ht , and assume Ht = 1. In this case, the equilibrium is
uniquely pinned down.
Lemma 1 (Equilibrium determinacy) Conditional on Ht , the equilibrium is unique.
Suppose that Ht is not fixed. Then, outside the ZLB, St and it are undetermined. According to the equilibrium in the domestic bond market (24), the excess return is affectedby the central bank’s optimal FX policy (through the supply of bCB and hence throughgf lt , as can be seen in Equation (22)). For a given expectation of St+1 , this equilibrium
excess return determines the equilibrium value for St (1 + it ), which is compatible withdifferent combinations of St and it . However, if Ht is fixed, then Ht /St = yt , since foreigners hold no money (hH∗= Ht /St − hHtt = 0) and households hold the minimum amountHof money (ht = yt ). This implies that the exchange rate St is determined, which meansthat the interest rate it is determined as well.
At the ZLB, it = 0, which implies that St is equal to the value that clears the domestic
bond market (24). Note that it does not clear the money market as Ht /St = yt no longer
holds, since now foreigners are willing to hold money. However, Ht and BtCB are notdetermined. The optimal FX interventions can be obtained with an infinite number of
combinations of sterilized and unsterilized interventions (Ht and BtCB ). However, if Ht isfixed, then the amount of BtCB that gives the optimal FX interventions is determined.

4.3

Implementation of the Optimum in a Decentralized Equilibrium

Here we discuss how the optimum is implemented in a decentralized equilibrium by analyzing households optimal choice. Consider the central bank’s foreign exchange interventions(sterilized or unsterilized). These interventions are relevant for the economy gross foreign
clt .
liabilities. Suppose that the optimal gross foreign liability position of the economy is gf
The households optimal portfolio allocation, characterized by Equation (17), can becompared with Equation (35). For Equation (35) to be implemented in the decentralized
equilibrium, we need thatλH − λF = −α0 Γ(40)where we used ηt = U ′ (ct ), ηt+1 = U ′ (ct+1 ) and Λ − Λ̃ = 0.
cl > 0, α0 is more likely to be negative,
In safe haven countries, where typically gft
so optimal foreign exchange interventions are only consistent with the households being
financially constrained when issuing domestic-currency bonds (λH > 0), since λF ≥ 0.
The central bank, as we have seen, does not fully exhaust the –private– marginal benefit
17

of going long in foreign currency and short in domestic currency. Households can beprevented from exploiting this residual marginal benefit only if their domestic-currency
borrowing constraint is binding.
In this case, the central bank desires fewer domestic liabilities than households. The
optimum is then easily implementable for the central bank by supplying just the rightamount of domestic liabilities to complement the existing public domestic supply andcl , through foreign exchange interventions. More precisely, the optimal foreign
reach gft
exchange intervention must satisfy
bbCBF = gf
clt − bG − hH
(41)
t
t
t
must close
The domestic currency bonds issued by foreign exchange interventions bbCBFtc
the gap between the optimal gross foreign liabilities gf lt and the existing real supply ofdomestic bonds, which is equal to the amount of government bonds that are not held byH
the central bank to back households asset holding bGt − ht . For that level of domestic
currency bonds, households would like to issue more domestic currency bonds (bHt < 0),
but they are prevented from doing so by their no-borrowing constraints. That way, theoptimum is implementable.17

4.4

Adding a Domestic Motive for FX Intervention

The focus of this paper is on the opportunity cost of FX intervention, so that we haveconsidered the benefits of interventions. The literature discusses numerous motives forintervention. To illustrate the cost-benefit analysis, we assume that the central bank has
an additional incentive for intervention.18 We now assume that households face shortselling constraints so that the real repayment on domestic debt −bH
t does not exceed
some limit. The constraint in (14) is replaced by:Et(1 + it )St H
bt ≥ b̄H
St+1

17

(42)

cl < 0 (α0 > 0), then Equation (40) would imply that λF > 0 (since λH ≥ 0),
Note that if gftmeaning that the household should be constrained in issuing foreign-currency bonds for the optimum tobe implementable. In that case, the central bank would typically save in domestic currency and borrowin foreign currency, but there would remain a private benefit of going short in foreign currency and longin domestic currency, which can only be consistent with a binding constraint on foreign liabilities ina decentralized economy. The optimum can be implemented by the central bank (or government) bycl .
supplying just the right amount of foreign-currency liabilities to reach −gft18
An alternative would be to assume that the central bank wants to stabilize the exchange rate, as
Amador et al. (2020). One way to rationalize exchange rate stabilization would be to introduce nominal
rigidities and imperfect substitution between domestic and foreign goods. In that case, variations in theexchange rate driven by financial shocks, such as shocks to Γ, shocks to σy , or noise trader shocks as
in Itskhoki and Mukhin (2022), would introduce undesirable variations in the terms of trade that would
warrant some exchange rate stabilization. Note that, in that case, open market operations would be
effective outside the ZLB, but ineffective at the ZLB, whereas both sterilized and unsterilized interventions
would remain effective at the ZLB. This would also rationalize the fact that, in practice, FX interventions
have been heavily used at the ZLB.

18

A non-zero level of b̄H gives an additional motive for monetary policy. We assume
that b̄H can be either positive or negative. A positive sign for b̄H implies an amount offorced savings.
This additional motive for intervention affects the last term in the first-order condition(38). Equation (39) becomes:
t
mt+1 SSt+1

Etα0Γ = −Γgf lt
ηt Et mt+1
E m E m∗
t

t+1

t



St
t+1 St+1

 +ΓΛηt E mt

t+1 Etb̄H

(1+it )St
St+1

Et



t )St
m∗t+1 (1+i
St+1



(43)
The second term on the right-hand side is active in the presence of a binding financial
constraint (Λ > 0), and is of the same sign as b̄H , so it is positive if households are forced
to save in domestic currency, that is, if b̄H > 0. This generates an incentive for a higher
interest rate (which improves the households intertemporal allocation by reducing therequired savings) and therefore an accumulation of FX reserves. Note that if b̄H < 0 (ifhouseholds face a borrowing limit), then the central bank would have a motive to depress
the real domestic interest rate (which improves the households intertemporal allocation
by increasing the allowed borrowing) and hence to reverse FX interventions.

5

A Linear-Quadratic Version of a Safe Haven Economy

In this section, we focus on the safe haven currency case. We assume that the SDF of
financial intermediaries is inversely proportional to the growth of a global factor yt∗ . The
safe haven feature is given by assuming that the domestic currency appreciates when the
global factor declines.19 We also assume that domestic output is only partly correlatedwith this factor. We consider an approximated version of the model and assume lognormal∗∗) and cov(mt+1 , Xt+1
) to evaluate theshocks. The objective is to derive cov(m∗t+1 , Xt+1
opportunity cost of FX interventions given in (29).
Assumption 1 (Specific case) Our specific case is characterized by
1. The utility is logarithmic: U (ct ) = log(ct );2. The SDF of international financial intermediaries is driven by the global factor:
y∗m∗t+1 = β y∗t ;t+1

19

There is a small literature which provides explanations for safe haven effects, but the focus is on the
US and the mechanisms do not apply to small countries. See Maggiori (2017) or Hassan et al. (2021).
Papers that model time-varying safe haven effects include Gourinchas and Rey (2022), Devereux et al.
(2022), and Kekre and Lenel (2021).

19∗) ∼ N (σy2 /2, σy2 ),3. The period-t + 1 global factor is log-normally distributed: log(yt+1with σy > 0;∗
4. Domestic output is correlated with the global factor: log(yt+1 ) = α log(yt+1) + (1 −α)σy2 /2 for some real α;∗

5. The exchange rate is correlated with the global factor St+1 = Heρ log(yt+1 ) for somereal ρ.
6. In period t domestic output and the global factor are normalized to 1: yt = yt ∗ = 1;
Assumption 1.6 normalizes the expected foreign stochastic discount factor to β under∗∗
log-utility: Et m∗t+1 = Et (β/yt+1) = β. Assumption 1.4 ensures that E(yt+1 ) = E(yt+1).
We make the following assumption on the parameters:
Assumption 2 (Safe haven) ρ > 0 and α < 1.
A positive ρ captures the hedging capacity of safe haven currencies: the exchange rate
appreciates when the global factor is weak, so the domestic currency is a good hedge
against fluctuations in the global factor. A low α reflects the small exposure of thedomestic output to global risk. Domestic output comoves with the global factor, so thedomestic currency is also a hedge for domestic households. However, since domestic output
is less volatile, in equilibrium domestic households are willing to be short in domestic
bonds.

5.1

The Utility Cost of Reserves

We first solve the equilibrium for given nf lt and gf lt to evaluate the planner’s optimality
conditions as a function of nf lt and gf lt . We will consider in turn the case where the
ZLB is not binding and the case where it is. We use second-order approximations. We
denote from now on the variables in log with a tilde: ỹ = log(y) and ỹ ∗ = log(y ∗ ). We
also define ĩ∗t = log(1 + i∗t ) and ĩt = log(1 + it ).
The foreign and domestic interest rates must satisfy∗∗

Et (em̃t+1 +ĩt ) = 1∗

Et (em̃t+1 +S̃t −S̃t+1 +ĩt ) = 1 + χ + Γgf lt

(44)
(45)

∗
Besides, we have S̃t+1 = ρỹt+1.
This yields:

1
β
1
1
2
(1 + it )St = (1 + χ + Γgf lt )e− 2 (1+ρ)ρσyβ
1 + i∗t =

20

(46)

See Appendix D.1 for a full derivation. These equations define respectively the foreigninterest rate i∗t as a function of the subjective discount factor and (1 + it )St as a functionof financial frictions (Γ and χ), of the hedging properties of the domestic exchange rate
and of gross foreign liabilities. This determines it outside the ZLB (since St = 1) and St
at the ZLB (since it = 0).
What are the implications for the cost of foreign exchange interventions? We can write
the difference in risk premia, which is a component of the utility cost of reserves U CF Xt(see Equation (29)), as follows∗∗
) cov(mt+1 , Xt+1)
cov(m∗t+1 , Xt+1
1
−∆cov
−
=
(1
+
χ
+
Γgf
l
)
1
−
e
t
Et m∗t+1
Et mt+1
β(47)where ∆cov = cov(m̃t+1 , S̃t+1 ) − cov(m̃∗t+1 , S̃t+1 ) is the difference between the covarianceof the log-linearized domestic currency excess return with the intermediaries SDF and thecovariance with the domestic SDF. See Appendix D.2 for a full derivation.
In Appendix D.2 we also show that∆cov = ρσy2 [1 − α(1 + nf lt + gf lt ) − ρgf lt ](48)

In a safe haven economy as defined in 2, if nf lt and gf lt are not too high, this covariancedifferential is positive. This implies that the difference in covariances introduces a gainfrom foreign exchange interventions. The domestic economy is less exposed to global
risk, so the planner benefits from going short in domestic bonds and long in foreign
bonds. Notice that this covariance differential is decreasing in gf lt . By increasing its
balance-sheet exposure to exchange rate risk, the planner would increase the risk exposureof the domestic household and the domestic covariance would catch up to the foreign
covariance.20

5.2

Optimal Allocations with no Domestic Motive for FX Interventions

The marginal utility cost of FX interventions U CF Xt , and especially the covariance differential, is only one component of the planner’s optimality condition (38). The following
lemma lays down explicit expressions for the optimal FX interventions. For the moment,
we abstract from the domestic motives of monetary policy by assuming that b̄H = 0.
cl
Lemma 2 Suppose that the economy is a safe haven as in Definition 2. Denote by gftdd
and nf lt the optimal gross and net foreign liabilities. We focus on solutions where nf lt <− log(β) + α2 . Then:
20

Note that, as shown in Equation (47), the covariance differential is proportional to 1 + χ + Γgf lt ,which corresponds to the risk-adjusted return on the domestic currency bond. The higher the average
return, the higher the covariances. As a result, stronger financial frictions also contribute to a higher
covariance differential.

21

dlt = min{bG − 1, nf l(gfclt )}, where nf l(gf lt ) is defined in Appendix D.3;
(i) nft
cl is implicitly defined by:(ii) gft
1 − (1 + χ + 2Γgf lt )e−∆cov = 0
where ∆cov is defined by Equation (48).
See the proof in Appendix D.3. Result (i) comes from the fact that the household may
HH
be financially constrained. In that case, nf lt = bGt − ht , where ht = 1. Otherwise,
the level of net foreign liabilities nf l equalizes the domestic and foreign discount factors
in expectations. Note that in that case, the household can only issue foreign bonds,
because, as we have seen, we must have λH > 0. Result (ii) derives from the optimality
condition with respect to gf lt , Equation (38). Interestingly, at the optimum, stronger
financial frictions (a higher Γ or a higher χ) result in a higher covariance differential.
Stronger financial frictions generate a more positive CIP, which limit the incentives toaccumulate reserves (see the definition for the utility cost of reserves (29)). With less FX
interventions, households are less exposed to currency risk, and the covariance differential
remains high. The extra term in Γ on the left-hand side reflects the fact that the planner
does not fully shut down the rent arising from the domestic currency excess return. The
planner intervenes enough to take advantage of the excess return, but takes into accountthe fact that interventions decrease the domestic excess return.
5.2.1

Comparative statics

Lemma 2 implies that, at the optimum,
cl =
gf
t

dlt )] − χρσy2 [1 − α(1 + nf
2Γ + ρ(α + ρ)σy2(49)

This is shown formally in Appendix D.4. Since we analyze safe haven economies, we focus
cl ≥ 0.
on the case where gf
tcdl . Higher leverage makes the economy more
Note that gf lt depends negatively on nftvulnerable to global risk and hence reduces the incentives of the central bank to take more
dl ? In the case where households arerisk on its balance sheet. What parameters drive nft
dl is positive
unconstrained, we show in Appendix D.4 that, under some conditions, nft2
if σy is high and α is low, while χ and Γ are not too large, even though the path ofdomestic output is the same as the foreign one on average. This is due to the fact thatdomestic households are less exposed to global risk than oreign investors, which lowers
the domestic discount factor relative to the foreign one. This is akin to an “inverse
precautionary saving motive”. This low risk exposure generates a borrowing motive thatcan drive domestic households to hit their no-borrowing constraint if the government debt
is too low (bGt − 1 < nf l t ).
22

dlt = bG − 1 is
In what follows, we suppose that households are constrained so that nftgiven. In that case, the optimal level of intervention is given by
bbCBF =tρσy2 [1 − αbG
t ]−χ
− (bG
t − 1)
2Γ + ρ(α + ρ)σy2

where we used equations (41) and (49).
The comparative statics for optimal FX intervention is given in the following proposition:
Proposition 2 Consider a safe haven economy as defined in 2 and assume that b̄H = 0.
clt ≥ 0 and nf
dl = bG − 1. Then optimal foreign exchange interventions,Suppose that gft
bbCBF :t(i) are increasing in risk measures σy and ρ;(ii) are decreasing in intermediaries financial frictions Γ and χ;(iii) are decreasing in the domestic output exposure to global risk α, as long as bG
t > 0;(iv) are decreasing in the supply of government bonds bGt .
Points (i) to (iv) can be shown by taking the derivatives of bbCBF
with respect to σy , ρ,Γ,t
χ, α, and bG
t . Risk tends to increase the covariance differential ∆cov, which generates anexcess benefit of foreign exchange interventions, while the intermediation frictions generatea cost. The exposure of domestic output to global risk decreases the covariance differential
and generates a cost. Point (iv) arises from the substitutability between government bonds
and the central bank liabilities (see Equation (41)). If the government issues more bonds,
then this reduces the need for the central bank to issue liabilities through FX interventions.
Interestingly, an increase in risk, which increases the optimal gf lt , typically generates
both a more negative UIP deviation and a more positive CIP deviation, as the financial
intermediaries have to absorb the excess domestic currency bonds. This is establishedformally in the following proposition:
cl ≥ 0 and nf
dl = bG − 1. Then:
Proposition 3 Suppose that gftt∗(i) Zt+1
is increasing in σy (it becomes more positive);∗(ii) Et Xt+1
is decreasing in σy (it becomes more negative) if Γ is not too large.

See the proof in Appendix D.5. The CIP deviation becomes more positive when riskincreases, as financial intermediaries need to absorb more capital inflows (more gf lt ). Asrisk increases, the UIP deviation becomes more negative, because it affects positively theforeigners risk premium. However, if the intermediation friction Γ is large, the increasein the CIP deviation can offset the increase in the risk premium, and the total impact on
the UIP deviation becomes ambiguous.
23

5.2.2

Numerical Illustration

We examine in more detail how the level of risk σy2 affects the economy through a numericalexample. Figure 2 shows the comparative statics of σy2 under a baseline specification ofparameters. We also consider two levels of bGt : 0.5 and 1.1. Panel a) shows the negative
relationship between the domestic interest rate and risk: the higher demand for domestic
bonds is accommodated through a decline in the domestic nominal interest rate. The ZLB
is attained at σy2 ≥ 0.62. Outside the ZLB, the interest rate declines to accommodate
the higher demand for domestic bonds (see Panel b)), while at the ZLB, the domestic∗∗
currency appreciates. Panel c) displays the deviations from UIP (Et Xt+1
) and CIP (Zt+1).
As we can see, an increase in risk leads to a more negative UIP deviation, and a morepositive CIP deviation, as explained in Proposition 3.
For Panels a) to c), a higher public debt bG
t reduces the domestic interest rate andthe domestic currency excess return (generating a less positive CIP deviation and a morenegative UIP deviation). With a higher level of net foreign liabilities nf lt , the central
bank targets lower domestic gross liabilities gf lt , as explained above (and as illustrated
in Panels a) and b) of Figure E.1 in the Appendix). The lower equilibrium interest ratethen results from the relative scarcity of domestic assets.
Panel d) shows that bbCBF
increases with risk because of the positive covariance differt
ential (see equation (48)) resulting from the assumption of safe-haven (α < 1 and ρ > 0).
An increase in risk raises the benefit of FX interventions, which the central bank takes
advantage of by buying FX reserves. However, the level of bbCBF
is only positive whentGGbt = 0.5. When bt is large, the central bank is long in domestic bonds rather than foreign
bonds, and short in foreign bonds rather than domestic bonds. In that case, an increasein risk pushes the central bank to sell domestic bonds and decrease its foreign currency
leverage. However, this is possible only if the central bank is allowed to be short in foreign
currency. Otherwise, the central bank cannot exploit its advantage. This perspective isconsistent with the experience of Switzerland and Japan. Swiss public debt has been
below 50% in the last 15 years, while it has been higher than 200% for Japan.
Note that households need not be constrained in their capacity to smooth consumption
between periods for the central bank interventions to be effective. In Figure E.1 in the
Appendix, we can see that in the case with a large public debt (dashed lines), the foreigncurrency no-borrowing constraint is not binding for most values of σy2 (see Panel c)), as
government debt helps households achieve their desired level of nf lt without having toborrow themselves. In fact, as risk increases, the discount factor of domestic households
decreases relative to the foreign one, leading to an increase in nf lt (see Panel b)). This net
foreign position is achieved by decreasing foreign currency bonds holdings bFt (see Panele)), as long as the foreign-currency no-borrowing constraint is not binding. Despite that,the central bank can achieve its target gf lt , because it can crowd out private domestic
savings bH
t by holding just the right amount of domestic bonds to achieve its desired gf lt .
As discussed above, households want to issue more gross foreign liabilities than the central24Figure 2: Comparative statics of σy2
~
b) St

a) it
4%

4%

0%

0%

ZLB

−4%

bG = 0.5

bG = 1.1

−4%

−8%

−8%

0.00

0.25

0.50

0.75

1.00

0.00

0.25

σ2y

0.75

1.00

0.75

1.00

σ2y

c) UIP and CIP deviations
4%

0.50

d) bCBF
t

∗
Zt+1

60%

0%

40%

−4%
20%

(

∗
E Xt+1

−8%

)
0%

−12%
0.00

0.25

0.50

0.75

1.00

σ2y

0.00

0.25

0.50

σ2y

Notes: Baseline parameters : β = 0.98, χ = 0.002, Γ = 0.5, α = 0.6, ρ = 0.2. We assume that b̄H = b̄F =0.

bank, but they are prevented from doing so by the domestic bond short-selling constraint.
We can see that households are still constrained in their capacity to issue domestic bonds(λH /ηt > 0, although it is very small, as shown in Panel d), and (1 + rt )bH
t remainsH
constant at b̄ , as shown in Panel f)).

5.3

Optimal Allocations with a Domestic Motive for FX Interventions

So far, we have examined the case where b̄H , households minimum domestic bond holdings,
is equal to zero. This assumption suppresses any “domestic” motive for FX interventions,as distorting the domestic interest rate cannot improve households consumption smoothing. In that case, the only motive for FX interventions stems from the utility gains ofholding FX reserves.
We now discuss the case where b̄H ̸= 0. In that case, as we can infer from Equations(38) and (43), the central bank can have an additional motive of buying or selling FX
reserves if Λ > 0. Note that, since λF = Λ, this means that this motive emerges whenhouseholds are unable to achieve their desired nf lt (both λH and λF are strictly positive).
If b̄H is positive, households are forced to save. In that case, because the forced savings
25

constraint bears on the total returns of savings, the central bank can reduce the amountof forced savings by maintaining a higher real interest rate. This is achieved throughmore FX interventions. If b̄H is negative, households have a limited borrowing capacity.
Here, on the opposite, the central bank would like to achieve a lower real interest rate to
generate a higher borrowing capacity for households, since the constraint bears on total
debt repayments. This is achieved through less FX interventions.
Figure 3 illustrates this. Panel d) shows that a positive b̄H shifts the central bank’s
FX reserve holdings upwards, while a negative b̄H produces a downward shift. The priceimplications are shown in Panels a), b) and c). In the former case, the equilibrium nominalinterest rate is higher (outside the ZLB) and the nominal exchange rate is more depreciated(in the ZLB), which generates a higher real interest rate, and hence a downward shift in
both CIP and UIP deviations. In the latter case, the opposite happens.
Figure 3: Comparative statics of σy2 with and without a “domestic motive”
~
b) St

a) it
4%

4%

0%

0%

ZLB

−4%

bH = 0bH = − 0.5

bH = 0.5

−4%

−8%

−8%

0.00

0.25

0.50

0.75

1.00

0.00

0.25

σ2y

1.00

0.75

1.00

d) bCBF
t
80%

∗
Zt+1

0%

60%

−4%

40%
∗
E(Xt+1
)

−8%

0.75

σ2y

c) UIP and CIP deviations
4%

0.50

20%

−12%

0%
0.00

0.25

0.50

0.75

1.00

σ2y

0.00

0.25

0.50

σ2y

Notes: Baseline parameters : β = 0.98, χ = 0.002 Γ = 0.5, α = 0.6, ρ = 0.2, bG = 0.5.

6Conclusion

The GFC was followed by significant changes in the international monetary system. We
have been observing systematic deviations from CIP, an increased demand for safe assets,
a strengthening role of the USD as a reserve currency and strong increase in central
26

bank balance sheets. In this context, there has been stronger demand for safe havencurrencies and more FX intervention by these countries central banks. In the case of the
Swiss National Bank, the spectacular increase in its balance sheet has occurred exclusively
through the purchase of foreign assets.
The objective of this paper is to provide a simple framework to clarify some aspectsof these developments. To explain UIP and CIP deviations in safe haven economies, wefollow the recent literature that gives a key role to constrained international financial
intermediaries. However, we assume that these intermediaries face exchange rate risk
and value the hedging properties of safe haven currencies. The increased demand forthese currencies may push the central bank to intervene and limit the extent of currency
appreciation.
We examine the opportunity cost of FX intervention when CIP and UIP deviations areof a different sign. We show that whether CIP or UIP matters depends on how domestic
residents value the hedging property of their currency compared to international investors.
If they give no value to its hedging property, UIP deviations should matter. This may
imply a benefit, and thus a higher incentive, for FX accumulation. We show that the
incentives to accumulate FX reserves in safe haven countries increase with the level ofglobal risk or of effective risk aversion of international intermediaries. In contrast, theincentive decreases with the level of debt.
We also attempt to estimate the opportunity cost of intervention for Switzerland andJapan. We find that in both countries, domestic households value the hedging propertiesof their currency less than international investors. Overall, the incentives for intervention
are stronger for Switzerland as the difference with international investors is larger andits public debt is much smaller than in Japan. While our analysis focuses on small safe
haven countries, it also sheds some light on the difference between the properties of safe
haven currencies and those of a reserve currency such the USD.

27

A

CHF and JPY as Safe Haven Currencies

The safe haven properties of the Swiss franc and the Japanese yen have been documented
by various authors, for example, Stavrakeva and Tang (2021), Ranaldo and Söderlind(2010), Grisse and Nitschka (2015), or Fink et al. (2022). We confirm this by relating
expected excess returns to various sources of risk.
We compute UIP deviations using short-term rates from Datastream and survey datafrom Consensus Economics.21 Table A.1 shows the correlation between expected excess
returns in CHF and JPY (Ex∗t+1 ) and different measures of risk. Since 2010, this correlation is systematically positive, suggesting that agents tend to expect the CHF and JPY
to yield excess returns at times of heightened uncertainty. When considering the entire
sample (from 1999 to 2021), the correlation is systematically weaker or negative, whichsuggests that the CHF and JPY have reinforced their perceived safe-haven properties
since 2010.
Table A.1: Correlation between UIP deviations and (global) risk variables
Corr(RiskV ariables, E(x∗t+1 ))A) CHF/USD

B) JPY/USD

Sample

USEPU

GEPU

WUI

USEPU

GEPU

WUI

1999-2021
2010-2021

-0.23
0.14

-0.29
0.26

-0.30
0.41

-0.11
0.14

-0.03
0.32

0.06
0.43

Notes: This table displays the correlation between Ex∗t+1 (at a 3-month horizon) and different risk
variables for the whole sample and a subsample starting in 2010. Panel A) displays this correlation
taking the CHF as the domestic currency and the USD as the foreign one. Similarly, Panel B)
considers the JPY as the domestic currency. USEPU is the US Economic Policy Uncertainty index
developed in Baker et al. (2016). GEPU is the Global EPU. WUI is the World Uncertainty Index
developed in Ahir et al. (2022). Since WUI is only available at a quarterly frequency, we take thequarterly mean of UIP deviations when computing the correlation.

To examine the dynamic impact of uncertainty shocks, Figure A.1 runs a localprojection regression (Jordà (2005)) of a Global Economic Policy Uncertainty (EPU)
shock on E(x∗t+1 ) for the period 2010-2021. The results show that, following an unanticipated shock to the Global EPU, E(x∗t+1 ) tends to increase both for the CHF and the JPY.
In other words, the CHF and the JPY are generally expected to appreciate following an
uncertainty shock.
21

See Kalemli-Özcan and Varela (2021) for a recent analysis of UIP deviations using Consensus Economics survey.

28

Figure A.1: Local Projections to a Global EPU shock
CHF
3.00%
2.00%
1.00%
0.00%
−1.00%−2.00%
JPY
3.00%
2.00%
1.00%
0.00%
−1.00%−2.00%
0

2

4

6

8

10

12

Months after the shock

Notes: This figure shows the results from the local projection of a Global EPU shock on the UIP deviations
over the sample 2010-2021, using the CHF and the JPY as the domestic currency, respectively. Formally,
we identify an uncertainty shock (shockt ) outside of the system by taking the residual of an AR(1) on
our Global EPU variable in the spirit of Stock and Watson (2012) who uses the VIX. We then run
E(x∗t+h ) = αh + βh shockt + ϕh xt + uht+h for h = 0, ..., 12 where xt are control variables made of p = 3
lags of the dependent variable. We then report β h at each horizon as well as the 90% confidence intervals
using the Newey-West estimator.

B

Computing excess returns and stochastic discountfactors

In this section, we discuss the construction of cov(x∗t+1 , m∗t+1 )/Et m∗t+1 and cov(x∗t+1 , mt+1 )/Et mt+1
considering either the CHF or the JPY as the domestic currency, and keeping the USD
as the foreign currency.

B.1

Excess returns

First, we compute excess returns. For it , we rely on the domestic (CHF or JPY) 3month risk-free rate, while i∗t is the US 3-month risk-free rate. For st we use nominal
spot exchange rate data expressed in the amount of domestic currency per unit of USD.
All data is from Datastream and is retrieved at the daily frequency. The daily datais aggregated to the quarterly frequency by taking the mean within each quarter. To
compute excess returns, we first compute quarterly excess returns according to (31). We
assume that what matters for the financial intermediaries is the moving excess returns of
this carry-trade over the past year by taking a moving sum of excess returns over thatof the current and last three quarters. This allows to have a smoother version of excess
returns.

29

B.2

Stochastic discount factors

International Financial Intermediaries We now discuss the construction of the SDF
of financial intermediaries, which is defined as m∗t+1 = β (N Wt+1 /N Wt )−γ . Similar to He
et al. (2017), we define N Wt+1 = ηt+1 × Wt+1 , where ηt+1 is a measure of the capital ratioof financial intermediaries and Wt is a measure of total wealth. The SDF is obtained byinteracting a measure of the growth rate of the capital ratio and total wealth. Below, we
discuss the construction of these growth rates.
We consider two measures of the capital ratio. The first specification (HKM) relieson the capital ratio measure from He et al. (2017) which is retrieved from Zhiguo He’s
website at a daily frequency and aggregated at a quarterly frequency by taking the mean.
The second specification (AEM) is based on Adrian et al. (2014) and is computed using
quarterly balance sheet data from the Federal Reserve Flow Of Funds (Table L.130). To
obtain an annual growth rate, we divide the residual of a regression of the capital ratio
in t on its one-year lagged value by the one-year lagged value of the capital ratio. This
gives rise to the intermediary capital risk factor. The two resulting measures are definedHKM
AEM
as ∆ηt+1
and ∆ηt+1, respectively.
For total wealth growth, we rely on a financial measure (MSCI US Equity Index) anda real measure (US GDP). For the financial measure, we consider moving annual excess
returns. Every quarter, they are obtained by summing up daily excess returns over the
past 4 quarters and subtracting the 3-month US risk-free rate. The resulting series is
M SCI
defined as ∆Wt+1. For the real measure, we compute moving annual growth everyGDP.
quarter. The resulting series is defined as ∆Wt+1i) × (1 +The SDF of financial intermediaries is then computed as m∗t+1 = β((1 + ∆ηt+1
j∆Wt+1 ))−γ for i ∈ {AEM, HKM } and j ∈ {M SCI, GDP }, with β = 0.99 and γ = 10.
This gives rise to 4 potential specifications of the SDF of financial intermediaries.
Domestic Households For Households (HH), the SDF is defined as mt+1 = β (Ct+1 /Ct )−γ .
Real consumption for Switzerland and Japan is retrieved from the FRED website at aquarterly frequency. As for the SDF of financial intermediaries, we compute a moving
annual growth rate and assume γ = 10 and β = 0.99.

C

Proofs - Constrained Planner Program

C.1

Other FOCs

We take the derivative with respect to hH
t :



St
F
H
+ ∆H
/ht : −Et ηt+1 itt − ∆ − Λ̃ = 0
St+1

(50)

Equations (50) and (36) then imply that ∆H
t = Λ̃ = Λ. Therefore, when Λ = Λ̃ = 0,
H
∆t = 0. This reflects the fact that, while households want to minimize their money30

holdings because they represent a cost (when it > 0), the amount of money held by thehouseholds is not relevant to the central bank when the economy is not constrained inits capacity to issue debt, since seigniorage is redistributed to households in period t + 1.
The cash-in-advance constraint is relevant only to the extent that it also restrains thecapacity of the economy to supply domestic assets to the rest of the world, just like theno-borrowing constraints.
We now take the derivatives with respect to prices:



Ht
St
H
gf lt +− ht
+ (1 + it )ξ
/it :−E ηt+1 (1 + it )
St+1
St


St
b̄H
∗
−α0 E mt+1 (1 + it )+ Λ̃ (1+it )St = 0
(51)
St+1
Et St+1



Ht
St
St H
/St :
−E ηt+1 (1 + it )gf lt − it
ht
− ∆F
St+1
St+1
St
 

H
b̄
St
−α0 E m∗t+1 (1 + it )+ Λ̃ (1+it )St = 0
(52)
St+1
Et St+1
Finally, we derive with respect to consumption:
U ′ (Ct ) − ηt = 0(53)

E (βU ′ (Ct+1 ) − ηt+1 ) = 0

(54)

/Ct :
/Ct+1 :′′
These equations imply that mCB
t+1 = ηt+1 /ηt = βU (Ct+1 )/U (Ct ) = mt+1 .

C.2

Monopolistic term and interest and exchange rate determinacy

Here we have to distinguish two cases. Either it > 0, and in that case ξ = 0, Ht /St = hHt
¯ F = 0.
and ∆F > 0. Or it = 0, and in that case ξ > 0 and ∆In the former case (if it > 0), Equation (51) yields


St
E
m
t+1
St+1
Λ̃
α0
b̄H
+
 


(55)
= −gf lt ηtηt E (1+it )St E m∗ (1+it )St
E m∗ Stt

t+1 St+1

St+1

t

t+1

St+1

where we have used E(ηt+1 /ηt ) = mt+1 . If Λ̄ = 0, α0 is of the same sign as −gf l, the grossexternal position in domestic currency. In that case, if the country is short in domestic
currency, then α0 is negative.
Using Equation (36), Equation (52) yields the same equation, so it is redundant. Thismeans that there is some nominal indeterminacy. This nominal indeterminacy does notcome from the future exchange rate, which is exogenously fixed, but from the amount ofexcess return adjustment that comes from it and St . In other terms, the optimal nominal
money supply Ht is undetermined. For instance, if the supply of money Ht is higher, then31

the exchange rate St will be higher (more depreciated), so that the optimal interest rate
it will have to be lower to generate a given excess return.
In the latter case (if it = 0), Equation (55) remains true. Note that in that case, theexchange rate is not undetermined, because it = 0.

D

Proofs - Linear-Quadratic Case

D.1

Equations (46)Equation (44) yields∗∗

Et (em̃t+1 +ĩt ) = 1
∗

1

∗

∗

∗

1

∗

∗

⇔

eE(m̃t+1 )+ 2 V (m̃t+1 )+ĩt = 1⇔

elog(β)+E(−̃yt+1 )+ 2 V (ỹt+1 )+ĩt = 1⇔

elog(β)+ĩt = 1∗

Similarly, Equation (44) yields∗

Et (em̃t+1 −S̃t+1 +ĩt +S̃t ) = 1 + χ + Γgf lt
1

∗

∗

⇔

eE(m̃t+1 −S̃t+1 )+ 2 V (m̃t+1 −S̃t+1 )+ĩt +S̃t = 1 + χ + Γgf lt⇔

elog(β)−E((1+ρ)ỹt+1 )+ 2 V ((1+ρ)ỹt+1 )+ĩt +S̃t = 1 + χ + Γgf lt

1∗elog(β)+⇔∗(1+ρ)ρ 2
σy +ĩt +S̃t2

= 1 + χ + Γgf lt

This yields (46).

D.2

Covariance differential

The difference in risk premia can be written as follows∗∗
cov(m∗t+1 , Xt+1
) cov(mt+1 , Xt+1)
1
cov(S̃t+1 ,m̃∗t+1 )−cov(S̃t+1 ,m̃t+1 )−= (1 + χ + Γgf lt ) 1 − e
Et m∗t+1
E(mt+1 )β
We used∗tcov(m∗t+1 , Xt+1
) = cov m∗t+1 , (1 + it ) SSt+1− cov(m∗t+1 , (1 + i∗t ))
|
{z
}
=0





St
St
∗
∗
= E mt+1 (1 + it ) St+1 − E mt+1 E (1 + it ) St+1
 ∗


 
∗
= E em̃t+1 +ĩt +S̃t −S̃t+1 − E em̃t+1 E eĩt +S̃t −S̃t+1
 ∗
h
i
∗
= E em̃t+1 +ĩt +S̃t −S̃t+1 1 − ecov(S̃t+1 ,m̃t+1 )|{z}
1+χ+Γgf lt

32

where we used (45), andE(m∗t+1 ) = β
which yieldsih∗cov(m∗t+1 , Xt+1)
1
cov(S̃t+1 ,m̃∗t+1 )= (1 + χ + Γgf lt ) 1 − e
Et m∗t+1
β(56)

Similarly:∗S
cov mt+1 ,(1+it ) S t −cov(mt+1 , (1 + it ))∗ )cov(mt+1 ,Xt+1=
E(m t+1 )
S
E mt+1 (1+it ) S t

t+1

{z

|

}

=0

E(mt+1 )


St
−
E
(1
+
i
)
t
E(mt+1 )
St+1




E em̃t+1 +ĩt +S̃t −S̃t+1
ĩt +S̃t −S̃t+1=−
Ee
E (em̃t+1 )t+1=− log(β)+ĩt +S̃t −E(S̃t+1 )+=e= β1 e

V (S̃t+1 )−cov(S̃t+1 ,m̃t+1 )
2

h

1−e

cov(S̃t+1 ,m̃t+1 )V (m̃∗
V (S̃t+1 )t+1 )ĩt +S̃t −E(S̃t+1 )++E(m̃∗t+1 )+−cov(S̃t+1 ,m̃∗t+1 )
2
2

h

i
∗

∗

ecov(S̃t+1 ,m̃t+1 )−cov(S̃t+1 ,m̃t+1 ) − ecov(S̃t+1 ,m̃t+1 )
 ∗
i
h
∗
∗
= β1 E em̃t+1 +ĩt +S̃t −S̃t+1 ecov(S̃t+1 ,m̃t+1 )−cov(S̃t+1 ,m̃t+1 ) − ecov(S̃t+1 ,m̃t+1 ){z}|
1+χ+Γgf lt

V (m̃∗)

t+1
where we used − log(β) = E(m̃∗t+1 ) +. This yields Equation (47).
2
Now note that∗m̃∗t+1 = log(β) − Ỹt+1
,

(57)

.
To obtain m̃t+1 , we can rewrite the resource constraints (25) as


nf lt
C t = Y t 1 + Yt


X∗
1+i∗t
Ct+1 = Yt+1 1 − nfYtlt 1+gt+1− gfYtlt 1+gt+1
t+1
with 1 + gt+1 = Yt+1 /Yt . We used the fact that, in equilibrium, (Ht /St − hHt )it St /St+1 isHequal to zero (either Ht /St − ht = 0 or it = 0). Taking logs and using a second-order∗approximation (assuming Ỹt+1 , nf lt /Yt , gf lt /Yt , Xt+1
and gt+1 are small), we obtainC̃t = Ỹt + nfYtlt − 12



nf lt
Yt

2

C̃t+1 = Ỹt+1 − nfYtlt (1 + i∗t − gt+1 ) + 21



nf lt
Yt

2

∗
(1 + i∗t ) − gfYtlt (Xt+1− gt+1 )Finally, we use the approximation gt+1 = Ỹt+1 − Ỹt along with the assumption that Yt = 1
and hence Ỹt = 0 to obtain the approximated household’s budget constraints:
1c̃t = nf lt − nf lt2
2



1
2
∗
c̃t+1 = ỹt+1 (1 + nf lt + gf lt ) − nf lt − nf lt (1 + i∗t ) − gf lt Xt+1
2
33

(58)

i

Using (58), we getm̃t+1 = C̃t − C̃t+1∗∗= log(β) − αỸt+1(1 + nf lt + gf lt ) + (nf lt − nf lt2 /2)(2 + ĩ∗t ) + gf lt (ĩt − ĩ∗t + S̃t − ρỸt+1
),
(59)
∗
∗
∗
∗
using Ỹt+1 = αỸt+1 , Xt+1 = ĩt − ĩt + S̃t − S̃t+1 and S̃t+1 = ρỸt+1 . Therefore, using (57)and (59), we find Equation (48).

D.3

Proof of Lemma 2

Another way to write Equation (29) is:t+ αη0tΓ=0
E (mt+1 (1 + i∗t )) − E mt+1 (1 + it ) SSt+1t(1 + i∗t ) E (mt+1 ) − (1 + it ) E mt+1 SSt+1+ αη0tΓ = 0| {z }| {z }
1
E(m∗
t+1 )

1+χ+Γgf lt St
E m∗
t+1 St+1

S
E mt+1 S t
1+χ+Γgf lt
t+1

1
−  ∗
E(m∗t+1 )E m

St
t+1 St+1

1 − (1 + χ + Γgf lt )E(mt+1 )

α0 Γ+ ηt E(mt+1 )=0

βα0 ΓE(m∗t+1 )ηt E(mt+1 )

=0



St
E mt+1St+1E(mt+1 )


St
E m∗
t+1 St+1
E(m∗
t+1 )

+

Equation (39) yields
St
E mt+1St+1E(mt+1 )


t
St
E m∗
t+1 St+1
E(m∗
t+1 )


St
E mt+1St+1E(mt+1 )


t
St
E m∗
t+1 St+1
E(m∗
t+1 )



St
E mt+1St+1E(mt+1 )t  ∗St
E mt+1
St+1
E(m∗
t+1 )

− Γgf l1 − (1 + χ + Γgf l )1 − (1 + χ + 2Γgf l )=0=0

Besides,S
E mt+1 S t
t+1

E(mt+1 )S
E m∗t+1 S t∗= ecov(m̃t+1 ,S̃t+1 )−cov(m̃t+1 ,S̃t+1 ) = e−∆cov

t+1

E(m∗t+1 )Hence result (ii) of Lemma 2.
Note that (34) implies thatΛηt

= 1 − E[mt+1 (1 + i∗t )]Λ = 0 is equivalent toE[mt+1 (1 + i∗t )] = 1
E(mt+1 )=βE(m̃t+1 )+ 12 V (m̃t+1 )e=βwhere we used (46) and where m̃t+1 is given by (59).
34

(60)

We have  2 2


ρ σy
σy2
1
2
2
E(m̃t+1 ) = log(β)−(1+nf l+gf l) +[1−log(β)] nf lt − nf lt + −− ρσy + χ + Γgf lt gf lt
2
22
where we used log(1 + χ + Γgf lt ) ≃ χ + Γgf lt , andσy2 ρ2 σy2
1
V (m̃t+1 ) = α2 (1 + nf l + gf l) +
gf lt
2
2
2
Therefore, Λ = 0 is equivalent to m̃(nf lt , gf lt ) = 0 with



σy2
1
2
2
m̃(nf lt , gf lt ) = −(1−α )(1+nf l+gf l) +[1−log(β)] nf lt − nf lt + −ρσy2 + χ + Γgf lt gf lt
2
2
(61)
2
m̃(nf lt , gf lt ) is increasing in nf lt if nf lt < − log(β) + α ). We consider only solutions
that satisfy this condition. In that case, the solution is unique. Denote by nf l(gf lt ) thisGGsolution. If nf l(gf lt ) > bG − hH
t = b − 1, then nf lt = b − 1 and Λ > 0.

D.4

clt and nf
dlt
Solutions for gf

For a given nf lt , gf lt is implicitly defined by2

1 − (1 + χ + 2Γgf lt )e−ρσy [1−α(1+nf lt +gf lt )−ρgf lt ] = 0
Using log(1 + χ + 2Γgf lt ) ≃ χ + 2Γgf lt , this yields
χ + 2Γgf lt − ρσy2 [1 − α(1 + nf lt + gf lt ) − ρgf lt ] = 0
After rearranging, we obtain (49).
If λ = 0, (49) and m̃(nf lt , gf lt ) = 0 jointly define nf lt and gf lt . If Λ > 0, then gf lt
is defined by (49) with nf lt = bGt − 1.
Consider the case where Λ = 0. As before, consider solutions where nf lt < − log(β) +2α ) and denote by nf l(gf lt ) the unique solution. Suppose additionally that 1+nf l(gf lt )+gf lt > 0. If σy2 is large, and α, χ and Γ are small, then m̃(nf lt , gf lt ) = 0 implies thatnf lt − nf lt2 /2 > 0. As long as nf lt < 2, this implies that nf lt > 0.
Special case with α=0 In the special case where α = 0, we can compute implicit
solutions for nf lt and gf lt when Λ = 0.
First, in that case, (49) implies
cl =
gf
t

ρσy2 − χ2Γ + ρ2 σy2

dl is the solution to the second-order polynomial equation m̃(nf lt , gf lt ) = 0 thatand nft
is on the increasing segment of the polynomial m̃(nf lt , gf lt ):


s

2
σy2
2
2
2
c
c
c
σy
σy
(1 + gf lt ) − (−ρσy + χ + Γgf lt )gf lt
dlt = 1 1 −

nf
−
1−
−4 2
2
2[1 − log(β)]
2[1 − log(β)]
1 − log(β)
35

D.5

Proof of Proposition 3

Note that the CIP deviation, as defined in (9), is increasing in gf lt (hence (i)), since= gf lt .
E(m∗t+1 ) = β is fixed, and aH∗t
Finally, note that the UIP deviation can be written as (we use (10), (56) and E(m∗t+1 ) =β as well):ih2∗
Et Xt+1= β1 χ + Γgf lt − (1 + χ + Γgf lt )(1 − e−ρσy )ih2= − β1 1 − (1 + χ + Γgf lt )e−ρσyclt and nf lt with bG −1, we obtain
where we used the results in D.2. Replacing gf lt with gf
t
i
h


2
ρσy2 [1−αbGt ]−χ∗Et Xt+1= − β1 1 − 1 + χ + Γ 2Γ+ρ(α+ρ)σ
e−ρσy
2
y
"
#
2
G
≃ − β11−eχ+Γρσy [1−αbt ]−χ22 −ρσy2Γ+ρ(α+ρ)σy∗with respect to σy2 is of the same sign as
The derivative of Et Xt+1

−ρ + Γ

2Γρ(1 − αbGt ) + χρ(α + ρ)[2Γ + ρ(α + ρ)σy2 ]2∗Therefore, Et Xt+1
is decreasing in σy if Γ is not too large (hence (ii)).

36

E

Additional Figures
Figure E.1: Comparative statics of σy2 - continueda) gfltb) nflt

12%
10%
8%
6%
4%
2%
0%

0%
−20%
bG = 0.5

−40%
0.00

0.25

0.50

0.75

1.00

0.00

0.25

σ2y

bG = 1.1
0.50

0.75

1.00

0.75

1.00

0.75

1.00

σ2y

c) λF ηtd) λH ηt

60%

60%

40%

40%

20%

20%
0%

0%
0.00

0.25

0.50

0.75

1.00

0.00

0.25

σ2y

0.50

σ2y

e) (1 + it∗ )bFf) (1 + rt)bH
4%
2%
0%
−2%
−4%

0%
0.00

0.25

0.50

0.75

1.00

0.00

σ2y

0.25

0.50

σ2y

Notes: Baseline parameters : β = 0.98, χ = 0.002 Γ = 0.5, α = 0.6, ρ = 0.2. We assume that b̄H = b̄F = 0.

37

References
Adler, G. and R. C. Mano (2021). The cost of foreign exchange intervention: Concepts andmeasurement. Journal of Macroeconomics 67, 103045.
Adrian, T., E. Etula, and T. Muir (2014). Financial intermediaries and the cross-section of asset
returns. The Journal of Finance 69 (6), 2557–2596.
Ahir, H., N. Bloom, and D. Furceri (2022). The world uncertainty index. Technical report,
National Bureau of Economic Research.
Amador, M., J. Bianchi, L. Bocola, and F. Perri (2020). Exchange Rate Policies at the Zero
Lower Bound. Review of Economic Studies 87, 1605–1645.
Bacchetta, P., K. Benhima, and Y. Kalantzis (2013). Capital Controls with International Reserve
Accumulation: Can This Be Optimal? American Economic Journal: Macroeconomics 5 (3),229–62.
Bacchetta, P., J. S. Davis, and E. van Wincoop (2023). Dollar Shortages, CIP Deviations andthe Safe Haven Role of the Dollar. mimeo.
Baker, S. R., N. Bloom, and S. J. Davis (2016). Measuring economic policy uncertainty. The
quarterly journal of economics 131 (4), 1593–1636.
Basu, S., E. Boz, G. Gopinath, F. Roch, and F. Unsal (2020). A Conceptual Model for the
Integrated Policy Framework. Working Paper 20/121, International Monetary Fund.
Bianchi, J., S. Bigio, and C. Engel (2022). Scrambling for Dollars: International Liquidity,
Banks and Exchange Rates. Mimeo.
Bianchi, J. and G. Lorenzoni (2022). The Prudential Use of Capital Controls and Foreign Currency Reserves. In G. Gopinath, E. Helpman, and K. Rogoff (Eds.), Handbook of International
Economics, pp. forthcoming. Amsterdam: North Holland.
Brunnermeier, M. K. and Y. Sannikov (2014, February). A macroeconomic model with a financial
sector. American Economic Review 104 (2), 379–421.
Cavallino, P. (2019). Capital Flows and Foreign Exchange Intervention. American Economic
Journal: Macroeconomics 11 (2), 127–70.
Cerutti, E. M., M. Obstfeld, and H. Zhou (2021). Covered interest parity deviations: Macrofinancial determinants. Journal of International Economics 130, 103447.
Devereux, M. B., C. Engel, and S. P. Y. Wu (2022). Collateral Advantage: Exchange Rates,
Capital Flows, and Global Cycles. Mimeo.
Du, W. and J. Schreger (2022). CIP Deviations, the Dollar, and Frictions in International CapitalMarkets. In G. Gopinath, E. Helpman, and K. Rogoff (Eds.), Handbook of International
Economics, pp. forthcoming. Amsterdam: North Holland.

38

Edwards, S. (1985). On the interest-rate elasticity of the demand for international reserves:
Some evidence from developing countries. Journal of International Money and Finance 4,287–295.
Fanelli, S. and L. Straub (2021). A Theory of Foreign Exchange Interventions. The Review of
Economic Studies 88 (6), 2857–2885.
Fang, X. and Y. Liu (2021). Volatility, intermediaries, and exchange rates. Journal of Financial
Economics 141 (1), 217–233.
Fink, F., L. Frei, and O. Gloede (2022). Global Risk Sentiment and the Swiss Franc: A Timevarying Daily Factor Decomposition Model. Journal of International Money and Finance 122,
102539.
Gabaix, X. and M. Maggiori (2015). International Liquidity and Exchange Rate Dynamics. The
Quarterly Journal of Economics 130 (3), 1369–1420.
Gertler, M. and N. Kiyotaki (2010). Chapter 11 - financial intermediation and credit policy inbusiness cycle analysis. Volume 3 of Handbook of Monetary Economics, pp. 547–599. Elsevier.
Gourinchas, P.-O. and H. Rey (2022). Exorbitant Privilege and Exorbitant Duty. CEPR Discussion Paper 16944, Centre for Economic Policy Research.
Grisse, C. and T. Nitschka (2015). On Financial Risk and the Safe Haven Characteristics ofSwiss Franc Exchange Rates. Journal of Empirical Finance 32, 153–164.
Hassan, T. A., T. M. Mertens, and T. Zhang (2021). A Risk-based Theory of Exchange Rate
Stabilization. Mimeo.
He, Z., B. Kelly, and A. Manela (2017). Intermediary asset pricing: New evidence from many
asset classes. Journal of Financial Economics 126 (1), 1–35.
He, Z. and A. Krishnamurthy (2011). A Model of Capital and Crises. The Review of Economic
Studies 79 (2), 735–777.
Itskhoki, O. and D. Mukhin (2021). Exchange Rate Disconnect in General Equilibrium. Journalof Political Economy 129 (8), 2183–2231.
Itskhoki, O. and D. Mukhin (2022). Optimal Exchange Rate Policy. mimeo.
Jeanne, O. and R. Rancière (2011). The optimal level of international reserves for emerging
market countries: A new formula and some applications. The Economic Journal 121, 905–
930.
Jiang, Z., A. Krishnamurthy, and H. Lustig (2021a). Dollar Safety and the Global Financial
Cycle. Mimeo.
Jiang, Z., A. Krishnamurthy, and H. Lustig (2021b). Foreign Safe Asset Demand and the Dollar
Exchange Rate. The Journal of Finance 76, 1049–1089.

39

Jordà, Ò. (2005). Estimation and inference of impulse responses by local projections. American
economic review 95 (1), 161–182.
Kalemli-Özcan, S. and L. Varela (2021). Five Facts about the UIP Premium. NBER Working
Paper 29238, National Bureau of Economic Research.
Kekre, R. and M. Lenel (2021). The Flight to Safety and International Risk Sharing. NBER
Working Paper 28923, National Bureau of Economic Research.
Maggiori, M. (2017). Financial Intermediation, International Risk Sharing, and Reserve Currencies. American Economic Review 107 (10), 3038–71.
Maggiori, M. (2021). FX Policy When Financial Markets are Imperfect. Working Paper 942,
BIS.
Ranaldo, A. and P. Söderlind (2010). Safe Haven Currencies. Review of Finance 14 (3), 385–407.
Rime, D., A. Schrimpf, and O. Syrstad (2022). Covered Interest Parity Arbitrage. The Review
of Financial Studies 35 (11), 5185–5227.
Stavrakeva, V. and J. Tang (2021). The Dollar During the Great Recession: The Information
Channel of US Monetary Policy and the Flight to Safety. manuscript.
Stock, J. H. and M. W. Watson (2012). Disentangling the channels of the 2007-2009 recession.
Technical report, National Bureau of Economic Research.
Valchev, R. (2020). Bond Convenience Yields and Exchange Rate Dynamics. American Economic Journal: Macroeconomics 12 (2), 124–66.
Yeyati, E. L. and J. F. Gómez (2022). Leaning-against-the-wind intervention and the “carrytrade” view of the cost of reserves. Open Economies Review , forthcoming.

40