Foreign exchange rate risk in a small open economy

Working Paper No. 365
Foreign exchange rate risk in a
small open economy
Bianca De Paoli and Jens Søndergaard
March 2009

---

Working Paper No. 365
Foreign exchange rate risk in a small open economy
Bianca De Paoli(1) and Jens Søndergaard(2)
Abstract
Resolving the forward premium puzzle requires a volatile foreign exchange rate risk premium that
covaries negatively with the expected depreciation rate. Earlier work has shown how models featuring
consumption habits can generate such premia when either trade costs or ‘deep habits’ are assumed.
We show that as long as consumption habits are slow-moving and shocks are highly persistent, a
standard small open endowment economy — without any additional features — can address the puzzle.
Moreover endogenising the labour supply decision in the small open economy can improve the model’s
ability to match risk premia observations so long as it makes business cycles less synchronised.
(1) Monetary Assessment and Strategy Division, Bank of England. Email: bianca.depaoli@bankofengland.co.uk
(2) Monetary Assessment and Strategy Division, Bank of England. Email: jens.sondergaard@bankofengland.co.uk
The views expressed in this paper are those of the authors, and not necessarily those of the Bank of England. We have
benefited from helpful comments from Mark Astley, Andy Blake, Hafedh Bouakez, Ester Faia, Adrien Verdelhan,
Peter Westaway, Pawel Zabczyk and seminar participants at the Bank of England, Dallas Fed, Danmark’s Nationalbank, the
2007 Computing in Economics and Finance conference and the 2007 Dynare workshop. All remaining errors are our own.
This paper was finalised on 5 February 2009.
The Bank of England’s working paper series is externally refereed.
Information on the Bank’s working paper series can be found at
www.bankofengland.co.uk/publications/workingpapers/index.htm
Publications Group, Bank of England, Threadneedle Street, London, EC2R 8AH
Telephone +44 (0)20 7601 4030 Fax +44 (0)20 7601 3298 email mapublications@bankofengland.co.uk
© Bank of England 2009
ISSN 1749-9135 (on-line)

---

Contents
Summary
3
1 Introduction
5
2 Model
7
2.1
The goods market
8
2.2
The asset market
11
2.3
Equilibrium
12
3 FX premia and the Fama puzzle
13
3.1
The failure of the risk-neutral UIP
13
3.2
Risk premium explanation of UIP failure
14
3.3
Time-varying risk aversion and FX risk premium
15
4 Model simulations results
17
4.1
Model calibration
17
4.2
Results
18
5 Conclusion
22
Appendix A: Deriving the FX premium
24
Appendix B: Cyclical properties of precautionary savings
26
References
28
Working Paper No. 365 March 2009
2

---

Summary
Investors require compensation (or a `premium') to hold risky nancial asset. So if some
currencies are perceived to be riskier than others, investors may demand a foreign exchange (FX)
premium to invest in those currencies. This paper presents a small open economy model that can
explain why FX premia arise in currency markets. We use this model to examine how well it
resolves the so-called uncovered interest rate parity (UIP) puzzle. UIP is simply a condition that
follows from nancial market arbitrage. It ensures that the interest rate return on a domestic
currency asset should equal the interest rate on each foreign currency assets, less the expected
appreciation of the domestic currency. The puzzle stems from the empirical observation that high
interest rate currencies tend to appreciate – contrary to what UIP would predict.
A key feature of our model is that households are assumed to have consumption habits, ie
households get used to a `habit' level of consumption, and only attain higher utility if actual
consumption rises relative to that level.
We demonstrate that our model will only resolve the UIP puzzle if it produces signi cant
precautionary savings effects, where savings rise in response to increased uncertainty. And these
savings effects will only occur if we assume quite persistent productivity shocks combined with
very slow-moving consumption habits.
In our model, changes in precautionary savings are a result of changes in households' attitude
towards risk, and changes in economic prospects. In the face of bad shocks, for example,
households increase their precautionary savings if they expect consumption to be low relative to
their habits level. Thus, the slower is the adjustment of habits to the shock, the larger will be the
revisions in precautionary savings. These revisions are also larger when the shocks are more
persistent.
To understand the combined role of slow-moving consumption habits and persistent shocks in
resolving the UIP puzzle, consider how a temporary fall in productivity in the rest of the world
works its way through our model. The drop in foreign productivity causes an ex ante excess
demand for foreign goods which is eliminated by a rise in the relative price of foreign goods, ie a
Working Paper No. 365 March 2009
3

---

domestic currency depreciation. But since this is ultimately a temporary shock, the domestic
currency is expected to appreciate back towards its initial steady state.
However, the same negative foreign shock also triggers a large increase in foreign precautionary
savings, putting downward pressure on foreign interest rates and hence causing domestic interest
rates to exceed foreign rates at the same time as the domestic currency is expected to appreciate,
thus potentially resolving the puzzle. But at the same time the increase in foreign households'
borrowing to smooth their consumption (known as `intertemporal substitution') will tend to put
upward pressure on foreign interest rates and hence cause domestic rates to lie below foreign
rates at the same time as the domestic currency is expected to appreciate (ie in line with the
predictions of UIP). So we can only account for the tendency for high interest rate currencies to
appreciate if the precautionary savings effects outweigh the intertemporal substitution effects.
This would be the case if the shock is very persistent and consumption habits are very
slow-moving.
We initially show our result at work in a model with xed labour supply. We then examine how
our result changes when we allow domestic households in our small open economy to vary their
hours worked. In our model, this extension makes domestic consumption less synchronised with
foreign consumption. To ensure that risk is ef ciently shared across countries, the real exchange
rate would have to uctuate more. We nd that a more volatile real exchange rate combined with
a stronger precautionary savings effect actually improves the model's ability to address the UIP
puzzle. But when we allow both domestic and foreign households to vary their hours worked,
consumption is both smooth and synchronised across countries. This dampens the FX premium
volatility and impedes the model's ability to resolve the UIP puzzle.
Working Paper No. 365 March 2009
4

---

1 Introduction
Would an investment strategy that borrows in a low interest rate currency and invests the
proceeds in a high-yielding currency be pro table? The naive answer is `yes' since the interest
rate differential on the borrowing-lending spread represents a potential pro t opportunity. The
traditional answer is `no' since any excess return from investing in a high interest rate currency
would be offset by an associated depreciation in that particular currency. But, actually, the
empirical evidence indicates that not only is this strategy pro table on average but it yields
returns that exceed the interest rate differential. This is because, in practice, high interest rate
currencies frequently appreciate over time. And therefore, rather than offsetting any interest rate
differentials, the exchange rate movements actually increase the pro t of this particular
investment strategy (see Cavallo (2006)).1 This nding is well known in the academic literature
and is known as the forward premium anomaly or the uncovered interest rate parity (UIP) puzzle
(see Fama (1984)).2
A traditional open economy model cannot replicate the forward premium anomaly as it typically
assumes linear UIP holds. When investors are assumed to be risk-neutral, any cross-country
differences in interest rates are associated with offsetting movements in expected depreciation. A
large literature has tried previously to account for the forward premium anomaly. One strand of
research, explored in Bekaert (1996), attributes the failure of UIP to the existence of time-varying
foreign exchange (FX) risk premia. When risk is allowed for, risk-averse investors may require
additional compensation to hold riskier assets. A risk-adjusted UIP condition breaks the tight link
between expected changes in the exhange rate and interest rate differentials. But as demonstrated
by Fama (1984), the FX risk premium embodied in this UIP condition needs to have certain
dynamics properties in order to resolve the forward premium anomaly. The challenge for this
strand of the literature has been to come up with an open economy model that generates an FX
risk premium with the time series properties that resolves this long-lasting anomaly.3
1For further studies assessing the pro tability of such strategies see Bilson (1981) and Della Corte and Tsiakas (2009).
2While the original forward premium anomaly is documented by Fama (1984), a number of papers have looked at the robustness of his
result (see Baillie and Bollerslev (2000), Bansal and Dahlquist (2000), Bansal (1997) and Flood and Rose (1996) for recent contributions
and Sarno (2005) for a comprehensive survey of the literature). Importantly, it has been shown that the puzzle is not robust to short-run
analysis (see Lyons and Rose (1995) and Chaboud and Wright (2005) for an assessment of intraday data). For this reason, the current
paper proposes a quarterly model of the exchange rate risk premium.
3There are other strands of literature that have tried to rationalise the forward premium anomaly using theoretical models. `Peso
problem'-type arguments and other explanations related to irrational market participant behaviour can be found in the early literature (see
Engel (1996) for a survey). More recently, Bacchetta and van Wincoop (2005) and Gourinchas and Tornell (2004) examine the role of
Working Paper No. 365 March 2009
5

---

In this paper, we re-examine the forward premium anomaly using a standard open economy
macro framework as in Gali and Monacelli (2005) and De Paoli (2009). Having consumption
habits in the model is crucial in order to resolve the UIP puzzle. These ensure that in periods
where the domestic currency is expected to appreciate, domestic investors are less risk-averse
than foreign households, and this translates into domestic interest rates being higher than foreign
interest rates. We are not the rst to explore the role of consumption habits in solving the UIP
puzzle. Both Verdelhan (2006) and Moore and Roche (2007) have shown how models featuring
consumption habits can help rationalise the UIP puzzle. These papers assume that investors are
subject to permanent shocks and have Campbell and Cochrane (1999) type preferences. The
former considers an endowment economy subject to trade costs. The latter presents a monetary
model which adds so-called `deep habits' to the Campbell and Cochrane (1999) setting. Unlike
these authors, we do not rely on such additional model features and instead we use a linear,
additive external habit speci cation which nests those in Uhlig (2004) or Smets and Wouters
(2007).4 This formulation enables us to demonstrate the precise role played by the persistence of
habit formation in resolving the UIP puzzle.
Our analysis shows a standard small open endowment economy can go a long way towards
addressing this puzzle while replicating key macroeconomic moments as long as consumption
habits are slow-moving and shocks are highly persistent. Consumption habits alone are not
suf cient to generate a constellation of movements in exchange rates and interest rates that
resolve the UIP puzzle. It is crucial that investors expect changes to economic conditions – as
captured by their excess consumption level – to persist.
We also show that moving towards a general equilibrium setting where the labour supply
decision in the small open economy is endogenous, improves the model's ability to match risk
premia observations so long as it makes business cycles less synchronised. Such a setting can
generate larger real exchange rate uctuations and would increase FX premium volatility.
Nevertheless, perhaps more realistically, when both the domestic and foreign labour supply
imperfect information. Alvarez, Atkeson and Kehoe (2005) look at the properties of the FX risk premium in a model with asset market
segmentation. In addition, Lyons and Rose (1995) proposes a framework in which, for small Sharpe ratios, the forward bias does attract
speculative capital and therefore persists until this ratio is suf ciently large (see Leon and Valente (2006) for an empirical test of this
`limits to speculation' hypothesis) .
4Campbell and Cochrane (1999) demonstrate that up to a rst-order approximation the habit formation process they impose is equivalent
to an autoregressive linear speci cation similar to the one we use.
Working Paper No. 365 March 2009
6

---

decisions are endogenous, the model has more dif culty matching observed risk premia. This
setting produces very smooth consumption dynamics, small uctuations in the FX premium and
behaves similarly to a model solved under certainty equivalence. Such a caveat is similar to the
ndings in the closed economy literature (see Jermann (1998) and Rudebusch and Swanson
(2008)) which have stressed the pivotal role of real rigidities – such as adjustment costs in labour
supply. Our results suggest that similar model features may be needed in an open economy
setting.
The paper is structured as follows: in Section 2 we present the model. Section 3 discusses how
our model could generate an FX premium consistent with the Fama puzzle. Section 4 presents
simulation results illustrating how model features affect the model's ability to address this
puzzle. Finally, the concluding remarks are presented in Section 5.
2 Model
As in Lucas (1982), we analyse the role of the foreign exchange rate risk premium in a
two-country general equilibrium context. But our theoretical framework assumes home bias in
consumption and, thus, incorporates deviations from purchasing power parity and uctuations in
the real exchange rate. The size of this consumption home bias depends on the degree of
openness and the relative size of the economy. This speci cation allows us to characterise the
small open economy by taking the limit of the home size to zero. Prior to applying the limit, we
derive the optimal equilibrium conditions for the general two-country model. After the limit is
taken, the two countries, Home and Foreign, represent the small open economy and the rest of
the world, respectively. This setup follows closely that of Gali and Monacelli (2005) or De Paoli
(2009). Moreover, apart from the case of endogenous labour supply, we also analyse the case of
inelastic labour supply, which translates into an endowment economy model (as in Lucas (1982)).
The original Gali and Monacelli (2005) or De Paoli (2009) speci cations feature a Calvo-type
sticky price-setting in order to address monetary policy issues. However, in the present paper we
abstract from such issues and consider a exible price version of these models, with the added
feature of a slow-moving external consumption habit formation.5
5The exible price allocation of such models is equivalent to the case where the central bank targets output price in ation.
Working Paper No. 365 March 2009
7

---

2.1 The goods market
The world economy is populated with a continuum of agents of unit mass, where the population
in the segment [0; n/ belongs to country H (Home) and the population in the segment .n; 1]
belongs to country F (Foreign). The utility function of a consumer j in country H is given by:
1
U j
s t
t D Et X
U .C js; Xs/ V .ys. j/; "t/ :
(1)
sDt
Households obtain utility from consumption U .C
.Cit hXt/1
1
t ; Xt / D
and contribute to the
1
production of a differentiated good y. j/ attaining disutility V .yt; "Y;t/ D t y1Ct :6 Productivity
1C
shocks are denoted by t. The parameter, , is the inverse of the intertemporal elasticity of
substitution while is the inverse of the Frisch labour supply parameter.
We assume that agents have external consumption habits Xt which follows an ARMA process
where
controls its persistence, ie
Xt D .1 /Ct 1 C Xt 1;
(2)
and, similarly, in the foreign economy
Xt D .1 /Ct 1 C Xt 1:
(3)
The consumption basket Ct is a constant elasticity of substitution (CES) aggregate over
domestic, CH;t, and foreign produced tradables CF;t:
1
1
C
1
t D h 1 CH;t C.1 / CF;t i 1 ;
(4)
with the corresponding domestic price index, Pt, de ned as
P
1
1
t D h PH;t C.1 / PF;t i 11 :
(5)
6This speci cation would be equivalent to one in which the labour market is decentralised. These rms employ workers who have
disutility of supplying labour and this disutility is separable from the consumption utility.
Working Paper No. 365 March 2009
8

---

The sub-indices CHt and CFt are Home and Foreign consumption of the differentiated products
produced in countries H and F. These are de ned as follows:
1
1
C
" 1
1
" 1
1
H;t D
Z nc dz# 1; C
Z 1c dz# 1 (6)
n
t .z/
F;t D
t .z/
0
1
n
n
where
> 1 is the elasticity of substitution across the differentiated products. The
consumption-based price indices that correspond to the above speci cations of preferences are
given by
1
1
1
1
1
1
PH;t D
Z np
; P
Z 1p
;
(7)
n
t .z/1
dz
F;t D
t .z/1
dz
0
1
n
n
We consider the case in which the law of one price holds, and thus, PH;t D St P and
H;t
PF;t D St P where S
F;t
t is the nominal exchange rate (the Home-currency price of Foreign
currency). However, as in Sutherland (2005), we assume a particular speci cation of
and
which implies that consumers at home and abroad have a consumption home bias. Thus, we
de ne the real exchange rate, Qt as the value of the domestic consumption basket per unit of the
foreign consumption basket, ie
S
Q
t Pt
t D
:
(8)
Pt
In order to characterise a small open economy setting, we assume that the parameter governing
domestic consumers' preference for foreign goods, .1
/; is a function of the relative size of
the foreign economy, 1
n; as well as the degree of openness, in the domestic economy:
.1
/ D .1 n/ :
(9)
Hence, a more open domestic economy (higher ) would – ceteris paribus – imply a larger share
of foreign goods in the domestic consumption basket. Similarly, the greater the size of the
foreign economy relative to the domestic economy (higher 1
n), the larger the share of foreign
goods in the domestic consumption basket.
Agents in the Foreign economy have preferences analogous to (1), (4) and (6). Moreover, foreign
consumers' preferences for home goods also depend on the relative size of the home economy, n,
and the degree of openness of the domestic economy ; that is
D n :
(10)
Working Paper No. 365 March 2009
9

---