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G99/2
Noise Trading and Exchange Rate Regimes
Olivier Jeanne and Andrew K. Rose
2 March 1999
JEL Classification Numbers: F33, C15
Abstract
1
Both the literature and new empirical evidence show that exchange rate regimes differ
primarily by the noisiness of the exchange rate, not by measurable macroeconomic
fundamentals. This motivates a theoretical analysis of exchange rate regimes with
noise traders. The presence of noise traders can lead to multiple equilibria in the
foreign exchange market. The entry of noise traders alters the composition of the
market and generates excess exchange rate volatility, since noise traders both create
and share the risk associated with exchange rate volatility. In such circumstances,
monetary policy can be used to lower exchange rate volatility without altering
macroeconomic fundamentals.
Olivier Jeanne, Research Department, International Monetary Fund, 700 19th Street
N.W. Washington D.C. 20431. Tel: (202) 623-4272; Fax: (202) 623-6334; E-mail:
ojeanne@imf.org
Andrew K. Rose, Haas School of Business, University of California, Berkeley, CA
USA 94720-1900. Tel: (510) 642-6609; Fax: (510) 642-4700; Email:
arose@haas.berkeley.edu
1
Jeanne is an economist at the Research Department, IMF, and CEPR Research Affiliate. Rose is
B.T. Rocca Professor of International Business, Haas School of Business at the University of
California, Berkeley, director of the NBER International Finance and Macroeconomics program,
and CEPR Research Fellow. Work for this research was started while Jeanne visited the
Department of Economics at UC Berkeley, and made progress while Rose visited the European
University Institute, Victoria University, and the Reserve Bank of New Zealand; we thank those
institutions for hospitality. We thank Jose Campa, Philipp Hartmann, Paul Krugman, Marcus
Miller, Matthew Spiegel, Richard Stanton, seminar participants at Auckland and Victoria
Universities, MIT, the Reserve Bank of New Zealand, and especially Robert Flood (with whom
we are engaged in ongoing related research) and Rich Lyons for comments, discussion and
encouragement. A current version of this paper and the STATA data set used to generate the
graphics are available at http://haas.berkeley.edu/-arose.
2
1 Introduction
Why are floating exchange rates so volatile?
In making his celebrated case for flexible exchange rates, Friedman (1953) argued:
“…instability of exchange rates is a symptom of instability in the underlying
economic structure. A flexible exchange rate need not be an unstable exchange rate. If
it is, it is primarily because there is underlying instability in the economic
conditions...”
Friedman's argument is that exchange rate instability is a manifestation of economic
volatility. Exchange rate regimes differ in the mechanisms through which this
underlying volatility is channelled. For instance, "liquidity" shocks may affect the
nominal exchange rate if the latter floats, but the money supply if the rate is fixed.
Underlying systemic volatility cannot be reduced by the regime, only channelled more
or less efficiently. The economy can be thought of as a balloon; squeezing volatility
out of one part (e.g., the exchange market) merely transfers the volatility elsewhere.
2
How then to explain the volatility of floating exchange rates? Flotations of fixed
exchange rates should lead only to temporary increases in exchange rate turbulence,
so long as the underlying economic volatility does not change.
3
But for over a decade
economists have known that exchange rate variability is much higher in flexible
exchange rate regimes than in floats; we provide references and more evidence below.
In theory, exchange rate variability could vary with the exchange rate regime because
of variations in underlying fundamental economic volatility. After all, the exchange
rate regime is chosen by the policy authorities. Unfortunately, there is remarkably
little evidence of a systematic relationship between the exchange rate regime and
macroeconomic phenomena. A number of researchers have shown that the variability
of observable macroeconomic variables such as money, output, and consumption do
not differ systematically across exchange rate regimes; again, we provide evidence
below.
Simply put, countries with fixed exchange rates have less volatile exchange rates than
floating countries, but macro-economies which are equally volatile, at least to a first
approximation. This finding is inconsistent with theories which model either a) the
exchange rate, or b) the exchange rate regime as manifestations of underlying
economic shocks. It is therefore unsurprising that both classes of theories work badly
in practice. The former has been well known at least since the work of Meese and
Rogoff (1983). But the latter is the focus of this paper. Not only do macroeconomic
models have no predictive value for floating exchange rates, they cannot even explain
the difference in exchange rate volatility between fixed and flexible regimes.
2
Much of the argument here is common with Flood and Rose (1998).
3
It is thus unsurprising that many were surprised and struck by the magnitude of the increase in
exchange rate volatility rates following the shift towards generalised floating in 1973, eg, Mussa
(1979) or Obstfeld (1995). Indeed, much of the most influential work in international finance
during the 1970s and 1980s was geared towards rationalising the apparently high level of floating
exchange rate volatility; Dornbusch (1976) is a classic example.
3
This set of observations motivates our paper. Our objective is to establish and account
for the stylised fact that exchange rate volatility differs systematically across exchange
rate regimes in the apparent absence of corresponding differences in macroeconomic
volatility. We are interested in developing a theoretical framework which can
rationalise this phenomenon.
How can one model exchange rate regimes without relying on (non-existent)
differences in macroeconomic fundamentals? Since the only obvious cross-regime
difference is in the behaviour of the exchange rate, we focus our attention on the
structure of the foreign exchange markets themselves. Our main theme is that a theory
of exchange rate regimes cannot ignore the micro-structure of the foreign exchange
market. Rather than assume that it is exogenous, we endogenise the structure of the
markets. Of course, since monetary policy lies at the core of any theory of the
exchange rate regime, macroeconomic fundamentals cannot be ignored altogether.
What is required is an integration of a micro-structural theory of market volatility and
a macroeconomic theory of exchange rate determination. In this paper we provide an
example of such a theory.
To develop a formal theory we need to give content to the notion of microstructure of
the foreign exchange market. The model that we propose is based on "noise trading",
that is trading based on whims, fads and non- fundamental influences. We make a
distinction between foreign exchange markets where a large fraction of traders are
noise traders and those where noise trading is absent or negligible. We identify the
microstructure of the market with the composition of the pool of foreign exchange
traders who operate in the market. Exchange rate volatility, as a result, has two
components: fundamentals and noise. The size of the second component depends on
the structure of the market.
To show that the exchange rate regime affects the presence of noise traders, we
compare two stances for the monetary authority. In a "target zone" the monetary
authorities commit themselves to maintain the volatility of the exchange rate below a
reference value. In a pure float, by way of contrast, monetary policy is set
independently from developments in the foreign exchange market. We demonstrate
that a pure float may give rise to multiple equilibria.
4
In particular, there is sometimes
an equilibrium with low exchange rate volatility and a low number of noise traders,
which exists along with a high exchange rate volatility equilibrium with many noise
traders. Since these equilibria exist for the same level of "fundamental"
macroeconomic volatility, our model is able to rationalise the stylised fact which
macroeconomic models cannot. The reason behind the multiplicity of equilibria is
that in equilibrium, noise traders tend to cluster in the same markets, as is standard in
many models of noise trading (e.g., Admati and Pfleiderer, 1988). The entry of noise
traders in the market for a particular currency changes the structure of risks and
returns in a way that makes it more attractive for other noise traders to join. This
4
The multiplicity of equilibria is also a feature of Flood and Marion (1996) and Flood and Rose
(1998), who use a more primitive stochastic portfolio-balance model with a regime-varying risk
premium and homogeneous agents. See also Hau (1998).
4
results in herd-like behaviour in the migration of noise traders across markets,
although their entry decisions are individually rational.
5
A target zone makes it possible to pin down the economy on the equilibrium with low
exchange rate volatility. A target zone implies a commitment to make monetary policy
responsive to the entry of noise traders in the foreign exchange market. The monetary
authorities offset any increase in exchange rate volatility induced by the arrival of
noise traders, by reducing the volatility of monetary fundamentals. This effectively
insulates exchange rate volatility from potential changes in the structure of the foreign
exchange market. By discouraging the entry of noise traders, the potential for multiple
equilibria disappears and the economy stays at an equilibrium with low exchange rate
volatility. Thus, the mere promise that the authorities will react to the entry of noise
traders, if it is believed, suffices to keep noise traders away.
De Long et. al. (1990) first formalised noise trading in a purely domestic context. A
few papers have subsequently introduced noise trading in the context of foreign
exchange.
6
Mark and Wu (1998) make some progress on the forward discount puzzle
by investigating uncovered interest parity in a model with noise traders. Faruqee and
Redding (1999) show that the entry of liquidity providers can accelerate the reversion
of the exchange rate towards its fundamental value in an environment with noise
traders. A closer precursor to our paper is Hau's (1998) analysis of the free entry of
traders with noisy expectations into a foreign exchange market. Hau finds that
temporary noise may result in higher exchange rate volatility and multiple equilibria
as we do, but abstracts from explicit consideration of macroeconomic or monetary
policy. More generally, these papers do not share our focus on exchange rate regimes.
The paper which is closest in spirit to our analysis of target zones is Krugman and
Miller (1993). Krugman and Miller argue that the real policymaker's' motivation in
instituting target zones is the hope that they will protect their currencies from pure
speculative movements that are not related to the fundamentals. They show that a
target zone may reduce exchange rate volatility in a model with stop-loss traders.
The paper is structured as follows. In section 2 below, we present the stylised
empirical facts of macroeconomic volatility and exchange rate regimes. We show that
standard macroeconomic models cannot be used to understand the data. We then
proceed to the core of the paper in section 3, which presents a model of the foreign
exchange markets with an endogenously determined number of noise traders. We use
the model to analyse monetary policy, and discuss the relevant empirical evidence.
The paper concludes with a brief summary and suggestions for future research.
5
The decisions of noise traders whether or not to enter a particular market are rational in the sense
that they are made on the basis of utility maximisation, and take into account the consequences of
noise.
6
There is a related literature which examines the (de-)stabilising nature of speculation in foreign
exchange markets which does not involve noise trading. For example, Carlson and Osler (1997)
show that rational speculation can be destabilising. Frankel and Froot (1990) argue that feedback
trading rules can increase exchange rate volatility.
5
2 A macroeconomic mystery
We have two objectives in this section of the paper. First, we establish one stylised
fact. We show that exchange rate volatility varies systematically and dramatically
across exchange rate regimes, while observable macroeconomic volatility does not.
Second, we show that macroeconomic models cannot allow one to understand this
finding.
The main purpose of this paper is to provide a theoretical framework consistent with
stylised facts, rather than to establish the latter with new empirics. Consequently, we
are at pains in this section to show that our interpretation of the data is consistent with
existing work and is not particularly sensitive to measurement issues. We use a
variety of sources from the literature to support our case. The evidence is univariate
and multivariate, structural and non-structural, and exploits differences across both
countries and time.
Mussa (1986) established convincingly that nominal and real exchange rate variability
varies substantially and systematically with the exchange rate regime. Mussa used
bilateral dollar exchange rates for a variety of industrial countries from 1957 through
1984. He showed that the variance of real exchange rates was an order of magnitude
greater in the floating period after the Bretton Woods period, than it was during the
Bretton Woods regime of pegged rates.
7
In his comment on Mussa, Black (1986)
argued that "empirical workers in the field of exchange rates will not regard this as
new information" and cites work which precedes Mussa's.
8
Mussa's evidence is
especially convincing to us for two reasons. First, it is essentially undisputed, at least
to our knowledge. Second, the objective of Mussa's paper is unrelated to ours: Mussa
was interested in rejecting exchange rate models with flexible prices.
Baxter and Stockman (1989) extended Mussa's work on exchange rates to other
macroeconomic variables. Using data for a variety of OECD and developing
countries, Baxter and Stockman examine the variability of output, trade variables, and
both private and government consumption, using different de-trending techniques.
They are “unable to find evidence that the cyclic behaviour of real macroeconomic
aggregates depends systematically on the exchange-rate regime. The only exception is
the well-known case of the real exchange rate.”
The evidence presented by Mussa, Baxter and Stockman is compelling, but
incomplete. It relies on differences in the behaviour of individual countries across
time. Time-specific effects may confound such empirical work. Examining the
behaviour of a cross-section of countries during a single time-period is a way to check
the stylised fact for consistency, and is also of intrinsic interest. Further, the analysis
is univariate. Models which link changes in exchange rate volatility explicitly to
changes in macroeconomic volatility are potentially useful adjuncts, since the latter
7
Mussa's “first important regularity” is “The short term variability of real exchange rates is
substantially larger when the nominal exchange rate between these countries is floating rather than
fixed.”
8
Certainly Stockman (1983) provides consistent evidence earlier. See also Aliber (1976) and other
references given by Black.
6
effects can be potentially subtle and difficult to uncover with univariate techniques.
Most importantly, it is only by using macroeconomic models that we will be able to
reveal their inability to explain the phenomenon with which we are concerned.
These objections have been addressed by the work of Flood and Rose (1995). They
begin with the conventional monetary model of the exchange rate. A simple money
market equilibrium is posited in the domestic “Centre” country, linking the natural
logarithm of the money stock (m) deflated by the (log of the) price level (p) to the
interest rate (i) at a point in time t; the same condition characterises the foreign
country (denoted with an asterisk). Prices are assumed to be perfectly flexible, and
purchasing power parity is satisfied at all times so that the (log of the) price of foreign
exchange (e) is simply the ratio of price levels. The model can be written:
ttt
ipm
α
−
=
−
(1)
***
ttt
ipm α−=− (2)
*
ttt
ppe −= (3)
so that:
).((
**
ttttt
iimme −+−= α (4)
The model's ability to explain exchange rate volatility can be tested by comparing the
characteristics of the left- and right-hand sides of equation (4).
Figure 1 contains quarterly time-series evidence on Deutschemark exchange rates
from 1959 through 1996 for twenty OECD countries; this represents the left-hand side
of equation 4. Comparable evidence for the right-hand side is portrayed in Figure 2;
we use MI and short maturity money market interest rates, and a consensus estimate
from the literature for the interest semi-elasticity of money demand (unity).
910
The message from the two figures is straightforward. Consistent with Mussa's finding,
Figure 1 shows that nominal exchange rate variability is low when exchange rates are
fixed (in the 1960s or for strict EMS peggers like Austria and the Netherlands), and
high when exchange rates float. But macroeconomic fundamentals (as dictated by
equation (4)) do not exhibit regime-varying volatility in Figure 2.
Comparable cross-section evidence is available in Figure 3. For each of the twenty
countries, the standard deviation of the exchange rate (estimated for each country over
time) is graphed against the standard deviation of the right-hand side of equation (4).
For generality, we use the United States in place of Germany as the reference country.
9
The data set is taken from the International Monetary Rind's International Financial Statistics CD-
ROM, and has been checked for errors. It is available as a STATA data set at
http://haas.berkeleyedu/-arose.
10
FIood and Rose (1995), show that the argument holds for a very wide range of reasonable
parameter values.
7
There is again no evidence of any clear relationship between macroeconomic and
exchange rate volatility.
11
It might be objected that the empirical rejection of equation (4) is hardly surprising
since it is derived from assumptions, in particular instantaneous PPP, that are
notoriously rejected by the data. The main result, however, turns out to be very robust
to changes in model specification. For instance, Flood and Rose (1995) extend the
model to include the effects of sticky prices, real income, random shocks, and a
variety of other issues without changing the results. The intuition behind this
insensitivity is simple: such extensions simply make the right-hand side of equation
(4) more complicated combinations of money, output, interest rates, and prices, and
lags. Flood and Rose found that they could not match the volatility characteristics of
exchange rates to those of structural economic fundamentals, even allowing for
stochastic structural disturbances. In particular, traditional economic fundamentals of
structural models do not have the regime-varying volatility needed to match the
regime-varying volatility of exchange rates. Where Flood and Rose provide structural
evidence across time for a number of countries, Rose (1994) provides comparable
cross-country data, with similar results.
3 A micro-structural theory of exchange rate regimes
The analysis in the preceding section makes us pessimistic about the ability of purely
macroeconomic models to explain regime-varying exchange rate volatility. An
alternative strategy is to consider models where the structure of the foreign exchange
market changes with the exchange rate regime.
The model we present mixes elements from two hitherto disparate branches of
economic theory, the macroeconomic theory of exchange rate determination, and the
noise trading approach to asset price volatility. As in chemistry, we make the
experiment illuminating by combining two components which are as pure as possible.
Thus, we pick simple conventional building blocks, uncontaminated by tangential
complications. On the macroeconomic side, we use the conventional monetary model
of the exchange rate. On the microstructure side we employ the model of noise trading
developed by De Long et al. (1990). As shown above, the macroeconomic part of the
model performs poorly by itself. We now show that one can improve the fit of the
model by, paradoxically, adding noise.
In the model we present, exchange rate volatility has two components:
macroeconomic fundamentals and noise which is unrelated to fundamentals. The size
of the noise component is endogenously determined; it depends on the decisions of
noise traders who decide whether or not to enter the foreign exchange market. Their
decisions to enter depend in turn, on the volatility of the exchange rate and the risk
premium on foreign bonds. Thus, monetary policy determines the exchange rate not
11
These ocular results can be verified more formally with statistical tests, as in Flood and Rose
(1995). For instance, the regression slope for the data portrayed in Figure 3 is slightly negative
with a t-statistic of 0.1.
8
only directly, by changing the relative money supplies, but also indirectly, by affecting
the composition of the foreign exchange market.
3.1 Macroeconomic fundamentals
We continue to maintain (l)-(3), so that simple monetary equilibria hold and
purchasing power parity is satisfied continuously. We further assume that the domestic
country is in a steady state with constant money supply, interest rate and price level.
Hence the expression for the exchange rate can be re-written dropping the time index
for domestic variables:
).()(
**
ttt
iimme −+−= α (5)
We initially assume that the difference between domestic and foreign money supplies,
*
t
mm − follows a stochastic i.i.d. normal process centred on zero. This variable will
assume the role of economic “fundamentals” in the remainder of the analysis.
12
For
the moment we assume that this policy variable is exogenous, as would be appropriate
if the exchange rate floats freely. We relax this assumption when we consider official
exchange rate policy below.
The interest rate is determined by equilibrium in the international bonds market. We
assume that investors in the international bonds market care about the return of their
portfolio measured in terms of domestic currency. The domestic currency may be
viewed as the international currency which serves as the standard of comparison in
evaluating portfolio returns.
13
Investors are risk averse and require a risk premium to
hold bonds denominated in foreign currency.
The quantity of foreign external liabilities results, in equilibrium, from the foreign
current account and the balance of payments. These external liabilities may take the
form of bonds denominated in either currency. The supply of bonds denominated in
foreign currency results from the foreign fiscal and monetary authorities' actions, in
particular the respective shares of domestic- and foreign currency-denominated bonds
on the asset side of the central bank's balance sheet. We assume hereafter that the
foreign authorities maintain the supply of foreign currency denominated bonds,
expressed in terms of domestic money, at a constant level
_
B
. This assumption is
made for the sake of analytical convenience, and can be relaxed without changing the
thrust of our results.
14
12
Since we maintain this structural equation throughout our analysis, our model cannot rationalise
the Flood-Rose (1995) mystery discussed above.
13
Implicitly we think of the domestic country as large and the foreign country as a small open
economy. One could generalise this assumption by assuming that some investors care about their
portfolio returns in terms of foreign currency, or that all investors evaluate their returns in terms of
a currency basket, but this would complicate the model without producing additional insights.
Note also that one does not need to make the distinction between nominal and real returns in terms
of domestic currency since the domestic price level is constant.
14
Some assumption is needed, since there is no natural way to endogenise the currency composition
of the foreign country's external debt. The assumption we make has the advantage of keeping the
9
3.2 Micro-structure: trading behaviour
Foreign exchange traders are modelled as overlapping generations of investors who
live for two periods and allocate their portfolio between domestic and foreign one-
period nominal bonds in the first period of their life. Traders have the same
endowments and tastes, but differ in their ability to trade in the foreign bonds market.
Some of them are able to form rational expectations on risk and returns costlessly,
while others have noisy expectations and must pay an entry cost to invest in foreign
bonds. We refer to the former as “informed” traders and the latter as “noise” traders.
Noise traders trade on the basis of fads which are unrelated to fundamentals; informed
traders do not (though they have no special information advantage). Noise traders also
have higher costs of market participation than informed traders.
At each period the new-born traders form a continuum of measure
∈
j
[0, 2]. Each
individual trader j receives an endowment of W units of domestic currency. She then
decides whether or not to enter the foreign bonds market. We denote by
j
t
δ the
dummy variable characterising the entry decision of trader j at time t; it equals one if
she enters, zero if not. Traders enter the market for foreign bonds if this increases
their utility. Trader j's entry decision is taken before the time t monetary policy shock
is revealed, on the basis of the information available at t - 1:
∀
j, t 1=
j
t
δ )0()1(
11
=≥=⇔
−−
j
t
j
t
j
t
j
t
j
t
j
t
UEUE δδ (6)
where
j
t
U is the utility of a new-born trader j at time t, and the expectations operator
bears the trader's index to allow for heterogeneity (the expectations operator without
index denotes the rational expectation).
A trader who has entered the foreign bonds market invests
j
t
b in foreign bonds so as
to maximise the expected utility of her end-of-life wealth, expressed in terms of the
domestic currency. We assume that trader j's portfolio allocation problem at time t is:
))exp((max
1
j
t
j
t
j
t
b
WEU
j
t
+
−−= α (7)
where
j
t
W
1+
is the end-of-life wealth of trader j. It is given by:
)()1(
11
j
j
t
j
t
j
t
cbWiW −++=
++ ρ
δ (8)
Trader j's end-of-life wealth is equal to the trader's initial endowment times the yield
of domestic bonds plus, if j enters, the excess return on foreign bonds minus a fixed
cost that must be borne in order to enter the foreign bonds market. The excess return
on foreign bonds between t and t + 1 is given by:
ieei
tttt
−−+=
++
)(
1
*
1
ρ (9)
model simple. It would not be very difficult to consider alternatives, such as a stochastic supply of
foreign currency denominated bonds expressed in terms of domestic currency.
10
The cost
j
c reflects the costs associated with entering the foreign market for trader
j.
15
We assume that foreign exchange traders are heterogeneous with respect to this
cost.
There are two types of traders: informed traders, located in the interval [0, 1], and
noise traders, in (1, 2]. Informed investors have an accurate knowledge of the way the
exchange rate is determined, and bear no entry cost. They are knowledgeable about
the economy, can process new information costlessly and make their decisions on the
basis of rational expectations about the future. Thus, for ]1,0[
∈
j one can write:
)()(
11 +−
=
ttt
j
t
EE ρρ (10)
)()(
11 ++
=
ttt
j
t
VarVar ρρ (11)
0
=
j
c (12)
where )(
1+t
j
t
E ρ and )(
1+t
j
t
Var ρ are the expected value and conditional variance of the
excess return on domestic bonds as evaluated by trader j at period t, and )(
1+t
j
t
E ρ and
)(
1+t
j
t
Var ρ are their mathematical counterparts.
Noise traders, by way of contrast, have imperfect knowledge of the determinants of
the exchange rate and bear a positive entry cost. We adopt the (standard) assumption.
that noise traders perceive the second moment of returns correctly, but allow their
perception of first-moments to be affected by noise that is unrelated to economic
fundamentals.
16
The noise is common across traders; there is no private information in
the model. Moreover noise traders bear a strictly positive entry cost. Formally we
assume that for j
∈
(1, 2]:
tt
j
t
vE +=
−
+
ρρ )(
1
(13)
)(
1+t
j
t
Var ρ = )(
1+tt
Var ρ (14)
γ
=
j
c (15)
where
−
ρ is the unconditional mean of the excess return (or average risk premium) and
the noise term
t
v is a stochastic i.i.d. normal shock common across j and uncorrelated
with
*
t
m . We interpret the noise term as a fad which is widespread but non-
fundamental. Unlike De Long et al. (1990), our noise traders do not make systematic
errors in their prediction of excess returns.
15
These costs are much discussed in the literature, and may include informational problems, tax
issues, and other phenomena. There is no presumption that they are small, given the size of the
"home market effect"; Lewis (1995) provides a survey.
16
For evidence of bias in exchange rate expectations, see Frankel and Froot (1987).
11
We link the size of noise traders' errors to economic uncertainty by assuming that the
variance of the noise is proportional to the true unconditional variance of the exchange
rate:
)()( eVarvVar
λ
=
(16)
where
λ
is a positive coefficient. (Assuming that )(vVar is constant is easier but less
plausible.)
3.3 Equilibrium
An equilibrium in this model consists of stochastic processes for the exchange
rate }{
t
e , the risk premium }{
t
ρ
, and individual traders' decision rules }{
j
t
δ and }{
j
t
b ,
such that at each period t, }{
j
t
δ satisfies the entry condition (6), }{
j
t
b is the solution to
the optimal portfolio allocation problem (7), and the market for domestic bonds is in
equilibrium:
∫
∞+
=
0
_
.djbB
j
t
j
t
δ (17)
This equilibrium appears to be difficult to determine, since it involves entry decisions
by a continuum of heterogeneous agents in a stochastic environment. However, we
exploit the assumption that the monetary shock is independently and identically
distributed, which suggests that the set of equilibrium individual decision rules takes a
simple stable form.
We solve the model with a "guess-and-verify" technique, first postulating its
properties, then checking that they are satisfied. We conjecture that:
(i) the fluctuations of the exchange rate are identically and independently
distributed around an average level
_
e ,
(ii) all informed traders, and a constant number of noise traders, n, enter the foreign
bonds market at each period.
We characterise the equilibrium by proceeding in two steps. First, we determine the
equilibrium exchange rate, taking the number of noise traders in the foreign market as
given. We then endogenise the number of noise traders by using the no-entry
condition.
3.4 Analysis with an exogenous number of noise traders
In equilibrium the foreign interest rate and the risk premium are identically and
independently distributed around average values that we denote
_*
i and
_
p respectively.
The average risk premium is equal to the average difference between the foreign and
domestic nominal interest rates:
12
iip −=
_*_
(18)
which taking the expectation of equation (5), implies:
__
pe α−+ (19)
A rise in
_
e corresponds to an appreciation of the foreign currency. Equation (19)
implies that a higher average risk premium, by decreasing the demand for the foreign
currency, leads to its depreciation.
The risk premium is determined by equilibrium in the market for bonds denominated
in foreign currency. If the excess return on these bonds is normally distributed (which
is true in equilibrium, as we show below), it is well-known that maximising (7) is
equivalent to maximising the mean- variance objective function:
)(
2
)(
11
j
t
j
t
j
t
j
t
WVar
a
WE
++
− (20)
and the demand for bonds denominated in the foreign currency by an individual trader
is given by:
)(
)(
1
1
+
+
=
t
j
t
t
j
t
j
t
aVar
pE
b
ρ
(21)
The equality of demand and supply in the bonds market implies:
)(
)(
)(
1
_
1
1
_
+
+
+
+
+=
tt
t
t
j
t
tt
aVar
vp
n
aVar
E
B
ρ
ρ
ρ
(22)
)(
)()(
_
1
eaVar
vnE
ttt
++
=
+
ρρ
Taking the expectation of (22) at t-1 then gives an expression for the average risk
premium:
)(
1
_
_
eVar
n
B
a
+
=ρ (23)
The average risk premium is increasing with the variance of the exchange rate, the
coefficient of absolute risk aversion and the quantity of bonds per trader. We can then
derive the equilibrium exchange rate by substituting the definition of
1+t
ρ
into (22)
and using (5) to substitute out the interest rate differential, which gives:
t
t
t
nv
mm
ee
α
α
α +
+
+
−
=
1
1
_
_
*
(24)
13
This expression confirms that the fluctuations of the exchange rate are i.i.d. normal in
equilibrium.
Taking the variance of (24) and using (16) to substitute out the variance of the noise
allows us to close the characterisation of equilibrium with an expression for exchange
rate variability:
17
222
)1(
*)(
)(
n
mmVar
eVar
λαα −+
−
= (25)
The variance of the exchange rate depends on fundamentals and noise. The
fundamental component is proportional to the variance of money supply. The novelty
in this model is the noise component, which is proportional to the square of the
number of noise traders active in the market. An exogenous increase in the number of
noise traders unambiguously increases the variance of the exchange rate, which tends
to raise the risk premium. On the other hand, it also increases the total number of
traders demanding foreign bonds, which lowers the risk premium. That is, noise
traders have two counter- acting roles in our model; they both a) create risk and b)
share risk. As a result, the impact of the extra noise traders on the equilibrium risk
premium is non-monotonic. The ambiguous effect of noise trading on the risk
premium is portrayed in Figure 4. This ambiguity - the fact that the risk premium can
be decreasing or increasing with the number of noise traders - lies at the heart of our
model.
18
3.5 Endogenous Entry
We now endogenise the composition of the pool of active traders. The entry decision
for informed traders is trivial: they bear no entry cost and always enter the foreign
bonds market in equilibrium. However, a noise trader enters only if the benefit of
diversifying her portfolio into foreign bonds exceeds her cost of entry. We show in
the appendix that this condition takes the form:
γρ ≥))(,(
_
eVarGB (26)
where )),(,(
_
eVarpGB the gross benefit of entry for noise traders, is given by:
λ
λ
ρ
++
+
= 1log(
2
1
)()1(2
))((
_
_
aeVara
eVarpGB (27)
17
Note that this expression yields a positive value for the variance of the exchange rate for all
∈
n
[0, 1] iff ,/)1(
22
ααλ +< a condition that we assume satisfied thereafter.
18
Figures 4, 5 and 6 were obtained for the following values of the parameters:
,
3
,
4
,
1
=
=
=
λ
α
a
_
B
= 4, and
γ
= 0.3. The variance of relative money supply, ),(
∗
− mmVar was set to 1 in figure
4.
14
The partial derivatives of equation (27) have an intuitive interpretation. The benefit of
entry, as assessed by noise traders, is increasing with the risk premium and decreasing
with exchange rate variability. But in equilibrium both the risk premium and the
variance of the exchange rate are functions of the number of noise traders that enter
the foreign bond market; this can be seen in equations (18) and (25). This circularity,
as we now show, can generate multiple equilibria.
We illustrate our result in Figures 5-7. These shows the net benefit of entry for the
marginal noise trader, for three different levels of the variance of (monetary)
“fundamentals”. The benefit depends on the number of noise traders, n, as well as the
impact that these noise traders have on exchange rate variability and the risk premium,
Var(e) and
_
ρ .
Figure 5 portrays a low level of fundamentals variance. It shows that the only possible
equilibrium is one in which noise traders do not enter the foreign bonds market. The
variance of macroeconomic fundamentals is so low that the benefit of entry is always
negative for the marginal entrant, however many noise traders are present.
Figure 6 is the more interesting case; it portrays an intermediate level of fundamental
variance. There are two stable equilibria in this scenario, corresponding to points A
and C (point B is unstable). Point A corresponds to an equilibrium with low exchange
rate volatility and a low risk premium. Here, the foreign market does not offer noise
traders a large enough gain to induce many of them to enter. But there is another
equilibrium at point C, which corresponds to a high volatility, high risk premium
equilibrium. In this equilibrium, more noise traders are attracted to the foreign bonds
market by the high risk premium that they themselves generate by entering the market.
Thus, our model can generate different levels of exchange rate volatility for the same
level of macroeconomic volatility. We can rationalise the stylised fact which
motivated this paper by simply labelling point A a “fixed exchange rate regime” and
point C a “floating exchange rate regime”.
Figure 7 is symmetric to Figure 5; fundamental volatility is so high that there is only
one equilibrium with high exchange rate volatility and noise traders present.
Figure 8 portrays the relationship between the variance of fundamentals and exchange
rate volatility. The lower branch corresponds to equilibria in which noise traders do
not enter the foreign markets (or only a small number of them do); the higher branch
to equilibria with entry; and the branch in the middle to unstable equilibria. If the
variance of fundamentals, Var(m - m*), is below a threshold there is a unique
equilibrium as in Figure 5; noise traders stay away from the foreign market. If this
variance is above a much higher threshold, the equilibrium is again unique since noise
traders always enter (the Figure 7 case). In between the two thresholds there is a “zone
of multiplicity.” If the variance of fundamentals falls inside this intermediate range
there are two stable equilibria. One has low exchange rate volatility and limited entry
of noise traders; the many noise traders who are present in the other make the
exchange rate more volatile.
Under a pure float, hence, there is no simple relationship between the volatility of
monetary fundamentals and the exchange rate. Two countries with similar
15
fundamentals may exhibit radically different levels of exchange rate volatility. In the
high volatility equilibrium, exchange rate volatility is excessive, in the sense that it is
higher than the level that can be ascribed to the traditional macroeconomic
fundamentals. This excessive volatility can be eliminated with a policy which
switches equilibria, as we now show.
3.6 Exchange Rate Policy
The purpose of this section is to analyse policies that reduce exchange rate noise.
These policies work by allowing the policymaker to co-ordinate activity to a low
volatility equilibrium. We consider an exchange rate “target zone,” following
Krugman and Miller (1993).
Krugman and Miller argue that the main cost of floating exchange rates, as perceived
by policy-makers, is that they leave currencies vulnerable to purely speculative price
movements that are unrelated to fundamentals. They interpret a target zone as a
mechanism designed to reduce exchange rate volatility by limiting the impact of these
non-fundamental influences. Our model is well suited to a discussion of such issues.
Suppose that the foreign monetary authorities wish to implement the following
monetary process:
~
*
tt
mmm ∆=− (28)
where
~
t
m∆ is an exogenous i.i.d. normal process.
19
In the absence of noise traders,
this process implies the following exchange rate process:
α+
∆
+=
1
~
_
~
t
t
m
ee (29)
This monetary regime can be characterised in two ways. Either the monetary
authorities can announce a) that the money supply will fluctuate around a constant
level with a given variance )]([
~
mVar ∆ , or b) that the exchange rate will fluctuate
around its mean with a given variance ])1/()()([
2
~~
α+∆= mVareVar . We identify the
first type of announcement with a floating exchange rate and the second with an
exchange rate target zone.
In the absence of noise traders, it does not matter whether the monetary regime is
expressed in terms of the money supply or the exchange rate. Since these variables
are linked by (29), specifying the process for }{
~
t
m∆ equivalent to specifying the
process for {
~
}{
t
e .
19
We take this process as given. Our model does not allow us to derive the optimal policy rule from
primitive policy objectives such as output stabilisation, given its lack of nominal frictions.
16
Things are different in the presence of noise traders, at least in the zone of
multiplicity. Suppose that )(
~
t
mVar ∆ is in the range where multiple equilibria can
arise. From Figure 8 we know that the monetary process is consistent with both a low
exchange rate volatility equilibrium (point A) and a high exchange rate volatility
equilibrium (point C). That is, specifying a monetary process leaves the composition
of the foreign exchange market indeterminate and allows for multiple equilibria.
Taking the monetary process (28) as an exogenous policy variable means that noise
traders may rationally decide to enter the foreign market. If enough of them enter, the
economy winds up at the high volatility equilibrium. Benign neglect of the exchange
rate can result in excessive volatility.
A solution to the multiplicity problem is to announce explicit bands for the exchange
rate - a target zone. This announcement, so long as it is credible, keeps noise traders
away and pins down the economy on the equilibrium with low exchange rate
volatility.
20
Of course, an exchange rate target zone has implications for
macroeconomic fundamentals, since (24) implies:
.
~
*
t
t
t
nvmmm α−∆=− (30)
Under an exchange rate target zone, the domestic monetary authorities commit
themselves to offsetting the impact of the entry of noise traders on exchange rate
volatility by changing the money supply. Knowing that the authorities would react in
this way to their entry, noise traders stay away from the markets. A target zone -
provided that it is credible - pins down the market to the low-volatility equilibrium.
There is a free lunch of exchange rate stability.
This policy analysis begs the question of why any country should care about exchange
rate volatility at all. Our model has been kept highly stylised; it abstracts from country
size, openness, and the nominal frictions that make exchange rate policy decisions
non-trivial. Still, it is interesting to note that in our model reducing exchange rate
volatility may not involve any sacrifice in terms of monetary autonomy. This violates
Mundell's “Incompatible Trinity” of fixed exchange rates, monetary autonomy and
capital mobility. A threat by the monetary authority (to react if noise traders enter)
changes the composition of the market. By discouraging the entry of noise traders, the
market is steered to a low volatility equilibrium where intervention is unnecessary.
Words speak loudly enough that actions are unnecessary.
21
20
Equation (29) ensures that (28) is satisfied because the exchange rate variance and the risk
premium resulting from (29) make noise traders prefer to stay out of the foreign bonds market.
21
By picking out a single equilibrium, the expectations "honeymoon" offered by the exchange rate
target zone in our model is stronger than in Krugman's (1991) model, where the exchange rate is
stabilised by the promise of interventions that have to be fulfilled in equilibrium.
17
3.7 Empirical Evidence
It is difficult to provide direct empirical support for our model. While it is possible to
estimate exchange rate volatility and risk premia, there are few data available on
foreign exchange trading volume. Information on disaggregated trading activity is
even more rare.
22
These would be critical components of any serious test of our
theory. However there are a few suggestive pieces of evidence which support our
argument.
One key part of our model is the prediction that an increase in trading volume is
associated with an increase in the level of exchange rate volatility, since the increase
in volume comes, at the margin, from noise traders. To our knowledge, there are only
two sources of data on exchange rate volume; both have problems.
The Chicago Mercantile Exchange has data series on volumes of trade in their futures
markets. However, there are problems with the data set. First, it only includes futures
market volume, ignoring spot markets, options and other derivatives. Second, the
rates are all bilateral dollar rates. Third, there are gaps in the series. Fourth, there are
only a limited number of currencies traded on the markets. Bearing these caveats in
mind, it is still instructive to examine the data set.
As our regressor we use annual data on CME trading volume for the years 1973
through 1989.
23
These are available for the following currencies (vis-à-vis the
American dollar): (British) pound sterling; Canadian dollar; (German) DM; (Italian)
lira; (Japanese) yen; (Mexican) peso; Swiss franc; (Dutch) guilder; French franc; and
Australian dollar. As our regressand, we use annual exchange rate volatilities, the
estimated standard deviation of the first-difference in the natural logarithm of the
monthly exchange rate (using the IFS end-of-month exchange rate series "ae"). We
use non-overlapping monthly data to arrive at a single estimate for annual exchange
rate volatility. We are left with a panel of annual data (spanning year and exchange
rates).
A simple OLS regression of exchange rate volatility on volume yields a positive slope
coefficient (as predicted) which is insignificantly different from zero at conventional
levels (the robust t-statistic is 1.5). However, as our model shows, exchange rate
volatility and volume are simultaneously determined, making OLS an inappropriate
estimator. As an instrumental variable, we use the natural logarithm of distance
between the USA and the foreign country. This variable is suggested by the literature
on the "gravity equation" of international trade. In our data set, distance is correlated
with volume (the slope coefficient is 4) and thus is an admissible instrumental
variable. When we compute IV estimates, the slope coefficient in our
volatility/volume regression remains positive, and grows in both size and statistical
significance; its t-statistic is 2.9. That is, greater exchange rate trading volume is
associated with more exchange rate volatility, as our model predicts. This panel
22
Indeed, disaggregated data on trading activity is non-existent over any reasonable span of time (eg,
a year).
23
Futures trading began in the middle of 1972 and a continuous data set covering the period after
1990 is not currently available to us. Some of the currencies were not traded throughout the entire
period, so that we have 129 annual observations.
18
evidence twins well with the case study of the Tokyo foreign exchange market by Ito,
Lyons and Melvin (1998) which found that extra trading was associated with higher
exchange rate volatility.
The Bank for International Settlements collects data on a wider range of foreign
exchange products, including spot trading and most derivatives. However, these data
are broken down into only a few bilateral markets, and only for trades involving either
the dollar or DM. Further, these data are currently only available for 1992 and 1995.
Thus, we are unable to perform a regression analysis. Still, there is some evidence
that increased volume is associated with greater exchange rate volatility. The 1996
survey shows that the vast majority of foreign exchange transactions occur between
floating exchange rate regimes; only one of the top ten exchange markets was a fixed
rate.
24
The 1993 survey shows that of the top thirteen foreign exchange markets, only
two were for fixed exchange rate regimes.
25
Again, this evidence is consistent with
our model.
There are a few other pieces of support for our approach. In their survey of market
practitioners, Cheung and Wong (1998) show that most traders believe that non-
fundamentals are of pervasive importance in foreign exchange markets, especially in
the short run. Market practitioners also believe that increased speculation raises both
volatility and liquidity Flood and Rose (1996) find that deviations from uncovered
interest parity (often interpreted as risk premia) are much smaller under fixed
exchange rate regimes than in floating rate regimes, again consistent with our model.
Mark and Wu (1998) also make progress in the same area using a model which relies
on noise traders. Rose (1996) finds that our model's focus on stated exchange rate
policy and the violation of the “Incompatible Trinity” twins with the
OECD data; Evans and Lyons (1999) show that the order flow which lies at the heart
of most micro-structure models is an important determinant of exchange rate
movements.
Individually, none of these pieces of evidence is convincing. Jointly, we think of them
as weak corroboration of our model, and a strong encouragement, to us and other
researchers, to develop new data sets. A more definitive test awaits better data.
4 Conclusion
Floating exchange rates tend to be volatile; fixed exchange rates are not. Does the
volatility in floating rates get transferred to another part of the economy when rates are
fixed? No. To a first approximation, countries with fixed exchange rates have less
volatile exchange rates than floating countries, but macro-economies which are
equally volatile.
24
Table F-4 indicates that the DM/FR rate was in seventh place in terms of volume in April 1995.
25
Tables 2-B and 2-C shows that the DM/Pound market in sixth place in terms of volume, while the
DM/FFY rate was in eleventh place
19
This well-known finding is inconsistent with theories which model the exchange rate
regime (or the exchange rate itself) as a manifestation of underlying macroeconomic
shocks. Unsurprisingly, such theories have performed poorly when applied to the data
In this paper we have presented a micro-structural model which can be used to
understand exchange rate volatility in floating exchange rates. Our model introduces
noise traders, who create exchange rate volatility if they choose to enter the foreign
exchange market in order to diversify their portfolios and buy foreign bonds. Noise
traders benefit from holding foreign bonds, but pay a cost from entering foreign
markets while also creating undesirable exchange rate volatility.
For a range of fundamental macroeconomic volatility, our model generates multiple
equilibria; the noise traders can either be present or absent from the markets. If they
are present, they generate exchange rate volatility; we think of this as being a floating
rate regime. But there is another, “fixed rate,” equilibrium without noise traders and
with a more stable exchange rate. With a suitable policy stance, the policy authorities
can co-ordinate activity to this equilibrium. In fact, an appropriate exchange rate
target zone can lower exchange rate volatility without any macroeconomic at all.
Since the policy reduces exchange rate volatility by ensuring that the fixed exchange
rate equilibrium is chosen, reducing exchange rate volatility is costless in our model.
In our model, exchange rate policy works by affecting the composition of the foreign
exchange market, not by the traditional mechanism of subordinating monetary policy
to an exchange rate target.
Our micro-structural model of exchange rate volatility is extremely stylised and
unable to capture many aspects of reality. But, as we have shown, much more
complicated macroeconomic models are even less capable of rationalising the facts.
Our model has not been directly validated with any empirics. Still, we think of it as a
useful new theoretical starting point to investigate exchange rate volatility.
20
Appendix
This appendix derives the net benefit of entering the domestic market for noise traders
(equation (27)). We proceed backwards, computing first the expected utility of a noise
trader j after she has entered the market, and then her expected utility before entry.
Noise trader j's expected utility after entry is given by:
))(exp())1(exp()1(
1+
−++−−==
t
j
t
j
tj
j
t
j
t
j
t
abEacWiaUE ρδ (31)
where the excess return
1+t
ρ
is perceived by the noise trader to be normally
distributed, with mean
tt
j
t
vpE +=
+
_
1
)(ρ and conditional variance )(eVar . To simplify
the algebra we adopt the notation
2
)(
e
eVar σ= and
2
)(
v
vVar σ= for the remainder of the
appendix.
It is then an exercise to compute:
))
2
)((exp()(exp(
2
11
j
tet
j
t
j
tt
j
t
j
t
b
a
EababE σρρ −−=−
++
. (32)
To prove (32), we denote the innovation in the excess return at t + 1 by
)(
111 +++
−=∈
t
j
ttt
E ρρ and make use of the identity:
−+∈+=
∈
+
++
+
+
j
tet
j
t
j
tt
j
t
ee
t
t
j
t
b
a
Eabbaab
2
1
2
1
2
3
22
2
1
1
2
)()(
2
1
2
σρσ
σσ
ρ (33)
Using the fact that the conditional distribution of
1+
∈
t
is normal with variance
2
e
σ , one
obtains:
( )
∫
∞+
∞−
+
+
++
∈
∈
−−=−
1
2
2
2
11
2
exp
2
1
)exp()exp(
t
e
t
e
t
j
tt
j
t
j
t
dababE
σ
πσ
ρρ
−−=
+
j
tet
j
t
j
t
b
a
Eab
2
1
2
)(exp σρ
∫
∞+
∞−
+
+
∈
∈+
−
1
2
2
1
2
2
)(
exp
2
1
t
e
t
j
te
e
d
ba
σ
σ
πσ
which gives (32) since the integral on the right-hand side is equal to unity, as it is the
integral of a normal density function.
Trader j’s portfolio allocation problem involves maximising the quadratic function of
j
t
b that appears in (32). The solution
j
t
b is given by (21) and the expected utility after
maximisation is:
21
.
2
)(
)1(exp)1(max
2
2
1
−++−−==
+
e
t
j
t
j
i
t
j
t
j
t
b
E
acWiaUE
j
t
σ
ρ
δ (34)
At the time of her entry decision, noise trader j does not know what her expectation of
the excess return will be after entry. However she knows that this expectation will be
given by
tt
j
t
vE +=
+
_
1
)( ρρ , where
t
v is normally distributed with variance
2
v
σ . Hence
the expected utility before entry is given by:
))1(exp()1(
1 j
j
t
j
t
j
t
acWiaUE ++−−==
−
δ
dv
vv
v
v
v
e
t
−
+
−
∫
∞+
∞−
2
2
2
2
_
2
exp
2
1
2
)(
exp
σ
πσ
σ
ρ
(35)
and using the decomposition
22
22
2
22
2
22
2
2
2
2
2
_
2
)(
)(222
)(
ev
ev
ev
v
t
evv
t
e
t
v
vv
σσ
σσ
ρ
σσ
σ
σσ
ρ
σσ
ρ
+
+
+
+
+
=+
+
(36)
One can compute the integral term on the right-hand side of (35) as:
+
−
+
=−
+
−
∫
∞+
22
2
22
2
2
2
2
2
(2
exp)
2
exp(
2
1
2
)(
(exp
evev
e
t
v
t
v
e
t
dv
vv
σσ
ρ
σσ
σ
σ
πσ
σ
ρ
which gives a closed-form expression for trader j’s expected utility under entry:
.
(2
)1(exp)1(
22
2
22
2
1
+
+
−+−
+
−==
−
j
evev
e
j
t
j
t
j
t
acWiaUE
σσ
ρ
σσ
σ
δ (37)
).)1(exp()0(
1
WiaUE
j
t
j
t
j
t
+−−==
−
δ (38)
It follows from equations (37) and (38) that trader j’s utility is higher under entry if:
.1
(2
exp
22
2
22
2
<
+
+
−
+
j
evev
e
ac
σσ
ρ
σσ
σ
(39)
Taking the logarithm of this inequality shows that entry occurs if equation (26) is
satisfied.
22
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Reserve Bank of San Francisco Economic Review 1, 19-30.
Rose, Andrew K (1996) “Exchange Rate Volatility, Monetary Policy, and Capital
Mobility: Empirical Evidence on the Holy Trinity” The Journal of
International Money and Finance.
Stockman, Alan C (1983) “Real Exchange Rates under Alternative Nominal
Exchange-Rate Systems” Journal of International Money and Finance 2-2,
147-166.
25
Figure 1: Time series evidence on exchange rates
Percentage changes of price of 1DM
Exchange Rate Volatility
26
Figure 2: Time series evidence on macroeconomic fundamentals
Percentage change in *)],(*)[( iimm
−
+
−
German Centre
Macroeconomic Fundamentals
27
Figure 3: A cross section of fundamental and exchange rate volatilities
28
Figure 4: Exchange rate volatility, the average risk premium, and noise trading
29
Figure 5: The (non-) incentives to enter with low fundamental volatility
30
Figure 6: The zone of multiplicity with moderate fundamental volatility
31
Figure 7: Incentives to enter with highly volatile fundamentals
32
Figure 8: The non-linear relationship between exchange rate and fundamental
variance
