On the credibility issue of exchange rate target zones
©1999
Diplomarbeit
64 Seiten
Zusammenfassung
Inhaltsangabe:Zusammenfassung:
Währungskrisen, wie die des Europäischen Währungssystems (EWS) 1992, können noch immer nur unzureichend vorausgesagt werden. Es wird jedoch anerkannt, dass die Glaubwürdigkeit eines jeden Währungssystems ausschließlich endogen bestimmt wird, d.h. durch Interaktion der einzelnen Akteure an den Wechselkursmärkten.
Die vorliegende Arbeit untersucht, ob ein bestimmtes Glaubwürdigkeitsmaß, eingeführt von Avesani, Gallo und Salmon (1995), geeignet ist, die große EWS Krise 1992 vorauszusagen. Dabei handelt es sich um ein dynamisches Spiel, in dem die beiden Akteure (Zentralbank und Finanzmarkt) miteinander interagieren und damit die Glaubwürdigkeit des Systems aushandeln. Es wird gezeigt, dass o.g. Glaubwürdigkeitsmaß ein geeigneter Indikator für Währungskrisen darstellt, was empirisch anhand des französischen Francs, der italienischen Lira und des niederländischen Guldens nachgewiesen wird.
Introduction:
The objective of the present study is to present the literature of exchange rate target zones and to explore empirically the Avesani-Gallo-Salmon credibility measure for selected currencies belonging to the Exchange-Rate Mechanism (ERM) of the European Monetary System (EMS). In the past decade the empirical literature on real world target zones mushroomed considerably, but still, its overall significance in explaining strains in the relevant foreign exchange markets remains relatively low. In this context, I will explore whether the Avesani-Gallo-Salmon (1995) credibility measure demonstrates more power than earlier studies in the analysis of the 1992/3 EMS crisis.
It is not my intention to pursue a normative analysis whether flexible or fixed exchange rates or intermediate regimes are superior to one another in terms of their economic implications. This question is beyond the scope of my analysis.1 Given that target zones are applied frequently in the real world, I purely want to figure out, whether their application is sensible on grounds of a firm commitment of the policy-makers.
Exchange rate bands have been discussed widely after the breakdown of the Bretton Woods System in 1973. Although experiences with the fixed exchange rate regime were disappointing in the 1960s, increased volatility and/or overshooting of the exchange rates in the 1970s let economists doubt the famous argument by Friedman (1953) that speculation would stabilize exchange rate movements in the floating system. McKinnon (1976) reckoned that […]
Währungskrisen, wie die des Europäischen Währungssystems (EWS) 1992, können noch immer nur unzureichend vorausgesagt werden. Es wird jedoch anerkannt, dass die Glaubwürdigkeit eines jeden Währungssystems ausschließlich endogen bestimmt wird, d.h. durch Interaktion der einzelnen Akteure an den Wechselkursmärkten.
Die vorliegende Arbeit untersucht, ob ein bestimmtes Glaubwürdigkeitsmaß, eingeführt von Avesani, Gallo und Salmon (1995), geeignet ist, die große EWS Krise 1992 vorauszusagen. Dabei handelt es sich um ein dynamisches Spiel, in dem die beiden Akteure (Zentralbank und Finanzmarkt) miteinander interagieren und damit die Glaubwürdigkeit des Systems aushandeln. Es wird gezeigt, dass o.g. Glaubwürdigkeitsmaß ein geeigneter Indikator für Währungskrisen darstellt, was empirisch anhand des französischen Francs, der italienischen Lira und des niederländischen Guldens nachgewiesen wird.
Introduction:
The objective of the present study is to present the literature of exchange rate target zones and to explore empirically the Avesani-Gallo-Salmon credibility measure for selected currencies belonging to the Exchange-Rate Mechanism (ERM) of the European Monetary System (EMS). In the past decade the empirical literature on real world target zones mushroomed considerably, but still, its overall significance in explaining strains in the relevant foreign exchange markets remains relatively low. In this context, I will explore whether the Avesani-Gallo-Salmon (1995) credibility measure demonstrates more power than earlier studies in the analysis of the 1992/3 EMS crisis.
It is not my intention to pursue a normative analysis whether flexible or fixed exchange rates or intermediate regimes are superior to one another in terms of their economic implications. This question is beyond the scope of my analysis.1 Given that target zones are applied frequently in the real world, I purely want to figure out, whether their application is sensible on grounds of a firm commitment of the policy-makers.
Exchange rate bands have been discussed widely after the breakdown of the Bretton Woods System in 1973. Although experiences with the fixed exchange rate regime were disappointing in the 1960s, increased volatility and/or overshooting of the exchange rates in the 1970s let economists doubt the famous argument by Friedman (1953) that speculation would stabilize exchange rate movements in the floating system. McKinnon (1976) reckoned that […]
Leseprobe
Inhaltsverzeichnis
ID 6735
Rau-Göhring, Matthias: On the credibility issue of exchange rate target zones
Hamburg: Diplomica GmbH, 2003
Zugl.: Bonn, Universität, Diplomarbeit, 1999
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CONTENTS
i
Contents
1 Introduction
1
2 The Basic Target Zone Model
4
2.1
Analytical Solution for the Free Float . . . . . . . . . . . . . . . . . . . .
4
2.2
Brownian Motion and Stochastic Calculus . . . . . . . . . . . . . . . . .
7
2.3
The Target Zone Solution . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.4
Empirical Evidence of the Basic Model . . . . . . . . . . . . . . . . . . .
15
3 Extensions to the Basic Target Zone Model
16
3.1
Intra-marginal Intervention
. . . . . . . . . . . . . . . . . . . . . . . . .
17
3.2
Sterilized Intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.3
Possibility of Realignments . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4 Measures for Credibility
23
4.1
Svensson's Simple Measure . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4.2
The Interest Rate Differential . . . . . . . . . . . . . . . . . . . . . . . .
24
4.3
The AGS Measures of Absolute and Relative Credibility . . . . . . . . .
26
5 Empirical Exploration of the AGS R-Credibility Measure
31
6 Conclusion
34
A General Solution of Differential Equation (17) in Ch. 2.3.
36
B The Algorithm for Approximation of the Fundamental Term and the
Calculation of R-credibility
38
C Figures
51
1
INTRODUCTION
1
1
Introduction
The objective of the present study is to present the literature of exchange rate target
zones and to explore empirically the Avesani-Gallo-Salmon credibility measure for se-
lected currencies belonging to the Exchange-Rate Mechanism (ERM) of the European
Monetary System (EMS). In the past decade the empirical literature on real world target
zones mushroomed considerably, but still, its overall significance in explaining strains in
the relevant foreign exchange markets remains relatively low. In this context, I will ex-
plore whether the Avesani-Gallo-Salmon (1995) credibility measure demonstrates more
power than earlier studies in the analysis of the 1992/3 EMS crisis.
It is not my intention to pursue a normative analysis whether flexible or fixed ex-
change rates or intermediate regimes are superior to one another in terms of their eco-
nomic implications.This question is beyond the scope of my analysis.
1
Given that target
zones are applied frequently in the real world, I purely want to figure out, whether their
application is sensible on grounds of a firm commitment of the policy-makers.
Exchange rate bands have been discussed widely after the breakdown of the Bretton
Woods System in 1973. Although experiences with the fixed exchange rate regime were
disappointing in the 1960s, increased volatility and/or overshooting of the exchange
rates in the 1970s let economists doubt the famous argument by Friedman (1953) that
speculation would stabilize exchange rate movements in the floating system. McKin-
non (1976) reckoned that there may be an insufficient amount of speculation due to
institutional constraints, other economists suspected speculators to be ill-informed and
therefore act in an irrational manner. However, the desire to avoid speculative pres-
sure and crises within the foreign exchange market resulted in the IMF's Guidelines for
Floating, which entailed the possibility that a country brings its exchange rate "within,
or closer to, some target zone of rates"(see IMF (1974, p.181). In 1979 the European
Monetary System was established and -although different in design- subsequently some
Latin American, Scandinavian and South-East Asian countries set up unilateral target
zones for their exchange rates, too. With the Plaza Communiqué of the G-5 coun-
tries in 1985 and its successor, the Louvre Accord of 1987, also the major industrial
1
The pros and cons for flexible and fixed exchange rates have been discussed widely throughout the
last three decades. I do not mention them here, but the reade should be aware that it is not clear from
a theoretical point of view whether for instance exchange rate target zones have a benefitial effect to
the economies which apply it, or not.
Strong policy conclusions may be only derived through incorporating sufficient economic detail in
some kind of general equilibrium model to tackle the welfare issues in choosing between exchange rate
regimes.
1
INTRODUCTION
2
countries returned to manage their exchange rates rather than either fix or float them.
Exchange-rate target zones (target zones from now on) were seen as a sensible third way
of exchange rate management. They are defined by a central parity for the exchange
rate and a fluctuation band around this parity.
2
The first comprehensive target zone proposal was made by Williamson (1983). He
presented a model that subsumes the following features:
3
1. The monetary authority is expected to alter its monetary policy when the exchange
rate approaches one of the edges of the zone.
2. The exchange rate should be kept within a published but wide zone.
3. This zone should be defined in terms of the real effective exchange-rate.
4. The central parity should be set according to the "fundamental equilibrium ex-
change rate", meaning the rate needed to reconcile internal and external balance
in the medium term.
5. The zone should be adjusted in response to relevant information that alter the
estimate of the equilibrium exchange rate.
In the following years this proposal for reform was discussed and improved exten-
sively. But the debate solely concentrated on the policy design and institutional charac-
teristics relating to target zones. It took nearly another five years until Paul Krugman
presented his seminal paper on the theoretical foundations of exchange rate behaviour
within target zones in end 1987, published as Krugman (1991).
Krugman's analysis has become classical and his main argument can be understood
intuitively. Figure 1 shows the exchange rate on the vertical axis, measured as the
logarithm of units of domestic currency per unit of foreign currency, and the logarithms
of a compound fundamental term on the horizontal axis. Given certain assumptions,
Krugman (1991) shows that the rational expectations equilibrium relationship between
the fundamentals and the exchange rate in a credible target zone exhibits the S-shape
2
There have been various target zone proposals in the literature, such as the adjustable peg, the
crawling peg and the flexible target zone. Unless otherwise noted I will refer to the standard target
zone model, which contains a fixed central parity with a relatively small fluctuation band. Thus, in a
strong sense, also the Bretton Woods System may be seen as a target zone.
3
See Williamson (1983) on a detailed description.
1
INTRODUCTION
3
illustrated in figure 1
4
. Contrary to the free float solution, as represented by the 45-
degree line, the target zone solution shows a bias toward the central parity or to put it
differently, the target zone creates a "honeymoon effect" that stabilizes the exchange rate
within the band. Some economists refer to this as a "free lunch"-gain of a target zone.
However, Bertola and Caballero (1992) show that imperfect or lacking credibility of the
authorities' policy may lead to an inverted S-shape for the exchange rate solution. This
implies that in absence of a credible policy, target zones may have a destabilizing effect
for the foreign exchange market rather than a "honeymoon effect". Credibility in this
context is the policy makers willingness, ability and emphasis to follow the announced
path of exchange rate policy rather than diverging from it.
The differences between the credible and the imperfect-credible target zone models
build the motivation for my analysis. Following Bertola and Caballero (1992), policy-
makers should not opt for a target zone when their commitment to such a policy is
relatively weak. In this context, I will focus my analysis on the aspects of the EMS,
notably the exchange rates of the Lira and French Franc vis-à-vis the Deutschmark.
Although there has been an intensive discussion on the credibility of the EMS
5
, little
focus was set on an endogenous evolution of credibility; merely all publications assume
a fixed stock of credibility that the authority could either deplete or maintain.
To come up with an alternative assessment, which explicitely includes the endoge-
nous evolution of credibility using the Avesani-Gallo-Salmon credibility measure, the
remainder of my thesis is organized as follows: Drawing on Krugman (1991), section 2
derives the equilibrium measure of the exchange rate within a target zone and discusses
both the model's assumptions and implications. Section 3 relaxes some of the most
critical assumptions and tries to infer the conclusions for the exchange rate behaviour.
Section 4 presents alternative measures for credibility and examines the usefulness of the
measures concerned. In section 5 the application of the Avesani-Gallo-Salmon (1995)
credibility measure for the exchange rates of the Italian Lira (Lit) and French franc (FF)
vis-à-vis the Deutschmark (DM) is outlined and results are interpreted. Finally, section
6 concludes.
4
Note that Krugman's basic target zone model will be derived and discussed extensively throughout
chapter 2.
5
See for instance Flood, Rose and Mathieson (1990), Weber (1991), and Campa, Chang and Reider
(1997).
2
THE BASIC TARGET ZONE MODEL
4
2
The Basic Target Zone Model
Regardless of the fact that already the Bretton Woods System has been a target zone
with its +/- 1 percentage fluctuation band, it required more than two decades until
the work of Halm (1965) on the band proposal was taken up again by Professor Paul
Krugman. Using the technical framework of regulated Brownian motion, his elegant
target zone model, published as Krugman (1991), has become both the standard and
the starting point for almost every contribution to the target zone literature followed
within the last seven years. The current chapter is dedicated to derive and explain the
model formally, followed by a verbal discussion of its implications. A brief overview of
empirical tests of the basic target zone model links up with chapter 3.
2.1
Analytical Solution for the Free Float
Krugman's analysis starts from the presumption that the exchange rate is priced as any
other financial asset.
6
He states that the exchange rate can "be viewed as the price of
a compound asset" (Krugman (1991, p. 677)), depending on current fundamentals plus
expectations of future values of the exchange rate. Changes in the exchange rate are
assumed to be purely stochastic (as will be seen by choosing the underlying stochastic
pattern), which is in line with empirical evidence.
7
Using a deterministic flexible price
monetary model, Krugman's basic target zone model can now be derived thoroughly:
8
The basic equations of the monetary model are
s
t
= p
t
- p
t
(1)
6
The asset market approach offers a realistic explanation of exchange rate variability in a rational
expectations based analysis. The "normal" flow analysis of demand and supply determines the price
as an outcome of the intersection of both schedules. Thus, price changes are caused by shifts in the
relevant markets' demand and/or supply functions. However, in an asset approach the market as a
whole changes its view of what the asset is worth. For instance changes may be due to the arrival of
new information in the market. It is important to note that prices can change without any trade taking
place just by changing views of buyers and sellers. This is very distinct from the analysis within a flow
demand and supply analysis.
7
Meese and Rogoff (1983) have left us without illusion by showing that a random walk model
outperforms forecasts of structural models of the exchange rate in almost any case. As DeGrauwe and
Dewachter (1990) have shown, this may be attributed to the factor that little errors in the estimation
of economic agents produce large changes in the exchange rate dynamics predicted by the underlying
structural model.
8
Note that all equations are denoted in natural logarithms unless otherwise noted. An asterisk
typifies foreign variables.
2
THE BASIC TARGET ZONE MODEL
5
E
t
[ds
t
|
t
] /dt = (i
- i
)
t
(2)
m
t
- p
t
=
1
y
t
-
2
i
t
,
1
,
2
> 0
(3)
where (1) denotes the purchasing power parity (PPP) relationship. Equation (2) is
the uncovered interest rate parity
9
, where E [· |
t
]
stands for the mathematical con-
ditional expectation operator, measuring expected instantanous exchange rate change
conditional on the time t information set
t
,
containing the structure of the model and
all variables dated t or earlier. Equation (3) stands for the money market equilibrium,
the right-hand side reflecting the demand for real balances, depending on real income
y
and the domestic interest rate i.
1
is the income elasticity of real money demand
and, since i is not the logarithmic value,
2
is the interest rate semielasticity of money
demand. Thus, the model ties together money market equilibrium, PPP and uncovered
interest rate parity.
Solving (1) for the domestic price level and the uncovered interest rate parity for
the domestic interest rate and including the resulting equations in (3), one yields
s
t
= m
t
- p
t
-
1
y
t
+
2
E [ds
t
|
t
] /dt + i
t
.
(4)
Creating v
t
-
1
y
t
- p
t
+ i
t
as a general purpose term reflecting money demand
shocks, the exchange rate equation takes the form
s
t
= m
t
+ v
t
+
2
E [ds
t
|
t
] /dt,
(5)
which is the basic equilibrium equation for the exchange rate in this monetary model.
For simplicity v is called the velocity term. Money supply and velocity are often put
together in a compound fundamental term k. The economic interpretation of equation
(5) is relatively easy: in equilibrium, the exchange rate equals the sum of economic
fundamentals plus a term proportional to the expected change in the exchange rate.
Bertola and Caballero (1992) state that this is in line with the assumption of forward
looking agents, who price a given asset not only on behalf of current factors but also
on expectations about future events. Whereas the money supply process is assumed to
be under perfect control of the monetary authority, the velocity term follows a Wiener
process or Brownian motion process at all times, s.t.
9
In this context the risk premium term in the uncovered interest rate parity is neglected. On a
further discussion of the interest parity relationship, see chapter 4.2.
2
THE BASIC TARGET ZONE MODEL
6
dv
t
= dt + dW
t
,
(6)
where denotes the drift (which Krugman assumes to be zero), the instantaneous
standard deviation and W stands for the Wiener process. Before I turn to a descrip-
tion of this stochastic process assumption, let me briefly introduce the exchange rate
behaviour under a free float: in absence of intervention of the monetary authority (i.e.
money supply is held constant at, for simplicity, zero level m = 0), the compound
fundamental term should follow a Brownian motion process just like the velocity term.
Froot and Obstfeld (1991b) show that for equation (5) there exists a unique bubble-
free saddlepath solution, which is simply the expected present value of the fundamental
path according to
s
t
=
Z
=t
exp
h
-( - t)
-1
2
i
E
t
[k
] d ,
(7)
where -( - t)
-1
2
discounts the stream of expectations of the evolution of the
fundamental term.
10
The highlight of this exchange rate path is that, as Froot and
Obstfeld (1991b) assert, it is valid for all types of exchange rate regimes and all types
of regime switches. This is a very important result is also used in Krugman (1991).
The latter author assumes that the exchange rate can be written as a twice-continuously
differentiable function of the compound fundamental term:
s
t
= G(k
t
).
(8)
In the free float case considered above, the general solution for the exchange rate is
simply
s
t
=
Z
t
h
exp
h
(t
- )
-1
2
i
E
t
[k
t
| k
]
i
d
=
Z
t
h
exp(t
- )
-1
2
i
-1
2
(k
t
+ (
- t))d
= k
t
+
2
for =0
=
k
t
.
(9)
Thus, referring to figure 1, in times of a free float the exchange rate process {s
t
}
should follow a random walk
11
among the 45-degree line, given that the exchange rate
10
Given that time t is the current time, this is simply k
t
plus the expectations about the future path
of the fundamental term.
11
The Brownian motion process is nothing else as the continuous time equivalent of a random walk
with deterministic drift.
Curly brackets denote sequences.
2
THE BASIC TARGET ZONE MODEL
7
depends linearly on the fundamental term and a zero expected exchange rate change.
12
2.2
Brownian Motion and Stochastic Calculus
Although Krugman (1991, p. 671) states that "there is no good economic reason for
assuming a random walk", the assumption of the continuous-time Brownian motion
matches empirical observations of floating exchange rates (see Meese and Rogoff (1983)
for an empirical exploration). The intuition behind this assumption can be explained
by two points strongly connected with a variable that follows a continuous-time random
walk:
1. There are no discrete jumps of the variable, i.e. it is continuous at all points in
time; this is an appealing feature, since arbitrage in efficient markets should rule
out such discrete jumps.
2. Changes in the variable are normally distributed for a given time interval with a
zero mean and a variance proportional to the time interval (which is helpful in
any empirical analysis).
This latter phenomenon will be explained in more detail in the following paragraphs.
Since probability theory is often a hurdle in understanding the target zone literatur, let
me now briefly introduce the definition of a stochastic process, the highlights of the
Wiener process/Brownian motion
13
and lastly Itô's lemma as a main tool for analyzing
the target zone solution within the following subsection.
Basically, a stochastic process is defined as a sequence of random variables, s.t.
Definition 1 (Stochastic Process)
{x
t
}, t T, on a probability space (,=,P)
where x
t
is a measurable mapping x : T × R
T
, s.t.
x () =
{x
() ,
T},
where T is the index set. The probability space contains the sample space and
the -field =, which is the set of subsets of the sample space with the usual properties
of -fields. P is the appropriate probability measure on the measurable space. For
12
The second factor is given through absence of intervention and the properties of the Wiener process,
which will be outlined in the next subsection.
13
Recall that these expressions can be used synonymously.
2
THE BASIC TARGET ZONE MODEL
8
generality of this definition, the Borel -field B has to be introduced. If there exists a
topology
14
on the sample space , a Borel -field B is defined as the -field generated by
the system of open intervals. This definition is important for the ability to measure the
mapping x : T × R
T
.
As Dothan (1990, p.162) denotes, this function is measurable
on the probability space' -field iff for every set from the Borel -field the inverse image
x
-1
(B)
belongs to =.
Though very technical, this definitions are extremely useful for analyzing the Brow-
nian motion process. Denote C as the space of continuous real-valued functions and
B
C
as the -field generated by the open intervals on C. Then, according to Davidson
(1994, p. 442n), the Wiener measure is the probability measure on (C, B
C
)
with the
properties
1. W (x
0
= 0) = 1.
2. W (x
t
a) =
h
(2t)
-1/2
iR
a
-
exp(
-
2
/2t)d,
where 0 < t 1 and denotes the
increments of the process.
3. Increments are independent on the -field of all decidable events.
This garantuees that the Wiener process starts in the origin a.s., entails independent
increments on the sample space, and increments follow the (Gaussian) normal distribu-
tion with x
t
j
N(0,t
j
)
and E
h
x
t
j
x
t
i
i
= t
j
t
i
.
15
For the process assumed by Krugman
(1991), i.e. dv
t
= dt + dW
t
,
the latter two conditions garantuee that the moments of
the independent increments are given by E
p
[dW
t
] = dt
=0
= 0
and E
p
[(dW
t
)
2
] =
2
dt.
Thus, although the process has a zero expected change, and it does not exhibit sta-
tionary behaviour
16
. According to Hendry (1995), the Wiener process is of unbounded
variance, s.t. erratic jumps are possible despite being continuous.
Lastly, as indicated by Dothan (1990, p.179), the probability that sample paths of
the Wiener process are differentiable at some point t 0 is zero. This implies the
14
Given a set M , a topology is defined as a set T of subsets of M that exhibits certain properties
not delineated here. The purpose of a topology, as a rough idea, is to examine the invariate properties
of distortions.
15
Note that "" is the mininum operator.
Doob (1953) shows, that a Brownian motion with independent increments has to be a martingale,
using that the increments are Gaussian distributed. This will be an important feature to analyze in
the empirical exploration.
16
The process does neither show strong stationarity, nor does it display weak stationarity. This holds
as a general result for a random walk.
2
THE BASIC TARGET ZONE MODEL
9
random walk behaviour, since the process becomes to be non predictable, which is in
line with the stylized facts of exchange rate dynamics as noted earlier.
Having introduced these salient features of the Brownian motion process, let me
know briefly derive Itô's lemma, which will be essential in the analysis of the target
zone solution for the exchange rate. Firstly, the term filtration has to be defined.
Definition 2 (Filtration)
Every increasing sequence
0
1
2
... of
-
fields of the sample space is a filtration to the underlying probability space.
Intuitively, a filtration is nothing else as the information structure of the process
(which can be assumed to expand as the process developes), where terminal information
in -field is complete. This is crucial for the Brownian motion process' characteristic
as a martingale, which is a kind of "local mean property". Shortly if (
n
)
is a filtration
generated by the process {x
t
}, the martingale property implies that
E [x
n
|
m
] = x
m
a.s. for n > m 0.
Thus, at any time and state the expectation of the future value of x is its current
value. Any martingale can now be represented as a sum of stochastic integrals of the
general form I(, t) =
R
t
0
f (, )dM (, ),
where t [0,1] and {M} is the quadratic
variation process.
One can show (see for instance Davidson (1994, pp. 504-507)) that for a determin-
istic quadratic variation process and orthogonal increments, I(t) is a square integrable
martingale, i.e. it shows L
2
convergence. If {M} is a Brownian motion, the integral is
referred to as the Itô integral, defined as
Definition 3 (Itô Integral)
Z
t
0
x
s
dW s := lim
m
Z
t
0
x
ms
dW
s
.
(10)
The Itô integral is thus a special stochastic integral with respect to the Wiener pro-
cess. Itô's lemma, however, describes the changes of variables in that specific stochastic
integral, which will give us the general solution for a certain Brownian motion process.
17
Itô's lemma is simply
17
See Dothan (1990) or Davidson (1994) for a comprehensive development of the Itô integral and
Itô's lemma in a continuous time context.
2
THE BASIC TARGET ZONE MODEL
10
f (W
t
)
- f (W
0
)
=
Z
t
0
f
0
(W
s
)dW
s
+
1
2
Z
t
0
f
00
(W
s
)ds
df (W
t
)
=
f
0
(W
t
)dW
t
+
1
2
f
00
(W
t
)dt.
(11)
This clearly links the considerations on probability theory and stochastic calculus
with the formal solution of the target zone exchange rate solution. Reconsider that
equation (8) shall be the solution to the Krugman model. Since the compound funda-
mental term follows a Brownian motion process, Itô's lemma can be used to derive the
saddlepath solution for G(k
t
)
.
2.3
The Target Zone Solution
Using the tools illustrated in the last subsection, the target zone solution for the ex-
change rate becomes now only a matter of routine.
To rule out multiple equilibria Froot and Obstfeld (1991a), Froot and Obstfeld
(1991b) and Flood and Garber (1983) reason that one has to define the exchange rate
rule thoroughly. In detail you need the exact intervention rule of the monetary au-
thority and the announced boundary conditions. Krugman (1991) makes the following
assumptions in the basic target zone model:
1. Interventions take place through a change in the money supply term m, i.e. inter-
ventions are assumed to be not sterilized.
2. Interventions are assumed to be infinitesimal only and occur only at the band's
margins rather than in the interior of the exchange rate band.
3. The exchange rate band's limits are preannounced by the authority and are per-
fectly credible in the market's view.
Svensson (1991a) shows that imposing a target zone for the exchange rate is syn-
onymous with a unique, pre-specified fundamental band, given a certain policy rule for
intervention. Thus, the monetary authorities purpose has to be a fundamental k
t
with
k
-
k
t
-
k .
(12)
To manage the money supply's target, Svensson (1991a) describes interventions as
2
THE BASIC TARGET ZONE MODEL
11
dm = dL
- dU,
(13)
where dL and dU represent the intervention at the lower edge and the upper edge
of the exchange rate band, respectively.
18
Since the velocity term is still assumed to
follow a continuous-time random walk, the compound fundamental term follows a so-
called "regulated Brownian motion". To state it differently, the exchange rate does
not necessarily follow equation (5) as the compound fundamental term does not evolve
according to (6). The new stochastic pattern can be described by
dk
t
= dt + dW
t
+ dL
t
- dU
t
.
(14)
Froot and Obstfeld (1991a) use a very elegant two step procedure to obtain the
saddlepath solution for the exchange rate, which I will scetch now. In the first step
the general solution for equation (5) is derived, given that fundamentals follow (6). In
a second step the authors show that there exists a unique saddlepath solution for the
exchange rate for a specific stochastic process and appropriate boundary conditions.
Step 1:
Froot and Obstfeld (1991a) assume that the exchange rate develops in line with the
twice continuously differentiable function s
t
= G(k
t
)
. Itô's lemma can be used to obtain
an expression for the expectations term in equation (5). Thus,
E [ds
t
|
t
] /dt = E[dG(k
t
)
|
t
]/dt
= G
0
(k
t
) +
2
2
G
00
(k
t
).
(15)
Including this result and subsuming the fundamental terms in the variable k
t
, one
yields a second-order differential equation
G(k
t
) = k
t
+
2
"
G
0
(k
t
) +
2
2
G
00
(k
t
)
#
.
(16)
The general solution to equation (16) is
G(k
t
) = k
t
+
2
+ A
1
exp(
1
k
t
) + A
2
exp(
2
k
t
),
(17)
18
dL equals zero as long as k >k
-
and dU equals zero as long as k <
_
k . Both regulators are non-
negative.
2
THE BASIC TARGET ZONE MODEL
12
where A
1
and A
2
are constants of integration that encompass the initial conditions,
such as the boundary conditions set by the policy maker.
1
and
2
are the characteristic
roots of the equation
2
+
·
2
2
¸
-
2
2
2
= 0.
One can find a detailed derivation of the general solution in Annex A. The discussion
of dynamic stability is excluded.
As one can now see, that the free float solution of equation (9) can be explained by
initial conditions A
1
= A
2
= 0.
As Obstfeld and Rogoff (1996) cite, the constants of
integration can be determined using the assumption that the authority sets an initial
boundary condition for exchange rate movements, the target zone. This leads to the
second step in the analysis of Froot and Obstfeld (1991a), where one can see that the
free float is nothing but a special solution of equation (18) below, where the fundamental
band's limits are assumed to increase to infinite.
Step 2:
As already noted in the context of (14), the authority is assumed to defend the
target zone with probability one. Krugman's assumption of infinitesimal and marginal
intervention implies that the fundamental process must have some "absorbing limits",
namely k
-
,
_
k.
19
As long as the exchange rate is within the band, it should float without
intervention of the monetary authority. But, as Froot and Obstfeld (1991a) denote,
expectations about future regime shifts in the underlying stochastic process have im-
portant influences on the saddlepath solution of the exchange rate. Equation (7) should
hold as long as the fundamentals are within their prespecified boundaries, where parallel
equations (5) and (6) hold.
The next step -according to Froot and Obstfeld (1991a)- has to be the imposition
of exact boundary conditions, to determine the constants of integration. Implementing
reflecting boundary conditions
·
k
-
,
_
k
¸
in (7) and using (17), they yield the complex
saddlepath solution
G(k
t
) = k
t
+
2
1 +
e
2
_
k+
1
k
t
- e
2
k
-
+
1
k
t
+ e
1
k
-
+
2
k
t
- e
1
_
k+
2
k
t
e
2
k
-
+
1
_
k
- e
2
_
k+
1
k
-
(18)
Although this condition seems to be very complicated, Svensson (1991b) and Weber
(1991) show that (18) is simply equation (17), where the solutions for the constants of
19
Reconsider Svensson (1991a)'s argument that a certain exchange rate band corresponds to a specific
fundamental band.
Details
- Seiten
- Erscheinungsform
- Originalausgabe
- Jahr
- 1999
- ISBN (eBook)
- 9783832467357
- ISBN (Paperback)
- 9783838667355
- DOI
- 10.3239/9783832467357
- Dateigröße
- 1 MB
- Sprache
- Englisch
- Institution / Hochschule
- Rheinische Friedrich-Wilhelms-Universität Bonn – Wirtschaftswissenschaften
- Erscheinungsdatum
- 2003 (April)
- Note
- 2,0
- Schlagworte
- wechselkurse wechselkurs-zielzonen europäisches währungssystem glaubwürdigkeit währungskrisen
