Optimal Exchange Rates: A Stochastic Control Approach
AHMED S. ABUTALEB
Currently with Cairo University, School of Engineering, Giza, Egypt, and
previously with MIT Lincoln Lab, Lexington, Massachusetts,
MA 02421, USA
MICHAEL G. PAPAIOANNOU
Visiting Scholar, International Monetary Fund,
700 19th Street, NW, Washington,
DC 20431, USA
Abstract: We present novel formulas for the determination of the optimal values of the exchange rate and the
external (foreign currency) debt. A stochastic differential equation is developed that relates the net worth of an
economy to both its exchange rate and level of external debt. Using the martingale optimality principle, the optimal
values for the exchange rate and the external debt are derived. Applied to Egypt, we find that its actual nominal
exchange rate has stayed above its optimal value since the early 1990s, except for two years, and that the actual
external debt-to-net worth ratio is lower than the optimal one for most of the time. Since 2013 the actual external
debt-to-net worth ratio is close to the risky levels of the optimal values.
Keywords: Exchange Rate, Martingales, Martingale Optimality Principle, Optimal Stochastic Control, Optimal
Debt
1 Introduction
We study the interaction between the real exchange
rate and external debt in an environment where both
the return on capital and the real rate of interest are
stochastic variables. Our model of such dynamic
interaction reveals that an "overvalued" exchange
rate, i.e., the cost of an identical basket of goods is
more expensive domestically than abroad at the
prevailing nominal exchange rate, leads to a steady
rise in the external debt. In turn, the accumulation of
debt due to ensuing trade account deficits and the
interest rate payments on the debt exert downward
pressure on the exchange rate, which may lead to a
currency (balance of payments) crisis. In particular, a
significant depreciation of the currency increases the
debt burden and increases the probability of a debt
crisis, [1].
The vast literature on exchange rates devotes
considerable attention to the determinants of the
equilibrium real exchange rate, e.g., ([2], [3], [4], [5],
[6], and [7]), focusing on three main methodologies:
(1) the fundamental equilibrium exchange rate
(FEER) or macroeconomic balance (MB) approach,
(2) the behavioral equilibrium exchange rate (BEER)
approach, and (3) the stock-flow approach to the
equilibrium real exchange rate. Further, long-term
equilibrium real exchange rates are assumed to
depend on exogenous and policy variables that need
to be sustainable, or on “long-run fundamentals.”
Regarding exogenous variables, their sustainability is
not under the control of domestic policymakers (the
empirical estimation of their sustainable levels and
the assessment of the effect of those levels on the
long-run equilibrium real exchange rates present
problems). With regard to policy variables, policies
are sustainable if they are optimal. In this case, long-
run equilibrium real exchange rates are also
“desirable” ones, as they are based on optimal
policies. However, if sustainable (optimal) policies
are unlikely to be implemented, i.e., if actual policies
are likely to be inappropriate and therefore the
dynamics of the real exchange rate would be
determined by these policies, the “desirable” long-
run equilibrium real exchange rates would provide a
misleading indication of where the real exchange rate
is heading. This will make the long-rum equilibrium
Received: July 23, 2021. Revised: November 13, 2022. Accepted: December 5, 2022. Published: December 31, 2022.
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DOI: 10.37394/23207.2022.19.182
Ahmed S. Abutaleb, Michael G. Papaioannou
E-ISSN: 2224-2899
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real exchange rate the relevant concept for
formulating exchange rate policy. In particular, if the
policies in place are not sustainable (non-optimal),
e.g., trade or capital account restrictions, or other
distortionary policies, then the long-term equilibrium
real exchange rate can be argued that it is not a true
equilibrium rate. Under these conditions, the
distinction between the true long-run equilibrium real
exchange rate and “desirable” long-run equilibrium
real exchange rate becomes indistinguishable.
Accordingly, the challenge is to identify the relevant
set of fundamental determinants of the long-run
equilibrium real exchange rate, determine the
appropriate sustainable policies (the optimal policy
set), and estimate the “desirable” long-run
equilibrium real exchange rate.However, although
the link between the real (and nominal) exchange
rate, productivity, and external debt is crucial for the
economic and financial stability of a country, we are
not aware of any studies that have dealt with this
issue.
1
Since both exchange rates and external debt
are dynamic concepts, representation of their
interlinks warrants a suitable dynamic approach.
Typically, the analysis of the dynamic interaction
between variables is carried out in optimal value
terms. This paper aims to develop such a framework
of dynamic behavior and to apply it to Egypt. This
framework, appropriately calibrated for specific
country conditions, can serve as a policy tool for
government authorities to assess the state of their
external debt and draw conclusions on the following
exchange rate policies.
In this novel study, we examine the determination of
the optimal real exchange rate that stems from the
maximization of a utility function as opposed to the
equilibrium real exchange rate. We find that the
optimal real exchange rate is proportional to the
return on domestic investments, and inversely related
to the U.S. interest rate. This result was obtained
using the Martingale Optimality Principle ([9], [10],
[11], [12], and [13]). The paper is organized as
follows: Section 2 presents the problem formulation
of the dynamic interaction between the exchange rate
and external debt, while section 3 highlights the
optimization problem and gives the optimal values of
the exchange rate and foreign debt. Section 4
1
[8] estimate optimal levels of foreign debt for Egypt
during 1985-2008, without taking explicitly into account
the dynamic interactions between foreign debt and the
exchange rate.
discusses the results and provides some concluding
remarks on the applicability and shortcomings of the
proposed framework and points out some areas for
future research. An Appendix presents all necessary
derivations.
2 Problem Formulation
Our prototype model is a simplification of a complex
economy that focuses on shocks (economic and
financial disturbances) that have led to crises. The
model is analytically tractable, with all derived
equations having an economic interpretation.
However, introduction of more realistic assumptions
generates a less transparent solution and economic
interpretation. The prototype model is proposed as a
“benchmark” model. We assume two sources of
uncertainty: one source concerns the value of GDP
and the return on capital, and the second concerns the
interest rate on loans and bonds. It is important to
recognize that there is a correlation between these
two sources of uncertainty.Adopting a stochastic
calculus formulation, the model is expressed in real
terms. The net worth or wealth, X(t), in nominal
terms is defined as [1]:
)()()( tLtKtX
(1)
Taking the derivative of both sides we get:
)()()( tdLtdKtdX
where K(t) is the capital owned by the residents of
the country, and
L(t) is the country’s external debt, denominated in
the $US
The change in capital, dK(t), is the rate of investment
I(t):
dttItdK )()(
(2)
Also, real consumption C(t), real investments I(t),
and real GDP Y(t) are related through the equation:
)()()()(
)()()(*)()()(
)()( tdLtedttItp
dttLtitptedttYtp
dttCtp
( 3)
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where p(t) is the domestic price index,
)(* tp
is the
US price index, e(t) is the nominal exchange rate
(domestic price of foreign exchange or units of
domestic currency per unit of $US). Note that the
formulation of this equation excludes government
expenditures and assumes that the external sector is
represented by the change in the external debt due to
the trade account, dL(t), and the interest rate
payments on the debt, i(t)L(t).
In particular, as we assume that the external debt L(t)
is denominated in $US, the term i(t)L(t) stands for
the interest payments in $US, at the rate of interest
i(t), on US dollar denominated loans and bonds.
Further, we assume that the accumulation of debt
refers to annual intervals.
By dividing eqn. (3) by p(t), we obtain the real
consumption C(t) (measured in domestic-goods
units) as:
)(
)(
)()(*
)(
)()(
)(
)(*)(
)(
)(
tdL
tp
tetp
dttI
dttLti
tp
tpte
dttY
dttC
(4)
Note that in this study we do not assume "Purchasing
Power Parity" (PPP) or the "Law of One Price", i.e.,
e(t)/p(t) = 1. As the PPP hypothesis states that the
prices of a good (or a basket of goods) at home and
abroad should be the same, when measured in a
common currency, this hypothesis in essence
assumes that the nominal exchange rate, e(t), adjusts
to equalize domestic and foreign prices. If we denote
the foreign price index as p*(t), then PPP states that
e(t)p*(t) = p(t). When p*(t) is normalized at p*(t) =
1, then PPP is reduced to e(t)/p(t) = 1.
Further, by rearranging terms in equation (4), we get:
dttYtItC
tetp
tp
dttLtitdL )()()(
)()(*
)(
)()()(
(5)
Now, we develop the SDE for the net worth, X(t):
)(
)(
)()(*
)()( tdL
tp
tetp
tdKtdX
dt
tYtItC
tetp
tp
tLti
tp
tetp
dttI
)()()(
)()(*
)(
)()(
)(
)()(*
)(
dt
tYtC
te
tp
tp
te
tLti
tp
te
dt
te
tp
tp
te
tI
)()(
)(
)(
)(
)(
)()(
)(
)(
)(
)(
)(
)(
1)(
dttYtCtLti
tp
tetp
)()()()(
)(
)()(*
(6)
By substituting
dttbtLtXdttbtKdttY )()()()()()(
in eqn. (6),
we get:
dt
tbtLtXtC
tLti
tp
tetp
tdX
)()()()(
)()(
)(
)()(*
)(
Dividing by X(t), we get:
dt
tb
tX
tL
tX
tC
tX
tL
ti
tp
tetp
tX
tdX
)(
)(
)(
1
)(
)(
)(
)(
)(
)(
)()(*
)(
)(
(7)
If we define
)(
)(
)( tX
tL
tl
,
)(
)(
)( tX
tC
tc
, (8)
then,
dt
tbtltc
tlti
tp
tetp
tX
tdX
)()(1)(
)()(
)(
)()(*
)(
)(
dttbtl
dttlti
tp
tetp
dttc
)()(1
)()(
)(
)()(*
)(
(9)
Assuming that the rate of interest i(t) on US dollar-
denominated loans and bonds can be represented by
the following process:
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)()( tdWidtdtti ii
(10)
The other source of uncertainty is the return on
investments, b(t), which can be represented by the
following process:
)()( tdWbdtdttb bb
(11)
By substituting equations (10) and (11) in equation
(9), we get:
)()(1
)()(
)(
)()(*
)(
)(
)(
tdWbdttl
tdWidttl
tp
tetp
dttc
tX
tdX
bb
ii
Collecting terms, we get:
)()(1)()(
)(
)()(*
)()(
)(
)()(*
)(
)(
)(
tdWtltdWtl
tp
tetp
dttbltl
tp
tetp
itcb
tX
tdX
bbii
(12)
Then, the unknowns are: (1) c(t)=C(t)/X(t), (2)
l(t)=L(t)/X(t), , and (3) e(t).
3 Problem Solution (The Optimization
Problem)
The objective of the policy maker is to find the
normalized consumption, the normalized foreign
debt, and the real exchange rate that maximizes the
expected value of the utility of consumption and the
utility of the networth at time T, the end of the
optimization period. Specifically, we need to find:
)())((
0
)(),(
max TxUdsscUeE x
Ts
Txsc
(13)
subject to the dynamic constraints:
)()(1)()(
)(
)()(*
)()(
)(
)()(*
)(
)(
)(
tdWtltdWtl
tp
tetp
dttbltl
tp
tetp
itcb
tX
tdX
bbii
(12)
and
xX )0(
where
is the discount rate,
))(( scU
is the utility
function of households, and
)(TxUx
is the utility
function of the final value of the economy’s net
worth. Thus, the two utility functions,
)(scU
and
)(TxUx
, reflect, respectively, the desire to
increase the public welfare or consumption, and the
desire to increase the net worth of the society at some
future time T.
We use the martingale optimality principle to get c(t),
l(t), and
)(* tp
e(t)/p(t).
After some manipulations (see Appendix A), we
obtain the optimal values as:
2
22
2
)(
)(
)(*
b
ib
ib
i
b
i
tp
te
tp
(A. 19)
Recall that p(t) is the price index of the home
country, e(t) is the nominal exchange rate between
the home currency and the US Dollar, i is the mean
value of the US interest rate, with variance
2
i
, and
b is the mean value of the return on investment for
the home country, with variance
2
b
.
(Note that the left-hand-side of the real exchange rate
equation (A.19) is e(t)p*(t)/p(t), with p*(t) being the
US consumer price index set at 1.) Further, equation
(A.19) states that as the return on investments in the
home country increases above the US interest rate,
indicating strength of an economy, the exchange rate
will appreciate, i.e., e(t)/p(t) will decrease. If the
variance
2
b
increases drastically, i.e., if the
uncertainty regarding the return on domestic
investment increases, the exchange rate will
depreciate, i.e., e(t)/p(t) will increase.
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The optimal value of the debt (bonds and loans) ratio
)(/)()( tXtLtl
is derived as:
1
)(
)()(*
/1
)( 2
b
xtp
tetp
ib
tl
,
10
,
0
x
(A. 16)
where
is a tuning parameter, controlled by the
policy maker, and
x
is the degree of risk aversion,
being small for a risk taker. Equation (A. 16) states
that the optimal value of debt (foreign-currency
denominated bonds and loans) is related to the
productivity of the economy, as represented by the
mean return on investment, b, with the economy
allowed to borrow more as its performance improves.
The optimal exchange rate is related to the optimal
external debt as follows:
)(1
11
/1
)(
)()(*
2tl
i
tp
tetp
i
x
,
0
x
,
10
(A. 17)
This equation can be used to calculate the optimal
external debt, l(t)=L(t)/X(t):
1
)(
)()(*
/1
)( 2
b
xtp
tetp
ib
tl
,
2
)(
)()(*
/1 bx tp
tetp
ib
,
10
,
10 x
-
with 𝜉 and 𝛾𝑥 used as tuning parameters.
An exact but complicated expression, for the optimal
exchange rate, is also obtained as:
1
4
2
1
2)(
)()(*
2
2
2
2
2
x
i
b
bx
bx
i
b
i
b
tp
tetp
,
10
,
0
x
(A. 21)
Using the approximate expressions, the results,
however, are close to the ones obtained from the
exact expression. The optimal value of the external
debt, l(t), is different from the value obtained from
the Hamilton-Jacobi-Bellman (HJB) equation-based
optimal control approach [8]. Note that [8] use the
PPP assumption, i.e., the actual exchange rate was
not taken into consideration.
4 Results and Conclusions
In this section, we apply the above derived optimal
values to the Egyptian economy. Thus, equation (A.
19) could be written as:
)(*
)(
/
)(
2
22
2tp
tp
b
i
b
i
te
b
ib
i
(14)
Taking the logarithm (ln) of both sides, we get:
22 lnln)(*ln)(ln)(ln
bi
bi
tptpte
(15)
Equation (15) is similar to other results reported in
the literature using empirical models [2] and [3].
Below, using data for Egypt, we present our model’s
findings about the return on domestic investment,
average interest paid on external debt, actual and
optimal debt ratios, and actual, optimal, and PPP
exchange rates. First, the return on investment b((t),
defined as
)(/)()( tKtYtb
, where K(t) is gross
fixed capital formation (GFCF) and Y(t) is the GDP,
both in current $US, is shown in Figure 1. We also
present the estimate of b(t) using a Moving Average
model of order 2.
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Fig. 1: Return on Investments b(t), True and Estimate
Further, Figure 2 shows that the interest paid on
Egypt’s foreign-currency denominated bonds and
loans had an elliptic shape.
Fig. 2: Average interest on new external debt
commitments (%)
In addition, Figure 3 shows the actual and
approximate optimal external debt ratios,
l(t)=L(t)/X(t), for Egypt. In particular, both
conservative (high value for ℽ) and risky (low value
for ℽ) estimates are displayed for the approximate
optimal external debt. It is observed that the actual
external debt-to-net worth ratio is lower than the
risky approximate optimal external debt-to-net worth
ratio for almost the entire period analyzed, 1985-
2017 (with the exception of 1991 and 2011).
However, the actual external debt-to-net worth ratio
is higher than the conservative approximate optimal
external debt-to-net worth ratio for the period 1985-
1995; is about equal between 1996-2010; and is
higher again during 2011-2017. These results
indicate that Egypt’s actual external debt-to-net
worth ratio was higher than the conservative
approximate optimal external debt-to-net worth ratio
for 18 out of the 33 years in the sample, implying
that the country had contracted more debt than it
should during these years.
Fig. 3: Actual, Optimal (conservative), and Optimal
(risky) debt ratio l(t)=L(t)/X(t)
Finally, Figure 4 depicts the estimated approximate
optimal exchange rate for Egypt, the nominal
(official) exchange rate and the purchasing power
parity (PPP) exchange rate (using WDI data for
annual inflation P(t), the PPP and the nominal
(official) exchange rates).
Fig. 4: Optimal, Official, and PPP exchange rate
As shown in Fig. 4, Egypt’s nominal (official)
exchange rate (LE/$US) was much less than its
optimal values during 1985-1994. Then, it stayed
0
10
20
30
40
1985
1988
1991
1994
1997
2000
2003
2006
2009
2012
2015
Percentage %
∆Y/K (MA2)
b(t)=∆Y/K
0,0
2,0
4,0
6,0
8,0
10,0
1985
1991
1997
2003
2009
2015
Percent %
Average
interest on
new external
debt
commitment
s (%)(759)
exp_interest
0
1
2
3
4
5
6
7
8
9
10
1985
1990
1995
2000
2005
2010
2015
Percentage %
l(t)_approximate
debt ratio (risky)
l(t)=L/X, actual
debt ratio
l(t)_approximate
debt ratio
(conservative)
0
5
10
15
20
1985
1990
1995
2000
2005
2010
2015
LE per $US
Optimal exchange
rate
(approximate)
PPP conversion
factor, GDP (LCU
per international
$)
Official exchange
rate (LCU per
US$, period
average)
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consistently above its optimal values during 1998-
2008, while, starting in 2009, the official exchange
rate is less than its optimal values, but converging in
2015-2016 and became almost equal in 2017.
Ultimately, for these results to be used for policy
analysis, further refining in our derived formulas
needs to be contacted and continuous update of the
empirical findings to be performed. In particular,
future research could focus on the relationship of
optimal exchange rates, based on the optimization of
various social utility functions, and optimal sovereign
debt paths, as well as on the applicability and
potential shortcomings of the proposed novel
framework.
Appendix A (The Martingale
Approach)
In this appendix we use the martingale optimality
principle [11] and [12] to find the optimal value of
the normalized external debt and consequently the
optimal level of the exchange rate.
First, we employ the SDE of the net worth, X(t), as
given by eqn. (14):
)()(1)()(
)(
)()(*
)(
)(
)()(*
)(
)(
)(
tdWtltdWtl
tp
tetp
dttl
tp
tetp
ibtcb
tX
tdX
bbii
(A. 1)
The unknowns are: (1) c(t)=C(t)/X(t), (2)
l(t)=L(t)/X(t), and (3) e(t). .
For the purposes of this analysis, we define the utility
function from normalized consumption as:
1
)( 1
c
tcU
,
1,0
(A. 2)
Also, we define the objective function from net worth
as:
)())(()0(
0
)(),(
max TxUdsscUeEXV x
Ts
Txsc
(A. 3)
Thus, we have two utility functions,
)(tcU
and
)(TxUx
. The first reflects the desire to increase the
public welfare or consumption, while the second one
reflects the desire to increase the net worth of the
society at some future time T.
In the martingale approach, we need to find the
process H(t) such that
tdsbscsXsHtXtH
0
)()()()()(
is a
martingale.
Assume that H(t) has the following SDE:
)()()()()(
)(
)( tdWttdWtdtt
tH
tdH iHibHbH
(A. 4)
where
)(tWb
and
)(tWi
are Wiener processes. The
drift and diffusion coefficients are unknowns to be
estimated.
Using Ito’s Lemma, we get:
dHdXXdHHdXHXd)(
)()(1)()(
)(
)(
)(
)(
)(
)(
)()()()(
)()()()(
)()(1)()(
)(
)(
)(
)(
)(
)(
)()(
tdWtltdWtl
tp
te
dttl
tp
te
ibtcb
tXtdWtdWdttH
tdWtdWdttHtX
tdWtltdWtl
tp
te
dttl
tp
te
ibtcb
tXtH
bbii
iHibHbH
iHibHbH
bbii
dttl
tp
tetp
dttltXtH
tdWtdWdttHtX
tdWtltdWtl
tp
tetp
dttl
tp
tetp
ibtcb
tXtH
HiiHbb
iHibHbH
bbii
)(
)(
)()(*
)(1)()(
)()()()(
)()(1)()(
)(
)()(*
)(
)(
)()(*
)(
)()(
Collecting terms, we get:
)()(1)()(
)(
)(
)(
)(
)(
)(1
)(
)(
)(
)(
)()(
)()(
tdWtltdWtl
tp
te
dt
tl
tp
te
tl
tl
tp
te
ibtcb
tXtH
tXtHd
bbHbiiHi
HiiHbbH
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Moving
)(tcb
dt to the left-hand side, we get:
)()(1)()()()(
)(
)()(*
)()(
)(
)(
)()(*
)(1
)(
)(
)()(*
)()(
tdWtltXtHtdWtl
tp
tetp
tXtH
dt
tl
tp
tetp
tl
tl
tp
tetp
ib
HXdtbtcHXHXd
bbHbiiHi
HiiHbbH
or
)()(1)()(
)()(
)(
)()(*
)()(
)(
)(
)()(*
)()(
)(
)()(*
)()(
tdWtltXtH
tdWtl
tp
tetp
tXtH
dttl
tp
tetp
tltl
tp
tetp
ibHX
dtbtcHXHXd
bbHb
iiHi
HiiHbb
HHbb
For
dsbscsXsHtXtH HHbb
)()()()()(
to be a martingale, we need the drift term to be 0.
This requires that:
0)(
)(
)()(*
)()(
)(
)()(*
Hii
Hbb
tl
tp
tetp
tltl
tp
tetp
ib
A possible solution is
0
H
,
or
0 HbbH
We can now divide
)(
)()(*
tp
tetp
i
into two parts:
)(
)()(*
)(
)()(*
1
)(
)()(*
tp
tetp
i
tp
tetp
i
tp
tetp
i
Thus,
0)(
)(
)()(*
)(
)(
)(
)()(*
1)(
)(
)()(*
HiiHbbtl
tp
tetp
tl
tl
tp
tetp
itl
tp
tetp
ib
Collecting terms, we get:
01)(
)(
)()(*
)(
)(
)()(*
itl
tp
tetp
tl
tp
tetp
ib
Hii
Hbb
Then, a possible solution is:
0
)(
)()(*
Hbb
tp
tetp
ib
, which
yields
b
Hb
tp
tetp
ib
)(
)()(*
(A.5a)
01)(
)(
)()(*
itl
tp
tetp Hii
which yields
i
Hi i
1
(A.5b)
and
)(
)()(*
)(
)()(*
tp
tetp
ib
tp
tetp
ib
b
bHbbH
(A.5c)
Substituting in the d(HX) equation above, we get:
)()(1)()(
)()(
)(
)(
)()(
)()(
tdWtltXtH
tdWtl
tp
te
tXtH
dtbtcHXHXd
bbHb
iiHi
(A. 6)
and the SDE for H(t) becomes:
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)(
1
)(
)(
)()(*
)(
)()(*
)(
)(
tdW
i
tdW
tp
tetp
ib
dt
tp
tetp
ib
tH
tdH
i
i
b
b
The new Optimization Problem
The new system dynamics become:
)()(1)()(
)()(
)(
)(
)(
)(
)(
tdWtltXtH
tdWtl
tp
te
dtbtc
HX
HXd
bbHb
iiHi
Integrating both sides, between 0 and T, we get:
TdsbscsXsHXHTXTH
0
)()()()0()0()()(
T
bbHb
iiHi
sdWsl
sdWsl
sp
se
sHsX
0)()(1
)()(
)(
)(
)()(
Taking the expectation of both sides and using the
fact that E{H(0)}=1, we get:
)0()0()()()()()(
0
XHEdsbscsXsHTXTHE T
)0()0( HEX
)0(X
(A. 7)
The optimization problem could now be stated as
follows:
Find c(t) and X(T) that maximize
)0(XV
:
)())(()0(
0
)(),(
max TXUdsscUeEXV x
Ts
Txsc
(A. 8)
subject to the constraint:
)0()()()()()(
0
XdsscsXsHTXTHE T
with
1
)( 1
c
tcU
,
1,0
(A. 2)
Using the method of the Lagrange multiplier, we
need to find:
)0()()()()()(
)(
1
)(
0
0
1
)(),(
max
XdsbscsXsHTXTHE
TXUds
sc
eE
T
T
x
s
TXsc
,
which has the form:
)0()()(
)(
)()()(
1
)(
0
1
)(),(
max
XTXTH
TXU
dsbscsXsH
sc
e
E
T
x
s
TXsc
where
is the Lagrange multiplier. Assuming that
the conditions for the exchange of derivative and
expectation are satisfied, taking the derivative for c(t)
we get:
0)()()(
1
)(
)( 0
1
TsdsbscsXsH
sc
eE
sc
i.e.,
0)()()(
1
)(
)(
1
bscsXsH
sc
e
sc
s
which yields:
0)()()(
sXsHsce s
i.e.,
)()()( sXsHesc s
Taking the ln of both sides, we get:
)()(ln)(ln sXsHssc
and
)()(ln
1
)(ln sXsH
s
sc
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Thus,
TttXtHetc t
0 ,)()()( /1/1
/
,
(A. 9)
with
being a constant deterministic value.
We can now find an expression for the optimal X(T):
0
)0()()(
)(
)()()(
1
)(
)( 0
1
XTXTH
TXU
dsbscsXsH
sc
e
E
TX
T
x
s
i.e.,
0)0()()()(
)(
XTXTHTXUE
TX x
Interchanging derivative and expectations, we get:
0)0()()(
)(
)(
)(
XTXTH
TX
TXU
TX x
By setting
x
x
x
TX
TXU
1
)(
)( 1
, we get:
0)0()()(
)(1
)(
)(
1
XTXTH
TX
TX
TX x
x
which yields:
0)()(
THTX x
i.e.,
)()( THTX x
Taking the ln of both sides, we get:
)(ln)(ln THTX
x
and
)(ln
1
)(ln THTX
x
Thus,
xx THTX
/1/1 )()(
(A. 10)
We assume that this optimal value is also valid for all
“t” [13].
An SDE for X(t)
Assume that the equation of X*(T) is partially valid
for X*(t) i.e. the drift part is not correct while the
diffusion part is correct. We know that
tdsbscsXsHtXtH
0
)()()()()(
is a
martingale, and changing the probability measure
does not change the diffusion part and only the drift
part will change. And since it is a martingale, it only
has a diffusion part. Thus, under the change of
measure, the SDE for
tdsbscsXsHtXtH
0
)()()()()(
stays the
same. This is why we work with only the diffusion
[13].
We now derive an SDE for the optimal
TtX )(0
. Since we are assuming that:
TttHtX xx 0 ,)()( /1/1
,
then,
TttHdtdX xx 0 )()( /1/1
(A.11)
This is only for the diffusion part.
We recall that:
)()(
)(
)( tdWtdWdt
tH
tdH iHibHbH
(A. 4)
If we define
Hy
, using Ito Lemma, we get:
2
2
2
2
1dH
H
H
dH
H
H
dy
Substituting the expression for dH(t), we get:
2
22
1
)()(1
2
1
)()(
tdWtdWdtHH
tdWtdWdtHHdy
iHibHbH
iHibHbH
and
)()()((.))( tdWtdWtHdttHd iHibHb
(A. 13)
where
(.)
has all the drift terms.
Setting
x
/1
, we get:
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)()(
)(/1(.))( /1/1
tdWtdW
tHdttHd
iHibHb
xxx
(A. 14)
Substituting “
x
tHd
/1
)(
” into the equation of
dX(t) of (A. 11), we get:
)()((.)
)(/1)( /1/1
tdWtdWdt
tHtdX
iHibHb
xxx
Since
TttHtX xx 0 ,)()( /1/1
,
then,
)()((.))(/1)( tdWtdWdttXtdX iHibHbx
(A.15)
Equation (A. 15) shows that the optimal net worth,
X(t), follows a Geometric Brownian motion with
linear trend. This equation is valid only for the
diffusion part.
We know from the setup of the problem that X(t) has
another SDE:
)()(1)()(
)(
)()(*
)(1)(
)(
)()(*
)(
)(
)(
tdWtltdWtl
tp
tetp
dttlbtl
tp
tetp
itc
tX
tdX
bbii
(A.1)
We recall that
Hb
and
Hi
could be positive or
negative and their signs do not matter because they
are multiplied by a Wiener process.
Equating both equations, only the diffusion terms, we
get:
Hbxb
tl
/1)(1
, or more
accurately
Hbxb
tl
/1)(1
,
Hb
>0
which yields:
b
b
b
x
b
bHbx
b
Hb
x
tp
tetp
ib
tl
)(
)()(*
/1
/1
1/1)(
1
)(
)()(*
/1 2
b
x
tp
tetp
ib
(A. 16)
and
Hixi
tl
tp
tetp
/1)(
)(
)()(*
i.e.,
)(
11
/1
)(
1
/1
)(
)()(*
2tl
i
tltp
tetp
i
x
i
Hi
x
,
0
x
,
Hi
>0 (A. 17)
This result indicates that as the economy improves,
the optimal value of l(t) increases, which in turn
leads to decreases in the exchange rate. Thus, an
increase in the l(t), reflecting an increase in the GDP
performance, results in an exchange rate
appreciation, i.e., e(t) is reduced.
Substituting l(t) of equation (A. 16) into equation (A.
17), we get:
itp
tetp
i
b
i
i
tp
tetp
ib
i
tp
tetp
bx
b
i
bx
bx
i
x
2
2
2
2
2
2
)(
)()(*
1
)(
)()(*
1
/1
)(
)()(*
itp
tetp
i
bbx
b
i
2
2
2
)(
)()(*
1
Or,
btp
tetp
b
i
b
i
tp
tetp
b
x
b
i2
2
2
)(
)()(*
1
11
)(
)()(*
(A. 18)
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Assuming
1
, such that
bbtp
tetp
b
ib
x
b
x
22 1
)(
)()(*
1
then,
b
b
i
tp
tetp
b
x
b
i2
2
21
1
)(
)()(*
Assuming that
1
and
1
x
Then,
2
22
2
)(
)()(*
b
ib
ib
i
b
i
tp
tetp
(A. 19)
This result indicates that as the mean rate of return on
investments, b, increases relative to the US interest
rate, i, the exchange rate e(t) will decrease, i.e., the
local currency will appreciate.
An exact expression for the optimal exchange rate
could be obtained if we use eqn. (A. 18) without
approximations Viz;
1
)(
)()(*
)(
)()(*
2
2
2
x
i
b
bx
tp
tetp
i
b
tp
tetp
i.e
0
1
)(
)()(*
)(
)()(*
2
2
2
2
x
i
b
bx
i
b
tp
tetp
tp
tetp
(A. 20)
Solving the quadratic equation we get:
1
4
2
1
2)(
)()(*
2
2
2
2
2
x
i
b
bx
bx
i
b
i
b
tp
tetp
,
10
,
0
x
(A. 21)
This is the exact expression for the optimal exchange
rate.
References:
[1] Fleming, W. H., and J. L. Stein, “Stochastic
Optimal Control, International Finance and
Debt,” Journal of Banking and Finance, Vol.
28(5), pp. 979–996, 2004.
[2] Fischer, C., “Equilibrium Real Exchange Rate
Estimates Across Time and Space,” Federal
Reserve Bank of Dalas, Globalization Institute
Working Paper No. 362, April, Dallas: Federal
Reserve Bank of Dallas, 2019
[3] Brixiova, Z., B. Égert, and T. Hadj Amor
Essid, “The Real Exchange Rate and External
Competitiveness in Egypt, Morocco and
Tunisia”, AFRICAN DEVELOPMENT BANK
GROUP, Working Paper No. 187, 2013.
[4] Bussiere, M., M. Zorzi, A. Chudik, and A.
Dieppe, “Methodological Advances in the
Assessment of Equilibrium Exchange Rates,”
European Central Bank, Working Paper No.
1151, 2010.
[5] Montiel, Peter J., “The Long-Run Equilibrium
Real Exchange Rate: Theory and
Measurement,” in Mohsin S. Khan, Saleh M.
Nsouli, and Chorng-Huey Wong (eds),
Macroeconomic Management: Programs and
Policies, Washington: International Monetary
Fund, 2002.
[6] MacDonald, R., “Concepts to Calculate
Equilibrium Exchange Rates: An Overview”,
Discussion Paper 3/00, Economic Research
Group, Deutsche Bundesbank, 2000.
[7] Hansen, J., and W. Roeger, “Estimation of
Real Equilibrium Exchange Rates,” European
Commission, Directorate-General for
Economic and Financial Affairs, Economic
Papers No. 144, September (ECFIN/534/00-
EN), Brussels: European Communities, 2000.
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2022.19.182
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E-ISSN: 2224-2899
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Volume 19, 2022
[8] Abutaleb, A., and M. G. Hamad, 2012,
“Optimal Foreign Debt for Egypt: A Stochastic
Control Approach,” Economic Modeling, Vol.
29(3), pp. 544-556, 2012.
[9] Karatzas, Ioannis, John P. Lehoczky, and
Steven E. Shreve, 1991, “Equilibrium Models
with Singular Asset Prices,” Mathematical
Finance, Vol. 1, No. 3 (July), pp. 11-29, 1991.
[10] Dumas, B., 1992, "Dynamic Equilibrium and
the Real Exchange Rate in a Spatially
Separated World," The Review of Financial
Studies, Vol. 5 (2), pp. 153-180.
[11] Zhang A., Stochastic Optimization in Finance
and Life Insurance: Applications of the
Martingale Method, Ph. D dissertation,
Department of Mathematics, University of
Kaiserslautern, Germany, 2007.
[12] Abutaleb, A., “The estimation of the
instantaneous amplitudes of sum of sinusoids
with unknown frequencies and phases: The
martingale approach”, Signal Processing, Vol.
93, No 4, 2013.
[13] Karlsson, E., Optimal Portfolio Allocation by
the Martingale Method in an Incomplete and
Partially Observable Market, M. Sc., kth Royal
Institute of Technology, School of Engineering
Sciences, Sweden, 2016.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Dr. Abutaleb is responsible for the technical
derivations and writings.
Dr. Papaiouannou is responsible for the conceptual
framework and the conclusions.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_
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