Government Debt Control: Optimal Currency Portfolio and Payments
Author(s): Ricardo Huamán-Aguilar and Abel Cadenillas
Source: Operations Research, Vol. 63, No. 5 (September-October 2015), pp. 1044-1057
Published by: INFORMS
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Operations Research
hilTïïïïïï
Vol. 63, No. 5, September-October 2015, pp. 1044-1057
ISSN 0030-364X (print) | ISSN 1526-5463 (online) http://dx.doi.org/10.1287/opre.2015.1412
©2015 INFORMS
Government Debt Control:
Optimal Currency Portfolio and Payments
Ricardo Huamân-Aguilar, Abel Cadenillas
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
{huamnagu@ualberta.ca, abel@ualberta.ca)
Motivated by empirical facts, we develop a theoretical model for optimal currency government debt portfolio and debt
payments, which allows both government debt aversion and jumps in the exchange rates. We obtain first a realistic stochastic
differential equation for public debt and then solve explicitly the optimal currency debt problem. We show that higher debt
aversion and jumps in the exchange rates lead to a lower proportion of optimal debt in foreign currencies. Furthermore,
we show that for a government with extreme debt aversion it is optimal not to issue debt in foreign currencies. To the best
of our knowledge, this is the first theoretical model that provides a rigorous explanation of why developing countries have
reduced consistently their proportion of foreign debt in their debt portfolios.
Keywords', debt control; optimal currency debt portfolio; optimal debt payments; stochastic control.
Subject classifications: finance: portfolio, management; probability: stochastic model applications.
Area of review. Financial Engineering.
History : Received September 2014; revisions received February 2015, May 2015; accepted June 2015. Published online
in Articles in Advance September 21, 2015.
1. Introduction a country can be of trillions of dollars and, as we men
Public debt is a key macroeconomic variable. In particu- tioned above' the government debt portfol
lar, the currency composition of public debt is an impor- the lar§est m the country-the theoretical literat
tant variable that can exacerbate a debt crisis, such as in almost 110 attention t0 currency government deb
Mexico in 1994. Since a government debt portfolio is, in As far as we know' Lmandro and Masolle
most cases, the largest financial portfolio in a country, high Giavazzi and Missale (2004) are the only
foreign currency debt (especially short term) exposes the dea' w'tk currency debt portfolios. These a
country to the fluctuations of the exchange rates, and this the following limitations: (1) the debt dynamic
becomes dramatic when unexpected and huge depreciations 'sl;'c because they consider only one-period
(or devaluations) of the domestic currency occur. jumps in the exchange rates are not consid
Panizza (2008) points out that developing countries have anc' (3) foe ro'e °f debt aversion is not i
been reducing consistently the proportion of foreign debt important elements of the currency debt analy
in their portfolios in favor of local currency debt. This been included.
empirical fact is happening in the context in which most To the best of our knowledge, for the fir
developing countries have access to the international capital literature, we present a model for debt m
markets. That is, although countries can borrow in exter- includes jumps in the exchange rates and
nal currencies, there is a deliberate tendency to borrow in We obtain explicitly the optimal currency
domestic currency. A natural question arises. What is the and optimal debt payments. We use this m
explanation for this type of government behavior? There is that the behavior of developing countries of r
a consensus that the development of the domestic capital proportion of foreign debt in their debt po
markets has played a key role. According to Borensztein tent with a high debt aversion. Moreover, w
et al. (2008), debt crises are the factors that have urged extremely high debt aversion can lead to
countries to pursue such strength in the domestic markets. ernment debt in local currency. That is, it wo
In other words, the underlying factor that explains this ten- for such a country to have no debt in foreig
dency is the goal of reducing the exchange rate vulnera- We model the currency debt problem as a
bilities and hence the chances of a debt crisis. This can trol model in continuous time with infinite h
be interpreted as the countries having become more debt believe that is the suitable framework to st
averse because of their past experiences. lios. In fact, Bolder (2002) shows that the gov
What does the theoretical literature on currency gov- problem can be conceptualized in this mann
ernment debt management say about the fact above? Sur- his problem is different from ours. He a
prisingly, although the order of magnitude of the debt of government issues only local currency debt, an
1044
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Huaman-Aguilar and Cadenillas: Optimal Currency Government Debt
1045
Operations Research 63(5), pp. 1044—1057, ©2015 INFORMS
to find the optimal proportion of the different terms of debt. 2. The Government Debt Model
In contrast, our focus is on finding the optimal currency jn tbjs section we derive the stochastic differential equation
debt composition. That is why we do not consider debt fof ^ debt of a country and> based on it> we then state the
in different terms; instead, we assume that the government government currency debt control problem. Our goal here
issues bonds in different currencies. t0 obtain a realistic debt dynamics, that considers debt in
To get a realistic debt dynamics, we extend the model a finite number of foreign currencies> and includes jumps
of government debt for a single currency, presented in in thg exchange rates
macroeconomic textbooks (see Blanchard and Fischer 1989,
for example), to a multicurrency setting. We model the 21 The Debt Ratjo Dynamics
exchange rates dynamics as a stochastic process that
presents jumps, and thus we consider a more general frame- The government debt is defined by
work than Zapatero (1995) and Cadenillas and Zapatero .
(1999). We succeed in finding a stochastic differential equa- := §ross Pubhc debt e
tion for government debt, in which it
on the present and past values of variables such as the inter- _ , ... , . . , ,.
r , , , , , . The gross pubhc debt is the cumulative total of all govern
est rates, exchange rates, debt payments, and the proportions , . , ..... ,
.. , i . . ment borrowings less repayments. That is, it includes the
of debt in different currencies. , ,, , , , , , , . ,
. ,, , j , . j. ... central and local government debt, and the domestic and
The running cost, or what we call here the debt disutil, , ,
pYtprnal Hpnt
ity of the government, depends on both the debt payments , . , ' . , .
and the debt itself. The existence of the former cost comes . Indlls ^section, our goal is to extend the fol
from the fact that, in order to get additional positive fiscal sion for one smgle currency debt presented m nia
,. , ... ., , , . c ,• textbooks (see, for instance, Blanchard and Fischer 1989):
results to repay debt, the government has to cut spending or v ' '
increase taxes. The rationale for considering the latter cost
X(t)
— Jo
X(0)Jo
+ f
stems from the fact that the government
acknowledges
that
r0X(s) ds — f p(
having high debt can lead to debt problems, which can end
up in a debt crisis, for instance. Since we are interested in where r0 is the (conti
analyzing the effects of debt aversion on the optimal cur- the process of deb
rency debt portfolio, we have to consider a disutility func- We consider a
tion in which the debt aversion is a parameter such that the an(i m foreign curre
greater the debt aversion, the higher the disutility generated bonds held in local
by the debt itself and debt payments. In our model, the ber of bonds held i
disutility function presents constant relative risk aversion The prices of t
(CRRA) in debt payments and the debt itself. We have two {0,1,..., m}. Thus,
reasons for such choice. First, we have a unique parameter in currency j. For
that fully characterizes government debt aversion. Second, this currency is a for
our model with constant relative risk aversion is a gener- A,(f)R](f) is the am
alization of the quadratic function, which is the disutility We require m exc
function most widely used in different areas of economics in local curren
(see, for example, Kydland and Prescott 1977, Taylor 1979, change rate of the cur
Cadenillas and Zapatero 1999, Cadenillas et al. 2013). rency. To be more
The goal of the government debt manager is to choose
both the currency debt portfolio and the payments that min- Qj(0 := 'oca' curren
imize the expected total cost (disutility). currency j at time t.
We succeed in solving the problem explicitly. Thus, we
can perform some comparative statics to analyze the effects Then, the total debt in terms of local currency X can be
of some parameters (such as the debt aversion and the written as
size and frequency of the jumps of the exchange rates) on m
the optimal currency debt portfolio and the optimal debt X(t) = A0(t)R0(t) + £]Aj(t)Rj(t)Qj(t);
payments.
The
for
paper
is
organized
government
orem that states a sufficient condition for a debt control
to be optimal. In §4 we find a candidate for solution and
then verify that this candidate is indeed the solution. In §5
;=1
debt
as
foll
control
in
X(*)=z0(0+£z/0ßy(0,
;=1
we perform some comparative economic analysis to show, where Z,(r) := A,■(/)/?,(?), Vi e {0,1,, m).
among other results, the role played by government debt To find the debt dynamics, given the evolution of the
aversion. We write the conclusions in §6. number of bonds in each currency, we require the dynam
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Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt
1046 Operations Research 63(5). pp. 1044-1057, © 2015 INFORMS
ics of the price of the bonds and the dynamics of the if a Poisson even
exchange rates. ...,m}, then it generates an effect on all the exchange
Since we are interested in studying a currency debt port- rates via tpiy The occurrence of thi
folio, in our model the source of randomness will come we know that the intensity of t
from the exchange rates. We will assume that the exchange This parameter measures the f
rates follow a process driven by Brownian motions and the exchange rate. The informat
Poisson processes. For technical reasons, we need to spec- jumps is contained in the matrix t
ify a suitable probability space in which these processes ment in the diagonal «pjj is positiv
are defined. with empirical currency crises (such as Mexico in 1994,
Consider a complete probability space (ft, W, P) en- Asia in 1997, and Russia in 1
dowed with a filtration F = {if t, t e [0, oo)}, which is rate went up dramatically. Th
the P-augmentation of the filtration generated by both rate j occurs, then an incre
an m-dimensional Brownian motion W = {Wu ... ,Wm}, (or devaluation) of size <p;;
and an independent m-dimensional Poisson process N — <pyi — 0.3 means that the e
with corresponding intensities {A,,..., A,„}. edly 30%. Thus, our model acco
Let Q = {(ß|(f),..., Qm(t)), t f 0)} be an F-adapted government that has foreign cur
process. Following Jeanblanc-Picqué and Pontier (1990), lio faces a risk of depreciatio
Cadenillas (2002), and Guo and Xu (2004), we generalize currency. Since we also wan
the model in Zapatero (1995) and Cadenillas and Zapatero size of jumps on the optima
(1999), and consider the following multidimensional setting parameters in matrix tp give
for the exchange rates.3 For j e {1, 2,..., m}: itive measure of the magnitude
reason why we choose the modeling given in Eq
dQj(t) = pijQj(t) dt + <TjQj(t) dW(t) over other jump processes m
+ <p Q (t~) dN(t) (3) assume that the prices of the bon
with initial exchange rates ßy(0) = > 0. Here £ry and dRj(t)— Rj(t
are the jth row of the m x /«-matrices a = [crj and where P; (0) = 1, and r,- e (0, o
(p = [<Pij\, respectively. The parameter fij e (—oo, oo). est rate on debt issued
Throughout the paper AT denotes the transpose of matrix A. that the process R = {(R
We will assume that aaT and the family of matrices {(pjtpj: F-adapted.
; e {l,...,m}} are all positive definite, where Çj stands In Appendix A, we der
for the /th column of matrix tp. discrete time model with one foreign currency. The con
Furthermore, we denote by N the compensated Poisson tinuous-time version with m foreign currencies is
process. That is, for j e (1,..., m}, rt
X(t) = X(0)+ A0(s~)dR0(s)
m
r'
['
where
Aj
>
0
and
NjAs
ous version of any process Y(t) is denoted by Y(r). We M 0 0
a
note that the exchange rate Qj has right-continuous with left- We assume that the process of rate paym
limits paths, a property that is determined by the stochas- m-dimensional process A = (A,,...,Am)
tic integral with respect to the jump process. For more right continuous with left limits. In add
information about stochastic differential equations like (3), nonnegative, and the technical condit
see, for example, Ikeda and Watanabe( 1981), Protter (2004), <oo for every t> 0 to be satisfied. He
Cont and Tankov (2004), and Applebaum (2009). the expected value of the random variab
Remark 1. Equation (3) implies that the process Q. jumps X(0) —x.
at time t > 0 if and only if some of the Poisson processes Usin8 both the dynamics of the exch
in N = {A,,..., Nm) do so. That is why Q, is rightcontin- the dynamics of the Price of the bonds
uous with left limits. Moreover, since the number of jumps we obtain the stochastic differential equati
of a Poisson process is finite on each finite interval [0, f], process X.
then the sample paths of both the process Q.(-) and its . , A, . . . . „ , . _ , . ,
if. .• ■ t -\ u a a « dX(t) = rnX(t)dt+ y^(r: +u,: —rçAA:{t)RAt)QAt)dt
left-continuous version Q;( • ) are bounded on any finite v ' IJ - ' ^ J
interval
[0,
f].
m
The model for the exchange ra
priate to describe sudden depr
the
local
rates
currency.
are
The
random,
i=i
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time
driven
o
b
Huamàn-Aguilar and Cadenillas: Optimal Currency Government Debt
Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS 1047
Here we point out that since the process Rj is continuous, Here
its left-continuous version coincides with the process itself, t t
and hence they are interchangeable. £(t):=exp|f ß(s)ds+f itt(s)<t dW(s)
The debt portfolio vector process 77 = (77,,..., 77m) is l 0 0
defined for j e {1,..., m} by
+ £_( lOg^l+ 7Tr(s
A,(t)Rj(t)Q,(t)
TTj(t)--=A
JytJ>
VX(0>
0, (7)
HI
where
from
which
we
deduce
that
_
"L
ß(s):=r0
\(r)Rj(t)Qj(r) ww,_x . ... ;='
TTj(r) = X(r) ' VX(r) > °" (8)
,
+
_
Tr
(s)&-£A
is the unique solution of the homogenous equation
For completeness, we define 77(f) := 0 if X(t) = 0, and ... . ... , , . T, ., , , . T. . .
. . a ■ c v / — \ ATT imm • r J . J \ d£(t) = £(t)rrldt + £(t)TT'(t)bdt + £(t)TT'(t)ad
77(f ) :=0 if X(r) = 0. Here 77 elRm is F-adapted and 77;- bW bW 0 bW w b
represents the proportion of debt issued in foreign currency + %(r)TTT(t~)tp dN(t),
j € {I, Obviously, the proportion of debt in local
currency 7r0 e 1R is given by 7r0(t) = 1 - rr/f). with initial condition £(0) = 1. Here log a st
Expressing Equation (6) in a compact manner, we have natural logarithm of a > 0. We note that the
the following government debt dynamics: be interpreted as the debt dynamics with initia
to one, and without any debt payments.
dX(t) = X(t)r0 dt + X(t)irT(t)b dt + X(t)vT(t)(T dW(t) We now discuss the nature of the jumps of th
w t / _x ,\ , /ax cess- Suppose the Poisson event Nt occurs at the random
+ X(t )77 (t)cp dN(t) - p(t) dt, (9) time j Then> Equation (12) imp]ies
where X(0) = x > 0, b = r + /jl — r0l, with p = )• Since
(p-i pm) and r = (r,,.,., rm), with 1 a vector of ones £(/) / r , j , y
in IRm. We require the following technical assumption: £(/-) ~ jJ0 ß(s)ds + 77
Ex[f0 ttt(T(tttt ds] < oo for every t > 0.
If we set 77/0 = 0 for all je{l,...,m} and t > 0 in + + ^(5-)^.)
Equation (9), we recover the dynamics of debt in one sin- Jo J /
gle currency given by Equation (1). Thus, Equation (9) is (\cJ rJ
indeed an extension to the multicurrency debt dynamics. ' ( exP j J ß(s)ds + J vT(s)a dW(s)
We state our result in the next proposition. ^
Proposition 1. The stochastic differential equation for the + j lo
government debt dynamics is given by 0 ' '
= (1 + 77 T(J~)(pi),
dX(t) = X(t)r0 dt + X(t)irT(t)bdt + X(t)7TT{t)cr dW(t)
+ X(r)vT (t~)<p dN{t) — p(t) dt, (10)
we have
, y/AX A , T1 / X X(J) = X(r)(l+TTT(j-)<Pi). (13)
where X(0) = x > 0; b = r + p — r0i; 77 = (77,,..., irm)
is the vector of proportions of debt in foreign currencies, By (j ^ the debt values before a
and p is the debt payment rate process, i.e., the debt pay- and x(J), respectively, have the
ment rate expressed in local currency. if the value before the jump is posit
We recall that <pj denotes the jth column of matrix (p. If never jump to a negative or zer
we impose the technical condition that for every s ^ 0 jumP in X can be upward or down
From Proposition 1, the debt at time t, X(t), d
1 _)_ 77r(s~)<p > 0, V /' € {1, 2,..., m}, (11) on the following variables: intere
(/•„, rx,...,rm), exchange rate depreciation (pl,.
the above linear stochastic differential equation for the debt portfolio currency composition pro
X possesses a unique explicit solution. Indeed, an applica- an(i debt payment rate process p. M
tion of Itô's formula to X(t)/Ç(t) gives on the realizations of the random
exchange rates, i.e., the Brownian motion and th
, x .. ./ [' , \„, x_i , \ w a /.ax son process, and their corresponding parameters a, w, and
(0 — ^(0^* Jo P(s)€(s) t ^ ( ) (A,,..., Am). Thus, we have a realistic debt dynamics.
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Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt
1048 Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS
From the perspective of a developing country, the for- have never su
eign interest rates and the randomness of the exchange rates and the United
are given. Furthermore, for a debt policy maker, the local have experien
interest rate is essentially exogenous.4 On the other hand, and Greece). T
the debt manager can exert control on the debt portfolio not only prov
process tt and, to some degree, on the debt payment rate terizes fully th
process p. They are precisely the control variables in our generalizes th
government debt problem that we will describe in the next economics, name
subsection.
Remark 2. For a utility function u(y)
2.2. The Debt Problem aversion is defined by —yu"(y)/u'(y) (see, for instance,
In reality there exists a debt problem as long as public debt MasCollel et al.1995 or Pratt 1964). Similar
is positive. Thus, we consider the following stopping time: tili£y function d(^ we define the relative r
yd"{y)/d'(y).
0(co).— inf{s ^ 0. X(s , (o) ^ 0}. (14) Since X(t) — 0 for t > 0, there is no debt problem after
Since after 0 there is no debt problem, we consider the
following: if 0(cu) is finite, we impose 2T0(o>) := X0-(cu)
and X,(co) := 0 for all t > 0(w). In view of (11) and (13),
we must have X0~ = 0. Thus, the dynamics of X is given
by Equation (10) for every t e [0, 0), X@ = X&- = 0, and
X(t) = 0 for every t > 0. We observe that X(?) > 0 for
0. Hence, we also impose p(t) = 0 for t > 0. Thus, the
total cost after the debt becomes zero is null. Hence,
~ 00
: 0.
/ e~s'(aXJ+l +pj+l)dt
t e jq Here 5 > 0 represents the discount rate.
Now we turn our attention to the running costs, or the As we discussed above, we will ass
disutility function of the government. This cost depends on policy maker can exert control on both
the debt payments and the level of the existing debt. The portfolio and the debt payment rate.
existence of the former cost comes from the fact that, in formal definition of control process.
order to get positive fiscal results, countries have to cut „ . . r„ , ,
,. j . _ ' , , ,• , Definition 1. Let u: [0, oo) x fi IRm x R be a process
spending and increase taxes. On the other hand, the cost , . ,, , , , . , ,
linked to the existing debt exists because high debt can lead e ne ^ , f' W
, , , ^ .. ■ ei. c . j r l* , . , folio debt process and p is a payment rate process, which
to debt problems
, are nght
for
We
in
in the future—a default or a debt cnsis, . r . ..... . , '
continuous with left limits and adapted to If. For a
example.
b
r
observe
that
the
different
pie,
and
re
and
Zapatero
/ e~Sth(X,, p,) dt
represents
has
areas
Kydland
7o
an
constant
Remark
2
effects
of
of
ec
Pres
< oo, (16)
1999,
agent
C
wit
relative
below).
Since
Vy 6 m), Vj>0, (17)
debt
aversion
folio, we consider a function in which the debt aversion is ^ f'^ 7rr(5-^ .)2(x+1
a parameter. Specifically, for y e (0, oo), we define the cost * [•'o '
(or loss) function by Vy € {1,..., m), Vf ^ 0. (18)
h(x,p):=axy+ + py+ , (15)
The set of all admissible controls wil
sd(x) = si.
where x represents the public debt and p the debt payment
rate. Here, a e (0, oo) is a parameter that represents the Remark 3. We observe t
importance that the government gives to the existing debt
x relative to the debt rate payment p. The function h has
[° p.
e~s'Xj+ldt
the property of CRRA equal to y in x and
Thus, the
Jo
< 00.
parameter y represents the aversion of the debt manager
with respect to the existing debt and the debt payment. jjien we ^
That is, for a given debt level and debt payment, the bigger
the parameter y the higher the disutility of the government.
oo
For instance, countries that have nevert —>fim£;t[e-*Z7+1/(K0)]
had
a default or
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= O. (19)
Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt
Operations Research 63(5), pp. 1044—1057, ©2015 INFORMS 1049
To complete this section, we state the debt problem. with
Problem 1. Consider the debt dynamics given in Proposi- m
tion 1. We want to select the admissible control û = (tt, p) ß := r0 — p + ttJ b — Y A;7rc (pj — aaTttc',
that minimizes the performance functional given by >=l
J{x\ u) = J(x\ 7t, p) from which we observe that X(t) > 0 for all t > 0. Hence
0 = oo. Then the discounted government disutility is
:=£.
[° e-s,(aXJ+1+pJ+l)dt
JO
(20)
The control û = (tt, p) will be called the optimal debt
-OO
J(x; ttc,p) = Ex / e~s,h(Xt) dt
Jo
control.
P CO
= £r /Jo e~s,(aXJ+l+py+1XJ+l)dt
From a mathematical point of view, Problem 1 is a sto
chastic control problem with jumps. This theory has been »
studied and/or applied, for instance, in Jeanblanc-Picqué = (a + p7+1) / e~s'Ex[Xy+
and Pontier (1990), Cadenillas (2002), Guo and Xu (2004), 0 ^
and Oksendal and Sulem (2008). = (a + py+1)xy+l f e~dy+Dp+0
To illustrate our framework, let us consider two exam- 0
pies. In the first one the debt manager chooses not to issue
(a + py+l)xy+1
if (y + l)p + £ > 0,
debt in foreign currencies and to pay only the interest (y
rate
+ i)p + £
of their current debt. The second example is a more general
if (y + l)p + C<0.
version of the first one.
Example 1. Suppose the debt manager chooses the fol- „ , ..
, . , , . ,. v. . ,. To get the fourth equality above, we have used the result
lowing debt policy: v(t) =0 and p(t) = rnX(t) for every ... „ , ,A rA, .,
. . n . ,. . c .. /IA. Vu ,i u. ) . , computed in Lemma B.l (Appendix B), with
t ^ 0. According to Equation (10), the debt at every point
in time equals the initial debt. That is, V t > 0
£:=S- i(y+ l)y7rc <rcr 7rc
X(t) = x.
m
~ E a7K1 + *1 <Pj)iy+l) -1 - (y +1 Wc<pj\
This implies that X(t) > 0 for all t > 0, and hence © = oo.
7=1
Thus, the total discounted government cost (disutility) is
-(y+l)(7rjfi + r0).
a + r7+1 +]
J(x, tt, p) — ^ x (^1) Thus, the debt policy given in Example 2 is admissible
We point out that the debt policy (tt, p) is admissible if ^ anc* on'^ if (7 + l)p + ^ > 0- It would be interesting to
and only if S > 0. Since the latter condition is satisfied, this comPare those Policies with the °Ptimal debt Policy that we
policy is admissible wiH obtain in §4. Certainly, we expect this type of arbitrary
debt policy not to be, in general, optimal. We will confirm
Example 2. Let vc be an arbitrary constant vector in [Rm qqs fact jn ^
whose components are positive, and such that condi
tion (17) is satisfied. Let p be an arbitrary positive real
number. Suppose the debt manager considers the following
debt policy: for every t ^ 0, ir(t) = 7rc 6 IRm and p(t) =
The Value Function and a
Verification Theorem
pX(t). Then, considering this debt policy in Equation (10), The main purpose of this section is to state a sufficient
the dynamics of the debt becomes condition that an optimal solution of the debt problem mus
,v,a , s w . , , v,s t, , , vf . t ..... . satisfy. The value function is a key instrument to achieve
dX(t) = (r0-p)X(t)dt + X(t)vJbdt + X(t)iri<rdW(t) that goal.
-\-X{t~)TTTc(pdN{t), We define the value function V: (0, oo) -> 1R by
with X0 = x. We note that the above stochastic differential yr \._ j* ^ \ ^31
equation (SDE) has the form of Equation (10), except for the (ir,p)es<
last term. Consequently, using Equation (12) with p(s) = 0
for s ^ 0, the solution to the above SDE is given by This represents the smallest expected cost that can be
achieved when the initial debt is jc > 0 and we consider all
X(t) = x exp If'ßds + J' 7wTc<rdW(s) the admissible debt controls.
m Proposition 2. The value function V is nonnegative and
+ Y f log(l + 7TT(p.)dN (s) (22) homogeneous of degree y + 1. Therefore, it is in
7=1
• Jo 1 ' ' convex, and V (0+) = 0.
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Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt
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Proof. See Appendix C. □ candidate for value function. At the end of this section, we
TXr . . . are going to apply Theorem 1 to prove rigorously that the
We require some notation to define the Hamilton-Jacobi- ,•, . r .. , . i . . , , ^ .. , . ,
. . m candidate ror optimal control is indeed the optimal control,
Bellman (HJB) equation. Let g: (0, oo) R be a function A a-a + * i - • a a +u i
_ ÖV' ' and the candidate lor value lunction is indeed the value
in C2((0, oo)). For 77 e IR"' and peU, let us define the functjon
operator 2(77, p) by y/e want to find a control (■ir
ing function v that satisfy the conditions of
2(77, P)s(x) According to Equation (27) in that theorem,
:= \ tttera7TTx2g"(x) + (irTbx + r0x - p)g'(x) - Sg(x)
TT = arg mm\x2 ttt era7 ttv" (x) + xttt bv' (x)
m
+ E Xj(s(x + 77T(PjX) - g(x) - g'(x)7rT<PjX), (24)
+ E^j(V(X + ^(PjX) - v'(x)TTT<PjX)
1=1
where we recall that <pj is the /th column of matrix <p.
For a function v: (0, oo) —>■ IR in C2((0, oo)), consider P — argmin[—pv'(x) + h(x, p)].
the
F1JB
equation
p
Let
(
min
j2(7T,
state
p)v(x)
Next
we
be
optimal
a
us
+
h(x,
sufficient
^
Theorem
conjecture
/?)}
condition
+
1.
that
=
v
°.
for
a
is
(25
debt
EXj(v'(x
Let
v
€
vex
function
on
(0,
satisfies
the
HJB
Equ
the
polynomial
„ ^ , dh(x,p)
growth
v (x) + — 0.
dp
v(x) ^ C(1 +xy+ ), (26)
Thus, if we know the value function v, we can character
for some constant C. Then, for every x G (0, oo), we have ize the candidate for optimal debt control (tt, p). Based on
the following two results. both the proof of Proposition 2 and the form of the disu
(a) For every {tr,p) G :Â(x): tility function h, we conjecture that the value function for
y G (0, oo) is given by
v(x)^J(x\ 7T,p).
v(x) = Kxy+\ (29)
(b) Suppose that the stochastic control û = (fr, p)
defined by for every x > 0 and some constant K > 0.
By means of this conjecture, using (28), we charac
Û = (tt, p) := arg min {2(77, p)v(x) + h(x, p)\, (27) 77 e as the vector that solves the
(7T,p)esâ
in
7T\
m
is admissible for X = X and 0 = 0. Then yacrT tt + ^ A;(p; {(1 + ttt (pfy — 1} = 0; (30)
i=i
v(x) = J(x\ 77, p).
and for p e IR we have
In other words, (tr,p) is the optimal debt control and p(t) — Kx/yX(t) (31)
V = v is the value function for Problem 1. Here, X is the
debt process generated by the control (77, p), and © is the where ^ is the debt dynamics generated by the debt pol
corresponding stopping time defined in (14). icy Since al] the gntries in a and ç are constantS)
Proof See Appendix DD , we observe that the process tt is indeed a constant vector
in IR'".
To complete the specification of the candidates, it re
4. Tn6 Explicit Solution mains to characterize the constant K. By Theorem 1, we
At the beginning of this section, we are going to make con- know that for the function v and the process û = (77, p)
jectures to obtain a candidate for optimal debt control and a to be suitable candidates for solutions, they should satisfy
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Huaman-Aguilar and Cadenillas: Optimal Currency Government Debt
Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS - 1051
the HJB Equation (25). Considering it along with Equa- Proof. To prove this th
tion (27), we must have conditions of Theorem 1 are satisfied. Regarding the can
didate for value function, we get immediately that v
2?(77, p)v(x) + h(x, p) = 0. (32) C2((0, oo)) and v(0+) = 0. Moreover, since
A£, Vr • ,, . .. « . • ., increasing and strictly convex on its domain. On the other
After simplifying the previous equation, we obtain the b J
equivalent form- hand, by construction, we observe that the cand
y Kiy+l)/y + £k-<x = 0, (33)
where
satisfies the HJB Equation (25),
E6(if, p)v(x) + h(x, p) = 0.
™ kT -r l Taking C — K, condition (26) also holds. It remains to ver
£,= ^ — 2-,^i'l(l + ™ — 1 —(y+l)7T (Pj\ jfy that conditions (16) and (18) are satisfied for (77,
~ Since 77 is a constant vector, condition (18) is immedi
— 5(7 + l)yfiT o-a7 fi — (y + \)(fi7b + r0). (34) ate. To show conditio
Appendix B, Equation (B.l), with p = Kl/y. W
Here we emphasize that the choice of K in (33) guaran- sinœ (^ ß) .g on
tees that v(x) = Kxy+l satisfies the HJB Equation (25) for ^ E ]e % we kno
(77, p). Let us prove that indeed K > 0. Let /: IR —> IR be
defined by
f e s,h(X„ p,)dt
f(z) := yz{y+l)h + fz - a.
Jo
rOO
We note that / is convex on (0, 00). Since /(0+) = —aI e~s'(aXy+l
<0 — Ex
+ K(y+l)/yXJ+l) dt
and /(+00) = +00, then there exists a unique real number
K > 0 such that f(K)~ 0, as required.
= (a + Kiy+l),y) j™ E[e~s'Xy+]]dt
To complete this section, we are going to prove rigor
ously that for y e (0,oo) the above candidate for optimal ,» p
debt control (30)—(31) is indeed the optimal control, and — (ot + K )x exp{ ((y + \)K
the above candidate for value function (29) is indeed <the
00,
value function. For such proof, we observe from (33) that
/ , ,\vi/Y , ? a , Ts-i/y « where the last inequality follows from (35), that is,
(y-f l)Kl/y + £ = — + Kl,y > 0 (35)
K
because K > 0 and a > 0. ^ ^ + £ > 0
Theorem 2. Suppose fieW' satisfies By virtue of Theorem 1
m Kxy+l is the value function for Problem 1. □
(i) yaaTfi+ b + J^\j(Pj{(l + fiT(pj)y- 1} =0, (36)
;=1 We observe that the explicit solution for the debt pro
/••s , , ~r f\ r ■ m 1 /,« cess X given in Theorem 2 is an extension of a geometric
(11) 1 -f- 77 CP,- > 0, for every j (37) . . . , , .
1 Brownian motion to include jumps. This model is general
Let p be the process defined by enough to reflect the debt evo
if ß > 0, then the theoretical trend generat
p(t):=Kl/yX(t), (38) is consistent with the recent trend of most countries in
which debt increases over time. Moreover, it is also a gen
where K is the unique positive real solution to Equa- eralization of the basic stochastic model for the deb
tion (33), and X denotes the debt process generated by the ^ ^ ^ Greiner and Fjncke {2m)
debt control u = (it, p), which is given by . r ^ -, . , ,
v 0 As an application of Theorem 2, we present below two
X, = x exp
particular cases in which the government debt portfolio can
ßt + 7TTcrW (t) + J] log (1 + TTT<Pj)Nj(t)
1=1
be obtained explicitly.
Remark 4. Let y = 1. Then the value function is V(x)
with
= Kx2, and the optimal debt control is given by
ß:=r0 — Kl/y + 77Tb — "Y^kjfi7<p;- — jfiTaaTfi.
i=i
-1
fi(s) =
Then û — (fi, p) is optimal, and V(x) — v(x) = Kxy+] is
(r0l -r-p),
1=1
the value function for Problem 1. p(s) = KX(s).
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Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt
1052 Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS
Here X is the debt process generated by the debt policy where X deno
{it, p). The parameter K is given by control m = (77,, p), an
tion
Vc2 + 4a — c
K := > 0,
:=
5
—
2r0
with
Remark
exchange
and
ir{s) =
equation
yK{y+i)/y
where
c
to
the
T
5.
+
[v(tt
Suppose
rates.
optimal
Then
debt
ilK-a
=
0,
where
bTY~xb,
=
+
+
Yl"j=
that
the
there
value
control
is
i
£,
are
no
function
given
:=
8
-
jumps
is
V{x)
=
i
K
by
Then (77,, p) is the optimal debt control, and V (x) =
(aaT)~] (r(ll — r — p) v(x) — Kxy+i is the value function for Problem 1.
?
y
Proof. Set m = 1 in Theorem 2. □
p(s) = K1>yX(s).
We know that developing countries hold positive propor
Here^ X is the debt process generated by the debt policy tions of foreign currency in their debt portfolio. W
(77, p). The parameter K is given by (33) with fact m we provide below a proposition that
c „ . . „ r T „ . . . „t, . a sufficient and necessary condition for a positive solution
£:=S-i(y
+ l)y77rcro-r7r-(y + l)(77rf> + r0). A . 3 v
2V/ '' \t >\ u/ 77, e IR to exist.
If y=l, then we have an explicit formula for K, namely
Proposition 3. Suppose all the a
Vi2 + 4a — 1 are sat'sfied- Then we
K := >0. a unique 77, > 0 such that (39) and (40) are satisfied. In
either case 77, € (0, rj), where q := (A<p — b)/
5. Economic Analysis
Proof. Let /: IR -> R be defined by
In this section we analyze the effects of some parameters
on both the optimal debt control and the value function. To 2
simplify the analysis and facilitate the interpretation of the /(<•)•—7°" <•
implications of the model, in this section we will consider
two currencies: the local and one foreign currency. In other Note first that f(0) =
words, throughout this section m = 1. Since / is strictly increasing an
We recall that the source of the jumps in our model unique 77, > 0 such that /(
comes from a Poisson process, whose jumps have size exists a unique 77, > 0 such
one. The parameter <p allows us to introduce arbitrary sizes Then, from
of the jump. In other words, anytime the Poisson process
jumps, the effect on the exchange rate depreciation (or de- _b = /( - } _b
valuation) is (p. Thus, to be consistent with empirical cur
rency crises, we are going to assume ip> 0.
we conclude that b < 0. To show the 77 is an u
5.1. Optimal Solution with Two Currencies observe that condition (3
Under the previous considerations, we have the following
straightforward corollary of Theorem 2. 7°" — — b) — y er 77
Corollary 1. Let m = 1 and y e (0,00). Suppose 77, g IR
satisfies Proposition 3 says that to capture the reality of devel
oping countries we need to assume b < 0. Otherwise, we
(i) yo"277, +b + A<p{(l + 77,<p)7 - 1} =0, (39) do not get the empirical fact that the proportion of foreign
(ii) 1 + 77, <p > 0. (40)
debt in the government debt por
ingly, from now on we assume
Let p be the process defined by is' the interest
than the interest rate of foreign
p{t) = Kl/yX(t), (41) rate of depreciation (or devaluation) of the exchange rate p.
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Huamàn-Aguilar and Cadenillas: Optimal Currency Government Debt
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Table
Table
1. Parameter
1. Parameter
values. values. Economic Result 1. This economic result shows the
f <p ro r0
r
r
5 S
er
0
0.20.2 0.09
0.05
0.05
0.09
0.05
0.05
aa
0.1
0.9
0.9 0.1
AA
bb
1 1 -0.02
-0.02
effects of jumps in the exchange rates on
rency debt portfolio. Let b < 0. Let 7r(A, cp
optimal proportion of foreign currency debt
sity of the Poisson process N is À, and the s
is (p. Then
debt
and
Table
Table
2. Disutility
2. Disutility
valuessome
of some
values
debt policies
policies
of
and
value
value
function.
function.
(i) for every tp> 0, Aj > A0 implies
Tr(kl,<p)<Tr(k0,<p)\
7<3)(x)
V(x)
jP\x)
Z'V) tf2\x)
J(2)(x) J(3Hx)
V(x)
0.11543a:1,8
0.11443a:1,8
(ii) for0.11443x18
every A > 0, <pï> ç0 implies
-y
7= =
0.8 0.8
0.11617TC18
0.1 0.11484jc18
1617a:1'8 0.11484jc18
0.11543x'8
0.11421a;2,0
0.10889a:20
y=
y=
1.0 1.0
0.11388x20
0.1 0.11272a;2,0
1388jc20 0.11272x20
0.1142U20 0.10889jc2°
£(A, <Pi) < tt(A, <p0).
0.1142ÖJC22
0.10225a:2,2
y=
y=1.2
1.2 0.11264X22
0.11264jc22
0.11166JC22
0.11166jc22
0.11426x22 0.10225jc22
Note.
1 values
for the
of the used
other
parameters
thesethat countries with debt in a
Note.
See See
TableTable
1 for the
of thevalues
other parameters
in these
The Economicused
Resultin
1 implies
computations. foreign currency, whose exchange rate shows recurrent and
computations.
big depreciations (or devaluations), will reduce its propor
Table 3. The optimal
optimal solution.
solution. tion °fthis ^ °f debt in their Portfolio in favor
7f(f)
n(t)
y
7 = 0.8
7=
7= 1.0
1.0
7= 1.2
0.59459
0.47058
0.38936
P(t)
0.06655 X(t)
0.10889X(f)
0.10889X(t)
0.14952
0.14952 X(t)
X(r)'
V(x)
0.11443a:1,8
0.11443x'8
0.10889a2,0
0.10889a:20
0.10225a:2'2
0.10225a2,2
Note. See Table 1 for the values of the other parameters used
used in
in these
these
computations.
in local currency.
Proof of Economic Result 1. (i) Let /: (R x IR+ -> R
be defined by
f(z, A) := ya2z + A<p(l + zip)7 - (Atp - b).
We note that / is strictly increasing in each of its argu
ments. By Corollary 1, and definition of 7r(A0, <p), we have
5.2. Optimal Policy Versus Arbitrary Debt Policies *>• « =0 Since is "Wincreasing
We compare numerically the optimal debt policy we have /(7r(A0, tp), A,) > /(tt(A0, cp), A0) = 0.
obtained with the ones given in Examples 1 and 2 of Sec
tion 2. We will use the parameter values in Table 1 for the On the other hand, we observe that
numerical computations.
Specifically, we will consider three debt policies chosen y(0, a,) = b < 0.
arbitrarily, and denoted by (7tw, p'Jl), with i e {1,2, 3}. For
every t^ 0 we have 7r(1)(t) = 0, Tra'(t) — 50%, ir<3>(t) = Applying the intermediate value theorem to /(•,
100%, and pw(0 = T0X(f), f°r '€{1,2,3}. The corre- exists a unique z0 e (0, 77(A0, cp)) such that f(z
sponding disutilities are given by J(l,(x) :—J(x', tt{,>, p{,}). By the uniqueness of the solution given in Pro
We will compare them with the optimal debt policy (7r, p) we must have z0 = tt(A1 , <p). Hence, the claim hold
given in Corollary 1. We recall that the value function V (ü) This claim can be established in a similar
represents the minimum disutility and is a function of the
initial debt. Hence, the cost J(x; u«) must be greater than For a sPecial case' we Provide below the exPl
or equal to V(x). °f A and <p on the optimal proportion of foreign debt.
Table 2 shows that indeed J([)(x) > V(x) for i e Remark 6. Let us consider y = 1 and m = 1. Then, using
{1,2, 3}. As shown in Table 3, the optimal currency debt Remark 4 of §4 we have
portfolio and the optimal payments are sensitive to the
degree of debt aversion y. In particular, the higher the dir bip2
degree of debt aversion, the lower the proportion of for- fff ~ (<j2 _)_ A(p2)2 <
eign currency debt in the government portfolio. Indeed, this „
result will be proved rigorously in the next subsection. — = — < 0.
After illustrating that the government gets the best result d(P (a2 + ^T2)2
by following the optimal debt policy given in Corollary 1,
we proceed with the economic analysis.
5.3. Economic Results
Economic Result 2. This economic assertion is con
cerned with the effects of debt aversion on the optimal cur
rency debt portfolio. Let b < 0. For y e (0, 00), let ir(y) be
the optimal proportion of foreign currency debt when the
state and
and prove
prove two
tw economic results. We will use degree of debt aversion is y. Then ir( ■ ) is strictly d
We will state
the intermediate value theorem to prove them. ing. Moreover, lim^^ ir(y) = 0.
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Huamàn-Aguilar and Cadenillas: Optimal Currency Government Debt
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The Economie Result 2 states the role played by the Acknowledg
degree of debt aversion on the optimal currency debt port- The authors grate
folio. We observe that an increase in debt aversion leads anonymous referee
to a decrease in the proportion of foreign currency in the paper was present
portfolio and hence an increase of the proportion of debt in San Francisco, a
local currency. In other words, an increase in the degree of national Conferenc
debt aversion discourages countries from borrowing debt in for their commen
foreign currency. In addition, it is interesting to note that Sciences and Eng
an extreme debt aversion leads to decide not to borrow in 194137-2010 and 0
foreign currency ities Research Council of Canada [Grant 410-2010-0651].
Proof of Economic Result 2. Let /: IR x (0, oo) —> IR Appendix A. Debt Dynamics in Discrete Time
be defined by
To motivate the derivation of the multicurrency debt dynamics in
f(z y) - y<J2z + A<p(l + z,(p)y — (\<p — b) continuous time, in this appendix we present a discrete-time exam
ple. Before doing so, we discuss the connection between the price
Consider y, > y0. We note first that, for each fixed z>0 of a bond and the interest charges on the corresponding debt. Sup
we have f(z, y, )>f{7 To) and lim f(z y) = oo We pose that at time r the government issues one unit of a bond in local
also note that for each'fiîed y > oJThe function /(-, y) currtney. That is, the debt issued is R0(t). We want to calculate
is strictly
Moreover,
by definition
of RJt-I-,\tï
tt(y0), the interest
charges =
ofRJA
this debt
at time ' + A;; From E<iuation (4)
Jincreasing.
6 J yr0',
we knnw
thai
- RJtï
rr0A<
we know
that
R0(t
+ Ar) -- i\~d
R0(t) = R0(t)(er°A
/(w(yo).7o) = 0- Then
Thus, the difference of the prices of the bond i
/(«"(To). Ti) > f(â(yo). To) = 0.
resents the interest charges on debt in local cur
that the government issues one unit of a bond i
On the other hand, /(0, y,) = b < 0. Now applying the J instead. We want to ca
intermediate value theorem to /(-, y,), there exists a unique time ' + A? expressed
real number z e (0, fr(y0)) such that /(z, y,) = 0. By the glven bg the diff
c., i .. • • n - . Suppose that m = 1 and the change of bond prices and debt
uniqueness of the solution given in Proposition 3, we must ~ „ , * ..
, \ TT payments occur only at times t e 10, 1,2[. By definition of total
have
z
=
7r(y,).
Hence,
,
..
V
n
v 'u debt given in Equation (2), at time t = 0,
0 < Tr(y,) < fr(y0).
X(0) = A0(0)R0(0) + A, (0)*, (0)ß, (0)
To prove the second claim, we note that from Proposition 3,
we have 0 < ff, (y) <rj. The proof concludes after observ- Considerin
ing that lim fj = lim [(A<p - b)/ya2] = 0. □ sible payment
We show below that, in a special case, the effect of the
X(l) = A0(0)R0(l) + A,(0)R,(l)ß,(l)-p(l).
degree of debt aversion can be computed explicitly.
Remark 7. Suppose m = 1 and no jumps. Using Remark 5 We note that the debt at time t = 1 is determined by the debt
of §4, we get issued at time 0, i.e., A0(0) and A, (0), the new prices of the
bonds R00) and *iO)> and dle exchange rate 0,(1). Moreover
dir _ b ^ q the payment /;( I ) reduces the total debt. Based
the payment p(l) reduces the total debt. Based on the pre
dy y2 CT2 two equations, we can recast the total debt at time t = 1 as
lim 7T(y)= lim(—V)=0. X(l)=X(0)+A0(0)[/?0(l)-*o(0)]
y-> oo y er J
6. Conclusion
+ A, (0)[/?, (1)0,(1) —R0(0)ß0(0)] —
This form states that the total debt at time 1 is the sum of
We have made two contributions in this paper. On the . . . . , , , r , , ,
, . 1 ., , r r . the initial total debt X(0), the interest of debt in local cur
theoretical side, we have developed for the first time a rency ^(0)[^(1)_Ä (0)]> a
model for government debt control (that includes jumps eign currency expre
m the exchange rates and debt aversion), to find explicitly *„(0)0,(0)]. minus the debt
the optimal currency debt portfolio and optimal debt pay- At time t = \ the number
ments. On the applied side, this model provides a rigorous be the same as at time 0. I
explanation of the consistent reduction in the proportion
of foreign currency in government debt portfolios in favor X(l) = A0(l)R0(l) +
of local currency. Specifically, we have found that high
debt aversion and jumps in the exchange rates explain that where we recall that A„(l) a
behavior. local currency and foreign currency held at time 1, respectively.
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Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt
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Considering the prices of the bonds, the exchange rates, and pos- Thus, using
sible payments, the total debt at time r = 2 is l,...,m} are independent, we have
X(2) = A0(l)Ä0(2) + A1(l)Ä1(2)ß,(2)-p(2). Ex[e-s'Xy+i]=xy+lcxV{{(y+l)ß-8)t)
•-E*|exp{(y+l)i7Vw(r)}]
Combining the previous two equations, we arrive at
■E,
exp
J2\og(l+irT<pj)y+lNj(t)
X(2) = X(l)+A0(l)[*0(2)-tf0(l)] l t,=l
+ Ai(l)[/?i(2)ßi(2) (1)0! (1)] p( 2). (A2) Computing the expected values on the right-hand side of the above
equation, we obtain
Using Equations (Al) and (A2), we see that the total debt at time
t-2 can be written as Ex[Qxp{(y+\)7TTaW(t)}]=exp{^(y+\)2iTTaa7irt};
2
X(2)=X(0)+£Ao(i-l)[Ä0(i)-Ä0(i-l)] and for ;e{l,...,m}:
1=1
2
^[exp{log(l-t-7r7'(p;.)1
+EA1(i-l)[Ä1 (Oßj(O-Äi(i-l)ßt(i-l)]
Consequently,
2
-Ep(
0-
(A3)
Ex[e-s,Xj+l\=x^ltxV
Appendix B. Lemma 1 Appendix C. Proof of Proposition 2
Lemma B.l. Let p be a positive real number, and let n be a Proof. V is nonnegative by definition
constant vector in Rm such that property of V, we note that Equation (12) i
(l + 7rr<p;) >0, Vy'e{l,...,m}. J(vx;v,vp) = vy+lJ(x;ir,p).
Consider the following debt policy: ir(t) — it and p(t) = pX(t) Consequently,
for every t~^ 0. Then
V(vx):= inf J(vx;ir,vp) = vy+i inf J(x;Tt,p)
£x[e-6'X7+1] = x?+1exp{-((y+l)p+£)4, (Bl)
= vy+l inf J(x;Tt,p) = vy+l V(x).
(ir,p)ed
where
i t t Taking 1/jc in the above equation, we obtain V(x) = xy+1 V(l).
2("Xl)"y7r era ir Since xy+l is strictly increasing and convex, the proposition
follows. □
- E {(1+^ fj)Y+1 -1 - (t+1 }
7=1
Appendix D. Proof of Theorem 1
-Cy+1 )(irTb+r0).
Proof. Let (7r,p) be an admissible control process whose cor
Proof. According to Equation (22), the solution of the SDE that ^ponding debt dynamics is given by Equation (1
j . / \ • u stopping time such that r ^ 0, and let us consider the nonnegative
corresponds to (Tr,p) is given by ^ • • • , -,
process {a,at: 0}. Since v is t
an
X(t)=xexp|j ßds+j TrTadW(s)
application
of
Itô's
[. tAT
e
~s('ATlv(XIAT) = v(X0) +Jo / e-*(Se(ir„ Ps)v{Xs))ds
+ if l0g(l + 7TT(Pj)dNj(s)\,
J=1 0 I
+M(t)+EMj(t), (Dl)
7=1
where
where 2 is the operator defined in (24), and
m
ß:=r0-p+TtTb-(Pj -\TTTaaTir.
7=1
p tAT
M(t)-= j e-Ssv'(Xs)7rJaXsdW(s),
0
Since all the entries in a, <p, and ir are constants,
p tAT
Mj{ty.= jf e-Ss(v(Xs-+nJi<pjXs-)-v(Xs-))dNj(s),
X(t) = *exp j ßt+TTTaW(t)+Elog( 1 + vT <f>f) Njft) J.
formul
for every j6 (1,2,..., m}.
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Huamän-Aguilar and Cadenillas: Optimal Currency Government Debt
1056 Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS
Let a, b satisfy 0<a<X0 = x<b<oo. We define ra:=
show that the following family of ra
is uniformly integrable. Since
inffs^O: Xs-^a}, v=inf{s^O: Xs~^b}, and rab:=TaATb. (x,oo)}
We observe that rab < 0. Hence, for s < rab we have that both Xs
and Xs- belong to the interval [a,b]. In view of (11) and (13), if lim
b—>00
E
Ta < oc, then we have 0 < XTa ^ a.
From now on, we set T=rab. We claim that the above stochas
[e^'AT-)W(X,AT-)] = £,[lime-4('AT-)i,(X<AT-)]
=Ex[e-*UM°)v{XtMaj\.
tic integrals M and Mj are martingales. Indeed, since Xse[a,b]
for all se[0,t), and v' is continuous, there exists a real number
Consequently, taking the limit as b f oo in inequality (D3), we get
£ > 0 such that
f r tATa
v(x)<Ex[e-s^v(XlMa)]+Ex jf e~Ssh(Xs,ps)ds . (D4)
f e~2Ss(v'(Xs))2TTj(TcrTTTsX*ds
J0
^Ex
I V'
Now letting a f 0, we have Ta 10. Recalling that i>(0+)=0,
taking the limit as a f 0, and proceeding as in the case b\oo,
a a it, ds
J0
we obtain
Hence, the integral with respect to the Brownian motion above is a
lim£"_
martingale. Similarly, we prove the other part of the claim, that is,
u-> 0
Mj is a martingale. It suffices to show that for each
f tATa "1 f , tA0
/
e~Ssh{Xs,ps)ds
=EX / e~Ssh
Jq
JO
and
/* ab
o
/ e~1Ss A j ( u (Xs- + ttJ- <PjXs- ) - v (Xs- ))2 ds
linEx [e~s^v(XIATa )} = EX [\ime~s^v(XIMa )]
= £,[e-s'u(X()/((<0}].
Indeed, recalling that Xs- € [a, b\ for all s e [0, t), and v is con
tinuous, we have that Ex[f^Tab e~2Ss X.j(v(Xs-))2ds] is bounded.
Taking the limit as a J, 0 in inequality (D4), we get
Using (18) and (26), we conclude that Ex[f'QMab e~2Ss\j{v(Xs- +
7tJ-<PjXs-))2ds] is also bounded. Hence, the previous inequality
holds. This proves that the integrals with respect to the compen
r /• /a©
rWai[c8XXl)/!(<ei]+£I Jf e~Ssh(Xs,Ps)ds
sated Poisson process are martingales. This completes the proof
of the claim. Consequently, for every t ^ 0, we have Ex\M(t)] = Q,
We note that (19) and (26) imply
and Ex[Mj(t)]=0 for each
Taking expectations in Equation (Dl),
hm£,[e-8yx,)/{,<e}]=0.
Ex[e-s^v(XtArj\
Now letting /->oo, by the monotone convergence theorem, we
f tATab -
conclude that
= v(x)+Ex / e s(%(TTs,ps)v(Xs))ds
JO
v(x)^Ex f° e~Ssh(Xs,ps)ds
Jo
Since the HJB Equation (25) implies that for s e [0, r)
2(tTs,ps)v(Xs)^-h{Xs,ps), (D2)
it follows that
t(x)a.[[c8(M",»(x,Arj]
r t^Tab _
+ EX
Jo
/ e h(Xs,Ps)ds
Letting b foo, we have rb\co, and hence rab fT0. By the
monotone convergence theorem,
lim Ex
b—*oo
(D5)
This proves part (a) of this Theorem.
To show part (b), we observe that, since (ir,p) satisfies
(27), the HJB Equation (25) implies that the inequality (D2)
becomes an equality when (Trs,ps) = (irs,ps) and X(s)=X(s).
Then, inequality (D5) becomes an equality as well. This com
pletes the proof of this theorem. □
(D3)
Endnotes
1. There exist numerical approaches as well. See Melecky (2007)
for a survey.
2. Besides, the consideration of different terms makes the prob
r tATab „ r C c
lem intractable from an analytical perspective. That is why Bolder
(2002) has to pursue a simulation approach.
I e h(Xs,ps)ds =EXU
e h(Xs,ps)ds
3. Cadenillas and Zapatero (1999) assume that, in the absence of
government interventions in the exchange markets, the exchange
On the other hand, since v is continuous,
letting
b motion.
f oo,
we
obtain
rate follows a geometric
Brownian
That
is, they
assume
Equation (3) without the component that corresponds to the com
v(XtATJ->v(XlAJ.
pensated Poisson process N.
4. From the country perspective, of course the government (but
not necessarily the debt
manager) cancondition
influence, to some(26),
extent,
Moreover, since u satisfies the polynomial
growth
local interest
rate. (2006, p. 158), one can
using the strategy in Fleming the
and
Soner
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Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt
Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS 1057
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