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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 Stable URL: https://www.jstor.org/stable/24540432 Accessed: 04-02-2021 02:20 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms = 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. This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt 1050 Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS 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 This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms 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). This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms 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. This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms Huamàn-Aguilar and Cadenillas: Optimal Currency Government Debt Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS 1053 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. This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms Huamàn-Aguilar and Cadenillas: Optimal Currency Government Debt 1054 Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS 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. This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS 1055 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}. This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms Huamân-Aguilar and Cadenillas: Optimal Currency Government Debt Operations Research 63(5), pp. 1044-1057, ©2015 INFORMS 1057 References Licandro FG, Masoller OA (2000) The optimal currency composition of Uruguayan public debt. Working paper, Central Bank of Uruguay. Applebaum D (2009) Levy Processes and Stochastic Calculus, 2nd ed. MasCollel A, Whinston MD, Green JR (1995) Microeconomic Theory (Cambridge University Press, London). (Oxford University Press, New York). Blanchard O, Fischer S (1989) Lectures on Macroeconomics (MIT Press, Melecky M (2007) Choosing the currency structure for sovereign debt, a Cambridge, MA). review of current approaches. World Bank Policy Research Working Bolder DJ (2002) Toward a more complete debt strategy simulation frame work. Working Paper 2012-13, Bank of Canada, Ottawa, Ontario. Paper 4246, World Bank, Washington, DC. Borensztein E, Cowan K, Eichengreens B, Panizza U, eds. (2008)Oksendal Bond B, Sulem A (2008) Applied Stochastic Control for Jump Difus sions, 2nd ed. (Springer, Berlin). Markets in Latin America (MIT Press, Cambridge, MA), 1-28. Cadenillas A (2002) A stochastic maximum principle for systems Panizza with U (2008) Domestic and external public debt in developing coun tries. Discussion Papers 188, United Nations Conference on Trade jumps, with applications to finance. Systems Control Lett. 47(5): and Development (UNCTAD). 433^144. Pratt JW (1964) Risk aversion in the small and in the large. Econometrica Cadenillas A, Zapatero F (1999) Optimal central bank intervention in the 32(1): 122-136. foreign exchange market. J. Econom. Theory 87(1):218—242. Cadenillas A, Lakner P, Pinedo M (2013) Optimal production manageProtter P (2004) Stochastic Integration and Differential Equations, 2nd ed. (Springer, New York). ment when demand depends on the business cycle. Oper. Res. 61(4): Taylor JB (1979) Estimation and control of a macroeconomic model with 1046-1062. rational expectations. Econometrica 47(5):1267-86. Cont R, Tankov P (2004) Financing Modelling with Jump Processes (CRC Zapatero F (1995) Equilibrium asset prices and exchange rates. J. Econom. Press, Boca Raton, FL). Dynam. Control 19(4):787—811. Fleming WH, Soner HM (2006) Control Markov Processes and Viscosity Solutions, 2nd ed. (Springer, New York). Giavazzi F, Missale A (2004) Public debt management in Brasil. CEPR Ricardo Huamân-Aguilar is a Ph.D. candidate in mathemat Discussion Paper 4293, London. ical finance in the Department of Mathematical and Statistical Greiner A, Fincke B (2009) Public Debt and Economic Growth. Dynamic Sciences at the University of Alberta. His research interests are Modeling and Econometrics in Economics and Finance Series, in the applications of stochastic processes and optimal stochastic Vol. 11 (Springer, Berlin). Guo W, Xu C (2004) Optimal portfolio selection when stock prices followcontrol to economics, finance, and risk management. a jump-diffusion process. Math. Meth. Oper. Res. 60:485-496. Abel Cadenillas is a professor in the Department of Mathe Ikeda N, Watanabe S (1981) Stochastic Differential Equations and Diffumatical and Statistical Science at the University of Alberta. He sion Processes (North-Holland Publishing Company, Amsterdam). was World Class University Distinguished Professor of Finan Jeanblanc-Picqué M, Pontier M (1990) Optimal portfolio for a small cial Engineering, awarded by the Korean Ministry of Education, investor in a market model with discontinuous prices. Appl. Math. Science and Technology. His research interests include opera Optim. 22:287-310. Kydland FE, Prescott EC (1977) Rules rather than discretion: The incon tions research, finance, economics, management sciences, and mathematics. sistency of optimal plans. J. Political Econom. 85(3):473^192. This content downloaded from 142.150.190.39 on Thu, 04 Feb 2021 02:20:51 UTC All use subject to https://about.jstor.org/terms