alternative foreign exchange management protocols: an application of sensitivity analysis

Journal of Multinational Financial Management

12 (2002) 1–19

Alternative foreign exchange managementprotocols: an application of sensitivity analysis

Antoine Gautier a, Frieda Granot b, Maurice Levi b,*a Faculte des Sciences de l’Administration, Uni�ersite La�al, Quebec, QC, Canada G1K 7P4.

b Faculty of Commerce and Business Administration, The Uni�ersity of British Columbia, Vancou�er,British Columbia, Canada V6T 1Z2.

Received 8 October 1999; accepted 4 August 2000

Abstract

This paper considers the choice between two foreign exchange management protocols,namely whether to hold currencies received until required, or whether to convert foreigncurrencies into the home reporting currency and back as needed. The alternative protocolsinvolve a tradeoff between saving transaction costs versus stability of home currency valuesand economies of scale in interest earned on working capital balances. Qualitative sensitivityanalysis is applied to the currency management problem in this form to investigatesensitivities and interdependencies in sources of and needs for currencies. The analysis revealsseveral implications which are not apparent without viewing the problem in such a context.© 2002 Elsevier Science B.V. All rights reserved.

JEL classification: C61; F31

Keywords: Foreign exchange management; Multinational finance; Optimization; Sensitivity analysis

www.elsevier.com/locate/econbase

1. Introduction

One way for a firm to deal with the foreign exchange cash management problemis to hold funds in all currencies in which business is done. This facilitates makingpayments in the different currencies and saves transaction costs, but is costly inforegone investment income; small amounts in each currency yield relatively lower

* Corresponding author. Tel.: +1-604-8228260, fax: +1-604-8228521.E-mail address: [email protected] (M. Levi).

1042-444X/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.

PII: S1042-444X(01)00023-8

A. Gautier et al. / J. of Multi. Fin. Manag. 12 (2002) 1–192

returns than large amounts in a single currency, especially if this single currency isan important currency such as the U.S. dollar with a variety of interest-earninginstruments. An alternative way of dealing with the multinational currency manage-ment problem is to convert everything into a major currency and then buy variousforeign currencies as needed. This has benefits as far as interest earnings and choiceof investment vehicles are concerned, but is costly in terms of transaction costs anduncertainties on timing of conversion when foreign banking systems are inefficient.As well as the trade-off between interest earnings and transaction costs, there areconsiderations involving uncertainties of changes in exchange rates; the centralizedalternative of keeping funds in (say) the U.S. dollar has more certain value indollars versus the holding of foreign-currency cash balances.

Understanding the trade-offs that exist in choosing between cash managementprotocols allows a company to choose a suitable system based upon its needs.Proper understanding requires knowing how events in one part of the global cashpicture impact on the situation elsewhere. The analysis of such interdependenciesbetween preferred positions in different currencies is the motivation of this paper.

We start by representing the multinational cash management problem as ageneralized network flow problem with arc-gains or losses. The attractiveness of thenetwork flow approach has long been recognized — see for example Christofides etal., (1979), Gautier and Granot (1992). We then apply qualitative sensitivityanalysis developed by Gautier and Granot (1996) for generalized network flowproblems to help money managers understand the interdependencies in the sourcesof and needs for different currencies.

The qualitative sensitivity analysis reveals how sales, purchases and holdings ofdifferent currencies respond to exogenous changes in cash flows from operations.Several interesting and revealing implications are discovered concerning, for exam-ple, how different currency trading and holding decisions interact. These implica-tions would be difficult to determine without engaging in the type of analysisperformed here. For example, we show that cash balances in two foreign currenciesare substitutes only if these have similar characteristics in terms of exchange ratesversus interest rate differences. The substitution property implies that if one isforced to, say, sell the first currency, the model will always recommend purchase ofthe other.

A striking finding is that minor modifications in some parameters of the modelmay have substantial consequences. For example, a small exogenous change in therequired range of amount within which a given currency must be held can result inchanges in levels of other currencies that are several times the magnitude of theimposed, exogenous change. This amplification effect is enhanced when foreignexchange transaction costs are small. Such an effect could help account for the highvolatility of trading volumes and prices on the foreign exchange markets, sincemodest ‘shocks’ may lead to large, rippling effects.

Section 2 introduces the currency management optimization model first concep-tually, then mathematically, and discusses several operational assumptions. Section3 looks at the model as a network optimization problem, summarizes the relevantsensitivity analysis theory and discusses some limitations to its application. Section

A. Gautier et al. / J. of Multi. Fin. Manag. 12 (2002) 1–19 3

4 applies the sensitivity analysis theory to the particular network problem to obtainsubstitutes, complements and ripple bounds. Finally, Sections 5 and 6 workthrough numerical examples.

2. A currency management model

We assume that cash inflows are received in a variety of different currencies andsubsequent cash outflows are also in a variety of currencies. As is also typical ofmany multinationals, we assume that all currencies are managed with concern forthe value of cash holdings and the volatility of cash holdings in terms of thereporting currency of the firm. The reporting currency, also called the base currency,is denoted by the index 0. For example, a U.S.-based multinational could bereceiving numerous foreign currencies, making payments in numerous currencies,and caring about the level and volatility of cash holdings in dollars, the company’sreporting currency.

2.1. Multinational firms and cash management

There are several reasons why multinationals maintain cash holdings in majorcurrencies, such as the U.S. dollar. For one thing, short-term financial instrumentsexist in the major currencies that are not found in less-widely traded and heldcurrencies. In addition, by pooling funds together, economies of scale are enjoyedin rates of return. If funds are managed as well as held centrally, economies of scaleare also enjoyed in operations; one person can handle additional amounts of fundswith little if any increase in labor costs. Coordination is also facilitated bycentralizing cash management. Reasons for holding funds in major currencies andfor centralizing cash management are discussed in Levi (1992a,b, 1996).

2.2. The planning period and assumptions

For simplicity we consider a single-period planning model. However, since theprincipal purpose is explanatory and to provide a sensitivity analysis, and becausethe planning period can be arbitrary long (with appropriate adjustments to vari-ance), we believe the model provides valuable insights. Consideration of short-termstability of translated values is consistent with the reported practice of some majormultinational corporations (Lewent and Kearney, 1990; Maloney, 1990). Profes-sional recommendations on the appropriate objectives of corporate treasury man-agement also emphasize stability of translated foreign currency values.

In correspondence with the practical problem faced by multinational firms, eachforeign currency received at the beginning of the period can either be entirely kept,partially kept, or entirely sold into the firm’s base currency. All allocations, onceset, remain unchanged until the end of the period. At the end of the period, thebase currency may be used to purchase foreign currencies as needed. For thepurpose of analysis, the requirements of each currency are assumed known for the


end of the period; with international sales often being associated with trade credit,sales subsidiaries of multinationals can predict cash flows for an upcoming periodof time. However, exchange rates at the end of the period are assumed to beunknown.1 The objective is to achieve optimal performance over the upcomingplanning period for given forecasted cash flows. It may be to maximize the expectedvalue of the remaining amount of the base currency at the end of the period, tominimize a (quadratic or otherwise convex) exchange-rate risk function, or somecombination of both. Volatility of income, while diversifiable by shareholders, maybe of concern if tax rates are progressive, if there are performance clauses in bonds,and so on. This makes volatility as well as the level of income (or wealth) alegitimate concern — see Levi (1996), Levi and Sercu (1991).

2.3. Variables, constraints and costs

We adopt the following notation:

initial number of units of currency i=0, 1, …, nui

si number of units of currency i sold at the beginning of the periodzi number of units of currency i unsold at the beginning of the period

number of units of currency i needed at the end of the perioddi

number of units of currency i purchased at the end of the periodpi

V� number of units of base currency left at the end of the period — a randomvariablethe expected value of V�V

Various exchange rates are also required since the base currency may be purchasedat the beginning of the period, and may be sold at the end.

number of units of currency i needed to buy one unit of currency 0S(i/0)in the spot exchange market at the beginning of the periodnumber of units of currency 0 needed to buy one unit of currency iS� (0/i )in the spot exchange market at the end of the period — a randomvariable

S*(0/i ) expected value of S� (0/i ) — that is, S*(0/i )=E(S� (0/i )).rate of return on investments in currency i over the period.�i

the variance VAR(S� (0/i ))� i2

�ij the covariance COV(S� (0/i ), S� (0/j )).

Each unit of foreign currency initially present is either kept throughout the periodor exchanged for base currency, i.e. for each currency i=1, …, n

1 While exchange rate uncertainties can be hedged for major currencies, this is often impractical forthinly-traded currencies received and spent by many multinational firms. The extent of substitutabilitiesand complementarities between currencies would be affected by hedging, but consideration of thequantitative effects is beyond the scope of the current paper.


si+zi=ui. (1)

Similarly, the end of period currency requirement (di) is met by a combination ofinvestments in the foreign currency (an amount zi at rate �i) and the amountpurchased later (pi) using the base currency, i.e. for each foreign currency i=1, …, n,

(1+�i)zi+pi=di. (2)

For the base currency, the beginning of period balance, z0, is the amount that isinitially available plus amounts from beginning-of-period sales of foreigncurrencies:

z0=u0+�n

1

si

S(i/0). (3)

At the end of the period the remaining base currency corresponds to the amountkept during the period to gather interest (z0 at rate �0) minus expenditures forend-of-period purchases of foreign currencies — pi at the exchange rate S*(0/i ):

V� = (1+�0)z0−�n

1

S� (0/i ) pi.

With si, pi and zi we associate a parameter t is, t i

p and t iz, respectively. By varying the

parameters one can simulate sensitivity analysis to events such as variations inupper and/or lower bounds on amounts of currencies bought/sold, variations inminimum/maximum currency holdings, changes in expected returns, exchange rates,rate volatility and more. The objective now is to minimize a total cost functiondefined as a sum of real or �-valued individual cost functions, each of whichdepends on a variable–parameter pair in the model. These individual cost functionsinclude both real (incurred) costs of currency management as well as finite penaltiesfor, say, risk factors, and infinite penalties when a variable is at a level forbiddenby (say) management-imposed bounds; costs of bankruptcy might motivate rigidbounds as a pragmatic device to avoid extremely risky/overly exposed positions.

Explicitly the costs are defined as follows. We associate a parametric currencyconversion/handling cost b(·,·) with the quantities si, pi and zi. These are the costsof managing the currencies. For instance, we denote the total cost associated withkeeping z0 units of base currency by b 0

z(z0, t0z) where t0

z is a parameter associatedwith the base currency. The parameter t0

z may represent anything that affectscurrency management cost at a firm’s currency center, including costs attached toexchanging and handling foreign currencies. By having non-negativity constraintson cash holdings we impose the assumption that the firm has non-negative networth in each of its currency holdings at all times. This assumption is rooted in themuch larger spreads between borrowing and lending interest rates than betweenbuying and selling rates on currencies. The non-negativity constraints which imposethe restriction on borrowing, as well as upper bounds on variables, can beincorporated into the cost-function by setting the corresponding cost to +� for allforbidden values.


2.4. Risk and foreign exchange exposure

It follows from the expression of V� above that the expected value of the portfoliofrom the parent company’s perspective is,

V= (1+�0)z0−�n

1

S*(0/i ) pi, (4)

and its variance is,

VAR�

(1+�0)z0−�n

1

S� (0/i ) pi�

=�n

1

� i2pi

2+2�n

1

�ijpipj, (5)

which represents a risk factor to be included in the objective function — see e.g.Mulvey, 1987. The incorporation of risk as the end-of-period variance of thehome-currency value of a portfolio of currencies assumes that the success ofcorporate treasury management is judged in part according to the stability of thetranslated consolidated value of global liquid assets at the end of the period.

2.5. Mathematical program

In the mathematical model of currency management, two objectives are worthpursuing: on the one hand it is natural to aim for high values of the remaining basecurrency V for given cash flows. On the other hand, managers will be interested inlower levels of risk as expressed in Eq. (5). We propose to minimize the following.

c(x, t)= −V+��

n

1

� i2pi

2+2�n

1

�ijpipjn

+b 0z(z0, t0

z)+�n

1

(b iz(zi, t i

z)+b is(si, t i

s)+b ip(pi, t i

p)), (6)

where x and t are, respectively the decision �ector (z0, si, zi, pi, V)i=1,…,n and theparameter �ector (t0

z, t is, t i

z, t ip, tV)i=1,…,n. The positive scalar � represents the com-

parative preferences of the decision maker over the value versus risk tradeoff. Theresulting formulation of the currency management problem is,

min{c(x, t): (1−4)}. (7)

Since the constraints are linear, the computational difficulty of solving Problem (7)depends mainly on the nature of the objective function. In practice, estimatedmeasures of correlation are seldom negative, e.g. Das (1993), and thus the covari-ance term does not introduce a non-convex element. Depending on the remainderof the objective function, one may resort to quadratic programming or moregeneral convex programming codes in order to solve Problem (7).


3. Elements of qualitative sensitivity analysis

When designing a currency management protocol, some of the parameters suchas exchange rate standard deviations, firm- or bank-imposed bounds on amountsheld in various countries/currencies, willingness to take risks, etc. must be set tosome initial estimated values. The methods presented here help in designing such aprotocol by indicating how sensitive the model and its solution are to changes in thesystem-design parameters. In particular, the model permits the study of differentscenarios without having to re-solve Problem (7) thereby improving the currencymanager’s understanding of the effects of unforeseen changes on optimal exchangemanagement policies.

In this technical section we review several relevant results about generalizednetworks and qualitative sensitivity analysis as they apply to the present model.

3.1. Paths, cycles and elementary �ectors

In order to study the qualitative variations of the optimal solution and theoptimal value due to variations in certain parameters we use some of the results forqualitative sensitivity analysis established by Gautier and Granot (1996). We wantto learn about the properties of the difference between two optimal solutions —one before and the other after a change. Let us call � the vector of those differences.In order for both solutions to be feasible, i.e. to verify Eqs. (1)– (4), someconditions must be met by � : for example Eq. (1) implies that if si changes then zi

should change as well (provided the right-hand-side ui remains unchanged). Possiblechanges (�) that are of the simplest form, in the sense that few variables are actuallychanged, are called elementary vectors. More complex changes are sums of suchsimple changes, so that a property that is valid for all elementary vectors, of whichthere exists only a finite number, and that is conserved by summation, will remaintrue of all possible changes. Thus, the essence of qualitative sensitivity analysis liesin the study of elementary vectors. The generalized network structure of theconstraints Eqs. (1)– (4) is important and will be exploited later.

In a Generalized Network the amount of flow picked up by an arc is multipliedby a (positive) gain while traversing the arc. The difference between the flowentering a node and the flow leaving that same node is equal to a fixed demandspecific to the node (Rockafellar, 1984; Ahuja et al., 1993). The constraints Eqs.(1)– (4) can be regarded as node balance equations of such a generalized network. Anexample is given in Fig. 1 for three foreign currencies and a base currency, the U.S.dollar. All variables correspond to flows lea�ing a node except for the pi variableswhich are entering a node (in which case a flow S*(0/i )pi is entering the arc).Problem (7) can therefore be solved with the aid of any non-linear generalizednetwork flow code, see e.g. More and Wright (1993).

A generalized network is characterized by a set of nodes (N), a set of arcs (E),each of which has a gain — denoted �ij�0 for the arc (i, j ). By a path p we meanan ordered set of arcs such that each pair of successive arcs shares a node and nonode is shared by more than two arcs. A path is also a cycle if, in addition, a node


is shared by only its first and last arcs. A cycle can be uniquely represented by avector c= (cij) where cij is 1 if (i, j ) is a forward arc, −1 if (i, j ) is a backward arcand 0 otherwise. If one were to send one unit of flow along some cycle c, this unitwould be multiplied by gains along forward arcs and divided along backward arcs.The flow delivered at the end of the cycle would thus correspond to the acti�ity ofthe cycle,

�cij=1

�ij× �cij= −1

1�ij

=� � ijcij. (8)

A cycle is neutral if its activity has value 1 and acti�e otherwise.Examining differences such as � involves considering a circulation space, that is,

the set of vectors x which verify the following system of homogeneous equationscorresponding to Eqs. (1)– (4),

si+zi=0 and (1+�i)zi+pi=0 (i=1, …, n),

z0−�n

1

si

S(i/0)=0,

(1+�0) z0−�n

1

S*(0/i ) pi−V=0. (9)

An elementary �ector of the circulation space is a non-zero vector with minimalsupport (set of non-zero entries, see Rockafellar (1984)). The following will pro-vide some intuition into the relevance of elementary vectors to sensitivityanalysis. A non-trivial result is that any currency allocation can be written asx=x0+�p=1

P �pfp where F={f1, …, fP} is the set of elementary vectors,�1, …, �P�R and x0 is some arbitrary allocation (Rockafellar, 1984, p. 455).Therefore, a change in allocation can be regarded as a modification of thevector �= (�1, …, �P) and, in the algebraic sense, the family F spans the space ofall possible allocation perturbations. Rich implications follow from this. Indeed, forthe simple case where �= (1, 0, …, 0), the allocation is modified by f1 resulting ina new allocation x �=x+ f1. That is, each entry k of the allocation (amounts

Fig. 1. The graph G.


purchased, sold or kept) is modified by an amount fk1, so that the ratio of

magnitude of change in one entry xj to that of another xk is �fk1/f j

1�. The Rippleresult to follow states that any bound on that last quantity is also a bound oncomparative rates of change.

3.2. Conditions imposed on the objecti�e function

We require that the cost function be separable, i.e. of the form c(x, t)=�j c(xj, tj)and con�ex in the decision variable x, that is

cj(axj1+ (1−a)xj

2, tj)�acj(xj1, tj)+ (1−a)cj(xj

2, t), (10)

for all xj1, xj

2�R, 0�a�1, and for all values of the parameter tj. Convexity ariseswhen currency management is subject to diminishing returns. We further requirethat cj (xj, tj) be lower semicontinuous, that is, that all its level sets {xj : cj(xj, tj)�a}, a�R, be closed, and that each cost cj (xj, tj) be submodular, that is, for any twopairs (xj

1, t j1) and (xj

2, t j2) in R2.

cj((xj1, t j

1)� (xj2, t j

2))+cj((xj1, t j

1)�(xj2, t j

2))�cj(xj1, t j

1)+cj(xj2, t j

2),

where � and � denote, respectively the coordinatewise minimum and maximumof two vectors. Note that if cj is twice-continuously-differentiable, submodularity isequivalent to negative crossed, second-order derivatives, that is, �2cj/�xj�tj�0. Ifcj (xj, − tj) is submodular then cj (xj, tj) is said to be supermodular. The convexityand submodularity assumptions are quite flexible as will be illustrated in exampleslater in this paper. For additional examples see also Granot and Veinott, (1985),Gautier et al. (1997).

3.3. The zero-co�ariance case

If the covariances �ij are all zero, then the risk factor in Eq. (5) becomes �1n � i

2pi2.

This leads to a simpler model in terms of solution procedures and opens thepossibility of using qualitative sensitivity analysis. The assumed absence of covari-ance between spot rates vis a vis the base currency requires that the source ofvolatility is in cross rates between pairs of foreign currencies rather than in the basecurrency. This depends on the extent that U.S. dollar exchange rates fluctuate fromU.S.-based events versus idiosyncratic events in the foreign countries (Engle et al.,1990). However, if exchange rates are driven by U.S.-based events so that the zerocovariance assumption is not valid, the sensitivity analysis can still provide insights.For example, we can still consider the complementarities and substitutabilitiesbetween currencies, the effects of minimum currency holding constraints, and so on.Of course, it is recognized that quantitative conclusions would be affected bycovariance between exchange rates. Subject to what can be learned in the simple,zero-covariance model, results could in principle be obtained with exchange ratecovariance, with magnitudes possibly based on historical values. It is also possiblein principle to judge sensitivity to the sizes of covariances. The purpose of thispaper, however, is to develop an initial, simple model that can be operationalized,


albeit subject to simplifying assumptions. If we assume zero covariance, then theobjective in Eq. (7) is separable and Eq. (6) yields

c(x, t)= −V+ (��02z0

2+b 0z(z0, t0

z))

+�n

1

b iz(zi, t i

z)+�n

1

b is(si, t i

s)+�n

1

(�� i2pi

2+b ip(pi, t i

p)).

Problem (7) is now a monotropic problem, i.e., a convex and separable problem withlinear constraints.

3.4. Qualitati�e sensiti�ity analysis for generalized network flows

Under certain conditions, the Ripple Theorem for generalized network flowproblems (Gautier and Granot, 1996) will guarantee that the variation of onedecision variable, say xk, will be within a known factor of the variation in anothervariable, say xj. This fixed constant, denoted by Mjk, is obtained via the elementaryvectors of the circulation space of Problem (7) as follows:

Mjk=max{�fk/fj �:fj�0, f any elementary vector}.

Specifically, suppose that the arc-costs are convex as expressed in Eq. (10) andthat the value of tj is changed. Both values of the parameter are taken to yieldfeasible mathematical programs. Then, if x is optimal before the change, thereexists an optimal solution x � after the change which verifies the Ripple Property,that for every variable xk

�x �k−xk ��Mjk �x �j−xj �. (11)

The Monotonicity Property asserts that the optimal value of a given quantity ismonotone in the parameters of certain other quantities in the model. In order forthis to hold we require assumptions on the relation between the correspondingarcs/variables. To that end we say that xi and xj are complements (respectively,substitutes) if and only if for all elementary vectors f= (fk), fi fj�0 (respectively,0). (The currency significance of substitutes and complements is explained later).

4. Properties of the generalized network

In this section we study various properties of the graph G in Fig. 1 and verify thehypotheses needed to apply the above theory to the currency management problemas given in Eqs. (1)– (4). We first list the elementary vectors which play a centralrole therein.

4.1. Elementary �ectors

For simplicity of exposition we denote the arcs in G associated with the variablesz0 (respectively, si, zi, pi, V) by e0

z (respectively, e is, e i

z, e ip, eV). Thus any cycle in G

other than the loop eV is either of the form


c(i )=e is+e0

z +e ip−e i

z or c(i, j )=e is−e j

s+e jz−e j

p+e ip−e i

z.

These cycles have activities �i and �ij, respectively, which can be computed usingEq. (8):

�i=1+�0

(1+�i) S(i/0) S*(0/i )and �ij=

�i

�j

. (12)

We can refer to �i as the swap gain/loss of the foreign currency i relative to thebase currency, and we assume for practical purposes that the non-base currenciesare indexed by increasing swap gain/loss (i.e. �i��j whenever i j ). This positivenumber has a simple interpretation if one considers the use of one unit of currencyi at the beginning of the period. This unit can be kept to gather interest �i, yielding1+�i units at the end of the period, or be converted into the base currency at theearly rate S(i/0), gather interest at rate �0, and then be converted back to theforeign currency at the expected exchange rate S*(0/j ). The second option yields(1+�0)/(S(i/0) S*(0/i )) units of currency i at the end of the period. The ratio ofthese two resulting values is precisely �i. Thus, if one were to leave aside riskfactors, the first option to remain in the foreign currency is preferable if �i1(swap loss), the second is preferred if �i�1 (swap gain) and the two are equivalentif �i=1 (swap neutral).

Clearly, the swap gain/loss is typically close to 1 since the various exchange ratesreflect interest differentials between currencies/countries. Indeed, in the absence oftransaction costs and convexities in yields, the Unco�ered Interest Parity Theorem(Levi, 1996) predicts 1+�i= (1+�0)/(S(i/0) S*(0/i ), meaning that �i has value 1and the cycle c(i ) is neutral. However, if there are costs of transacting in the spotforeign exchange markets only we have 1+�i� (1+�0)/(S(i/0) S*(0/i ), so that�i1 although very close to 1. When there are also economies of scale frompooling funds and investing larger amounts, such as there are in bank depositswhich are a common vehicle for investing working capital, then �0 could besufficiently higher than the interest parity level to make �i�1. Clearly the baseinterest rate �0 may increase when the amount z0 kept at the money center reacheshigher levels. We assume here that �0 is fixed and, through bounds present inb0

z(·, t0z), impose z0 to be in the appropriate range. In the case where the optimal

value of z0 is close to the bounds, one may want to reoptimize in the next interestrange for �0.

The possibility that �0 could more than offset the transaction costs of going intoand out of the base currency is supported by the existence of currency managementcenters because funds would be maintained in the currencies received if there wereno compensation in yields earned at the center. The fact that some funds are keptlocally and some are returned suggests that for some currencies i, �i1, while forothers �i�1. Indeed, it is currencies for which there are well-developed financialmarkets — the British pound, Euro and so on — where we expect �i1, while forcurrencies without developed financial markets we expect �i�1, prompting the useof currency centers in major financial centers. Of course, if managers chose to buyall foreign exchange requirements forward there is no foreign exchange risk and we


have the special case of maximizing V. For generality and to reflect the fact thatforward hedging possibilities are limited in many of the thinly-traded currencieshandled by multinationals, we use the uncertain expected spot rate.

Given the simple topology of G we can now list all the elementary vectors of thecirculation space by their respective supports as follows:

f(i, j, 0)f(i ) f(i, 0, V) f(i, j ) f(i, j, V)Elementary vectorc(i ), eV c(i, j ) c(i, j ), eVc(i ), c( j )Cycles and arcs in support c(i )

The non-zero entries of each elementary vector are given below, where the entryof the vector f corresponding to the decision variable xj is denoted by f [xj ]. (Aselementary vectors are defined up to a multiplicative constant, any given non-zeroentry may be set equal to 1, with the remaining entries following from the systemEq. (9).) For each swap-neutral currency i (�i=1), we have

f(i )[z0]=1 f(i )[pi ]=1+�0

S*(0/i )f(i ) [si ]= − f(i, j, 0) [zi ]=S(i/0). (13)

For pairs of currencies i and j such that �i=�j, the cycle c(i, j ) is neutral, andtherefore is the support of the following elementary vector,

f(i, j )[zi ]= − f(i, j )[si ]=1 f(i, j )[sj ]= − f(i, j, 0)[zj ]=S( j/0)S(i/0)

f(i, j )[pj ]=S( j/0)(1+�j)

S(i/0)f(i, j )[pi ]=

−�i

(1+�j)�j

.

For each pair of currencies for which 1��i��j�1, the union of the supports ofc(i ) and c( j ) include that of c(i, j ) and yield the following elementary vector

f(i, j, 0)[z0] =1

f(i, j, 0)[pi ] =(�j−1) (1+�0)(�j−�i) S*(0/i )

f(i, j, 0)[si ] = − f(i, j, 0)[zi ]=(�j−1)�i S(i/0)

�j−�i

f(i, j, 0)[pj ] =(1−�i) (1+�0)(�j−�i) S*(0/j )

f(i, j, 0)[sj ] = − f( j, i, 0 )[zj ]=(1−�i) �j S( j/0)

�j−�i

. (14)

If �i�j1, then the signs of f(i, j, 0) are such that it can be thought of as a sumof (forward) flows along two cycles, c(i ) and c(i, j ). This results in the pattern ofsigns given in the second row of Table 1. The remainder of Table 1 follows fromsimilar considerations. We note that when �i�1, there exists an elementary vectorcorresponding to the active cycles c(i ) and eV, whose non-zero entries are given by


Table 1Signs of elementary vector entries (ij )a

zi pi sj zjElementary vector pjExists when,… z0 Vsi

− +�i=1 + +f(i )�i�j1 − + − + − + +f(i, j, 0)

− + − ++ −1�i�j +f(i, j, 0)+�i1�j − + + − + +f(i, j, 0)

f(i, j ) +�i=�j − + − +−− ++f(i, 0, V) + +�i�1

−�i1 + − − +f(i, 0, V)+ − + − + +f(i, j, V) �i�j −

a The first row gives the signs of the entries of the elementary vector f(i ), see Eq. (13), that exists onlyfor currencies i with unit swap factor, i.e. �i=1. The following three rows give the signs of f(i, j, 0)obtained from Eq. (14), an elementary vector that exists for all pairs of currencies i and j for which1��i�j�1. For such pairs of currencies, exactly one of three cases may occur, leading to differentsigns of �i−1 and �j−1 in Eq. (14) and therefore, to one of the three patterns. The remaining four rowsare obtained in a similar fashion.

f(i, 0, V)[V ]=1 f(i, 0, V)[si ]= − f(i, j, V)[zi ]=1

(1+�i)(�i−1)S*(0/i )

f(i, 0, V)[pi ]=1

(�i−1)S*(0/i )f(i, 0, V)[z0]=

�i

(�i−1)(1+�0).

If �i�1 (respectively, 1) then f(i, 0, V) can be thought of as a forward(respectively, backward) flow sent along c(i ) together with a flow along the loop eV.Finally if �i��j the following elementary vector corresponds to the active cyclec(i, j ) and the loop eV,

f(i, j, V)[V ] =1

f(i, j, V)[pi ] =−�j

(�j−�i)S*(0/i )f(i, j, V)[pj ]=

�i

(�j−�i)S*(0/j )

f(i, j, V)[si ] = − f(i, j, V)[zi ]=−�j

(�j−�i)(1+�i)S*(0/i )

f(i, j, V)[zj ] = − f(i, 0, V)[sj ]=�i

(�j−�i)(1+�j)S*(0/j ).

4.2. Substitutes and complements

One can now determine whether a given pair of arcs in G are substitutes,complements, or neither, by looking at their signs in the elementary vectors. Thesigns are given in Table 1 and the substitutability/complementarity relations inTable 2.

Table 2 shows a rich collection of qualitative implications. The top entries showthe results of the buying, selling and holding identities of individual currencies. For


example, the more of currency i is sold at the beginning of the period as a result ofchanges in the corresponding parameter t i

s, ceteris paribus, the more is bought at theend: these are same-currency effects. The middle entries show the implications ofbuying, holding and selling decisions of one currency on another, and on the firm’sultimate position: these are cross-currency effects. It shows, for example, that anevent directly affecting sales sj of currency j causing them to increase simply reducessales of currency i when transaction costs in both currencies more than offset thecurrency pooling advantages on yields. The connections between the amounts sold,kept and bought of currency j (sj, zj and pj) then follow from the individualcurrency buying, holding and selling conditions. Implications for holdings of thebase currency and end of period holdings are also shown in the table.

Table 2 reveals that only two patterns occur. The first is exhibited by the threevariables (sell, hold and purchase) of a same currency. Each variable is a comple-ment with itself, the sell (si) and purchase (pi) variables are complements as well,and the remaining four pairs are substitutes. We express this fact by saying that anycurrency is a complement with itself. The second pattern only occurs between twodifferent currencies i and j for which �i and �j are both �1 or �1. Such pairs arecomposed of either two swap-gain-currencies (�i, �j�1) or two swap-loss-curren-cies (�i, �j�1). As seen in Table 2, such pairs of currencies exhibit a patternexactly opposite to the first pattern. We therefore call them substitute currencies.

Intuitively, a position on a swap-gain currency is clearly aimed at increasing theexpected value V, while that of a swap-loss currency is mainly to contribute to alower risk. This is the case as long as risk and expected profit are negatively

Table 2Substitutes and complements in the currency management problema,b

si zi pi z0

C(1)CSCsi

Szi S(1)S CSC C C(1)pi

sj C(2) S(2) C(3)S(2)

S(3)C(2)S(2)zj C(2)

S(2) C(3)pj S(2) C(2)

C(1) S(1)z0 C(1) CC(4) or S(5) C(4) or S(5)V S(4) or C(5) S

a (1), if �i=1, �a or �b ; (2), unless �i1�j ; (3) if �j=1, �a or �b ; (4) if �i=�n�1; (5) if�i=�1�1; where �a=min (1, max {�k : k=1, …, n, �k1}); and �b=max (1, min {�k : k=1, …, n, �k�1}).

b This information is derived from Table 1. For example, the two columns that correspond to si andzi in Table 1 show opposite signs in all rows, and therefore, the two variables are substitutes.Considering the sales of two currencies si and sj, Table 1 shows that if �i=�j only one elementaryvector, f(i, j ), has entries on both. Since these entries have opposite signs, si and sj are the substitutes.Otherwise, if �i�j, the two vectors f(i, j, 0) and f(i, j, V) have entries on both si and sj. These haveconsistently opposite signs unless �i1�j. Therefore, Table 2 expresses that si and sj are substitutesunless �i1�j. The other entries are obtained in similar fashion.


correlated among currencies, which should result from market efficiency. Therefore,if a portfolio is modified with, say, more of a certain swap-gain currency, both totalrisk and expected return increase. Since the initial available funds are given, themanager should reduce holdings of at least one other currency. It is reasonable toassume that some other swap-gain currency will be chosen in order to compensateby reducing both risk and expected value. Clearly, increasing the holdings ofanother swap-gain currency would result in more of what one is trying to fix bymodifying the portfolio. When increased sales of currency j cause reduced sales ofcurrency i the currencies are substitutes; they are alternative ways of achieving thecurrency objective. For example, the currencies in the now defunct Exchange RateMechanism (ERM) of the European Monetary System (EMS) might from a U.S.perspective have been considered substitutes.

4.3. Ripple bounds

In order to obtain some bounds on the variations in the amounts bought, keptand sold of the various currencies, the Ripple Property first requires the computa-tion of the magnitudes Mij. We obtain these by exploration of the elementaryvectors derived above. One can verify that

Msizi=Mzisi

=1 Mzipi=Msipi

=1+�i Mpizi=Mpisi

=1

1+�i

Msiz 0=Mziz 0

=1

S(i/0)max

�1,

��i−�j ��i �1−�j �

: �j�1

Mpiz 0=

Msiz 0

1+�i

Mz 0si=Mz 0zi

=S(i/0) max�

1,�i �1−�j ��i−�j �

: �j��i

Mz 0pi= (1+�i)Mz 0si

MVsi=MVzi

=1

(1+�i)S*(0/i )max

� 1�1−�i �

if �i�1,�j

��j−�i �: �j��i

MVpi

= (1+�i)MVsi

MsiV=MziV

= (1+�i)S*(0/i ) max��1−�i �,

��j−�i ��j

MpiV

=MsiV

1+�i

Msisj=Mzisj

=Msizj=Mzizj

=S( j/0)S(i/0)

max�

1,�j �1−�i ��i �1−�j �

: if 1��i��j�1

Mpipj=

1+�j

1+�i

MsisjMsipj

= (1+�j) MsisjMpisj

=Mpizj=

Msisj

1+�j

MVz 0=

11+�0

max� �i

�1−�i �: �i�1

Mz 0V= (1+�0) max

i

�1−�i ��i

.

In the next section these bounds on ratios of rates of changes are used inillustrative examples. For instance, under the hypotheses of the MonotonicityProperty, a variation in (say) the lower bound on the amount z0, increasing z0 by(say) �, will modify the optimal value of the end-of-period amount V by at most�×Mz 0V=� max

i�1−�i �(1+�0)/�i.


5. Qualitative sensitivity analysis

In the following we use the qualitative results obtained above in several specificexamples.

5.1. Changes in base currency requirements

Suppose that management increases the minimum amount to be kept in basecurrency, forcing that quantity to increase by � units. From the convexity hypothe-ses, �=0 if the new lower bound l�z0, and �= l−z0 otherwise. The followingwill be observed:1. The amount sold (si) for all currencies i such that �i=1, �a or �b may increase.

For these and other currencies the change in si will be at most �×Mz 0sj. For

each currency we denote the (positive or negative) variation in the amount soldby mi.

2. A variation (respectively, a decrease) of −mi in the amount of currency i(respectively, currencies i with �i=1, �a or �b) kept, and a variation (respec-tively, an increase) of (1+�i)mi in the amount purchased.

3. A decrease of at most �×Mz 0V in the amount V of the base currency remainingat the end of the period.

Note that the theory does not make predictions as to the direction of changes ofthe variables. Various scenarios can be devised in which the same cause, as here animposed bound, will result in either increases or decreases in such variables.

5.2. Change in maximum amount to be kept in a foreign currency

Suppose management imposes an upper bound on the amount of a currency ikept throughout the period. Such a change might be dictated by a reducedwillingness to face individual currency exposure. If this amount is less than thecurrent optimum amount (zi) the holdings in currency i will naturally be decreaseddown to the new upper bound. Let � be the magnitude of that decrease. Thefollowing will be observed.1. Sales of currency i will increase by �, facilitating the lower zi, and the amount

bought back will increase correspondingly by (1+�i)�.2. For all other currencies ( j ), the amount kept will vary by some (positive or

negative) magnitude at most equal to �×Mzizj. Accordingly, the amount sold

will vary by minus this amount, and the amount bought by − (1+�j) times thisvariation. Further, if �i, �j�1 or �i, �j�1, then i and j are substitutes, andtherefore holdings in currency j will increase while sales and purchases willdecrease.

3. The amount kept in the base currency will vary by at most �×Mziz 0. If further,

currency i is such that �i=1, �a or �b, then this will be an increase.4. The amount V of the base currency left at the end of the period will vary by at

most �×MziV. If currency i further has the property that �i=�n�1 (respec-

tively, �i=�1�1), then we can predict that the amount V will increase(respectively, decrease).


Table 3Numerical example

Estimated spot Local interestSpot rate S(i/0)Currency i �i

100×[(1+�i)12−1](forward) 1/S*(0/i )

Swap-loss-currencies(pairwise substitutes)

1.4650 1.4482 10.875000 0.982878Irish Punt9.500000 0.990452Swedish Krona 7.8802 7.8417

23.500000 0.992007229.2300Greek Drachma 227.70007.343750 0.998885Malaysian Ringett 2.5980 2.60308.187500 0.9990091.6690Deutsche Mark 1.6645

117.0500117.1500 3.375000 0.999247Japanese Yen11.000000 0.9997451619.00001610.0000Italian Lira

1.41641.4150 4.875000 0.999891Australian Dollar0.6906 0.6919 5.937500 0.999927Great Britain Pound

5.062500 0.9999961.52791.5260Swiss Francs

Swap-gain-currencies(pairwise substitutes)

2.937500Hong Kong Dollar 1.0001317.7365 7.73404.875000 1.0001781.25261.2510Canadian Dollar

5.69635.6580 11.750000 1.000355French FrancsSpanish Pesetas 15.937500118.8000 1.000677120.0100

15.625000 1.001319155.3200153.6900Portuguese Escudos

These are rich implications which have prescriptive value when planning currencymanagement strategies and selecting currency protocols as we illustrate in thenumerical example below.

6. A numerical example

For an example of the usefulness of the model we gathered data for 15currencies, including ‘strong’ currencies and ‘weak’ ones. For a given day (March16, 1993) we recorded spot selling and forward buying rates with respect to the U.S.dollar — taken here as base currency. We also recorded (annualized) 1 month localinterest rates (the U.S. dollar annualized 1 month rate on that day was 3.5% forlarge deposits). The relevant magnitudes, as provided by the Bank of Montreal, aregiven in Table 3.

While in a practical application of the methods presented in this paper a firmmight use its own predictions of future spot rates, our example assumes thatexpected future spot rates can be proxied by forward rates.

In Table 3 the currencies are sorted by their increasing swap gains/losses whichrange from 0.982878 for the Irish punt (low local interest rates, high transactioncosts, i=1) to 1.001319 (high local interest rates, low transaction costs, i=15) for


the Portuguese escudo. Observe that the two currencies whose swap gains/losses lieon either side of the value 1 are the Swiss francs (�a=0.999996, swap loss) and theHong Kong dollar (�a=1.000131, swap gain).

As described in Table 2, two currencies whose swap gains/losses lie on the sameside of the value 1 are substitutes. No pairs of currencies are complements; this isnot a consequence of the particular data used in this numerical example but is adirect result of the analysis of the model. Of interest is the fact that substitutabilitybetween currencies does not appear to be based on what intuition would expect:some degree of similarity or likeness.

Suppose that the amount to be kept in U.S. dollars by a U.S. dollars reportingcompany is at its imposed lower limit, and that the limit is raised by $1000. Thensales of Swiss francs and Hong Kong dollars will either remain fixed or increase.Moreover, the sales of each currency will �ary (increase or decrease) by at most1000×Mz 0si

, an amount computed in Table 3. For example:1. the number of French francs sold will �ary by at most

1000×Mz 0s FF=1000×S(FF/0) max

�1,

�FF�1−�j ��FF−�j �

: �j��FF

=1000×5.6580 max�

1,1.000355�1−�j ��1.000355−�j �

: �j�1.000355

=56581.000355�1−1.000677��1.000355−1.000677� =FF11, 909

(the maximum is reached for j=Spanish Peseta), the equivalent of 11909/5.658=U.S. $2105.

2. Sales of Hong Kong dollars will increase by at most HK $29 098 (the equivalentof U.S. $3761).

3. The amount remaining in U.S. dollar at the end of the period will �ary by atmost 1000×Mz 0V=$175.

These bounds on the change are equivalent to respectively 2.105, 3.761 and 0.175times the U.S. $1000 variation, resulting in an amplification/dampening effect. Thepossible magnitudes are an indication of the great sensitivity of the currencyallocation to external influences. Further applications of the model to different andalternative conditions or management protocols can provide other findings ofpotential value to the currency manager.

As mentioned, the expressions for the Ripple Bounds given in Section 4.3 aresensitive to currency exchange transaction costs. With substantial increases inforeign exchange transaction costs, or the imposition of the sometimes proposedtransactions tax, a situation could be reached where all exchanges of currenciescause losses, i.e. no swap-gain currencies remain and all �i ’s are well under 1. Onecan then verify that all Ripple Bounds are reduced and therefore the amplificationeffect may disappear. In that respect, the present paper could help study theimplications of transaction cost variations, including the impact of a ‘Tobin’ tax.

The amplification/dampening effect exhibited above and the specific numericalvalues would clearly not follow from the application of simple intuition.Similarly,


some of the currencies revealed to be substitutes or complements in the examplemay not concur with many financial managers’ intuition. Determination of substi-tutes/complements and various amplification/dampening effects for all plausiblechanges would warrant a systematic exploration of the model.

Acknowledgements

This research was partially supported by the Natural Sciences and EngineeringResearch Council of Canada Grant 5-83998, OGP0121627 and by the Center ofInternational Business Studies, University of British Columbia. We have benefitedfrom the helpful comments of a referee and the editor of this journal.

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alternative foreign exchange management protocols: an application of sensitivity analysis

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