int. j. production economics · ﬁnancial hedging is preferred for low volatility exchange rates...

Operational hedging strategies and competitive exposureto exchange rates$

Lingxiu Dong a,1, Panos Kouvelis a,1, Ping Su b,n,1

a Olin Business School, Washington University in St. Louis, Campus Box 1133, St. Louis, MO 63130-4899, USAb Department of Management, Entrepreneurship & General Business, Hofstra University, Hempstead, NY 11549, USA

a r t i c l e i n f o

Article history:Received 15 July 2013Accepted 3 March 2014

Keywords:Currency exchange rateOperational flexibilityOperational hedgingAllocationCompetitionEconomic exposure

a b s t r a c t

This paper investigates the impact of operational flexibility on firms' economic exposure to currencyfluctuations in the presence of global competition. We consider a global firmwho sells as a monopolist inthe domestic market, and also sells to a foreign market facing competition from a local competitor ofcertain capacity. We compare the effects of two operational strategies of the global firm, namely,matching currency footprints (“natural hedge”) and the capacity pooling strategy with allocationflexibility. For a two-stage stochastic model, we derive the optimal capacity and selling decisions forthe global firm, and from the comparative statics analysis of our model we infer useful managerialinsights. (1) We find that operational flexibility enables the global firm to exploit the possible highexchange rate (i.e., devalued home currency) realizations for profit improvement, and thus increases inthe long run the firm's expected profit. (2) Furthermore, operational flexibility allows for downside riskcontrol as the exchange rate becomes more volatile, since the resource pooling option cleverly exercisedminimizes the impact of unfavorable currency realizations. (3) The global firm's operational flexibilityincreases its competitor's downside risk, but may also benefit the competitor's expected profit since aflexible global firm may decide not to compete when the exchange rate is not favorable. In conclusion,our paper substantiates that “natural hedge” is not effective from a profit maximization perspective.We clearly illustrate the robust profit maximizing performance and reasonable downside risk controlof operational hedging approaches, which rely on the clever exercising of operational flexibilityoptions such as resource pooling and allocation, in handling competitive exposure to fluctuatingexchange rates.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

1.1. Problem motivation

As firms are globalizing their supply chains, they are facingincreasing exposure to currency fluctuations. The risks associatedwith currency exposure can be significant. Currency fluctuationscan often be of 20–40% within a year, with the 1997 Asian Crisisresulting in even higher devaluations overnight for some of theweaker currencies (e.g., Indonesian Rupiah). Such strong macro-economic shocks can lead to drastic deviations from expected profitperformance and lead to heightened risk exposure. For example,

Honda Motor Co., the Japanese auto maker, with 80% of its profitsgenerated in the U.S. market, was predicted to reduce its income by$1.22 billion in 2004, because of the depreciations of dollar sales(Business Week 2004). German automakers suffered similar mag-nitude negative exposure effects in the face of the appreciating Euroof the last few years. In 2012, the global company executives had tomake managing currency risks (particularly risks related to euro-denominated earnings) their top priority, due to the recent fiscalcrisis in the eurozone peripheral countries.

Unfortunately, it has always been extremely hard to predict notonly the magnitude but even the direction of change (positive ornegative) of economic exposure to exchange rate shocks, as we havewitnessed even earlier in the days of “Endaka” and “Super-Endaka”periods (of consistently appreciating Yen relative to the dollar of the80s and early 90s), with the auto-industry again the example (HarvardBusiness School case 9-796-030: Japan's Automakers Face Endaka).An exchange rate regime heavily influenced by the US governmentactions to favor the domestic car makers ended up leading to erosionof their own market shares, competitive position and profitability.

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journal homepage: www.elsevier.com/locate/ijpe

Int. J. Production Economics

http://dx.doi.org/10.1016/j.ijpe.2014.03.0020925-5273/& 2014 Elsevier B.V. All rights reserved.

☆This research was in part supported by a summer research grant from the FrankG. Zarb School of Business, Hofstra University.

n Corresponding author. Tel.: þ1 516 463 5730; fax: þ1 516 463 4834.E-mail addresses: [email protected] (L. Dong), [email protected] (P. Kouvelis),

[email protected] (P. Su).1 The authors contributed equally to this work.

Please cite this article as: Dong, L., et al., Operational hedging strategies and competitive exposure to exchange rates. InternationalJournal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.002i

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Exchange rate shocks substantially affect the relative competitiveposition of firms as they are changing their cost structures, with theexact magnitude depending on their supply and demand networkstructures and the product lines of the competing firms. As a result,economic exposures to exchange rate shocks are hard to counteract.

Generally speaking, there are two types of exposure to currencyexchange rate, transaction (contractual) and competitive (operating)exposure (see Lessard and Lightstone, 1986 for a basic exposition ofthe concepts). Transaction exposure is caused by contracts denomi-nated in a foreign currency. Competitive exposure explicitly capturesthe effects that currency fluctuations have on a company's futurerevenues and costs, as a result of the overall effect of such macro-economic changes on the competitive position of the firm. Transac-tion exposure is easy to identify and estimate its magnitude as itappears in financial statements and contractual agreements, andunder reasonable assumptions on the distribution of the futureexchange rates, it can be effectively handled through financialhedging approaches (O'Brien, 1996). Competitive exposure is heavilydependent on the global supply chain structure and product marketsof the competing firms, and it is hard to estimate, as we arguedabove. Managing competitive exposure needs a longer-term perspec-tive, and cannot be dealt with solely through the use of financialhedging techniques. It is well recognized, and mostly anecdotallyadvocated, in the finance and international business literature (seeLessard and Lightstone, 1986 and Hertzell and Caspar, 1988 amongothers, with a more comprehensive listing left for our literaturereview, Section 1.2) that operational hedging, such as operationalflexibility of a global network of production facilities, can be aneffective long-term way to handle it. However, a counter arguingliterature has been built around risk avoidance and variance reduc-tion strategies predicated on “natural hedges” (and their implicationsfor structuring the network of global facilities of the firm) andfinancial hedging approaches for handling such exposures.

A frequently adopted operational approach by global firms is tocreate a “natural hedge” (Pringle and Connoly, 1993; Pringle, 1995;Logue, 1995), which is also called “matching the currency footprints”(Harris et al., 1996), by locating production facilities in the countrywhere sales are generated. This way, costs and revenues are matchedin the same currency and the firm's competitive exposure tocurrency fluctuation is eliminated. In an empirical study by Bodnarand Marston (2002), the authors document very low currencyexchange exposure for a majority of firms in their sample that haveused “natural hedges.” “Natural hedge” regains its popularity in thelast few years. “Toyota Motor Corp., Renault SA (RNO) and NissanMotor Co. are among carmakers widening their global productionfootprint to limit exposure to currency risk (Business Week, March2013).” Meanwhile, US auto makers such as Ford and GM haveincreased their production in overseas markets despite their export-favoring weaker home currencies, but probably creating “naturalhedges” in markets with increasing sales. However, the effectivenessof natural hedges is a point of heated debate (see Harris et al., 1996)to which our paper will add its unique, and reasonably conclusivefrom a profit maximizing objective, viewpoint. The main thrust ofour argument is as follows.

The “matching the currency footprints” approach typically resultsin inflexible national or regional networks. As a result, it forgoes theoperational flexibility of a global supply/production/distribution net-work in exchange for the false security of minimal risk of currencyexposure of the firm's own (preplanned, and non-anticipation ofcompetitive reactions) cash flows. Consequently, a firm that pursuesthis “natural hedge” approach is less capable in improving its profitand controlling its downside risk. We will formally advance this pointwith a stylized model of a global firm competing with a local firm in aforeign market in the presence of exchange rate uncertainties.

The main focus of our paper is on the effects of operationalflexibility, such as allocation flexibility, on the performance of

firms facing global competition in the presence of currencyuncertainties. “Allocation” flexibility (defined in Ding et al.,2007) refers to the ability to supply two markets from one flexiblefacility and make market commitment ex post. We consider aglobal firm selling to both domestic and foreign markets. In theforeign market, the global firm encounters competition from alocal supplier. We consider two distinct operational settings: theglobal firm (a) employs the “natural hedge” approach and(b) introduces allocation flexibility. For these operational settings,we explore (1) What are the effects of allocation flexibility on theglobal firm's capacity strategy, and its deployment tactics, inresponse to the realized exchange rate shocks? and (2) How doallocation flexibility affect firms' profits and risks in response tofluctuating exchange rates?

Our results show that operational flexibility improves the globalfirm's expected profit and reduces its downside risk. Furthermore,the global firm's flexibility increases the local firm's downside risk,although it does not always decrease its expected profit. Moreover,we find that allocation flexibility mitigates the adverse effect ofincreasing competition in the foreign market on the global firm.These effects become more prominent as the volatility of exchangerate increases. Our analysis substantiates that allocation flexibilityhas robust performance in terms of both expected profit anddownside risk in the presence of fluctuating exchange rates andforeign competition. Our results clearly disprove the effectiveness of“natural hedges” for dealing with competitive exposure for profitmaximizing firms.

1.2. Literature review

Our work is related to the research of operational hedging in theoperations management literature. In this literature, emphasis isgiven to modeling different types of operational hedging strategies,and the optimal operating (capacity, technology selection, inventoryetc.) policy under operational flexibility in a single firm setting.Boyabatli and Toktay (2004) provide a survey of frequently usedoperational hedging strategies in operations management andidentify two definitions of operational hedging in the literature:real options view and counterbalancing-action view. The realoptions view (Huchzermeier and Cohen, 1996) considers opera-tional hedging strategies as real (compound) options that areexercised in response to demand, price and exchange rate con-tingencies. These real options have different forms: postponementof the allocation on foreign markets in Ding et al. (2007) and Kazazet al. (2005), acquisitions in Hankins (2011), holding excess capacityin Cohen and Huchzermeier (1999), and switching options in Kogutand Kulatilaka (1994), Cohen and Huchzermeier (1999), Dasu and Li(1997), Li and Kouvelis (1999), and Aytekin and Birge (2004).

Specifically, the early influential work of Huchzermeier andCohen (1996) values global manufacturing strategy options for asingle firm (i.e., no competitive exposure considerations) underexchange rate risk. The operational hedge in their study is the useof excess capacity and production switching options. Their numer-ical study shows that operational hedging reduces the firm'sdownside risk. Kogut and Kulatilaka (1994) develop a stochasticdynamic programming model to investigate option value of thecostly switching production between two manufacturing plantslocated in different countries under exchange rate uncertainty.Aytekin and Birge (2004) generalize this work and show thatfinancial hedging is preferred for low volatility exchange rates andoperational hedging is preferred for high volatilities. Kouvelis et al.(2001) study the effects of real exchange rates on the choice anddynamic adjustment of ownership strategies of production facil-ities of global firms supplying foreign demand. Kazaz et al. (2005)examine two complementary forms of operational hedging, pro-duction hedging where the firm deliberately produces under

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capacity before the realization of exchange rate uncertainty, andallocation hedging where the firm under-serves a market underunfavorable exchange rate realizations. They show that productionand allocation hedging are features of a robust optimal policyunder exchange rate uncertainty for a profit maximizing firm.

The counterbalancing-action view (Van Mieghem, 2007),which is much closer to the etymological meaning of the word“hedge”, defines operational hedging as “mitigating risk by coun-terbalancing actions in a processing network that do not involvefinancial instruments.” Strategies such as dual-sourcing, compo-nent commonality, transshipping, holding safety stock are subjectsof study in an extensive literature (summarized in the abovepaper), often in a general supply chain setting with the main risksdriven by demand uncertainties and without considering theeffect of currency fluctuations. For the interested readers, seerepresentative works of Harrison and Van Mieghem (1999) andVan Mieghem (2003), and more recent works of Chod et al. (2012)and Anupindi and Jiang (2008). Our paper, even though it bearsconceptual similarities to the above literature, also differs from itin substantive ways. We study operational hedging approaches ascounterbalancing acts to exchange rate shocks, with subsequentdemand risks resulting from competitor reactions to the alteredcost structure of the competing firms after the exchange raterealization. Within our work, the two viewpoints on operationalhedging are reasonably unified, as our firm “hedges” its competi-tive exposure via appropriate exploitation of operational flexibilityresulting from the exercise of operational options (resource pool-ing and allocation).

Our work is also related to research on firms' economic exposurein the international business (finance/accounting) literature. Thisliterature (Adler and Dumas, 1984; Lessard and Lightstone, 1986;Flood and Lessard, 1986; Hertzell and Caspar, 1988; Marston, 2001)defines and measures the firm's different types of exposure toexchange rates, and investigates the hedging policies to mitigate it.The researchers have noticed that under global competition, thefinancial hedging restricted in pre-planned contracts based onexchange rate estimates, and isolated from strategic exploitationof real time managerial flexibility and relevant operating options, isnot adequate to maintain firms' competitive advantage. The aboveliterature has insightfully recognized the need for a joint use ofoperational hedges and financial hedges, but often advocates its usebased on anecdotal stories of effectiveness for certain firms incertain instances. A recent research by Bartram et al. (2010) showsempirically that firms jointly use financial and operational hedges.Financial hedges decrease exposure by 40% and operational hedgesreduce exposure by 10–15%.

A parallel stream of literature, with a strong preference foroperational hedges, even though of very specific nature, to financialhedges advocates the use of the “matching currency footprints”approach (see Pringle and Connoly, 1993, Pringle, 1995, and Logue,1995) as the way to manage large foreign currency exposures. Suchan approach has implications on the structure and location of theglobal facility network for the firm, but after that it requires relativelysimple currency contracts to hedge the risk associated with anyremaining exposure. Bodnar and Marston (2002) provide empiricalevidence supporting such an approach by showing that matchingforeign currency revenues and costs would have been important inreducing the foreign exchange rate exposure of many U.S. firms.Harris et al. (1996) and Melumad et al. (2009) argue informally vianumerical examples and anecdotal stories that global firms takingactions to match currency footprints to reduce variability in profit-ability may lose their strategic flexibility and lower expected profit,especially when their competitors can adjust quantities or prices. Inour paper, we will formally illustrate, and prove within the confinesof our stylized model, that the latter thesis is both correct and ofserious implications for global firms. Furthermore, we will rigorously

show that firms employing operational hedging via appropriateexploitation of operational flexibility (and exercise of appropriateoperational options) can achieve both robust profit performance andcontrol of downside risk in the presence of volatile exchange rates,while at the same time penalizing inflexible competitors.

Recently, researchers started to explore the integration offinancial and operational hedging tools (for example, Dong andLiu, 2007, Caldentey and Haugh, 2006, Ding et al., 2007, Goel andGutierrez, 2004, Gaur and Seshadri, 2005, Wong, 2007, and Zhuand Kapuscinski, 2011 and references therein). Most of this work isconfined within models of single firm reacting to pricing, demandand/or currency risks, and is not concerned on the competitiveexposure aspects of volatile exchange rates. Our paper models theimpact on firm profit performance and risk of the competitiveexposure to exchange rate shocks, and proves the effectiveness ofthe operational hedging approach in handling them.

The structure of the paper is as follows: in Section 2, weprovide models of the two operational settings and the corre-sponding optimal capacity and selling decisions. In Section 3, wecompare firms' expected profits and downside risks associatedwith the two settings. In Section 4, we conduct sensitivity analysiswith respect to the exchange rate volatility and the severity ofcompetition in the foreign market (in terms of the establishedcapacity). We conclude findings in Section 5.

2. Model: two operational settings

Consider a global firm (firm 1) selling to two markets: market 1,the domestic market, and market 2, a foreign market. In market 1,firm 1 is a monopolist; in market 2, firm 1 competes with an local firm(firm 2) that sells a perfectly substitutable product to market 2 exclu-sively. Each firm evaluates its own profit in its home currency.

We assume iso-elastic demand functions in both markets. Theiso-elastic demand function is frequently used in econometric andempirical marketing research, due to its consistency with theconsumer-utility-maximization theory, its unambiguous economicinterpretation, and good statistical fit with available sales data. Werefer readers to Monahan et al. (2004) for a thorough discussion ofthis assumption within a newsvendor setting and further refer-ences. The inverse demand function of the iso-elastic demand formarket i is

PiðQiÞ ¼ θiQ�aii ϵi; ð1Þ

where Pi is the market clearing price for quantity Qi sold in marketi, θi indicates market potential, �1=ai with aiA ð0;1� represents theelasticity of demand, and random variable ϵi represents the demandcurve uncertainty, i¼1, 2. We assume that ϵi's are independentlydistributed with known cumulative distribution function (cdf) Gi,and probability density function (pdf) gi, i¼1, 2. One interpretationof ϵi given by Petruzzi and Dada (1999) is that the demand curve isdeterministic while the scaling parameter representing the marketsize is random. Thus, we refer to ϵi, i¼1, 2, as demand scalingfactors. We also assume that ϵi is independent of the market pricePi and the currency exchange rate s. Without loss of generality wealso assume that E½ϵi� ¼ 1, i¼1, 2. We make no distinction betweenthe real exchange rate and the nominal exchange rate by assumingzero inflations in both countries. We assume that exchange rate s israndomly distributed, according to a known cumulative distributionfunction (cdf) F, and corresponding probability density function(pdf) f. In order to deal with the currency exchange rate uncertaintyand the demand uncertainty in both markets, firm 1 has thepostponement option on the production decisions (defined in VanMieghem and Dada, 1999). That is, the manufacturing process canbe decomposed into two steps: Step 1 involves long-leadtime coreproduction activities such as procuring and/or producing complex

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components/subassemblies, and is performed long before therealization of uncertainties; and step 2 involves short-leadtime finalassembly and configuration, and can be performed after therealization of uncertainties to meet the demand.

Firm 1 can employ different operational strategies dependingon its product-process features. We consider two operationalstrategies (see Fig. 1):

Strategy 1, natural hedge (N): Firm 1 has one dedicated facility ineach market serving exclusively that market. Strategy 1, locatingmanufacturing where the revenues are generated and therebymatching revenues and costs with little net exposure to exchangerate changes (also known as “matching currency footprints”), is acommon practice among multinationals to handle the currency risk(Harris et al., 1996). Both facilities can perform the step 2 productionafter demand and exchange rate uncertainties are realized.

Strategy 2, capacity pooling (C): Firm 1 adds more operationalflexibility upon strategy 1 through resource sharing between thetwo markets. Firm 1 has a single facility located in market 1,serving both markets. Firm 1's products share the vanilla-middle-products and its production facility can delay the localizationoperation until the realization of uncertainties.

In practice, firms can also invest in various operational flex-ibilities such as multiple production facilities and switchingoptions. In the scope of this paper, we consider allocation optionand compare it with the natural hedge strategy (strategy 1) thathas no allocation flexibility but creates natural hedge against theexchange rate fluctuations.

We divide the unit production cost of firm 1's products into twoparts: the time-consuming step 1 costs C1 per unit for bothmarkets, and the quick step 2 costs τi for market i product, i¼1, 2.We assume C1 and τi are the same for both strategies, whichallows us to focus on the benefit of operational flexibility andtheir effects on firms' exposures to exchange rate changes andprofit risks. Clearly, under this assumption, the model offersupper bounds of the benefit of strategy. The benefits can thenbe compared to additional cost that might be required for itsimplementation. Without loss of generality, step 1 costs aremeasured in market 1 currency and the step 2 costs τi measuredin market i currency.

Following the lead of Dong et al. (2013), we assume that thelocal competitor, firm 2, in the foreign market pursues a localstrategy and focuses on the local market only. Its local knowledgeand close connection with local suppliers enable it to act moreswiftly to capitalize the market opportunities. Therefore, theglobal firm, firm 1, and the local competitor, firm 2, are endowedwith different flexibilities to cope with changes in market condi-tions (see Hill, 2012, Chapter 13: The Strategy of InternationalBusiness). Firm 2's production can be decomposed into two steps.We assume that step 1 production cost is C2 per unit and step2 production cost is negligible. Firm 2, being a local firm in market2 without overseas operations or trade, has an established pro-duction facility of a fixed capacity. At step 2, firm 2 always sells fullcapacity to market 2, since the marginal revenue associated withthe iso-elastic demand is always positive. The two firms' joint

production quantity determines the market clearing price at step2 in market 2.

We use the following notation:

s foreign market currency exchange rate at time 1, i.e., thevalue of one unit of the foreign currency measured in thedomestic currency;

ϵi demand scaling factor for market i, i¼1, 2;μ expected exchange rate, μ¼E½s�;K1i firm 1's time 0 production output level in market i, i¼1,

2 under strategy 1;K firm 1's time 0 production output level, K � ðK11;K12Þ

under strategy 1, K � K1 under strategy 2 ;K2 firm 2's existing capacity;C1 firm 1's step 1 production cost per unit, measured in

market 1 currency;τi firm 1's step 2 production cost per unit of producing

market i product, measured in market i currency, i¼1, 2;qi firm 1's time 1 selling quantity to market i, i¼1, 2;Fð�Þ; f ð�Þ the cumulative distribution function (cdf) and probabil-

ity density function (pdf) of foreign exchange rate s;Gið�Þ; gið�Þ the cdf and pdf distributions of ϵi, i¼1, 2;π1i firm 1's realized time 1 profit from market i, i¼1, 2.

We compare the two strategies in a two-stage stochasticprogram setting of time interval [0, 1]. For simplicity, we set thetime discount rate to be zero. At time 0, firm 1 decides productionoutput level K and performs the step 1 production in the presenceof the exchange rate and demand uncertainties. At time 1, after therealization of exchange rate and demand uncertainties, firm1 determines selling quantity qn

i to each market (where qn

i is firm1's optimal selling quantity to market i, i¼1, 2.), given that firm2 sells its full capacity to market 2.

Let Π i represent firm i's time 0 expected profit, and π1, π2 therealized time 1 profit of firm 1 and firm 2, respectively. Expecta-tion, with respect to the demand and exchange rate distributions,is denoted by E. For each strategy, firm 1's two-stage decision canbe formulated as

Time 0 : maxKZ0

Π1ðK ;K2Þ;where Π1ðK ;K2Þ ¼ E½π1ðs; ϵ1; ϵ2;K;K2; qn

1; qn

2Þ��C1K1

andTime 1 : π1ðs; ϵ1; ϵ2;K;K2; qn

1; qn

2Þ ¼maxq1 ;q2

fπ1ðs; ϵ1; ϵ2;K;K2; q1; q2Þg;

ð2Þwhere K � ðK11;K12Þ under strategy 1 and K � K1 under strategy 2.

Firm 2 also has a two-step production. At step 1, firm 2 has anestablished facility of fixed capacity K2. At step 2, firm 2 alwayssells full capacity to market 2, since the marginal revenueassociated with the iso-elastic demand is always positive. Themarket price P2 is jointly determined by the total productionquantity supplied by both firms. Firm 2's expected profit undereach strategy is expressed as follows:

Π2ðK ;K2Þ ¼ E½π2ðs; ϵ2;K ;K2; qn

2Þ�;where π2ðs; ϵ2;K;K2; qn

2Þ ¼ ½P2ðqn

2þK2Þ�C2�K2: ð3ÞWe now analyze the two strategies by deriving firm 1's optimal

time 1 selling decisions qn

1 and qn

2, and optimal time 0 capacitydecision, Kn

1.

2.1. Strategy 1: natural hedge, one domestic and one foreignfacility (N)

Firm 1 has two dedicated facilities in each country, servingmarkets 1 and 2 separately. At time 0, firm 1 produces middle

Strategy 1 Natural Hedge (N)

0

1

Time

Market 1 Market 2

Strategy 2 Capacity Pooling (C)

Market 1 Market 2

Fig. 1. Illustration of firm 1's operational strategies.






products in quantities K11 and K12 for markets 1 and 2, respec-tively; at time 1, after the realization of the exchange rate anddemand uncertainties, firm 1 decides the selling quantity to thetwo markets and produces finished products.

Firm 1's two-stage problem can be formulated as

Time 0 : maxK11 ;K12 Z0

ΠN1 ðK11;K12Þ

where

ΠN1 ðK11;K12Þ ¼ �C1ðK11þK12ÞþEϵ1 ½πN

11ðϵ1;K11Þ�þEs;ϵ2 ½πN12ðs; ϵ2;K12;K2Þ�

and

Time 1 : πN11ðϵ1;K11Þ ¼ max

0rq1 rK11

ðP1ðq1Þ�τ1Þq1

πN12ðs; ϵ2;K12;K2Þ ¼ max

0rq2 rK12

fs½P2ðq2þK2Þ�τ2�q2g:

Lemma 1 summarizes the firm 1's time 1 optimal sellingdecisions to markets 1 and 2.

Lemma 1. At time 1, given capacities K11 and K12 for markets 1and 2, respectively, (1) firm 1's optimal selling quantity to market 1 is

qnN1 ¼

k11ðϵ1Þ if ϵ1rϵ1ðK11ÞK11 otherwise

(;

where

ϵ1ðK11Þ �Ka111τ1

θ1ð1�a1Þand k11ðϵ1Þ �

θ1ϵ1ð1�a1Þτ1

� �1=a1:

(2) Firm 1's optimal selling quantity to market 2 is

qnN2 ðs; ϵ2;K2Þ ¼

0 if ϵ2rt1ðK2Þq2U if t1ðK2Þoϵ2otN2 ðK12;K2ÞK12 if tN2 ðK12;K2Þrϵ2

8><>: ; ð4Þ

where

t1ðK2Þ �Ka22 τ2θ2

; tN2 ðK12;K2Þ �τ2ðK2þK12Þa2 þ1

θ2ðK2þð1�a2ÞK12Þ; and q2U

solves equation

½θ2ðð1�a2Þq2UþK2Þðq2UþK2Þ�a2 �1�ϵ2 ¼ τ2: ð5Þ

Lemma 1 shows that the postponement option allows firm 1 torespond to the demand uncertainty. As illustrated in Table 1, formarket 1: when demand is low in market 1, i.e., ϵ1rϵ1ðK11Þ, firm1 sells under-capacity at k11ðϵ1Þ; when demand is high in market 1,firm 1 sells at capacity K11. For market 2, when ϵ2 is small, it is notworthwhile to pursue market 2 because the low revenue cannoteven recoup the localization cost τ2. As the value of ϵ2 increases andreaches threshold t1ðK2Þ, firm 1 starts to sell q2U to market 2, andthe selling quantity increases in ϵ2. When the value of sϵ2 exceedsthreshold tN2 ðK12;K2Þ, firm 1 sells at capacity K12 to market 2. Notethat firm 1's selling decision in market 2 is not affected by theexchange rate under “natural hedge” because cost and revenue arematched in the foreign currency.

Proposition 1. Firm 1's expected profit ΠN1 ðK11;K12Þ is concave in

K11 and K12. The optimal capacity for market 1, KnN11 ; solves equationZ þ1

ϵ1ðKnN11 Þ½θ1ϵ1ð1�a1ÞðKnN

11 Þ�a1 �τ1�g1ðϵ1Þ dϵ1 ¼ C1; ð6Þ

and the optimal capacity for market 2, KnN12 ; solves equation

Es

Z þ1

tN2 ðKnN12 ;K2Þ

s½ϵ2θ2ðK2þKnN12 Þ�a2 �1ðK2þð1�a2ÞKnN

12 Þ�τ2�g2ðϵ2Þ dϵ2" #

¼ C1:

ð7ÞThe optimal total capacity investment is KnN

1 ¼ KnN11 þKnN

12 .

Proposition 1 states that the market-specific optimal capacityshould equalize the time 0 marginal production cost and expectedtime 1 market-specific marginal profit.

The following Proposition 2 states that firm 1's optimalcapacity for market 2, KnN

12 , is not monotone in firm 2's establishedcapacity. The rationale is similar to that under strategy 1.

Proposition 2. Firm 1's optimal capacity for market 2, KnN12 , is

concave in K2.

2.2. Strategy 2: capacity pooling strategy (C)

Firm 1 has a single facility located in his home country servingboth markets. At time 0, before the realization of demand andexchange rate uncertainties, firm 1 produces vanilla-middle-products at quantity K1. At time 1, after uncertainties are resolved,firm 1 decides selling quantities q1 to market 1 and q2 to market 2,with the total selling quantity being constrained by the capacity K1.

Firm 1's two-stage problem can be formulated as

Time 0 : maxK1 Z0

ΠC1ðK1Þ

where

ΠC1ðK1Þ ¼ Es;ϵ1 ;ϵ2 ½πC

1ðs; ϵ1; ϵ2;K1;K2Þ��C1K1

and

Time 1 : πC1ðs; ϵ1; ϵ2;K1;K2Þ ¼ max

q1 ;q2 Z 0;q1 þ q2 r K1

ðP1ðq1Þ�τ1Þq1þs½P2ðq2þK2Þ�τ2�q2� �

:

Lemma 2 summarizes firm 1's time 1 optimal selling decisionsto markets 1 and 2.

Lemma 2. At time 1, given capacity K1, (1) If market 1 has a lowdemand realization, ϵ1rϵ1ðK1Þ, then firm 1's optimal selling quan-tities to markets 1 and 2 are

ðqnC1 ; qnC

2 Þ ¼ðk11ðϵ1Þ;0Þ; ϵ2rt1ðK2Þðk11ðϵ1Þ; q2U Þ; t1ðK2Þoϵ2otC2ðϵ1;K1;K2ÞðK1�q2B; q2BÞ; tC2ðϵ1;K1;K2Þrϵ2

8><>: ;

where

tC2ðϵ1;K1;K2Þ �τ2ðK1�k11ðϵ1ÞþK2Þa2 þ1

θ2½K2þð1�a2ÞðK1�k11ðϵ1ÞÞ�;q2U is as defined by (5) in Lemma 1, q2B is the solution to equation

ð1�a1Þθ1ϵ1ðK1�q2BÞ�a1 �τ1 ¼ ½ðK2þð1�a2Þq2BÞθ2ðq2BþK2Þ�a2 �1�sϵ2�sτ2:

ð8Þ(2) If market 1 has a high demand realization ϵ14ϵ1ðK1Þ, then firm1's optimal selling quantities to markets 1 and 2 are

ðqnC1 ; qnC

2 Þ ¼ðK1;0Þ; ϵ2rt3ðϵ1; s;K1;K2ÞðK1�q2B; q2BÞ; t3ðϵ1; s;K1;K2Þoϵ2

(;

Table 1Time 1 selling decisions under strategy 1.

Market 1 Market 2

ϵ1rϵ1ðK11Þ qnN1 ¼ k11ðϵ1Þ ϵ2rt1ðK2Þ qnN

2 ¼ 0

t1ðK2Þoϵ2o tN2 ðK12;K2Þ qnN2 ¼ q2U

ϵ14ϵ1ðK11Þ qnN1 ¼ K11 ϵ2ZtN2 ðK12 ;K2Þ qnN

2 ¼ K12






where

t3ðϵ1; s;K1;K2Þ �θ1ϵ1ð1�a1ÞK �a1

1 �τ1þsτ2sθ2

Ka22 :

Lemma 2 states that when two markets share one productioncapacity, the selling decisions to the two markets are jointly madeand are affected by the demand uncertainties in both markets andthe exchange rate uncertainty. In particular, as illustrated inTable 2, when market 1 has a low demand realization (scenario1: S1), i.e., when ϵ1oϵ1ðK1Þ, market 1 only needs k11ðϵ1Þ, which isa portion of capacity K1. The capacity does not become a scarceresource until the value of ϵ2 exceeds threshold tC2ðϵ1;K1;K2Þ,when firm 1 has to reduce market 1 sales in order to pursue themore profitable market 2. The allocation decision under bindingcapacity, q2B, satisfies Eq. (8) which states that the allocation willequalize the marginal revenue from the two markets. Whenmarket 1 has a high demand realization (scenario 2: S2), i.e., whenϵ1Zϵ1ðK1Þ, market 1 needs the entire capacity K1 and capacity isbinding regardless of market 2 condition. Firm 1 starts selling tomarket 2 and allocating the capacity between the two marketswhen the value of sϵ2 exceeds threshold t3ðϵ1; s;K1;K2Þ.

Proposition 3. Firm 1's expected profit ΠC1ðK1;K2Þ is concave in its

capacity choice K1. The optimal capacity KnC1 solves equation

Es

Z ϵ1

0

Z þ1

tC2 ðϵ1 ;K1 ;K2Þs½ϵ2θ2ðK2þð1�a2Þq2BÞðK2þq2BÞ�a2 �1�τ2�

"

�g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1þZ þ1

ϵ1

Z þ1

t3ðϵ1 ;K1 ;K2Þs½ϵ2θ2ðK2

�þð1�a2Þq2BÞðK2þq2BÞ�a2 �1�τ2�g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1

þZ t3ðϵ1 ;K1 ;K2Þ

0½θ1ϵ1ð1�a1ÞK �a1

1 �τ1�g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1�#

¼ C1:

ð9Þ

3. Strategy comparison

In this section, we study the effect of operational flexibility onexpected profits and profit risks of both the global firm and thelocal firm in the foreign market.

First, we consider firm 1's effective time 1 profit for givencapacity levels K11 and K12 for strategy 1, and K1 ¼ K11þK12, forstrategy 2. Straightforward from analysis in Section 2, firm 1'seffective time 1 profit can be expressed as

Strategy 1:

πN1 ðK11;K12;K2Þ ¼ max

0rq1 rK11

fðP1ðq1Þ�τ1Þq1gþ max0rq2 rK12

fsðP2ðq2þK2Þ�τ2Þq2g:

Strategy 2:

πC1ðK1;K2Þ ¼ max

q1 ;q2 Z 0;q1 þ q2 r K1

ðP1ðq1Þ�τ1Þq1þsðP2ðq2þK2Þ�τ2Þq2� �

:

It is clear that πN1 rπC

1 for any given ðs; ϵ1; ϵ2Þ and ðK11;K12Þ.Hence, πN

1 r FSDπC1, where r FSD represents the first-order stochas-

tic dominance. Intuitively, comparing strategies 1 and 2, thepooling of capacity in strategy 2 allows firm 1 to respond betterto situations in which one market is more profitable than the othermarket by giving priority to the more profitable market whenallocating the central capacity. Thus, strategy 2 dominates strategy 1.Immediately, we have the following proposition.

Proposition 4. (1) Given capacities K11 and K12, firm 1's profitsunder the two strategies are ordered as: πN

1 ðK11;K12;K2ÞrFSDπC

1ðK11þK12;K2Þ.(2) For any increasing function u, EuðπN

1 ÞrEuðπC1Þ and the ranking

of u-values under optimal capacities follows the same order.(3) Under risk-neutral utility, the expected optimal profits are

ordered as: ΠN1 ðKnN

1 ÞrΠC1ðKnC

1 Þ.Proposition 4 states that firm 1 will always prefer the strategy

with more operational flexibility regardless of the utility functionused. In particular, under risk neutral utility, firm 1's optimalexpected profit increases with more operational flexibility. How-ever, the allocation option in strategy 2 does not necessarily leadto increased optimal capacity, and the reason is as follows. Weconsider firm 1's optimal capacity under strategy 1, KnN

11 þKnN12 , and

we compare firm 1's marginal revenue of capacity under strategy1 and strategy 2 at time 0, which is the expectation, across allexchange rate and demand realizations, of his time 1 marginalrevenue. Assuming demand is deterministic for simplicity, we firstcompute firm 1's time 1 marginal revenue. When the exchangerate is relatively low and market 2 is not profitable enough for firm1 to sell at large quantity, under strategy 2 firm 1 will sell morethan KnN

11 to market 1 and less than KnN12 to market 2. Thus, the

capacity that would be unused under strategy 2 would beallocated to market 1 to earn profit under strategy 2, and themarginal revenue of capacity is higher under strategy 2 than understrategy 2. When the exchange rate is relatively high, firm 1 willsell less than KnN

11 to market 1 and more than KnN12 to market 2. Since

firm 1's marginal revenue decreases in the selling quantity inmarket 2, in this case, the marginal revenue under strategy 2 islower than that under strategy 2. In expectation, the time0 marginal revenue of capacity under strategy 2 can be lower orhigher than under strategy 2 depending on the relative effect ofhigh and low exchange rates.

We use two measures to evaluate the downside risk associatedwith an operational strategy: VaR (value-at-risk) and EDR(expected downside risk). VaR, a widely used risk metric byfinancial institutions, regulators, nonfinancial corporations (e.g.,multinationals), and asset managers (see Jorion, 2001), is definedas measuring the maximum expected loss over a given horizonunder normal market conditions at a given confidence level α.Specifically, if we define V as PrðπrV Þ ¼ 1�α, then VaR¼ �V . Forexample, if a bank says its daily VaR of its trading portfolio is $35million at the 99% confidence level, it means there is 1% chance,under normal market conditions, for a loss greater than $35million to occur. Given confidence level, the larger is the VaRvalue, the higher the downside risk. The EDR measure is definedby Huchzermeier and Cohen (1996) as the expected deviation ofthe firm's profit over the planning horizon from a profit level z,EDRðzÞ ¼ E½ðz�πÞþ �. Given target profit level, the higher is the EDRvalue, the higher the downside risk.

Proposition 5. If under strategy 1 firm 1 invests K11 and K12 formarkets 1 and 2, respectively, and invests K11þK12 under strategy 2,then firm 1's VaRs are ordered as VaRNZVaRC , and EDRs are orderedas EDRNZEDRC .

Table 2Time 1 selling decisions under strategy 2.

S1 (market 1 low demand) : ϵ1rϵ1ðK1Þ S2 (market 1 high demand) :ϵ14ϵ1ðK1Þ

ϵ2rt1ðK2Þ qnC1 ¼ k11ðϵ1Þ ϵ2rt3ðϵ1 ;K1 ;K2Þ qnC

1 ¼ K1

qnC2 ¼ 0 qnC

2 ¼ 0

t1ðK2Þoϵ2otC2 ðϵ1 ;K1 ;K2Þ qnC1 ¼ k11ðϵ1Þ

qnC2 ¼ q2U

ϵ2ZtC2 ðϵ1 ;K1 ;K2Þ qnC1 ¼ K1�q2B ϵ24t3ðϵ1 ;K1 ;K2Þ qnC

1 ¼ K1�q2BqnC2 ¼ q2B qnC

2 ¼ q2B






Proposition 5 states that for a given capacity, more operationalflexibility enables firm 1 to respond more effectively in undesir-able demand and exchange rate scenarios, and thus firm 1'sdownside risk decreases. Numerical study (see examples inSection 4) shows that even when firm 1 adjusts capacity invest-ment optimally, the downside risks associated with the twostrategies follow the same ranking order.

The operational strategies employed by firm 1 also have impacton firm 2's downside risks.

Proposition 6. Assume that firm 1 under strategy 1 invests K11 and K12

for markets 1 and 2, respectively, and invests K11þK12 under strategy 2.Firm 2's profit is denoted as πC

2ðϵ2Þ ¼ ½ϵ2θ2ðq2Bðϵ2ÞþK2Þ�a2 �C2�K2.

1. Demand in market 1 is constant and low ðϵ1rϵ1ðK1ÞÞ,(1) If the confidence level for VaR satisfies αZG2ðtN2 Þ, firm 2's

VaRs are ordered as VaRNrVaRC .(2) If the profit level for EDR measures zrπC

2ðtN2 Þ, firm 2's EDRsare ordered as EDRNrEDRC .

2. Assume that there is no demand uncertainty, and demand inmarket 1 is high ðϵ14ϵ1ðK1ÞÞ. Let

sK12 ¼ðK1�K12Þ�a1θ1ð1�a1Þ�τ1

θ2ðK2þK12ð1�a2ÞÞðK2þK12Þ�a2 �1�τ2:

(1) If the confidence level for VaR satisfies αZFðsK12 Þ, firm 2'sVaRs are ordered as VaRNrVaRC .

(2) If the profit level for EDR measures zrπC2ðsK12 Þ, firm 2's EDRs

are ordered as EDRNrEDRC .

Proposition 6 states that if market demand uncertainty iscontrolled, when the confidence level of VaR, α, is high and whenthe benchmark profit level for EDR measure is low, firm 2'sdownside risk increases as firm 1 employs more flexibility in itsoperations. Note that under strategy 1, when firm 1 matches hisrevenue and cost currencies, firm 2 is immune to the exchangerate fluctuation and her profit is only subject to the risk brought byhigh realization of market demand 2. Strategy 2 allows firm 1 toexploit the opportunity associated with the weakening of market1's currency by competing aggressively in market 2. Therefore, themore flexible is firm 1's operation, the more aggressive he cancompete in market 2, and the higher the loss firm 2 maypotentially suffer.

Propositions 5 and 6 illustrate that a firm's employment ofoperational flexibility affects both the firm's and its competitor'scompetitive positions in global markets. In particular, operational

flexibility reduces a firm's downside risk, and increases its com-petitor's downside risk in the presence of the exchange rateuncertainty. The effect of operational flexibility on firm 2'sexpected profit is more complex. Given that firm 2 always sellsat capacity K2, her profit is only affected by the market 2 pricewhich in turn is affected by firm 1's selling quantity to market 2.Under strategy 1, firm 1's selling quantity to market 2 is onlyaffected by market 2 demand realization ϵ2, and it increases in ϵ2until reaching the capacity level. Under strategy 2, firm 1'sallocation to market 2 is jointly affected by the demand realiza-tions in both markets and also increases in the exchange rate s(Fig. 2).

4. Comparative statics

In this section, we examine, for each strategy, effects ofvariability of exchange rate and the severity of competition(reflected by the size of the established capacity K2) in the foreignmarket on firm 1's optimal capacity decisions, and on both firms'profits and corresponding profit downside risks.

First, we adopt the definition of variability by Rothschild andStiglitz (1970). We say that Y has more variability than X in the senseof RS if EðXÞ ¼ EðYÞ and there exists a random variable Z such that Yand XþZ have the same distribution and EðZjX ¼ xÞ ¼ 0 for all x.That is, Y has the same distribution as X plus a zero mean noiseterm, i.e., Y can be obtained from X by a sequence of mean-preserving spread of X, denoted as XrMPSY . Intuitively, Y has fattertails on both ends than X. For some families of distributions, suchas uniform, normal, and lognormal distributions, mean preservingspread can be obtained within the distribution family simply byadjusting distribution parameters.

The concept of mean-preserving spread is closely related to theconcept of the second-order stochastic dominance (SSD), as shownby the following lemmas.

Lemma 3 (Rothschild and Stiglitz (1970)). Suppose X and Y arepositive random variables with equal means. Then X dominates Yunder SSD iff Y has more variability in the sense of RS than X iff Y canbe obtained from X by a sequence of mean preserving spreads.

Lemma 4 (Milne and Neave (1994)). Suppose EðXÞ ¼ EðYÞ. Then Xdominates Y under SSD iff EuðXÞZEuðYÞ for every concave function u.

We provide a corollary that is useful for the subsequentanalysis.

Corollary 1. If XrMPSY , then EvðXÞrEvðYÞ for every convex func-tion v.

4.1. Exchange rate risk

The effect of changes in the exchange rate volatility on a firm'sperformance is largely determined by how effectively the firmresponds to various exchange rate scenarios. Lemma 5 shows thepartial sensitivity of firm 1's time 1 profit to the exchange rate sunder each of the three operational strategies.

Lemma 5. (1) πN11 is independent of s and πN

12 is linear nondecreas-ing in s. (2) πC

1 is convex nondecreasing in s.

Under strategy 1, by matching the currency footprints, firm 1'stime 1 profit increases at a constant rate as the exchange rate sincreases. In contrast, under strategy 2, firm 1's profit increases ins at an increasing rate. It shows that one of the important effects ofallocation flexibility is to enable a firm to exploit favorableexchange rate realizations. Alternatively, one can view firm 1'stime 1 profit as a preference function over the exchange rate s, the

Strategy 2(C)Strategy 1 (N)

Fig. 2. An example illustrating the effect of firm 1's operational flexibility on firm2's profit. Parameter settings are: ϵ1 ¼ 1, ϵ2 �U½0:5;1:5�, a1 ¼ a2 ¼ 0:5,θ1 ¼ θ2 ¼ 100, C1 ¼ 8, C2 ¼ 10, τ1 ¼ 3, τ2 ¼ 4, and K2 ¼ 25. ln s�Nðμ;s2Þ,E½s� ¼ expð0:325Þ, with s changing from 0.10 to 0.50.






convex increasing property suggests a risk seeking preference overs under strategy 2. Consequently, Proposition 7 shows the effect ofexchange rate volatility on firm 1's optimal capacity decision andoptimal expected profit.

Proposition 7. As the variability of the exchange rate s increases inthe sense of RS, (1) under strategy 1, firm 1's optimal capacities andexpected profits in both markets remain constant; (2) under strategy 2,firm 1's optimal capacity and expected profit increase.

Under strategy 1, matching of currency footprints decouplesfirm 1's operational decision from the effect of the exchange ratefluctuations, thus firm 1's optimal capacity decision and expectedprofit do not change when the exchange rate becomes morevolatile. Under strategy 2, operational flexibility allows firm 1 tosell aggressively in market 2 when the exchange rate realization isfavorable, thus the more volatile is the exchange rate, the moreoccurrences of favorable exchange rate that firm 1 can exploit, andthe more profit firm 1 can gain from market 2. Therefore, firm1 increases its optimal capacity as the exchange rate volatilityincreases and receives higher expected profit.

Proposition 8. Assume that there is no demand uncertainty inmarkets 1 and 2. Consider two exchange rates s½i� � F ½i�, i¼1, 2, withs½1� having more variability than s½2� in the sense of RS. sc is the pointsuch that F ½1�ZF ½2� for srsc and F ½1�oF ½2� for s4sc. At a givencapacity, (1) under both strategies, firm 1's VaR is higher (resp. lower)under s½1� if the confidence level of VaR αZ1�F ½1�ðscÞ (resp.αo1�FðscÞ); (2) under Strategy 2, firm 2's VaR is higher (resp. lower)under s½1� if the confidence level for VaR, αZFðscÞ (resp. αoFðscÞ).

Proposition 8 (1) states that for both strategies, given acapacity, under a reasonably high confidence level of VaR, α, theincrease of the exchange rate volatility increases firm 1's downsiderisk. This is because low realizations of exchange rate occur more

often as the exchange rate becomes more volatile and firm 1'sprofit increases in the currency exchange rate.

Under strategy 1, firm 2 has no exposure to the currency fluctua-tions. Proposition 8 (2) states that under strategy 2, for a reasonablyhigh confidence level of VaR, α, when firm 1's capacity is fixed, theincrease of the exchange rate volatility also increases firm 2's down-side risk. Firm 2's downside risk is associated with the presence ofhigh exchange rate realizations under which firm 1 aggressively sellsto market 2. Themore volatile is the exchange rate, themore prevalentare occurrences of high exchange rate realizations.

Fig. 3 compares changes in firm 1's optimal capacity decision,expected profit, and corresponding VaR and EDR as the exchangerate volatility increases, assuming there is no demand uncertaintyin either market. Several observations are in order. First, firm 1'soptimal expected profit is higher under strategy 2 than understrategy 1. Second, firm 1's downside risks, measured by VaR andEDR, are higher under strategy 1. Third, as the exchange ratevolatility increases, firm 1's optimal capacity and expected profitunder strategy 1 stay the same, and they increase under strategy 2,as predicted in Proposition 7. Downside risks under both strategiesincrease. Finally, both VaR and EDR increase more slowly understrategy 2, because firm 1 can respond better at low exchange raterealizations by employing the allocation option. Fig. 4 compareschanges in firm 2's expected profit and downside risks as theexchange rate volatility increases, given that firm 1 adjusts capa-city optimally and assuming that there is no demand uncertaintyin either market. Interestingly, firm 2's downside risks, both VaRand EDR, increase at a decreasing rate under strategy 2 and remainconstant under strategy 1. This is a reflection of the changes of themarket 2 clearing price at high exchange rates as the volatility ofthe exchange rate increases. At high exchange rates, understrategy 1, firm 1 sells to market 2 at full capacity Kn

12, whose valueremains constant in the exchange rate volatility (as shown inFig. 3). Similarly the market 2 clearing price is constant in thevolatility of exchange rates. Under strategy 2, although the full


Fig. 3. Firm 1's optimal capacity, profit performance and downside risks as a function of the exchange rate volatility: (a) optimal Kn

1, (b) optimal expected profit, (c) VaR withconfidence level α¼ 99%, and (d) expected downside risk with z¼ πnN

11 ðKnN11 ;K

nN12 Þ. Parameter settings are: ϵ1 ¼ ϵ2 ¼ 1, a1 ¼ a2 ¼ 0:5, θ1 ¼ θ2 ¼ 100, C1 ¼ 8, C2 ¼ 10, τ1 ¼ 3,

τ2 ¼ 4, and K2 ¼ 25. ln s�Nðμ;s2Þ, E½s� ¼ expð0:325Þ, with s changing from 0.10 to 0.80.






capacity Kn

1 is also increasing convexly in the exchange ratevolatility, firm 1's selling quantity to market 2 is strictly less thanthe full capacity (due to the market 1 effect) and is increasingconcavely in the exchange rate volatility; consequently, the market2 selling price increases concavely in the exchange rate volatility.

4.2. Firm 2's capacity K2

Firm 2's capacity level K2 represents the severity of thecompetition in market 2. The larger the value of K2, the moresevere the competition in the foreign market for global firm 1.

Strategy 2 (C)Strategy 1 (N)

Fig. 4. Firm 2's profit performance and downside risks as a function of the exchange rate volatility: (a) expected profit, (b) VaR with confidence level α¼ 99%, and(c) expected downside risk with z¼ πnN

11 ðKnN11 ;K

nN12 Þ. Parameter settings are the same as for Fig. 3.


Fig. 5. Firm 1's optimal capacity and profit performance in firm 2's capacity K2: (a) optimal Kn

1, (b) optimal expected profit, (c) VaR with confidence level α¼ 99%, and(d) expected downside risk with z¼ πnN

11 ðKnN11 ;K

nN12 Þ. Parameter settings are: ln s�Nð0:2;0:52Þ, K2 changes from 1 to 39, other parameters are the same as for Fig. 3.






Proposition 9. Under both strategies, firm 1's expected profit isconvex nonincreasing in firm 2's capacity K2.

Proposition 9 states that the intuitive result that firm 1's optimalprofit decreases as competition in market 2 becomes more severe.Fig. 5 shows that for both strategies, the optimal capacity firstincreases then decreases in K2 with the underlying rationalealready given in the discussion of Proposition 2. Consequently,VaR first increases then decreases in K2. Interestingly, strategy 2'sVaR starts decreasing earlier than for strategy 1. Fig. 6 shows thatfirm 2's expected profit is first increasing then decreasing in K2,while VaR first decreases then increases in K2. Firm 2' VaR understrategy 2 starts increasing earlier and faster than under strategy 1.

To study how operational flexibility mitigates the severity ofmarket 2's intensified competition on firm 1's performance, wechoose two values of K2, 5 and 35, and plot the decrease of theoptimal expected profit and the increase of VaR from K2 ¼ 5 toK2 ¼ 35 as functions of the exchange rate volatility in Fig. 7. Theincrease of VaR decreases under both strategies as the exchangerate volatility increases. The decrease of expected profit does notchange under strategy 1, but decreases under strategy 2. Thissuggests that in environments of increasing volatility in exchangerates, operational flexibility becomes more effective in helpingfirm 1 to deal with the foreign market competition, as it enablesfirm 1 to effectively exploit the currency fluctuations.

5. Summary

In this paper, we establish a stylized model to advocate the useof operational hedging approach, such as allocation of shared

resources among markets, in mitigating risks and enhancing longrun profitability of global firms exposed to fluctuating exchangerates. we examine two different operational strategies of a globalfirm selling in the domestic market as a monopolist and compet-ing in a foreign market against a local competitor. In the base casestrategy, we use the often advocated paradigm of a “naturalhedge” approach with the firm operating dedicated facilities foreach market. The strategy provides little operational flexibility, butminimizes risks by matching currency footprints of revenues andcosts. We introduce operational hedging via allocation option instrategy 2. Strategy 2 adopts risk pooling flexibility by allowing thetwo market products to share a vanilla middle product (or sharecapacity) and postponing the allocation of the middle product (orcommitting the capacity to market specific production) to after therealization of the exchange rate.

The operational flexibility, via the smart exercise of allocationoption, enables the global firm to sell conservatively to the foreignmarket when the exchange rate is unfavorable and to sell aggres-sively when the exchange rate is favorable. Consequently, opera-tional flexibility results in improving the firm's expected profit anddecreasing its downside risk, while at the same time exposing thelocal competitor to increased downside risks. Furthermore, forhigher volatility exchange rate environments, operational flexibil-ity effectively mitigates the adverse effects of increasing competi-tion (mostly captured through increased capacity commitments ofthe local firm) in the foreign market on the firm's expected profitand downside risks.

The global firm's exchange rate driven selling decisions inthe foreign market enabled by allocation option weaken theeffectiveness of competitive responses in the, pp. 325-336foreign market when the exchange rate is high (i.e., the firm's


Fig. 6. Firm 2's profit performance in its capacity K2: (a) expected profit and (b) VaR with confidence level α¼ 99%. Parameter settings are the same as for Fig. 5.


Fig. 7. Firm 1's profit performance change when firm 2 increases its capacity from K2 ¼ 5 to K2 ¼ 35, as a function of the exchange rate volatility: (a) optimal expected profitdrop and (b) VaR increase with confidence level α¼ 99%. Parameter settings are the same for Fig. 3.






home currency is depreciated). However, the global firm is lesswell positioned to weather the adversity of competition whenthe exchange rate is low (i.e., appreciated home currency), butstill operational flexibility can reduce the impact by divertingshared resources to the domestic market or even abandoningtemporarily the servicing of the foreign market. As a result, the

global firm's flexibility not only improves its performance butalso can adversely affect the local competitor's performance.More specifically, it increases the local firm's downside risk, andit can increase or decrease the local firm's expected profitdepending on the relative strength of effects from the low andhigh exchange rates.

Appendix A

Proof of Lemma 1. Omitted. See Proof of Lemma 2. □

Proof of Proposition 1. Take the first-order derivative of ΠN1 w.r.t. K11 and K12, we have

∂ΠN1 ðK11;K12Þ∂K11

��K11 ¼ KnN

11

¼Z þ1

ϵ1ðKnN11 Þ½θ1ϵ1ð1�a1ÞðKnN

11 Þ�a1 �τ1�g1ðϵ1Þ dϵ1�C1 ¼ 0;

∂ΠN1 ðK11;K12Þ∂K12

��K12 ¼ KnN

12

¼ Es

Z þ1

tN2 ðKnN12 ;K2Þ


12 Þ�τ2�g2ðϵ2Þ dϵ2" #

�C1 ¼ 0:

The concavity in K11 and K12 is straightforward by checking the second-order derivatives. □

Proof of Proposition 2. Firm 1's best response to K2, KnN12 ðK2Þ, is given in Eq. (7):

C1 ¼ Es

Z þ1

tN2 ðKnN12 ;K2Þ


12 Þ�τ2�g2ðϵ2Þ dϵ2" #

:

We have

∂KnN12

∂K2¼ �

EsR þ1tN2 ðKnN

12 ;K2Þ ϵ2θ2g2ðϵ2Þ dϵ2½ðK2þKnN12 Þ�a2 �2ðK2�a2K

nN12 Þ�

EsR þ1tN2 ðK

nN12 ;K2Þ ϵ2θ2gðϵ2Þ dϵ2½ðK2þKnN

12 Þ�a2 �2ð2K2þð1�a2ÞKnN12 Þ�

¼ � K2�a2KnN12

2K2þð1�a2ÞKnN12

∂2KnN12

∂K22

¼ �ða2þ1ÞðK2þKnN12 ð1�a2ÞÞðKnN

12 þK2Þð2K2þð1�a2ÞKnN

12 Þ3o0: □

Proof of Lemma 2. The Lagrangian of firm 1's stage 2 optimization is L1 ¼ πC1�λðq1þq2�K1Þ. Take the partial derivative of the

Lagrangian, we have

∂L1∂q1

¼ q�a11 θ1ϵ1ð1�a1Þ�τ1�λ¼ 0 ) qnC

1 ¼ q1ðs; ϵ1; ϵ2;K1;K2Þ;∂L1∂q2

¼ s θ2ϵ2ðq2þK2Þ�a2 1� a2q2q2þK2

� ��τ2

� ��λ¼ 0 ) qnC

2 ¼ q2ðs; ϵ1; ϵ2;K1;K2Þ;

0¼ λðq1þq2�K1Þ: ð10ÞIt is easy to see that the optimal allocation (qnC

1 ; qnC2 ) equalizes the marginal profits from two markets, i.e.,

ðqnC1 Þ�a1θ1ϵ1ð1�a1Þ�τ1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}marginal profit in market 1

¼ s θ2ϵ2ðqnC2 þK2Þ�a2 1� a2qnC

2

qnC2 þK2

!�τ2

" #|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

marginal profit in market 2

¼ λ: ð11Þ

If qnC1 þqnC

2 oK1, then (qnC1 ; qnC

2 ) can be obtained from (11) by setting λ¼ 0. That is, qnC1 ¼ k11ðϵ1Þ; qnC

2 ¼ q2U . Moreover, qnC1 is non-increasing in ϵ2,

and qnC2 is non-decreasing in ϵ2. If qnC

1 þqnC2 ¼ K1, (qnC

1 ; qnC2 ) can be obtained from (11) by setting qnC

1 ¼ K1�qnC2 . That is, qnC

1 ¼ q1B and qnC2 ¼ q2B.

(1) Scenario 1 (S1) – ϵ1rϵ1ðK1Þ: Firm 1 is not capacity constrained when he only sells to market 1, qnC1 ¼ k11ðϵ1Þ. Firm 1 will not sell to

market 2 until ϵ24t1ðK2Þ, where t1ðK2Þ solves ϵ2 by setting λ¼ 0 and qnC2 ¼ 0. As ϵ2 increases, firm 1 increases the sales in market 2.

Before firm 1 is binding in capacity, firm 1 treats the two markets separately, qnC2 ¼ q2U is given in Eq. (5). Firm 1's capacity will be

binding when ϵ2ZtC2ðϵ1;K1;K2Þ, where qnC2 ðϵ2 ¼ tC2Þ ¼ K1�k11ðϵ1Þ, and tC2ðϵ1;K1;K2Þ solves ϵ2 in Eq. (11) by setting λ¼ 0 and

qnC2 ¼ K1�k11ðϵ1Þ. For ϵ24tC2, q

nC1 ¼ K1�q2B, where q2B solves Eq. (11).

(2) Scenario 2 (S2) – ϵ14ϵ1ðK1Þ: Firm 1 is capacity constrained even when he only sells to market 1. qC1 ¼ K1 when ϵ2ot3ðϵ1; s;K1;K2Þ. Asmarket 2 becomes more attractive, firm 1 starts to allocate to market 2 at ϵ2 ¼ t3ðϵ1; s;K1;K2Þ, where t3ðϵ1; s;K1;K2Þ solves ϵ2 inEq. (11), by setting qnC

2 ¼ 0 and qnC1 ¼ K1. As ϵ2 increases further, qnC

1 oK1, and qnC1 þqnC

2 ¼ K1, thus, similar to (1), q2B can be obtained bysolving Eq. (11) with qnC

1 ¼ K1�q2B. □






Proof of Proposition 3. We use the superscript Si (i¼1, 2) to denote firm 1's time 1 profit π1 under Scenario i. Under Scenario 1,

πS11 ðs; ϵ1; ϵ2;K1;K2Þ ¼

�C1K1þðθ1ϵ1ðqnC1 Þ�a1 �τ1ÞqnC

1 ; ϵ2rt1ðK2Þ�C1K1þðθ1ϵ1ðqnC

1 Þ�a1 �τ1ÞqnC1

þsðϵ2θ2ðK2þq2UÞ�a2 �τ2Þq2U ; t1ðK2Þoϵ2otC2�C1K1þðθ1ϵ1ðK1�q2BÞ�a1 �τ1ÞðK1�q2BÞþsðϵ2θ2ðK2þq2BÞ�a2 �τ2Þq2B; ϵ2ZtC2

8>>>>>>><>>>>>>>:

;

and

∂πS11 ðs; ϵ1; ϵ2;K1;K2Þ

∂K1¼

�C1; ϵ2rtC2�C1þθ1ϵ1ð1�a1ÞðK1�q2BÞ�a1 �τ1; ϵ24tC2ð ¼ �C1þsϵ2θ2ðK2þq2BÞ�a2 �1ðK2þð1�a2Þq2BÞ�sτ2Þ;

8>><>>: ;

and

∂2πS11 ðs; ϵ1; ϵ2;K1;K2Þ

∂K21

¼0; ϵ2rtC2

�a1θ1ϵ1ð1�a1ÞðK1�q2BÞ�a1 �1 1�∂q2B∂K1

� �; ϵ24tC2

8><>: :

By implicit function theorem,

∂q2B∂K1

¼ a1θ1ð1�a1ÞðK1�q2BÞ�a1 �1

a1θ1ð1�a1ÞðK1�q2BÞ�a1 �1þsa2θ2ðq2BþK2Þ�a2 �2ð2K2þð1�a2Þq2BÞAð0;1Þ:

Hence,

∂2πS11 ðs; ϵ1; ϵ2;K1;K2Þ

∂K21

o0; for ϵ2ZtC2 :

Therefore,

ðEs;ϵ1 ;ϵ2 ½πS11 ðs; ϵ1; ϵ2;K1;K2Þ�Þ0K1

¼ Es

Z t1ðK2Þ

0

Z þ1

tC2 ðϵ1 ;K1 ;K2Þ

∂πS11 ðs; ϵ1; ϵ2;K1;K2Þ

∂K1g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1;

ðEs;ϵ1 ;ϵ2 ½πS11 ðs; ϵ1; ϵ2;K1;K2Þ�Þ″K1

¼ Es

Z t1ðK2Þ

0

Z þ1

tC2 ðϵ1 ;K1 ;K2Þ

∂2πS11 ðs; ϵ1; ϵ2;K1;K2Þ

∂K21

g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1o0:

Similarly, under Scenario 2,

πS21 ðs; ϵ1; ϵ2;K1;K2Þ ¼

�C1K1þðθ1ϵ1K�a11 �τ1ÞK1; ϵ2rt3

�C1K1þðθ1ϵ1ðK1�q2BÞ�a1 �τ1ÞðK1�q2BÞ; ϵ24t3þsðϵ2θ2ðK2þq2BÞ�a2 �τ2Þq2BÞ;

8><>: ;

∂πS21 ðs; ϵ1; ϵ2;K1;K2Þ

∂K1¼

�C1þθ1ϵ1ð1�a1ÞK �a11 �τ1; ϵ2rt3

�C1þθ1ϵ1ð1�a1ÞðK1�q2BÞ�a1 �τ1; ϵ24t3ð ¼ �C1þsϵ2θ2ðK2þq2BÞ�a2 �1ðK2þð1�a2Þq2BÞ�sτ2Þ;

8><>: ;

and

∂2πS21 ðs; ϵ1; ϵ2;K1;K2Þ

∂K21

¼�a1θ1ϵ1ð1�a1ÞK �a1 �1

1 ; ϵ2rt3

�a1θ1ϵ1ð1�a1ÞðK1�q2BÞ�a1 �1 1�∂q2B∂K1

� �o0; ϵ24t3

8><>: ;

Therefore,

ðEs;ϵ1 ;ϵ2 ½πS21 ðs; ϵ1; ϵ2;K1;K2Þ�Þ0K1

¼ Es

Z þ1

ϵ1ðK1Þ

Z þ1

tC2 ðϵ1 ;K1 ;K2Þ

∂πS11 ðs; ϵ1; ϵ2;K1;K2Þ

∂K1g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1;

ðEs;ϵ1 ;ϵ2 ½πS21 ðs; ϵ1; ϵ2;K1;K2Þ�Þ″K1

¼ Es

Z þ1

ϵ1ðK1Þ

Z þ1

tC2 ðϵ1 ;K1 ;K2Þ

∂2πS21 ðs; ϵ1; ϵ2;K1;K2Þ

∂K21

g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1o0:

At time 0,

∂ΠC1ðK1;K2Þ∂K1

¼ �C1þZ ϵ1

0ðEs;ϵ2 ½πS1

1 ðs; ϵ1; ϵ2;K1;K2Þ�Þ0K1g1ðϵ1Þ dϵ1þ

Z þ1

ϵ1ðK1ÞðEs;ϵ2 ½πS2

1 ðs; ϵ1; ϵ2;K1;K2Þ�Þ0K1g1ðϵ1Þ dϵ1; ð12Þ

∂2ΠC1ðK1;K2Þ∂K2

1

¼Z ϵ1ðK1Þ

0ðEs;ϵ2 ½πS1

1 ðs; ϵ1; ϵ2;K1;K2Þ�Þ″K1g1ðϵ1Þ dϵ1þ

Z þ1

ϵ1ðK1ÞðEs;ϵ2 ½πS2

1 ðs; ϵ1; ϵ2;K1;K2Þ�Þ″K1g1ðϵ1Þ dϵ1

þðϵ1ðK1ÞÞ0K1ðEs;ϵ2 ½ðπS1

1 ðs; ϵ1; ϵ2;K1;K2ÞÞ0K1jϵ1 ¼ ϵ1ðK1Þ��Es;ϵ2 ½ðπS1

1 ðs; ϵ1; ϵ2;K1;K2ÞÞ0K1jϵ1 ¼ ϵ1ðK1Þ�Þ






¼Z ϵ1ðK1Þ

0ðEs;ϵ2 ½π″S1

1 ðs; ϵ1; ϵ2;K1;K2Þ�Þg1ðϵ1Þ dϵ1þZ þ1

ϵ1ðK1ÞðEs;ϵ2 ½π″S2

1 ðs; ϵ1; ϵ2;K1;K2Þ�Þg1ðϵ1Þ dϵ1o0:

Hence, firm 1's profit is concave in K1 at time 1. It is trivially true that firm 1's profit at time 0 is concave. □

Proof of Proposition 4.

(1) Straightforward from the arguments in Section 3.(2) It follows Milne and Neave (1994).(3) The expected profit comparison follows (2). □

Proof of Proposition 5. Omitted. It trivially holds by Proposition 4. □

Proof of Proposition 6. For the domestic firm 2, π2 ¼ ½ϵ2θ2ðq2þK2Þ�a2 �C2�K2. π2 is non-increasing in s and ϵ2. Firm 2's downside risk isassociated with realization of large s or ϵ2.

1. Demand in market 1 is constant and low (ϵ1rϵ1ðK1Þ). Under strategy 1, πN2 ¼ ½ϵ2θ2ðK12þK2Þ�a2 �C2�K2, for ϵ2ZtN2 . Under strategy 2,

(1) If ϵ1ðK1ÞZϵ1, qnC2 ZK12 for ϵ2ZtN2 . Therefore, we have πN

2 ðϵ2ÞZπC2ðϵ2Þ for ϵ2ZtN2 . The definition of VaR, with VaR¼ �V and

Prðπ2ðϵ2ÞrV Þ ¼ 1�α is equivalent to Prðπ2ðϵ2ÞZVÞ ¼ α. Since π2 is non-increasing in s, it follows that α¼ G2ðπ�12 ðVÞÞ, where G2 is the

CDF of ϵ2. If αZG2ðtN2 Þ, then π�12 ðVÞ ¼ G�1

2 ðαÞZtN2 for both strategies, which implies VNZVC and VaRNrVaRC . (2) It follows thatEDRðπ2Þ ¼ Es;ϵ2 ½ðz�π2ðϵ2ÞÞþ � ¼ Es½

Rπ2ðϵ2Þr zðz�π2ðϵ2ÞÞg2ðϵ2Þ dϵ2� ¼ Es½

Rϵ2 Z ðπ2Þ � 1ðzÞðz�π2ðϵ2ÞÞg2ðϵ2Þ dϵ2�. Moreover, since πN

2 ðϵ2ÞZπC2ðϵ2Þ

for ϵ2ZtN2 , it follows that ðπC2Þ�1ðzÞr ðπN

2 Þ�1ðzÞ. Thus, we have for zrπC2ðtN2 Þ, EDRðπN

2 Þ ¼ Es½Rϵ2 Z ðπN

2 Þ� 1ðzÞðz�πN

2 Þg2 dϵ2�rEs½Rϵ2 Z ðπN

2 Þ� 1ðzÞðz�πC

2ðϵ2ÞÞg2ðϵ2Þ dϵ2�rEs½Rϵ2 Z ðπC

2 Þ� 1ðzÞðz�πC

2ðϵ2ÞÞg2ðϵ2Þ dϵ2� ¼ EDRðπC2Þ.

2. There is no demand uncertainty and market 1 demand is high (ϵ14ϵ1ðK1Þ), πC2ðsÞ ¼ ðϵ2θ2ðq2BþK2Þ�a2 �C2ÞK2, where q2BZK12 for

sZsK12 , where sK12 solves s by setting q2U ¼ K12 in equation (7). The rest of the proof follows the Proof of Proposition 6.1. □

Proof of Lemma 3. Omitted. □

Proof of Lemma 4. Omitted. □

Proof of Corollary 1. Omitted. □

Proof of Lemma 5. (1) Under strategy 1, firm 1's market 1 profit π11N is independent of s. Firm 1's time 1 profit in market 2 is as follows:

πN12 ¼

0; t1ðK2ÞZϵ2s½θ2ϵ2ðq2UþK2Þ�a2 �τ2�q2U ; t1ðK2Þoϵ2otN2 ðK1;K2Þs½θ2ϵ2ðK2þK12Þ�a2 �τ2�K12; ϵ2ZtN2 ðϵ1;K1;K2Þ

8><>:

Therefore, firm 1's profit π12N is linear in s. (2) under strategy 2, for the low demand scenario of market 1 (S1), by Envelope Theorem:

∂2πC1

∂s2¼

0; t1ðK2ÞZϵ20; t1ðK2Þoϵ2otNC ðK1;K2Þ:θ2ϵ2ðq2BþK2Þ�a2 �1ðq2BþK2�a2q2BÞ

∂q2B∂s

40; ϵ2ZtC2ðK1;K2Þ

8>>><>>>: ð13Þ

For the high demand scenario of market 1 (S2):

∂2πC1

∂s2¼

0; ϵ2rtC3ðϵ1; s;K1;K2Þθ2ϵ2ðq2BþK2Þ�a2 �1ðq2BþK2�a2q2BÞ

∂q2B∂s

40; ϵ24tC3ðϵ1; s;K1;K2Þ:

8><>:

Therefore, π12N and π1C are convex nondecreasing in s. □


(1) Under strategy 1, firm 1's capacity and production decisions are made based on expected exchange rate and demand. As long as theexpectation remains the same, optimal capacity level and expected profit remain the same for strategy 1.

(2) Under strategy 2, by Lemma 5 (2), firm 1's time 1 profit is convex nondecreasing in s:

∂ΠC1

∂K1¼ �C1þEs

Zϵ1

Zϵ2

∂πC1

∂K1

� �For the low demand scenario of market 1 (S1):

∂2πC1

∂K1∂s¼

0; ϵ2rtC2ðϵ1;K1;K2Þθ2ϵ2ðK2þq2BÞ�a2 �1ðK2þq2Bð1�a2ÞÞ�τ2; ϵ24tC2ðϵ1;K1;K2Þ:

(ð14Þ






For the high demand scenario of market 1 (S2):

∂2πC1

∂K1∂s¼

0; ϵ2rtC3ðϵ1; s;K1;K2Þθ2ϵ2ðK2þq2BÞ�a2 �1ðK2þq2Bð1�a2ÞÞ�τ2; ϵ24tC3ðϵ1; s;K1;K2Þ:

(ð15Þ

For ϵ24t3ðϵ1; s;K1;K2Þ under S1 and ϵ24t3ðϵ1; s;K1;K2Þ under S2:

∂3πC1

∂s2∂K1¼ a1θ1ð1�a1ÞðK1�q2BÞ�a1 �1∂q2B

∂sBy Lemma 2, q2B solves

Hðq2B; sÞ ¼ s½ðK2þð1�a2Þq2BÞθ2ðq2BþK2Þ�a2 �1ϵ2�τ2��½ð1�a1Þθ1ϵ1ðK1�q2BÞ�a1 �τ1� ¼ 0

By Implicit function theorem,

∂q2B∂s

¼ �∂H∂s∂H∂q2B

∂H∂s

¼ θ2ϵ2ðq2BþK2Þ�a2 �1ðK2þð1�a2Þq2BÞ�τ240

∂H∂q2B

¼ �½sa2θ2ϵ2ðq2BþK2Þ�a2 �2ð2K2þð1�a2Þq2BÞþa1θ1ϵ1ð1�a1ÞðK1�q2BÞ�a1 �1�o0

Therefore, ∂q2B=∂s40 and ∂3πC1=∂s

2∂K140. ∂πC1=∂K1 is convex in s.

Es½1�Zϵ1

Zϵ2

∂π1

∂K1

� �ZEs½2�

Zϵ1

Zϵ2

∂π1

∂K1

� �;

which implies KnC1 ðs½1�ÞZKnC

1 ðs½2�Þ. □


(1) Without loss of generality,we prove the case of αZ1�F ½1�ðscÞ. By Muller and Stoyan (2002, Definition 1.5.25),if F ½1� differs from F ½2� by amean preserving spread,then they cross at s¼ sc with F ½1�ZF ½2� for sosc and F ½1�oF ½2� for s4sc. For a given confidence level α,we have1�α¼ Pr½i�ðπ1ðsÞrV ½i�Þ ¼ Pr½i�ðsrπ�1

1 ðV ½i�ÞÞ ¼ F ½i�ðπ�11 ðV ½i�ÞÞ,i¼1,2. If αZ1�F ½1�ðscÞ,then F ½1�ðscÞZ1�α¼ F ½1�ðπ�1

1 ðV ½1�ÞÞ ¼ F ½2�ðπ�11 ðV ½2�ÞÞ.

It follows that π�11 ðV ½1�Þoπ�1

1 ðV ½2�Þ, V ½1�oV ½2�, and VaR½1�4VaR½2�.(2) Without loss of generality, we prove the case of αZF ½1�ðscÞ. For a given confidence level α¼ Pr½i�ðπ2ðsÞZV ½i�Þ ¼

Pr½i�ðsrπ�12 ðV ½i�ÞÞ ¼ F ½i�ðπ�1

2 ðV ½i�ÞÞ, i¼1,2. If αZF ½1�ðscÞ, then F ½1�ðπ�12 ðV ½1�ÞÞ ¼ F ½2�ðπ�1

2 ðV ½2�ÞÞforαZF ½1�ðscÞ. It follows thatπ�11 ðV ½1�Þ4π�1

1 ðV ½2�Þ, V ½1�oV ½2�, and VaR½1�4VaR½2�. □

Proof of Proposition 9. We use strategy 2 as an example. The proof for strategy 1 follows the same type of analysis. Taking the firstderivative of the firm 1's optimal expected profit with respect to K2, we have

∂ΠnC1

∂K2¼ Es

Z ϵ1

0

Z tC2

t1�a2sϵ2θ2ðq2UþK2Þ�a2 �1q2Uþ

Z þ1

tC2

�a2sϵ2θ2ðq2BþK2Þ�a2 �1q2B

!g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1

"

þZ 1

ϵ1

Z þ1

t3�a2sθ2ϵ2ðq2BþK2Þ�a2 �1q2Bg2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1

�

I� ∂½�a2sϵ2θ2ðq2BþK2Þ�a2 �1q2B�∂K2

¼ a2sθ2ϵ2ðq2BþK2Þ�a2 �2 ða2þ1Þq2Bþða2q2B�K2Þ∂q2BK2

� �;

J � ∂½�a2sϵ2θ2ðq2UþK2Þ�a2 �1q2U �∂K2

¼ a2sθ2ϵ2ðq2UþK2Þ�a2 �2 ða2þ1Þq2Uþða2q2U�K2Þ∂q2UK2

� �;

where

∂q2B∂K2

¼ �sa2θ2ðq2BþK2Þ�a2 �2ða2q2B�K2Þ∂2π1ðq2B;K2; sÞ

∂q22B

:

Since

∂2πC1ðq2B;K2; sÞ∂q22B

o0;∂q2B∂K2

ða2q2B�K2Þ40;

it follows HZ0. Similarly JZ0. Therefore,

∂2ΠnC1

∂K22

¼ Es

Z ϵ1

0

Z tC2

t1JþZ þ1

tC2

I� !

g2ðϵ2Þg1ðϵ1Þ dϵ2 dϵ1þZ 1

ϵ1

Z þ1

t3I � g2ðϵ2Þg1ðϵ1Þ dϵ2dϵ1

" #Z0: □






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int. j. production economics · ﬁnancial hedging is preferred for low volatility exchange rates...

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