We study how to manage commodity risks (price and consumption volume) via physical inventory and

Transcription

1 MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 15, No. 3, Summer 2013, pp ISSN (prit) ISSN (olie) INFORMS Maagig Storable Commodity Risks: The Role of Ivetory ad Fiacial Hedge Paos Kouvelis Oli Busiess School, Washigto Uiversity i St. Louis, St. Louis, Missouri kouvelis@wustl.edu Rog Li Oli Busiess School, Washigto Uiversity i St. Louis, St. Louis, Missouri 63130; ad Nakai Busiess School, Nakai Uiversity, Tiaji, Chia, lir@wustl.edu Qig Dig Lee Kog Chia School of Busiess, Sigapore Maagemet Uiversity, Sigapore , digqig@smu.edu.sg We study how to maage commodity risks (price ad cosumptio volume) via physical ivetory ad fiacial hedge i a multiperiod problem (with a iterperiod utility fuctio) for a risk-averse firm procurig a storable commodity from a spot market at a radom price ad a log-term supplier at a fixed price. The firm also has access to fiacial cotracts writte o the commodity price, such as futures cotracts ad call ad put optios. We examie differet cases of fiacial hedgig, for example, sigle-cotract ad multicotract hedges. For each case, we dyamically maximize the mea-variace utility of the firm s cash flow ad characterize a optimal itegrated policy of ivetory ad hedgig, which is easy to compute ad implemet. We fid that as log as futures are used i each period, aloe or ot, the optimal ivetory policy is myopic. The optimal hedgig policy, however, is ever myopic, but depeds o all the future optimal decisios. This is cotrary to fidigs of the literature usig itraperiod utility fuctios, which fids myopic hedgig to be optimal. Moreover, we fid that hedgig may lead to ivetory reductio i multiperiod problems. Thus the isights from the sigle-period studies i the literature hedgig leads to ivetory icrease do ot apply. Fially, isights are offered o the role ad impact of ivetory ad fiacial hedge o profitability, variace cotrol, ad service level, usig both aalytical ad umerical results. Key words: stochastic ivetory; commodity markets; futures; optios; risk maagemet; hedgig; risk aversio History: Received: September 11, 2010; accepted: Jauary 13, Published olie i Articles i Advace May 3, Itroductio The sourcig, ivetory storig, ad processig of storable commodities, which are evetually sold i the form of differetiated goods to ed-product markets, are corerstoe activities of may busiess strategies. Examples of commodities, storable ad tradable o exchages, are oil, steel, precious metals, cor, sugar, dyamic radom access memories, etc. However, commodity risks ca jeopardize eve the best thought-out strategies (Tevelso et al. 2007). These days, commodity price risks are eve more proouced ad uexpected tha before because of shifts i supply-ad-demad dyamics ad global fiacial turmoil (Fisher ad Kumar 2010). Prices of may commodities are ow fluctuatig as much i a sigle day as they did i a year i the early 1990s (Wiggis ad Blas 2008). For compaies that rely o such commodities as productio iputs ad caot pass cost icreases to their customers, such volatility substatially icreases their workig capital eeds ad risks of fiacial distress. As a result, procuremet orgaizatios are playig pivotal roles i the fiacial success of such firms, ad purchasig maagers are expected to have skills ever required before. For example, food compaies are i search of procuremet maagers with commodity tradig skills (Wiggis 2008). As argued ad show empirically by may fiace researchers, hedgig to reduce costly cash flow variability ca icrease the firm value (Froot et al. 1993). Commodity iputs cotribute to firms cash flow volatility i terms of ot oly the price risk (the price volatility), but also the cosumptio volume risk (the volatility of cosumed commodity volume to meet the ucertai ed-product demad). Traditioally, these two types of risks are treated ad hedged separately; the former is hedged by commodity derivatives, ad the latter is hedged by physical ivetories. This soiled approach, however, is a problem, especially i a large multibusiess orgaizatio (Fisher ad Kumar 2010). As argued i 507

2 508 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS Kleidorfer (2009), itegratig ivetory ad fiacial derivatives for a effective hedgig of commodity risks creates research challeges. I particular, simultaeously optimizig ivetory levels ad derivative portfolios is a difficult problem, with oly limited aswers, yet mostly for ostorable commodities like electricity. This paper attempts to offer some aswers to this itegrated risk maagemet problem for storable commodities. The fudametal settig of our problem is that a firm procures a storable commodity iput from a log-term supplier ad a commodity exchage (spot market), where sellig the commodity is also possible. The firm also has access to fiacial cotracts writte o the commodity spot price. The firm processes the commodity iput to make a differetiated ed product, facig ucertai demads. The firm aims to hedge its cash flow volatility through both physical ivetory ad fiacial hedge for the commodity. For example, Emerso Motor Techologies, headquartered i St. Louis, Missouri, ad sellig a large product lie of electromechaical motors for various applicatios, has a log-term cotract (with a typical cotract duratio of three years) with U.S. Steel o purchasig various grades of steel with egotiated terms: a fixed quatity ad a fixed price. To meet its regioal eeds (it has factories i the Uited States, Mexico, ad Chia), the compay works with various metal exchages for spot purchase ad sell. These exchages iclude the Lodo Metal Exchage, the CME Group ad the Chia-based Shaghai Futures Exchage. Facig the risig commodity risks, Emerso Motor Techologies is iterested i hedgig usig commodity derivatives or fiacial cotracts, also available at these exchages, with futures cotracts beig the most commoly offered. The above iformatio was obtaied from discussios with Ray Keefe, vice presidet of maufacturig, ad Ke Poczekaj, vice presidet of global supply chai of Emerso, i Jue I this paper, we dyamically maximize the firm s cash flow uder a mea-variace (MV) criterio to determie joitly optimal policies for ivetory ad fiacial hedgig. Note that futures ad forward cotracts, although possessig practical differeces, are treated the same aalytically (Gema 2005). Ulike forward cotracts, which are typically custom bilateral cotracts with baks, futures cotracts are offered i stadard forms ad are tradable at exchages ad thus have better liquidity. Therefore, we cosider futures rather tha forward cotracts i this paper. We ivestigate the iteractio betwee ivetory ad fiacial hedgig ad their impacts o the firm s ivetory level, mea profit, profit variace, ad MV utility. 2. Literature Review Our work falls uder the geeral themes of itegratig physical ad fiacial risk maagemet i supply chais ad hedgig commodity risks i supply maagemet, which are both expertly reviewed by Kleidorfer (2009, 2010). For a earlier literature review o supply cotracts ad spot markets, please see Kleidorfer ad Wu (2003). The more geeral field of supply cotracts is of passig relevace to our work, ad we refer the readers to Cacho (2003). I this sectio, we first briefly review the research o itegrated log-term ad short-term (spot market) cotracts. Note that the log-term cotract terms (quatity ad price) i this paper are exogeously give to reflect the practice at Emerso ad similar busiesses. We the review i detail the works most relevat to our paper ad highlight our cotributios to the literature. A geeral sigle-period framework is preseted i Wu ad Kleidorfer (2005) for itegratig log-term ad short-term cotract decisios for mostly ostorable goods. Please see refereces therei for a more extesive review of the related work o ostorable commodities. For storable commodities, much of the existig literature addresses various types of sourcig issues for risk-eutral decisio makers, ad thus hedgig is uecessary. For example, Lee ad Whag (2002) were the first to itegrate spot market cosideratios after sales withi a ewsvedor framework, ad thus effectively edogeize the salvage value used i these models. Martíez-de-albéiz ad Simchi-Levi (2005) use a multiperiod model to address the optimal creatio of a portfolio of log-term cotracts (icludig fixed commitmet ad flexibility cotracts) itegrated with potetial spot market purchases. With rich istitutioal details of the fed-cattle supply chai, Boyabatli et al. (2011) offer a lucid picture of a beef processor s problem i these eviromets via a stylized sigle-period model, i the presece of spot market trasactio costs, ecoomies of scale i processig, quality differeces, ad correlated ed-product demad. The work by Devalkar et al. (2011) is a example cocerig risk aversio. Motivated by soybea processig, it aalyzes the itegrated procuremet, processig, ad sellig decisios for a risk eutral/averse commodity processor i a multiperiod settig. Iput commodities are procured from a spot market, processed, ad the sold as commodities i a futures market. I cotrast to ours, this work covers more operatioal details, but lacks cocers of fiacial hedgig ad demad ucertaity. There is limited research o joit optimizatio of operatioal ad hedgig decisios. Most of it uses sigle-period settigs to study the iteractio betwee operatioal decisios (capacity ad/or

3 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS 509 ivetory) ad hedgig decisios uder ucertaities i currecy exchage rate, demad, weather, commodity prices, etc. For example, Dig et al. (2007) study the optimal policies for capacity ivestmet ad hedgig o currecy exchage rates for a risk-averse multiatioal ewsvedor ad fid that the futures cotract is the optimal hedge; Gaur ad Seshadri (2005) study the optimal hedgig for a risk-averse ewsvedor with ay give ivetory level, based o the empirically prove assumptio that the ewsvedor s demad is correlated with the price of a fiacial asset. Both works show that fiacial hedgig drives up ivetory/capacity. Caldetey ad Haugh (2006) exted the work of Gaur ad Seshadri (2005) by allowig cotiuous tradig i the fiacial market; Chod et al. (2010) exted the work of Gaur ad Seshadri (2005) by implicitly characterizig the optimal multidimesioal capacity ivestmet ad focusig o the complemetarity/substitutio effect betwee the operatioal (postpoemet ad product) flexibility ad fiacial hedgig. Similar to our paper, Oum et al. (2006), Bodily ad Palacios (2007), ad Caldetey ad Haugh (2009) study commodity procuremet with fiacial hedgig, but for ostorable commodities like electricity ad liquid atural gas. Note that sice such commodities caot be carried over to a ext period, sigle-period models are sufficiet for study. I cotrast to the sigle-period works reviewed above, which all assume a fair-priced fiacial hedgig, we study risk maagemet for storable commodities usig a multiperiod model without the fair-priced fiacial hedgig assumptio. We ext review the research o joit optimizatio of operatioal ad hedgig decisios i multiperiod settigs. To the best of our kowledge, the works by Smith ad Nau (1995), Che et al. (2007), ad Zhu ad Kapusciski (2011) are the oly three i this stream. Amog them, the oe by Smith ad Nau (1995) is most geeral i terms of joit optimizatio of operatioal ad hedgig decisios, but i a geeric project valuatio settig; the other two have specific operatioal settigs. Che et al. (2007) study a ivetory problem with multiple supply cotracts ad simply apply the optimal hedgig results of Smith ad Nau (1995). Zhu ad Kapusciski (2011) study a complex capacity allocatio problem, but the optimal hedgig results of Smith ad Nau (1995) are still applicable. To effectively cotrast these works to ours i terms of model formulatio, we first itroduce two differet types of utility fuctio for multiperiod riskaverse problems: iterperiod ad itraperiod utility fuctios (for more detailed defiitios ad comparisos, see Alexader ad Sobel 2006). We the compare our work to the three works metioed above to demostrate our cotributios i terms of model formulatio ad maagerial isights. Briefly speakig, iterperiod utility fuctios cout i the cash flow correlatios across periods, whereas itraperiod utility fuctios do ot. A example of itraperiod utility is the additive expoetial utility used by all the multiperiod works metioed above, where additive meas that the utility of the total cash flow i multiple periods is a sum of the discouted utility of each period s cash flow. I cotrast, we use a iterperiod utility fuctio. More importatly, as show by Alexader ad Sobel (2006), itraperiod utility fuctios uder frequetly ecoutered coditios may imply risk eutrality, ad thus iterperiod utility fuctios are the most appropriate oes to use for risk-averse decisio makers i multiperiod settigs. Thus, although more challegig to hadle, our formulatio is ew ad richer i capturig the cash flow correlatios across periods ad fully reflectig risk aversio i the multiperiod cotext. Smith ad Nau (1995) study profitable itegratio of optio pricig ad decisio aalysis methods i project valuatio, where, however, the ivestmet decisios for a project to be valued ad the fiacial decisios are coceptually equivalet to the ivetory decisios ad the hedgig decisios i our problem, respectively. They prove the separatio theorem. This theorem is directly applied by Che et al. (2007). Zhu ad Kapusciski (2011) study a multiperiod joit optimizatio problem of operatioal (iitial capacity allocatio ad periodic productio ad trasshipmet) ad hedgig (derivatives writte o exchage rates) for a multiatioal ewsvedor. Ulike our paper, it is assumed that o ivetory is carried from period to period, ad thus the oly lik betwee differet periods is the exchage rates; the iteractio betwee the optimal operatioal ad hedgig decisios is ot studied aalytically. Despite the complex operatioal decisios, the separatio theorem is also applicable to their model. I cotrast, we show that the separatio theorem is ot applicable to our model as we use a iterperiod utility fuctio (a differet, but more appropriate, type of utility fuctio). Note that applicatio of the separate theorem to our model would have implied that the optimal hedgig is myopic, which differs from our fidigs. I summary, our model, whe restricted to its oe-period versio, is cosistet with previous sigle-period ewsvedor-like results, while offerig sharper isights o the relatioship betwee fiacial hedge ad ivetory. However, our model elucidates the importace of treatig such problems by fully reflectig multiperiod cosideratios. The sigle-period isights are ot trasferrable to the multiperiod cotext i the presece of risk aversio ad hedgig cocers. I geeral, ivetory ad fiacial hedge might ot be separable, ad this is defiitely the case if futures cotracts are ot used.

4 510 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS Whe separable, the optimal ivetory policies are myopic; the optimal fiacial hedgig policies are ot myopic (which is differet from the sigle-period isights as well as the past multiperiod isights). Previous multiperiod results are heavily drive by the use of itraperiod utility fuctios, which limits the derived isights whe cash flow correlatios across periods ad geuie risk-aversio cocers of decisio makers eter the picture. Our work cotributes to the literature ad practice for maagig storable commodity risks with tractable optimal policies ad ew maagerial isights. Last, we recogize methodological similarities to a recet pure fiace paper. Basak ad Chabakauri (2010) employ a iterperiod MV utility model, i cotiuous ad discrete time, for a asset allocatio problem, where the asset icludes a risky stock ad a bod. Because of our cosideratio of additioal factors icludig physical ivetory, demad ucertaity, ad fiacial hedgig, the exact methodology ad results of this paper are ot applicable to our problem. However, by followig a similar approach of dyamically solvig a iterperiod MV utility fuctio, our model also avoids the time-icosistecy issue; this is the extet of overlap of the two papers. 3. Problem Descriptio A firm procures ad processes (or maufactures) a sigle commodity to make a ed product, which is the sold at a exogeous market price. We first list the sequece of evets for each period. At the start of each period, the firm procures the commodity from the log-term supply ad trades (buys or sells) i the spot market. The commodity is the processed i make-to-order fashio to meet the ucertai demad. Umet demad is assumed lost, ad excess ivetory of the commodity is carried over to the ext period. To accout for the sigificat amout of setup ad processig time i each period, we do ot allow spot tradig i the middle of a period. Each period, the firm eeds to make two operatioal decisios: (1) the fial ivetory level of the commodity ad (2) how much commodity to process (or how may ed products to produce). If profitable, that is, if the gross margi (the reveue less the procuremet ad processig cost) is positive, the firm should process to best meet the ed-product demad ad procure the commodity primarily for productio. However, if the gross margi is zero or egative, the firm should ot produce, but may still procure the commodity for a better utility. We will elaborate o this case i 4.1. We ext list our otatio ad assumptios for period, = 1 N. The decisio variables are described i the last two bulleted items. We follow the covetio of deotig radom variables by uppercase letters ad their realizatios by correspodig lowercase letters. = 1/ 1 + r f 0 1 : the period discout factor for the firm s cash flow, where r f is the risk-free iterest rate for each period. We assume that the firm is usig the risk-free iterest rate to discout its cash flows ad the periods are of equal legth. The former assumptio will be relaxed i 8; the latter assumptio ca be easily relaxed by adjustig r f accordig to the actual duratio of each period. D S 0: the ed-product demad i period ad the commodity spot price at the start of period, respectively. They are defied o probability space F Q, where ay state i ca be writte as a vector of a state of demad ad a state of spot price, ad F is geerated by D 1 N ad S 1 N +1. We defie the probability measure, Q, by followig the assumptios for the partially complete market Smith ad Nau (1995) itroduced, i which the market ad private ucertaities are equivalet to the spot price ad demad ucertaities i our paper, respectively. I other words, we assume that Q is ideed a combiatio of the real-world probability measure o demads ad the risk-eutral probability measure (or equivalet martigale measure) o spot prices. Note that the risk-eutral probability measure is commoly used i the literature (see Chod et al. 2010). We will relax this Q-measure assumptio i 8 ad cosider the real-world probability measure, P. S : We assume that S 1 N +1 is Markovia ad the spot market operates with zero bid ask spread ad zero trasactio cost. Although S ad D are allowed to correlate, give S = s, D is assumed idepedet of S +1 S N +1. I other words, S ca serve as a sufficiet statistic for the etire past whe predictig D ad S +1. I additio, kowig D will ot alter the firm s forecast o S +1 S N +1. This is cosistet with the partially complete market assumptio i Smith ad Nau (1995). To uderstad the feasibility of such a assumptio, let us look at a example. Suppose S 1 N +1 follows geometric Browia motio (GBM), that is, S +1 = S / e sb s 2/2 for all, where s is the volatility ad B 1 B N are idepedet ad idetically distributed (i.i.d.) stadard Normal radom variables. Note that B is idepedet of S for all. Suppose D ad S are correlated because they both deped o B 1, say D = G B 1, for all, where G is a arbitrary positive valued fuctio. Thus, although there is correlatio betwee D +1 ad S +1, betwee S +1 ad S, ad betwee S ad D, we still have D +1 = G B beig idepedet of D = G B 1 (as B ad B 1 are idepedet). D : We assume that D 1 D N are mutually idepedet for techical tractability. The cumulative distributio fuctio (cdf) of D, deoted by F d s, is a icreasig fuctio of d for ay s, where

5 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS 511 s is used to reflect the correlatio betwee S ad D. Note that a icreasig cdf is oly assumed for the strictly icreasig or decreasig properties of the optimal ivetory levels. For ay cdf, these properties should be modified to odecreasig or oicreasig, respectively. 0: the absolute risk aversio of the firm used i MV utility fuctios (U = E /2 V, where E ad V stad for the expectatio ad the variace of cash flows, respectively). w q 0: the exogeous wholesale price ad fixed order quatity, respectively, specified i the log-term cotract. Note that although the firm would prefer a low wholesale price, fial cotract terms should be egotiated with the supplier. r 0: the exogeous uit reveue of the edproduct sold i period, excludig the processig cost. I other words, we assume the firm is a price taker i the ed-product market. h 0: the uit ivetory holdig cost for the commodity iput i period. K i i > 0: the strike price ad the o-arbitrage price paid upo trasactio, respectively, for hedgig cotract (or hedge) i available for tradig at the start of period, where i = c (call optio), p (put optio). A call (put) optio is the right, but ot obligatio, to buy (sell) the commodity at the strike price o the expiratio date. We assume for model simplicity that the fiacial cotracts cosidered expire i exactly oe period. For iterested readers, please fid the discussio, results, ad formal proof for the relaxed problem after the proof for Propositio 5.1 i the olie supplemet (available at msom ). i S +1 : the payoff fuctio for hedge i, i = f (futures cotract), c, p, where, for the futures cotract, f S +1 = S +1 E Q S +1 s = S +1 s / ; for the call optio, c S +1 = S +1 K c + c /, where c = E Q S +1 K c + s ; for the put optio, p S +1 = K p S +1 + p /, where p = E Q K p S +1 + s. E i S +1 s / i : the risk premium for hedge i traded at the start of period, i = c p. The risk premium for hedge f traded at the start of period is E f S +1 s /s (Dotha 1990). Note that uder the Q-measure (i 3 7), the risk premium of ay fiacial hedge is zero; uder the P-measure (i 8), however, the risk premium for some fiacial hedges may be positive or egative. z 0: the fial ivetory level of the commodity (i.e., the maximum ed-product ivetory level available to fill demad D ) at the start of period. Ulike equity markets, spot markets do ot allow short sellig, ad thus z is assumed to be oegative. z i : the optimal commodity level whe oly hedge i, i = f, c, p, is used. z : the optimal commodity level whe a multicotract hedge is used. y i : the quatity of hedge i, i = f, c, p, traded at the start of period, where y i < 0 if cotract i is sold ad y i > 0 if cotract i is purchased. Without loss of geerality, we assume a commo uit, for example, tos of steel, is used for z ad y i. yi 1 : the optimal quatity of hedgig cotract i whe a sigle-cotract hedge is used. yi : the optimal quatity of hedgig cotract i whe a multicotract hedge is used. At the start of period, = 1 N, the firm, upo observig s, optimizes the ivetory level, z, ad the quatity of hedge i, y i, i = f, c, p, by maximizig the MV utility of the et preset value of the profit-togo or the total cash flows eared i periods N. At the ed of the horizo, we assume that the firm receives o log-term supply. Havig o demad to fill, the firm should simply sell all the excess ivetory to the spot market. Please ote that aalytical tractability of such a complex multiperiod problem is maily eabled by (1) the firm s ability to sell back to the spot market with o trasactio cost ad zero bid ask spread, ad (2) the firm s choice of the commoly used MV utility fuctio. 4. Optimal Policy for Sigle-Cotract Fiacial Hedgig We first study the ivetory ad hedgig policies for the firm whe it chooses to employ a sigle-cotract hedge every period, a hedge cotaiig a sigle fiacial cotract, such as the futures cotract or a call or a put optio. The same type of cotract is used across all periods. The aalysis for this case with siglecotract hedges is importat because it reflects the curret practice i which the futures cotract is the most popular sigle cotract used for hedgig The Firm s Utility Fuctio with Sigle-Cotract Fiacial Hedgig At the start of period, observig the curret spot price, s, the firm eeds to make the ivetory decisio, z 0, ad the hedgig decisio, y i, i = f, c, p. It the processes the commodity i make-to-order fashio to meet the ucertai demad D if profitable. Recall that r represets the uit reveue excludig the processig cost. Thus the firm s gross margi is r s, based o which the firm decides whether to produce to fill its customer demad. Note that the moey paid to the log-term supplier each period is a suk cost. Let i z s y i 1 deote the firm s (radom) profit fuctio i period (if the firm uses hedge

6 512 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS i, i = f, c, p every period) with commodity ivetory level z 0, give ay observed spot price s ad quatity of hedge i traded i the previous period y i 1. Although z ad y i are determied simultaeously, they have oe-period differece for their effective times, due to the oe-period time lag betwee the trasactio ad the exercise of the hedgig cotracts. Whe the gross margi r s is positive, the firm should process to best meet the demad, that is, the productio level should be z D. Because of the assumptio of zero bid ask spread ad trasactio cost for the spot market, we ca rewrite the model as if all the excess commodity at the ed of each period were sold to the spot market; the discouted payoff of this sale ca be couted as part of the profit eared i the curret period. As a result, the firm will always start the ext period with zero o-had ivetory: i z s y i 1 = y i 1 i s + s w q + r s z D + S +1 s h z D + (1) Whe the gross margi r s is zero or egative, sice productio is ot profitable, the firm should ot produce, but may make speculative purchases of the commodity oly to improve its utility. I this case, the firm s profit fuctio becomes i z s y i 1 = y i 1 i s + s w q + S +1 s h z (2) At the ed of the horizo, facig o customer demad, the firm simply sells all the excess commodity to the spot market (i.e., z N +1 0). Thus, regardless of the sig of r s, we have i N +1 z N +1 0 s N +1 y i N = y i N i s N +1 (3) Usig i as a short otatio for the profit eared i period, we formally defie the firm s MV utility fuctio for period, = 1 N + 1, by U z y i s y i 1 [ N +1 ] = E k i k s 2 [ N +1 ] V k i k s (4) k= k= where +1 i = i +1 Zi +1 S +1 y i ad i k = i k Zi k S k Y 1 i k 1, k + 2. Note that Zi k = zi k S k ad Y 1 i k = y 1 i k S k are radom variables represetig the optimal decisios for period k, k + 1 (radomess comig from S k oly, ot from S +1 S k 1, D D k 1, due to the assumptios of Markovia price process ad zero bid ask spread ad trasactio cost for the spot market). At the ed of the horizo, facig o risks, the firm should ot cosider hedgig, that is, yi 1 N +1 = 0, ad thus, usig (3), the firm s utility fuctio ca be simplified to U N +1 z i N +1 = 0 y1 i N +1 = 0 s N +1 y i = y i i s N +1 (5) 4.2. Optimal Policy for Each Period We characterize optimal ivetory ad hedgig policies for period i this sectio usig the iterative method. Usig the assumptio of zero bid ask spread for the spot market ad the MV utility form, we are able to first prove that the utility fuctio U z y i s y i 1 is cocave i y i ad thus characterize the optimal solutio yi 1, for ay give z. We the prove that U z yi 1 s y i 1, as a fuctio of z, displays a property that is weaker tha cocavity but guaratees a sigle optimum, z i (the solutio to d/dz U z yi 1 s y i 1 = 0). For readers coveiece, we preset the optimal policies by first simplifyig some otatio. First, we deote the future profit for period, < N, for the case of usig hedge i, i = f, c, p, as the sigle-cotract hedge by i = i +1 Zi +1 S +1 y i y 1 i i S +1 + N +1 k=+2 k 1 i k Zi k S k Y 1 i k 1 (6) Note that i is radom, but i N 0. Secod, the firstorder coditio (FOC) d/dz U z yi 1 s y i 1 = c u z s F z s c o z s F z s = 0, where F = 1 F, displays similarities to that from which the stadard ewsvedor solutio is derived. Thus, we defie c o z s ad c u z s as the overage cost ad the uderage cost, respectively: c o z s = c o1 s + c o2 z s where for i = f, c, p, c u z s = c u1 s + c u2 z s ad c o1 s = h + 2 Cov S +1 i s c h1 s c o2 z s = 2 V S +1 s E z D + s c h2 z s c u1 s = r s c u2 z s = r + h s V S +1 s E z D + s i which the hedgig-related terms i the overage costs are c h1 s = Cov S +1 2 i S +1 s Cov i S +1 i s V i S +1 s c h2 z s = 2 Cov2 S +1 i S +1 s V i S +1 s E z D + s (7) (8) (9)

7 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS 513 I cotrast to the stadard ewsvedor solutio, the overage cost (the uderage cost) i our problem cosists of two parts, a costat cost, c o1 s (c u1 s ), ad a variable cost, c o2 z s (c u2 z s ). Ideed, c o2 z s icreases i z (except whe i = f, c o2 z s = 0), whereas c u2 z s decreases i z. More iterpretatio of these costs is provided later i this sectio. Last, let z u s ad z o s deote the uique oegative values of z satisfyig c u z s =0 whe r s >0 ad c o z s =0 whe c o1 s 0, respectively. We show that a base-stock policy is optimal for every period. Propositio 4.1. Give spot price s, N, for i = f, c, p, the optimal quatities are as follows: Hedgig: y 1 i = E zi D + s 1 r >s +z i 1 r s Cov S +1 i S +1 s V i S +1 s Cov i S +1 i s V i S +1 s y 1 i N = E zi N D N + s N 1 rn >s N + z i N 1 r N s N Cov S N +1 i S N +1 s N V i S N +1 s N y 1 f N y1 j N j = c p ad Ivetory: Whe r s > 0, we have the followig cases: If c o1 s > 0, z i 0 zu s is the uique solutio to the FOC, that is, c o zi s F z i s = c u zi s F z i s. If c o1 s 0, the if z u s > z o s, z i zo zu s is the uique solutio to the FOC; if z u s z o s, z i zu s z o s satisfies the FOC or z i = +. Whe r s 0, the firm is speculative, ad z i = 0 c o1 s ( ) 2 V S +1 s Cov2 S +1 i S +1 s V i S +1 s We start by iterpretig the hedgig result. Fiacial hedgig cotributes othig to the mea profit because of the Q-measure assumptio. For the last period, the hedgeable profit risk is reflected by the profit term S N +1 z N D N + (part of the curret period profit eared from sellig the excess commodity of quatity z N D N + ). Thus, the firm should hold a short positio, that is, should sell the futures cotract or a call or buy a put, of quatity y 1 i N = E z i N D N + s N Cov S N +1 i S N +1 s N V i S N +1 s N if r N > s N This meas that for each uit of the excess commodity, the absolute hedgig amout is Cov S N +1 i S N +1 s N V i S N +1 s N 1 = 1 whe i = f Note that i 6 we show that the futures cotract is ideed the best sigle-cotract hedge for the last period. For ay other period, however, the firm hedges also the risk i the future profit i. Thus the optimal hedgig quatity yi 1 cosists of two parts, used to hedge S +1 z D + i the curret period profit ad the future profit i, respectively. Sice i is idepedet of z, the hedgig decisio oly iteracts with the ivetory decisio z for the same period through the term S +1 z D +. As discussed previously, our model ca be treated as the firm startig each period with zero o-had ivetory. Thus the curret period ivetory decisio has o impact o the future decisios. Coversely, if the firm s utility fuctio icludes the mea profit oly, the future decisios should ot ifluece the curret ivetory decisio. This, however, o loger holds whe the profit variace is also icluded i the utility fuctio. Note that the curret ivetory decisio ad the future decisios are liked or iteracted via the term S +1 z D +. Ideed, their depedece is captured by the covariace betwee the curret period profit ad the future profit 2 Cov S +1 z D + i s. Thus, i geeral, the optimal policy is ot myopic ad requires the kowledge of optimal decisios for all future periods. It is importat to ote that sice the futures cotract perfectly hedges the covariace, the correspodig optimal ivetory policy is myopic. More discussio of this myopic ivetory policy is provided i 5 for the multicotract hedge case. Note that because of the use of the multiperiod MV utility ad icorporatio of fiacial hedgig, our ivetory solutio is more complex tha the stadard ewsvedor solutio. It, however, reveals similar (but ew) isights o the trade-off betwee the overage cost, c o z s, ad the uderage cost, c u z s, defied by (7) (9). Note that the uderage cost is the profit loss, r s, less the utility gai or profit variace reductio due to uderage ivetory (less ivetory leads to a smaller profit variace). The overage cost equals the holdig cost, h, plus two types of utility loss, where oe is ivetory idepedet (showig the effect of overage ivetory ad hedgig o the mea ad covariace terms ivolvig z ) ad the other is ivetory depedet (showig the effect of overage ivetory ad hedgig o the variace terms ivolvig z ). Note that fiacial hedgig affects the overage cost oly as it is effective oly whe excess commodity exists ad is sold to the

8 514 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS spot market. Ideed, for the last period, ay fiacial hedge reduces the overage cost, resultig i a ivetory icrease (as show i Propositio 7.2). For ay other period, because the future profit also requires hedgig, fiacial hedge may ot reduce the overage cost itself, but it does reduce the icreasig rate of the overage cost, / z c o z s. Fially, whe the firm is risk eutral ( = 0), the ivetory solutio reduces to a stadard ewsvedor solutio (c o z s = h ad c u z s = r s ). The uderstadig of the overage ad uderage costs helps us ucover ad classify the firm s motives (productio ad/or speculatio) behid its actios. Specifically, these motives are explaied by its gross margi, r s, ad the costat part i the overage cost, c o1 s. The firm has pure productio motives whe r s > 0 ad c o1 s > 0, productio ad speculatio motives whe r s > 0 ad c o1 s 0, ad pure speculatio motives whe r s 0 ad c o1 s 0. Oly with a positive gross margi would the firm fid it profitable to produce. If beeficial, the firm may also speculate ad stock more tha eeded for productio. If the gross margi is, however, zero or egative, the firm makes speculative purchases oly whe it improves its utility (i.e., whe c o1 s is egative). I this case, fiacial hedges also serve as speculative tools. 5. Optimal Policy for Multicotract Fiacial Hedgig We ow cosider the case i which the firm adopts all fiacial cotracts available i the market, referred to as the multicotract hedge case. It is importat to ote that sice the multicotract hedge always icludes the futures cotract (always available i the market), the sigle-cotract case is ot a special case of the multicotract case. Note that ay fiacial cotract with twice differetiable payoff fuctios ca be replicated by bods, futures cotracts, ad call ad put optios (see details i Oum et al. 2006, 3.2; Carr ad Mada 2001). Thus a theoretical optimal multicotract hedge should iclude futures cotracts ad call ad put optios with a cotiuum of strike prices; a practical optimal multicotract hedge should ivolve these cotracts available i the market. I this sectio, we focus o the practical optimal multicotract hedge. We first defie some additioal parameters ad decisio variables eeded for this sectio. Let K c i ad c i S +1, i = 1 c, ad K p j ad p j S +1, j = 1 p, deote the strike prices ad payoff fuctios for the call ad put optios available at the start of period, respectively. Without loss of geerality, we assume these fiacial cotracts are ot replicatig each other. Let y = y f y c 1 y c c y p 1 y p p deote the array of the correspodig hedgig quatities. Let z s y 1,, ad U z y s y 1 deote the firm s curret period profit, future profit, ad MV utility fuctio for period, respectively. They are similar to the correspodig fuctios i the sigle-cotract hedge case, except that the hedgig payoff is ow the sum of the payoffs from each hedge; that is, y i 1 i S is replaced by y f 1 f S + c j=1 y c 1 j c j S + p j=1 y p 1 j p j S. Our aalysis shows that the derivatives of the utility fuctio with respect to y are U z y s y 1 y ad 2 U z y s y 1 y 2 = 2 s y z s = 2 s Note that s is the coditioal (o s ) covariace matrix for S +1 S +1 K c 1 + S +1 K c c + S +1 K p 1 S +1 K p p ; it is ivertible as these correspodig fiacial cotracts are ot replicatig each other. Also, z s = f c 1 c c p 1 p p, where, for example, c i = E z D + s Cov c i S +1 S +1 s Cov c i S +1 s. It is ot difficult to see that the Hessia matrix of the utility fuctio, 2 s, is egative semidefiite, ad thus the utility fuctio is cocave i y. This evetually leads to the optimality of the base-stock policy, as explaied i the sigle-cotract hedge case. Propositio 5.1. Give spot price s, N, the optimal quatities are as follows: Hedgig: y = s 1 z s ad yn = E z N D N + s N 0 0. Ivetory: z zf ad the overage cost c o z s = h > 0. Whe r s 0, z = 0; whe r s > 0, z is the uique solutio to h F z s = r s r +h s V S +1 s E z D + s F z s (10) This propositio implies that for the last period, the optimal multicotract hedge cotais futures oly. For ay other period, however, it also cotais call optios with strike prices lower tha the futures price ad put optios with strike prices higher tha the futures price. Like the sigle-cotract hedge case, the optimal quatity of each fiacial cotract i the hedge also cosists of two parts, which are used to hedge the curret period profit ad the future profit, respectively. It is importat to ote that the optimal ivetory level, z, is myopic ad idetical to zf (the optimal ivetory level whe usig futures aloe). As discussed i 4.2, as log as the futures cotract is utilized, aloe or ot, the covariace betwee

9 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS 515 the curret period profit ad the future profit is perfectly hedged, ad thus the ivetory policy is myopic. As a result, fiacial hedgig helps completely elimiate both types of utility loss i the overage cost; thus we have c o z s = h > 0. Sice the overage cost is always positive, the firm has o speculative motives it procures for productio oly whe the gross margi is positive. Moreover, the property z = zf is because the covariaces betwee the hedgig payoff ad the hedgeable curret period profit, S +1 z D +, for the correspodig two cases are equal, that is, y f V S +1 s + c i=1 y c i Cov c i S +1 S +1 s + p j=1 y p j Cov p j S +1 S +1 s = y 1 f V S +1 s. However, sice the hedgig payoff also affects other parts of the utility fuctio, usig the futures cotract aloe is ot optimal for ay period besides the last. Last, we cosider a special case: the ifiite-horizo case with the futures cotract icluded i the multicotract hedge. Our aalysis for the fiite-horizo case idicates that the above results of the optimal ivetory (myopic) ad hedgig policy are ideed derivable without requirig ay specific property of the utility fuctios for the future periods. This observatio implies that the optimal policy for the fiite-horizo case is also optimal for the ifiite-horizo case. 6. How to Select a Sigle-Cotract Hedge We ow compare ad rak hedges, sigle-cotract or multicotract, based o their cotributio to the utility, with emphasis o sigle-cotract hedges. Let ad S +1 deote ay fiacial hedge, siglecotract or multicotract, traded i period ad its payoff fuctio, respectively. Let f c Kc, ad p Kp deote the futures cotract, the call optio with strike price K c, ad the put optio with strike price K p, respectively. Let U z y 1 s y 1, similar to U z y f s y f 1, deote the firm s utility fuctio, give that hedge 1 is traded i period ad optimal hedges (which may or may ot iclude the same type(s) of cotract as 1, ulike the siglecotract hedge case) are applied for all future periods. Formally we defie the orderig of hedges based o their cotributio to the firm s utility. For ay two hedges 1 ad 2, we say 1 2 (i.e., 1 is ot worse tha 2 ) if U z 1 y 1 s y 1 U z 2 y 2 s y 1. It is importat to ote that such a defiitio o orderig, assumig the use of same optimal hedges for all future periods, promotes a fair compariso for the hedges. We say 1 2 (i.e., 1 is better tha 2 ) if U z 1 y 1 s y 1 > U z 2 y 2 s y 1 ad 1 2 (i.e., 1 ad 2 are equivalet) if U z 1 y 1 s y 1 = U z 2 y 2 s y 1 We ow compare sigle-cotract hedges that iclude the futures, call, ad put optios. Ulike the futures cotract, there are may tradable call ad put optios with a commo strike time but differet strike prices. Applyig the defiitio of orderig, we ca characterize a optimal call (with the best strike price Kc ) or a optimal put (with the best strike price Kp ). The followig compariso results are maily based o the observatio that the use of ay hedge i period reduces the profit variace by Cov 2 S +1 S +1 z D + + s /V S +1 s, where represets the correspodig future profit ad is defied similarly to i i (6). Propositio 6.1. For period, we have the followig: (1) f c 0 p ; 1 2 if ad oly if (iff ) Cov 2 S N +1 1 S N +1 s N V 1 S N +1 s N for = N ad iff Cov2 S N +1 2 S N +1 s N V 2 S N +1 s N Cov 2 1 S +1 S +1 z 1 D + + s V 1 S +1 s Cov2 2 S +1 S +1 z 2 D + + s V 2 S +1 s for < N. (2) For = N, f, ; c Kc1 c Kc2 iff K c1 < K c2 ad thus K c N = 0; p K p1 p Kp2 iff K p1 > K p2 ad thus K p N = + ; ad 1 2 iff z N 1 z N 2 whe r N > s N ad c o1 N S N > 0. For the last period, this propositio implies that the optimal hedge is the futures cotract or its equivalet cotracts, such as the call with the lowest strike price ad the put with the highest strike price (referred to as the deep-i-the-moey optios). Ituitively, the cash flow risk exists i the curret period profit oly ad is reflected by the term S N +1 z N D N +. Sice D N is idepedet of S N +1 ad thus caot be hedged at all, perfect hedges for the cash flow risk do ot exist. But a optimal hedge exists ad should result i the maximum reductio of the profit variace, where the reductio is expressed by Cov 2 S N +1 S N +1 s E 2 z N D N + s V S N +1 s Sice the maximum reductio occurs whe the hedge s payoff fuctio is perfectly correlated with S N +1, the futures cotract ad its equivalet optios are optimal. The equivalece betwee the futures ad the deep-i-the-moey optios remais for ay other period. Moreover, we prove for the last period a mootoic relatioship: the lower (higher) the strike price,

10 516 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS the better the call (put) optio. Furthermore, we show that the use of a better hedge leads to a higher ivetory level ad a bigger profit variace reductio. These results, however, do ot apply to ay other period. This is because the profit variace reductio may ot be mootoic to the ivetory level or the strike price if a call or put is used as a sigle-cotract hedge. Thus, a better hedge, though improvig the utility, may or may ot lead to a higher ivetory level. Our umerical study shows that i some cases the futures cotract is ot the best sigle-cotract hedge, but results i the highest ivetory level. 7. Role of the Operatioal ad Fiacial Hedges Sice the physical ivetory, well kow for effectively dealig with the supply-demad mismatch risk, also cotributes to hedgig the firm s cash flow i our model, we refer to it as the operatioal hedge. I this sectio, we study the role ad iteractio of the operatioal ad fiacial hedges i dealig with various risks. I our model, ivetory comes from two sources: the log-term supply (with fixed quatities ad prices) ad the spot market procuremet (buyig or sellig). By ature, the log-term supply with locked prices protects the firm agaist the commodity price risk, whereas the spot market procuremet protects the firm agaist the commodity cosumptio risk (or the ed-product demad risk). Sice the moey paid for the log-term supply is a suk cost, it ca be removed from the MV utility without affectig the optimizatio. Thus the real effective operatioal hedge is the ivetory procured from the spot market. Differet fiacial hedges we study iclude sigle-cotract ad multicotract hedges, where the cotracts are futures, call, ad put optios. It is well kow that they hedge the commodity price risk directly. I our paper, however, they also hedge the demad risk idirectly via their ifluece o ivetory levels (which is made through hedgig quatities ad correlatios betwee demads ad spot prices). We ext examie, aalytically ad umerically, the impact of differet fiacial hedges o the effectiveess of the operatioal hedge. The impact is measured by the ivetory level chage: more effective operatioal hedges lead to higher ivetory levels. We the study how the operatioal hedge ad differet fiacial hedges affect the firm s fiacial performace, characterized by the mea, variace, ad utility of the cash flows. Specifically, we compare some differet practical scearios, which iclude SS (sigle sourcig from log-term supply), DS (dual sourcig from logterm supply ad spot market), DS + i (dual sourcig with hedge i), i = f (futures), c (call), p (put), ad DS + HP (dual sourcig with a hedgig portfolio) Numerical Study Setup To supplemet the aalytical results provided i this paper, we performed a two-period umerical study with idetical periods (ad thus, i this sectio, we remove the subscript from the relevat cost parameters) ad correlated spot prices ad demads. Both spot prices ad demads follow Logormal distributio or GBM whe treated as cotiuous time stochastic processes. Specifically, let S +1 = S / e sb +1 s 2/2 ad D = e dw d 2/2, where B s ad W s are i.i.d. stadard Normal radom variables, ad B W is bivariate Normal with correlatio 1 1 (used to represet the correlatio betwee S ad D ), = 1 2. To choose reasoable values for parameters ad s, we fit the spot price distributio to the U.S. hot-rolled coil steel spot prices from Jauary 2009 to Jue The fittig idicates that the steel spot price is approximately GBM with mothly discout factor = ad mothly drift parameter s = We set = to reflect the Q-measure, that is, to guaratee E S +1 S = S. I particular, our umerical study parameters are as follows: = ; w = 3; q = ; s 1 = 3; r = ; h = 0 6; s = d = 0 114; = ; = 100; ad = , 0.01, 0.02, 0.06, 0.1. Note that we set the risk aversio,, appropriately small eough to avoid a pathological behavior that may occur whe maximizig the MV utility: reducig ivetory whe seeig a higher uit reveue (which magifies the profit variace s egative effect o utility). Aalytically, we ca show that z icreases i r if 1 2 r + h s E [( F 1 ( r s r+h s ) D ) + ] Applyig this to our umerical study setup, we should set below or at 0 1 to prevet the pathological behavior Impact o the Service Level Let z 0 deote the firm s optimal ivetory level for period for case DS, where the superscript 0 stads for o hedgig; it satisfies the FOC with the overage ad uderage costs modified by omittig the hedgig-related terms i (8). Below we show the impact of fiacial hedgig ad risk aversio ( ) o the optimal ivetory (service) level. For its sigificace, we preset a more detailed sesitivity aalysis result for z = zf. Propositio 7.1. For period, N, we obtai the followig: (1) For risk-eutral firms, that is, whe = 0 ( ) z i = F 1 r s = z i = f c p r + h s

11 Maufacturig & Service Operatios Maagemet 15(3), pp , 2013 INFORMS 517 (2) Sesitivity aalysis results are as follows: z = zf is decreasig i, h, ad V S +1 s ; z i, i = c p, is decreasig i if Cov S +1 i s Cov S +1 i S +1 s Cov i S +1 i s V i S +1 s z 0 is decreasig i if Cov S +1 s 0 (automatically held for = N ). (3) Service level compariso is z N z N zc N, zp N > z0 N. Ituitively, a more risk-averse firm usually keeps a lower ivetory level despite the optio of fiacial hedgig. A risk-eutral firm has the highest ivetory level, z. Our umerical study idicates that, with or without fiacial hedgig, risk-averse firms with risk aversio carry as much ivetory as riskeutral firms (see Figures 1 3). Whe futures cotracts are used, aloe or ot, as holdig ivetory becomes more expesive, a risk-averse firm lowers its ivetory level to icrease its mea profit ad at the same time decrease its profit variace. Facig a higher spot price volatility, a risk-averse firm keeps a lower ivetory level to reduce its profit variace. For the last period, our results are cosistet with the results of the curret literature for sigle-period problems (Dig et al. 2007, Gaur ad Seshadri 2005). I particular, risk-averse firms stock less tha riskeutral firms; the use of fiacial hedgig helps raise risk-averse firms ivetory levels. I additio, we show that a better hedge (e.g., a call with a lower strike price or a put with a higher strike price) leads to a higher ivetory level because of its better cotrol of the profit variace. Figure Optimal Ivetory Level = 0 5 Agaist /2 = Hedgig lower ivetory SS DS DS + F DS + C DS + P DS + HP Figure Optimal Ivetory Level = 0 Agaist /2 = Risk averse buyer uses risk-eutral iv 5.4% lower tha risk-eutral iv Hedgig lower iv SS DS DS + F DS + C DS + P DS + HP For ay other period, however, our results differ from the results of the curret literature for multiperiod problems (Smith ad Nau 1995, Che et al. 2007, Zhu ad Kapusciski 2011). As metioed i 2, all of these works use a itraperiod utility fuctio, which igores the cash flow correlatios across periods. They show that operatio ad hedgig decisios are separable ad the optimal fiacial hedgig is myopic (meaig that the optimal hedge is the futures whe applied to the cotext of our problem). I cotrast, usig a iterperiod MV utility fuctio, we prove that the optimal fiacial hedgig is ot myopic ad also icludes various call ad put optios. I additio, we observe that the Figure Optimal Ivetory Level = 0 5 Agaist /2 = Hedgig raises ivetory SS DS DS + F DS + C DS + P DS + HP