Financial Hedging and Optimal Procurement Policies Under Correlated Price and Demand

Financial Hedging and Optimal Procurement Policies under Correlated Price and Demand

Ankur Goel Analytics and Portfolio Management, PNC Financial Services, Pittsburgh, Pennsylvania 15222, USA, [email protected]

Fehmi Tanrisever Faculty of Business Administration, Bilkent University, Bilkent, 06800 Ankara, Turkey, [email protected]

e consider a firm that procures an input commodity to produce an output commodity to sell to the end retailer. W The retailer’s demand for the output commodity is negatively correlated with the price of the output commodity. The firm can sell the output commodity to the retailer through a spot, forward or an index-based contract. Input and output commodity prices are also correlated and follow a joint stochastic price process. The firm maximizes shareholder value by jointly determining optimal procurement and hedging policies. We show that partial hedging dominates both perfect hedging and no-hedging when input price, output price, and demand are correlated. We characterize the optimal financial hedging and procurement policies as a function of the term structure of the commodity prices, the correlation between the input and output prices, and the firm’s operating characteristics. In addition, our analysis illustrates that hedging is most beneficial when output price volatility is high and input price volatility is low. Our model is tested on futures price data for corn and ethanol from the Chicago Mercantile Exchange. Key words: integrated risk management; financial hedging; inventory management; yield uncertainty; myopic optima History: Received: January 2014; Accepted: March 2017 by Panos Kouvelis, after 3 revisions.

1. Introduction processor in a multi-period model when commodity demand and price are correlated. We characterize the Commodity price fluctuations create challenges for optimal financial hedging and inventory policies as a commodity processors in determining production, function of the term structure of the commodity procurement, and risk-mitigation strategies. The prices, the correlation between the input and output impact of price risk is more profound on processors prices, and the firm’s operating characteristics. We that have limited market power to pass a price show that partial hedging dominates perfect hedging2 increase in raw material to end customers. Many for a firm when input and output commodity prices firms, including agricultural processors, steel manu- are positively correlated. facturers, and energy producers, are susceptible to It is well established that in the absence of market the commodity price risk because both their input frictions, corporate-level risk management is a value- and output prices are market determined. The effect neutral proposition, and operating and financial of commodity price fluctuations is twofold; it creates hedging decisions are separable (Modigliani and uncertainty in the margins, and affects demand if Miller 1958). Financial theory explains the use of price and demand of the output product are corre- financial derivatives through capital market imperfec- lated. In this regard, financial hedging1 can play a piv- tions (e.g., transaction costs, information asymme- otal role in mitigating price risk and maximizing firm tries, and taxes) and agency problems (Froot et al. value. In addition to price risk, commodity processors 1993, Jin and Jorion 2006, Smith and Stulz 1985). In also face demand risk, and optimize their operating this study, we consider a publicly traded commodity policies to manage this risk. Since price and demand processor that operates to maximize shareholder for various commodities are correlated, this results in value, and experiences a correlated demand with the an interaction between hedging and operating poli- price of its output commodity due to logistical fric- cies. In this study, we jointly optimize the financial tions. These frictions result in the breakdown of the hedging and operating decisions of a commodity Modigliani and Miller (1958) framework and requires the joint optimization of hedging and operating poli- *This research was conducted when the author was a cies. A negative correlation between demand and faculty at the Case Western Reserve University. price results in the concavity of the objective function, 1924 Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1925 which creates an incentive to reduce price volatility, inventory into the subsequent period. In this study, and hence, a need to develop and integrate an optimal we model the negative correlation between ethanol hedging policy with inventory decisions. price and demand, as well as the inventory dynamics This research is motivated by the operations of an in a multi-period framework. ethanol producer that procures corn in the spot mar- Over the past few years, the effect of fossil-fuel sup- ket to produce ethanol to sell to the local retailer. The ply chains on the environment has become a central retailer’s demand for ethanol is negatively correlated issue in determining environmental policy across with the price of ethanol. Our firm, the ethanol pro- many countries. Environmental concerns over methyl ducer, contracts with the retailer to sell ethanol using tertiary butyl ether (MTBE), a substance blended with spot, forward or index-based contracts. The firm is a gasoline to raise the octane number, resulted in the price taker, where corn and ethanol prices are corre- US Environmental Protection Agency banning the lated and follow a joint stochastic process. In each substance in 2006. Ethanol now replaces MTBE as a period, the firm determines the procurement and pro- means to improve the octane performance of gasoline. cessing policies for corn and the hedging policy for This change had a substantial impact on the eco- ethanol sales. Excess inventory is carried over into the nomics of growing corn: ethanol production now next period, and excess demand is backlogged. We represents the highest use of corn in the US, followed show that the expected base stock policy is optimal by its use for feed. As ethanol production consumes under yield uncertainty, and it is characterized as a the largest portion of the corn supply, the prices of function of the term structure of the futures prices. ethanol and corn have begun to affect each other. As a When there is no yield uncertainty, we establish the result, we model corn and ethanol prices with a conditions for the optimality of the myopic policy. mean-reverting correlated stochastic process. Further- Renewable energy is a strategic issue in the United more, the conversion process of corn into ethanol is States (US) and in many other countries around the subject to yield uncertainty, and therefore, we also globe. In this regard, ethanol is considered a partial incorporate yield uncertainty when determining opti- substitute for gasoline, reducing reliance on fossil mal procurement and hedging policies. fuels. In 2016, according to the US Department of Another important issue in the supply chain of Energy, the US is expected to process 5.28 billion ethanol is its distribution cost. Ethanol has an affinity bushels of corn, generating a record 14.54 billion gal- for water, rendering it unsuitable for transporting lons of ethanol. Ethanol producers buy and process through pipelines. At present, the only possible corn to produce and sell ethanol to downstream modes of transportation for ethanol are trucks and retailers (jobbers).3 Ethanol is mixed with gasoline by trains, which result in transportation costs almost 10 the jobber in accordance with environmental regula- times higher for ethanol compared to gasoline (Wake- tions and the economics of the process. The retailer ley et al. 2009). This cost factor limits the economic (the ethanol user) optimizes its gasoline blend based feasibility of transporting ethanol over long distances. on ethanol prices and the chemical properties of the According to Wakeley et al. (2009), “Long-distance gasoline when it is mixed with ethanol. Maintaining transport of ethanol to the end user can negate ethanol’s state regulations, the retailer blends more ethanol as potential economic and environmental benefits relative to ethanol prices decrease and reduces the ethanol con- gasoline.” Therefore, the ethanol producer in our tent of its product as prices increase. This situation model prefers to carry excess inventory into subse- leads to a negative correlation between the retailer’s quent periods rather than shipping this inventory to demand and the commodity price.4 Due to the corre- an end user outside the local market. These circum- lation of demand and price, under high price realiza- stances entail that the price elasticity of ethanol at the tion, the ethanol producer observes a low demand for retailer’s end is transferred to the ethanol producer its output commodity and may not be able to sell all due to the inability to sell excess inventory outside of its inventory to local retailers, and also may not the local market because of high transportation costs. clear the remaining inventory in the exchange market The ethanol processor procures corn in the spot due to logistical frictions. This mismatch in produc- market and produces ethanol to sell in the local mar- tion and demand of ethanol results in firms carrying ket. The price of ethanol in the local market is excess inventory into subsequent periods. According perfectly correlated with the price of ethanol on the to the US Energy Information Administration, there Chicago Mercantile Exchange (CME). Ethanol pro- were about 20.9 million barrels of ethanol inventory ducers sell ethanol to jobbers through a variety of con- on 8/26/2016. This situation is consistent with a clas- tracts, including spot, forward, and index-based sic paper on the behavior of commodity prices by agreements (Dahlgran 2010, Franken and Parcell Deaton and Laroque (1992), according to which, in a 2003). In our model, we propose an index-based con- multi-period setting, the optimal price does not neces- tingent contract whose price and volume are deter- sarily clear the market and a firm carries positive mined as a function of spot and futures prices. The Goel and Tanrisever: Financial Hedging and Procurement 1926 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society sales price of the contract is a weighted function of For example, Secomandi (2010) evaluates the value of the futures price observed today for the contract that storage for natural gas in the presence of operational matures in the next period, and of the spot price to be constraints. Berling and Martinez-de Albeniz (2011) observed in the next period. The expected price of this develop operating policies for commodity processors contract is always equal to the futures price; however, under stochastic price and demand. Goel and Gutier- the weight on the futures contract vs. the spot contract rez (2009, 2011) postulate the significance of dynami- can change the price volatility. Since this index con- cally updating operating policies in the presence of tract reduces price volatility compared to the pure stochastically evolving convenience yield. Devalkar spot contract, we refer to this case as financial hedging. et al. (2011) obtain optimal commodity processing In particular, we show that for an ethanol producer, and storage decisions under capacity constraints. Sim- an index-based contract performs better than either ilar capacity and risk management problems for agri- spot or forward contracts, and we dynamically opti- cultural commodities are addressed by Boyabatli mize the contract in each period. et al. (2016) and Noparumpa et al. (2015). More Our contributions to the literature are as follows: recently, Devalkar et al. (2016) consider commodity (1) We show that partial hedging dominates perfect processing and risk management in partially com- hedging and no-hedging for a publicly traded firm plete markets in the presence of financial distress when input and output prices are positively corre- costs. lated, and demand is negatively correlated with out- Our study is also closely related to Plambeck and put price. (2) We identify three sufficient conditions Taylor (2011, 2013), who consider the dynamics under which a myopic policy is optimal for a price- between input and output prices for a commodity taker firm when output price and demand are nega- processor in the absence of financial hedging. Plam- tively correlated. (3) We characterize the optimal beck and Taylor (2011) explore the value of opera- policy for inventory procurement and the policy for tional flexibility, and show that the value of hedging the sales of the output commodity as a func- feedstock-intensity flexibility decreases with variabil- tion of the term structure of the futures prices, and ity in feedstock cost or output price. Plambeck and show that the expected base stock policy is optimal Taylor (2013) study the trade-off between input effi- under yield uncertainty. (4) Our numerical analysis ciency and capacity efficiency, and conclude that the shows that the value of hedging increases with output flexibility to adjust between these two types of effi- price volatility, but decreases with input price volatil- ciencies decreases with variability in input and output ity. We also observe that yield uncertainty has a non- prices if the expected margin is thin. Our study con- monotone effect on the value of hedging. Our model tributes to the literature by (1) dynamically integrat- is motivated by ethanol processing but it can be ing financial hedging with operating decisions in a applied to many other commodity-processing scenar- multi-period model, (2) considering the stochastic ios, such as steel, wheat, and chemicals. We test our dynamics of both input and output prices, as well as model on the futures price data obtained from the the associated effect of correlation on hedging deci- CME for corn and ethanol contracts traded from 4/1/ sions, and (3) exploring the effect of yield uncertainty 2005 to 12/31/2011. on hedging and operating decisions.

2. Literature Review 2.2. Hedging Under Utility/Profit Maximization In the economics literature, Rolfo (1980) derives an Our study is related to the literature on commodity optimal futures hedging strategy under both price processing, as discussed in section 2.1. In our study, and production uncertainties in a mean-variance we bridge the gap between commodity processing framework. He shows that the optimal hedge ratio is and financial hedging in a multi-period framework. equal to one in the absence of production risk, and is We discuss in detail the literature on financial hedg- less than one in the presence of production risk. A ing in operations and finance in sections 2.2 and 2.3, similar result is later provided under a continuous- respectively. Agricultural commodity processors are time model by Ho (1984), under a constant absolute constantly confronted with yield uncertainty in the risk-aversion (CARA) utility function of consumption. challenge of matching supply and demand. In this In contrast to Rolfo (1980) and Ho (1984), we show context, in section 2.4, we discuss how our research that even if there is no production risk, the optimal relates to the literature on managing yield uncertainty hedge ratio is less than one for a value-maximizing by commodity processors. firm when input and output prices are positively correlated, and demand is negatively correlated with the 2.1. Commodity Processing output price. There is a growing body of research on commodity Recently, financial hedging has received growing processing and trading in the operations literature. attention in the operations management literature. Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1927

In the context of risk-averse decision makers, Gaur in the following respects: (1) We establish our results and Seshadri (2005) address the problem of hedg- in the presence of logistical frictions, and they are ing inventory risk when demand is correlated with driven by the correlation between the input and the price of a financial asset. Similarly, Chen et al. output prices and the output price and demand. (2) (2007) show how to hedge operational risk through Our model incorporates inventory and price dynam- financial instruments in a partially complete mar- ics into the firm’s hedging plan, and clearly delin- ket in a multi-period model. Dong and Liu (2007) eates the role of demand, and holding and shortage derive a bilateral forward contract in a Nash bar- costs. In addition, input and output prices as well as gaining framework, and justify its prevalence due convenience yield are essential ingredients of the to the hedging benefits in spite of the presence of a optimal policy. (3) Our hedge ratio is nonlinear in liquid spot market under a utility-maximization the correlation term due to the inventory and price framework. Again in a risk-averse setting, Ding dynamics in our model. Overall, we provide a et al. (2007) show how a multi-national firm can dynamic hedging policy that can be easily imple- use a real option to partially hedge against demand mented. Similar to the finance literature, in this study, uncertainty, and use financial options on the cur- we explore the value of hedging under a value-maxi- rency exchange rate to hedge against currency risk. mization framework. There is also a stream of research showing the rele- vance of risk management for risk-neutral decision 2.4. Yield Uncertainty makers. For example, Caldentey and Haugh (2009) The economics literature has typically considered the explore the value of flexible supply contracts in impact of yield uncertainty in a single-period setup conjunction with financial hedging, and Turcic cast in a utility-maximization framework. Rolfo et al. (2015) examine the value of hedging input (1980) shows that the optimal hedge ratio is not equal costs in a decentralized supply chain with risk-neu- to one in the presence of yield uncertainty when indi- tral agents. Our study differs from the above vidual preferences are represented either by logarith- papers in the following respects: (1) Our model is mic or quadratic utility functions. Losq (1982) cast in a value-maximization framework, which is extends the results of Rolfo (1980) to a general utility- appropriate for well-diversified, publicly traded maximization framework and shows that when yield firms. (2) We integrate the dynamics of both input and price are independent, the firm should hedge and output commodity prices into the firm’s oper- less than the expected output, provided that the util- ating and hedging decisions. (3) We examine hedg- ity function shows decreasing absolute risk aversion. ing in a multi-period model in conjunction with Moschini and Lapan (1992, 1995) explore the effect of inventory dynamics. Next, we briefly discuss the correlation among yield, price, and basis risks on the finance literature on value maximization and finan- optimal hedge ratio when agents’ preferences are of cial hedging. a CARA type and in a mean-variance framework. Our paper focuses on the effect of yield uncertainty 2.3. Hedging Under Value Maximization on hedging, and is cast in the value framework Since the seminal paper of Modigliani and Miller of finance. We also consider a multi-period model (1958), it is now well known that trading financial that closely captures the inventory dynamics of the derivatives is a value-neutral proposition for a firm problem. under perfect capital markets. Financial theory In the operations literature, the structure of opti- explains the use of financial derivatives, in practice, mal operating policies has been well studied under through capital market imperfections (e.g., transac- yield uncertainty. Henig and Gerchak (1990) show tion costs, information asymmetries, and taxes) and that under yield uncertainty, order-up-to policies are agency problems (Froot et al. 1993, Jin and Jorion not optimal in a periodic-review inventory model. 2006, Smith and Stulz 1985). Financial hedging may More recently, Sobel and Babich (2012) prove the also create value when risk-averse agents that con- optimality of myopic policies with an order-up-to tract with the firm cannot fully diversify their claims structure in a multi-echelon model with an auto- (Bessembinder 1991). In a stylized single-period set- regressive demand, under the assumption that yield ting, Froot et al. (1993) investigate an investment and uncertainty is independent of the lot size. In our hedging problem in the presence of costly external model, we show that under the stochastically pro- funds. They show that positive correlation between portional yield model an expected base-stock policy the availability of investment opportunities and the is optimal. We also contribute to the literature on supply of internal cash flows creates a natural hedge; yield uncertainty in operations management by and hence, the firm underhedges in the financial introducing financial hedging. To the best of our market. Although this finding is similar to our under- knowledge, this aspect has never been explored in hedging result, our analysis and findings are different this literature. Goel and Tanrisever: Financial Hedging and Procurement 1928 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society

o Q delivery at time t + 1, is given by f ; ¼ E ½ptþ1. 3. Commodity Processing with t tþ1 PjPt Q Hedging The superscript denotes that the expectation is taken under the risk-neutral measure. Future cash In this section, we present the mathematical model in flows evaluated under a risk-neutral measure are dis- section 3.1, then we characterize the optimal policy counted at a risk-free rate rf, such that the discount D and determine the value of hedging in section 3.2. We rate b ¼ erf t, where Dt = 1. For notational brevity first assume deterministic yield to understand the we denote the one-period-ahead futures prices simply dynamics between commodity processing and finan- o i by ft and ft for the output and input commodities, cial hedging, and in section 4 we discuss the effect of respectively. yield uncertainty. Assumption 1 holds when the monopoly price in the local market, sm, is larger than the spot price plus 3.1. Mathematical Model k the logistical costs, t, to access the exchange market. We consider a commodity processor that procures an This is a reasonable condition, since it is well estab- input commodity to process and sell it to a retailer lished in the economics literature that monopoly whose demand is negatively correlated with the price prices are higher than competitive market prices of the processed output commodity. We assume that (Bresnahan 1982). The firm of our interest, the ethanol producer, is geographically located at a distance, such ASSUMPTION 1. The firm is a price taker for both input k that it costs t to transport the output commodity to and output commodities. the exchange, where it is traded at price pt. Our firm’s ASSUMPTION 2. Both input and output commodities are customer is located locally, such that transportation traded on an organized exchange that offers no arbitrage costs between the two are negligible. If our firm opti- opportunities. mizes the price in the local market then it can charge the monopoly price sm to its customers. However, ASSUMPTION 3. Without loss of generality, the proces- m ≥ + k since s pt t, it is profitable for the customer to sing of the output commodity has a lead time of one procure the commodity from the exchange if the pro- period. ducer offers the monopoly price. Therefore, the maxi- ASSUMPTION 4. Excess demand is backlogged. mum price our firm can charge is determined by the + k cost of the customer’s outside option, pt t. On the ASSUMPTION 5. The contract design between the firms is other hand, since the revenues are concave, there is credible (see, e.g., Boyabatli et al. 2011, Kouvelis et al. + k no economic reason for charging a price below pt t. 2013). Credibility of the contracts are ensured through As a result, the equilibrium spot price in the local collateral mechanisms similar to forward contracts. + k market is pt t. ASSUMPTION 6. It is cheaper to hold inventory in the It is common for commodity processors to sell an upper echelon. output commodity through spot and forward agreements. Also, in practice, there are index-based price In each period, the commodity processor (1) contracts that are a combination of spot and futures observes the spot prices of the input and output com- o prices. We define Wtðbt; ptþ1Þ¼btf þð1 btÞ modities and the inventory of the output commodity, t ptþ1 þ tþ1 as the index price that the customer agrees and (2) determines how much input commodity to + at time t to pay at time t 1, where bt [0, 1] is the procure and process, as well as determining the hed- hedging decision and pt+1 is the realized price of the ging policy for the sales of the output commodity. output commodity at time t + 1. Simultaneously, at We respectively denote the spot prices of input and time t, the buyer commits to the quantity, ; output commodities as st and pt, where ðst ptÞ Pt, and = c dt+1 At+1 Wt(bt, pt+1), as a function of the index Pt contains all the information related to the state of price Wt(bt, pt+1), where At+1 is the maximum possible the economy, demand and supply dynamics, and demand of the retailer (also called the market size) developing new technologies. The evolution of com- and c is the retailer’s demand elasticity with respect modity prices follows a mean reverting process of the to price (see, e.g., Inderfurth et al. 2014, Kazaz 2004). – Ornstein Uhlenbeck type, as outlined in the finance This contract is agreed upon at time t, but its value is literature (see, e.g., Schwartz 1997, Schwartz and realized at time t + 1, after the realization of the spot Smith 2000). This stochastic price process offers no price of the output commodity. That is, the contract risk-free arbitrage opportunities, such that Pt evolves between the processor and the retailer specifies the under a risk-neutral measure where the futures price price and quantity (Wt(bt, pt+1), dt+1)inacontingent of the input commodity i at time t, for delivery at time manner (see, e.g., Samuelson 1986, Bazerman and Gille- i Q t + 1, is given by f ; ¼ E ½s þ . Similarly, the t tþ1 PjPt t 1 spie 1998, and Biyalogorsky and Gerstner 2004 on con- futures price of the output commodity o at time t, for tingent contracts). In summary, the processor and the Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1929 retailer commit to a menu of price and quantity con- commodity through forward contracts, but since tracts at time t to be delivered/realized at time t + 1. procurement costs are linear in price and there are Hence, there is a one-to-one mapping between the price no financial or logistical frictions at the procurement and the quantity through the menu of contracts. The end, forward procurement of input commodity is retailer is in a binding contract to get the quantity as value-neutral. See Appendix B for a detailed proof agreed upon through the menu of contracts at time t, of this result. and does not have the leeway to adjust the quantity We denote xt as the current inventory of the output after the realization of the price at time t + 1. commodity, which can be used to satisfy customer If the firm decides to completely hedge the price demand. If the current inventory of the output com- = risk, then it chooses bt 1, such that modity is insufficient to satisfy demand, then the ; o Wtðbt ptþ1Þ¼ft þ tþ1. On the other hand, if the firm incurs a backlog cost of r per unit per period; on = firm decides not to hedge, then bt 0, such that the other hand, if the demand is less than the current = + k Wt(bt, pt+1) pt+1 t+1. As a result, the expected inventory, then the firm incurs a holding cost of h o price is always ft þ tþ1, irrespective of the hedging per unit per period. We assume that excess demand policy. The hedging decision does not influence the is backlogged, and the holding cost is small relative expected price of the contract, but controls the vari- to the sales price, such that it is economical for the ≤ ≤ ance of the price. Constraint 0 bt 1 ensures that firm to carry inventory rather than to reduce the the price of the index contract is always positive. We price. One unit of input converts to a units of output. define the revenue function as a function of the index We define h0 as the holding cost per unit of input ~ 0 price, such that Rtðbt; ptþ1Þ¼½Atþ1 cWtðbt; ptþ1Þ commodity, and from Assumption 6 we have h > h /a, ; Wtðbt ptþ1Þ for the realized spot price, pt+1, and the which is in accordance with the multi-echelon hedging policy bt. inventory literature. In each period, the commodity In our model, the retailer, unlike the commodity processor observes the inventory of the output pro- processor, is a relatively small company that wants to duct xt, the spot price for the input commodity st, avoid risk because of capital market frictions. We do and the spot price for the output commodity pt, and not explicitly model these frictions, but costs such as then jointly determines the echelon stock zt and the bankruptcy or financial distress motivate the firm to hedging amount bt of the output commodity. Given avoid risk in the market (cf. Chod et al. 2010). We the echelon stock zt, the actual input commodity assume that these costs are high enough to compen- bought in the spot market to be processed is deter- sate the retailer for the reduction in profits resulting mined by (zt xt)/a. The unit processing cost is from hedging the risk. Therefore, the retailer prefers denoted by ct. to hedge the price risk through an index-based con- The objective of the firm is to maximize shareholder tract with the processing firm. The retailer also has value in the presence of input and output commodity access to the exchange market. However, procure- price risks. As suggested by Seppi (2002), we use the ment from the exchange is subject to a number of dis- risk-neutral measure Q—originating from arbitrage advantages, including basis risk. First of all, there is pricing theory—to discount for the systematic risk in an inherent variability associated with the quality of the cash flows. In section 5, we estimate the parame- the commodity supplied from the exchange. Resolv- ters of the risk-neutral price process using a Kalman ing quality- and delivery-related issues is also harder filter. In our model, the value of the firm is repre- when procuring from a distant exchange compared to sented by Vtðxt; PtÞ, which is defined by the following a local firm. In addition, exchanges trade contracts stochastic dynamic program: with standard delivery dates; for example, the CME ; ; ; ; delivers wheat in March, May, July, September, and Vtðxt PtÞ¼ max Jtðxt zt bt PtÞð1Þ z ;b A ðx Þ December only. If the procurement cycle of the retai- t t t t ler does not match the delivery cycle of the exchange, n Q ~ J ðx ; z ; b ; P Þ¼ bE R ðb ; p Þ this results in basis risk which creates motivation to t t t t t PjPt t t tþ1 procure from the local producer. Due to the above ðst þ ctÞ þ þ disadvantages of buying from the exchange, the retai- ðzt xtÞrðxtÞ hðxtÞ a o ler prefers buying from a local processor. Neverthe- Q þ bE V ðz d ; PÞ less, anticipating these benefits, the processor may PjPt tþ1 t tþ1 charge its customer a premium over the exchange market price. Indeed, this premium is a part of the where, A ðx Þ¼fz x ; 0 b 1g, and V ðx ; P Þ k t t t t t T T T transaction cost t in our model. þ ¼ðpT TÞxT ðpT þ TÞxT . For expositional purposes, we consider that the input commodity is procured from the spot market. The first term in the objective function repre- It is also possible for the firm to procure input sents the expected revenue from commodity sales. Goel and Tanrisever: Financial Hedging and Procurement 1930 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society

The second term includes the cost of procurement (a) if Cov(st+1,pt+1) > 0 and processing. The third and fourth terms are 8 < s þbhabfiþc bc penalty and holding costs, respectively. These t t t tþ1 for t ¼ 1; 2; ...; T 2; W bðrþhÞa tðMtÞ¼ b o b costs are incurred at the end of the period, and : st aft þctþ a T b for t ¼ T 1; arecountedatthebeginningofthesubsequent 2 a T period. The last term is the cost-to-go function ð2Þ under the risk-neutral measurement. According to 8 < ðp ;s Þ=aþu ðz;bÞ the terminal condition, any inventory left over is Cov tþ1 tþ1 t t t ; ;...; ; 1 r2 for t¼1 2 T2 k R 2 o sold at a discount, pT T, and any shortages are bt ¼ M :1 T t o w ; + k 2 þ r2 0 ðft ptþ1Þ tþ1ðpÞdp for t¼T1 bought at a premium, pT T. Notice that, in the 0 above model, all the input commodity purchased ð3Þ is processed and none is stored as inventory for \ [ the purpose of consumption in future periods. if bt 0 then set bt ¼ 0 and if bt 1 then set This is indeed an optimal processing policy bt ¼ 1, and compute (2) to obtain zt . ≤ because due to Assumption 2, marginal conve- (b) if Cov(st+1,pt+1) 0, then bt ¼ 1 5 b i b c o c nience yield is positive, and, as a result, com- (i) if st ft ra then zt ¼ Atþ1 ft tþ1 modity storage with no economic use is and, imprudent (see, e.g., Goel and Gutierrez 2011, b i [ b Williams and Wright 1991). We now characterize (ii) if st ft q ra then zt ¼ xt. the optimal policies. The above theorem outlines the algorithm to com- 3.2. Characterization of Optimal Policies pute the procurement and hedging policies. The The lemmas below first establish the concavity of the strict joint concavity result from Lemma 2 ensures value function in the inventory level, and the joint that Equations (2) and (3) describe a unique solution: concavity of the objective function in the decision zt and bt . The procurement policy zt described by variables. Equation (2) has a newsvendor-like structure. The right-hand side of Equation (2) describes a critical ; LEMMA 1. (CONCAVITY OF VT). Vtðxt PtÞ is concave in ratio and the left-hand side is the probability that the xt for every realization of Pt. output price is less than the threshold Mt, which is a function of the procurement policy zt . The firm \ incurs a backlogging cost if ptþ1 Mt and a holding ; [ PROOF. It can be easily shown that VTðxT PTÞ is cost if ptþ1 Mt. A key observation from Theorem 1 concave in xT. Then, using concavity preservation is that for bt ¼ 1, there are no penalty and backlog- under maximization, we establish that VT1(xT1, ging costs since perfect hedging eliminates demand PT1) is concave in xT1. The final step of the proof uncertainty. As a result, no safety stock is required, involves an induction argument to show that but this policy is not necessarily optimal. Neverthe- h Vt(xt, Pt) is concave. less, when the firm decides to partially hedge, \ bt 1, it has to then estimate the expected backlogging and holding costs, which are a function of both ; ; ; LEMMA 2. (JOINT CONCAVITY OF JT). Jtðxt zt bt PtÞ is the hedging policy bt , and the procurement policy ; strictly jointly concave in zt and bt. zt . Function utðzt bt Þ calculates the expected value of the backlog plus holding costs, and it is instru- These results are critical in solving the dynamic mental in determining the hedging policy, as program because the simultaneous solution of described in Equation (3). Therefore, hedging and first-order conditions with respect to bt and zt will procurement decisions are jointly determined as a ensure a global maximum. We respectively define function of market information on prices and firm- w Ψ t(p) and t(p) as the probability and cumulative specific parameters. Moreover, these decisions are density functions of the output commodity price dynamically updated in response to changing input at time t. The following theorem characterizes the and output prices, as observed on the organized optimal policy and shows that the myopic policy commodity exchanges. There are three main drivers is optimal. All proofs appear in Appendix A, in determining the optimal hedging policy. First, cor- unless otherwise indicated. relation between the input and output prices provides a natural hedge, reducing reliance on the THEOREM 1. The myopic policy is optimal, such that financial hedge. Second, expected overstocking and optimal bt and zt are obtained by solving the following understocking costs due to demand uncertainty dri- equations: ven by stochastic prices provide an impetus for Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1931 hedging. Finally, the concavity of revenues in the prices, along with the presence of inventory across output price is a motivation for financial hedging, as time periods, motivates the firm to underhedge and outlined in the following lemma. results in hedging having less value. Buying and selling forward contracts for the input commodity do not o Q affect this result as these transactions do not affect the LEMMA 3. (JENSEN’S INEQUALITY). Rtðft Þ EPjP ½Rtðptþ1Þ. t inherent correlation between the prices in the market. According to Lemma 3, the firm’s revenue function In the finance literature, Froot et al. (1993), in a styl- decreases in the presence of price volatility. If the firm ized single-period setting, argue that correlation decides to sell to the end retailer through a forward between investment opportunities and cash flows of a contract then it eliminates the variance in the revenue firm results in underhedging. In a different context, function, enhancing the expected revenue from for- our analysis yields a similar result for underhedging. ward sales. In other words, since the revenue function Our results are established in the presence of logisti- is concave in the realized price of the output com- cal frictions, and they are driven by the correlation modity, the revenue decreases nonlinearly for high- between the input price, output price, and demand. and low-price scenarios. Selling a forward contract to In addition, our model incorporates inventory and the retailer eliminates the risk of low and high price price dynamics into the firm’s hedging plan, and realizations, enhancing revenue. The optimal hedging clearly delineates the role of demand, as well as the roles of holding and shortage costs. Our findings are strategy is a perfect hedge, bt ¼ 1, when the correlation between the input and output price is zero, also related to Ho (1984) and Rolfo (1980), who show because in the absence of such correlation there is no that a firm will underhedge only if there is output benefit of underhedging. On the other hand, when uncertainty in addition to price uncertainty. In con- input and output prices are correlated, the optimal trast, our model shows that a firm will underhedge hedging policy balances the expected cost of backlog- when the input and output commodity prices are pos- ging and holding with the expected benefits of the itively correlated, even when there is no production correlation between the input and output prices as uncertainty. Our results differ from Ho (1984) and well as with the benefits of hedging due to the concav- Rolfo (1980) because we have an integrated view of ity of the revenue function in the output price. The the firm, which includes the dynamics of both input effect of the correlation between the input and output and output commodity prices in determining the opti- prices is summarized in the following corollary. mal hedge for the output commodity. Ho (1984) and Rolfo (1980) consider only the output commodity to obtain the optimal hedge ratio. In our model, a posi- COROLLARY 1. Effect of correlation in input and output tive correlation between input and output commodity prices: prices provides an operational hedge that motivates \ (a) If Cov(pt+1,st+1) > 0 then bt 1. the firm to underhedge the output price risk. = (b) If Cov(pt+1,st+1) 0 then bt ¼ 1. THEOREM 2. (VALUE OF HEDGING). Suppose Cov(pt+1, ; ; When input and output prices are positively corre- st+1) > 0, and let Vtðxt PtÞ and Vtðxt PtÞ denote the lated, selling in the spot market provides a natural optimal value functions under no-hedging and perfect hedge, resulting in higher profits by reducing the hedging, respectively. Then, Vtðxt; PtÞVtðxt; PtÞ expected procurement cost. In our model, if the real- \ Vtðxt; PtÞ for every realization of Pt. ized output price is high, this results in low demand and high inventory of ethanol (after meeting the The above theorem elucidates the value of hedging demand). Due to a positive correlation between input for a value-maximizing firm when selling to a retailer and output prices, it is likely that the realization of the that faces demand that is negatively correlated with input price is also high. This case implies that when price. The value of hedging comes from two sources: the price of the input is high, the firm needs to pro- (1) increased expected revenue from the sales of an cure less since it has a high inventory of ethanol. Simi- index-based contract to the retailer by reducing price larly, if the realized price of the output commodity is risk, and (2) better operational planning by obtaining low, then it leads to high demand and low ethanol advanced demand information through the forward inventory (after meeting the demand). In this case, the sale and reducing holding and backlog costs. Notice firm needs to procure more (due to low inventory), that partial hedging dominates perfect hedging when but also faces a low input price. To summarize, the the input and output prices are positively correlated. positive correlation between input and output prices The covariance between the prices provides a natural controls the procurement cost, either through a lower hedge that renders the strategy of perfect hedging input price or a lower procurement quantity. There- sub-optimal. In addition, hedging elicits future fore, a positive correlation between input and output demand information from the downstream retailer. Goel and Tanrisever: Financial Hedging and Procurement 1932 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society

Using this information, the commodity processor store input. When input prices are in contango, i.e., b i \ makes a processing decision to reduce operational st ft 0, this situation may create an opportunity costs related to penalty and holding costs, thus creat- for the firm to process input into output for storage ing value through advanced demand information. We purposes. In particular, this storage cost efficiency can also would like to note that without frictions, corpo- allow the firm to trade the benefits of the contango of b i \ rate-level risk management is irrelevant (see, e.g., the input prices, st ft 0, with the holding cost of Froot et al. 1993, Jin and Jorion 2006, Smith and Stulz the output to determine the optimal stocking quan- b 1985). In our case, this friction is a form of transaction tity. It is important to notice that zt will be finite k cost, i.e., the logistical costs, t, to access the exchange because it trades the contango of the input commod- market (similar transaction costs have also been used ity price curve with the holding cost of the output in in Goel and Gutierrez (2011) when justifying the use subsequent periods. Carrying inventory for storage ≥ of forward contracts in commodity procurement). purposes can result in violating the constraint zt xt, These logistical costs are also the driver of downward rendering myopic policies sub-optimal. Following, sloping demand in this study, which makes hedging we discuss some properties of the myopic policy relevant. Note that downward sloping demand in described in Theorem 1. itself is not a market friction; it is a consequence of the logistical costs, k . More specifically, if k = 0, then the t t PROPOSITION 1. (PROPERTIES OF THE MYOPIC POLICY). processor can economically clear all the inventory in c the exchange market and does not face a downward (a) If Cov(pt+1,st+1) > 0 then zt is decreasing in , k = sloping demand. Hence, when t 0, hedging the and it is non-increasing otherwise. output price would be value-neutral since there are (b) If Cov(pt+1,st+1) > 0 then bt is increasing c \ G [ G no other frictions in our model. (decreasing) in when zt zt ðzt zt Þ, and it is G c o Theorem 1 also establishes the optimality of the constant otherwise, where zt ¼ Atþ1 ft c myopic policy under three conditions: (1) absence of tþ1. yield uncertainty, (2) linearly decreasing demand in (c) If Cov(pt+1,st+1) > 0 then bt is increasing in r the price of the output commodity, and (3) h > h0/a. and h, and it is constant otherwise. The absence of yield uncertainty is by construction, (d) If Cov(pt+1,st+1) > 0 then zt is increasing in r and once this assumption is later relaxed, we show and decreasing in h, and it is constant otherwise. that a myopic policy is not optimal. Considering a general demand function that decreases in price is not As the price elasticity of demand c increases, the sufficient to show the optimality of the myopic policy. expected demand will decrease, resulting in lower Furthermore, the condition h > h0/a ensures that it is amounts of the commodity being processed. If the G not economical to convert input into output for stor- processing quantity is below the mean demand, zt , age purposes. Notice that the positive marginal con- then as c increases, mean demand decreases, hence, 0 b i venience yield, st þ h ft 0, ensures that it is the overage risk of the firm increases. In this case, if \ never optimal to store input without an economic use the covariance is positive, then bt 1, and it is judi- 0 in the current period. However, when h < h /a,it cious for the firm to increase bt and reduce the vari- could be optimal to benefit from a negative spot-for- ability in demand to mitigate the overage risk. On the b i \ ward spread, st ft 0, and convert input into out- other hand, if the processing quantity is higher than G c put for storage purposes. This scenario does not lead the mean demand, zt , then as increases, mean to arbitrage, as explained in the discussion following demand decreases and the risk of overage increases. Theorem 3, but results in the sub-optimality of the To reduce the overage risk, the firm increases the vari- myopic policy due to a higher stocking level of out- ance of demand by decreasing bt . As holding and put. The optimality of the myopic policy for a price penalty costs increase, it is optimal to hedge more to taker when demand is dependent on price is a unique reduce the expected underage and overage costs. On and significant result, particularly since it is in con- the other hand, lower holding and higher penalty trast to the results for the price setter in the literature costs lead the firm to process more and vice versa. If (Federgruen and Heching 1999). The following theo- the covariance is non-positive, then it is optimal to rem characterizes the optimal policy when h < h0/a. completely hedge the price risk, and the optimal solution is insensitive to changes in r and h. 0 THEOREM 3. (STORAGE COST DIFFERENTIAL). If h < h /a In summary, the integrated approach to commodity then the optimal processing policy is given by a base- risk management proposed in this study has signifi- b stock level zt for a given bt and Pt. cant managerial implications. Hedging policies based only on the output commodity price risk can lead to The condition h < h0/a implies that the firm has the sub-optimal results. This scenario occurs because such ability to store output more efficiently than it can sub-optimal policies can disentangle the natural hedge Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1933 provided by a positive correlation between the output ^ ^ ^ Vtðxt; PtÞ¼ max Jtðzt; bt; xt; PtÞð4Þ ^ ; and input commodity prices, resulting in lower profits. zt bt BtðxtÞ Our analysis illustrates that firms need to understand ^ ^ ^ Q ~ ðzt xtÞ the dynamics between input and output prices across Jtðzt; bt; xt; PtÞ¼ bE Rtðbt; ptþ1Þðst þ ctÞ PjPt a the supply chain when developing hedging policies. þ þ b Q So far, we have assumed that there is no yield rðxtÞ hðxtÞ þ EE o PjPt uncertainty during the conversion process of the ^ V ðx ; PÞ input commodity to the output commodity. In the fol- tþ1 tþ1 ^ lowing section, we incorporate yield uncertainty into ^ ðzt xtÞ ^ where, xtþ1 ¼ zt þ a dtþ1, BtðxtÞ¼fzt commodity processing decisions to ascertain its ; ^ ; þ xt 0 bt 1g, and, VTðxT PTÞ¼ðpT TÞxT impact on optimal processing and hedging policies. ðpT þ TÞxT . The following theorem characterizes the optimal policy for procurement and hedging in the presence 4. Yield Uncertainty of yield uncertainty. In many agricultural processing environments, yield THEOREM 4. (OPTIMAL POLICY UNDER YIELD UNCERTAINTY). uncertainty is a challenge that production managers = ... must deal with to ensure a regular flow of output For t 1, ,T 1, let bt and zt be determined by products. The yield from processing agricultural raw simultaneously solving the following two equations: materials such as corn and wheat depends upon @ ^ ; 2 Q Vtþ1ðxtþ1 PÞ o 2r ð1 b ÞþbEE ðf p Þ grain quality, storage and handling, and processing o t PjPt t tþ1 @bt parameters. We now extend our model by incorpo- ¼ 0 and ð5Þ rating yield uncertainty to understand the value of @ ^ ; financial hedging with respect to yield risk. In our Q Vtþ1ðxtþ1 PÞ ðst þ ctÞþbEE ¼ 0; ð6Þ PjPt ^ model, yield uncertainty is exogenous in nature, but @zt ^ depends on the quantity processed. The purpose of ^ ðzt xtÞ where xtþ1 ¼ zt þ a dtþ1. Then the firm’s opti- this section is to: (1) characterize the optimal pro- mal processing policy is given by an expected base-stock curement policy, and (2) explore the impact of yield level zt , and the hedging policy is described by bt , such uncertainty on the optimal hedging policy. ; that (i) if bt 2½0 1 then zt ¼ zt and bt ¼ bt and (ii) if According to Sobel and Babich (2012), yield uncer- ; ; ; bt 62½0 1 then bt ¼ maxfminfbt 1g 0g, and zt is tainty is primarily modeled in three ways: via con- obtained by solving Equation (6). stant variance, stochastically proportional, and binomially distributed yield. In the constant variance Incorporating yield uncertainty into our analysis model, the randomness in yield is independent of the results in the sub-optimality of the myopic policy as it processing quantity. In the binomial model, the out- ^ may violate the constraint zt xt. Our approach in come of yield uncertainty is revealed as a sequence of defining an expected base-stock level is similar to binary outcomes. In our research, we follow the Sobel and Babich (2012), who define echelon-like base- stochastically proportional yield model, which closely stock levels. Nevertheless, the main distinction reflects the operational dynamics of ethanol proces- between our model and theirs is that we model sors. We define the yield of the output commodity, stochastically proportional yield (while they model a (yt), as a function of the quantity of the input com- yield with a constant variance) and our model incor- a + e modity processed, yt, such that (yt) (a )yt, porates stochasticity in prices. In our case, any e r where N(0, ). We denote the probability distri- attempt to model yield with a constant variance will e / e bution function of by ( ). If there is no variability in not result in the optimality of the myopic policy, as in the yield, then our model reduces to the model in sec- Sobel and Babich (2012) does, for two key reasons. tion 3, where one bushel of corn exactly converts to a First, Assumption 2 in Sobel and Babich (2012) is not gallons of ethanol. Furthermore, we assume that the applicable because demand in our model is price risk of yield uncertainty is completely idiosyncratic dependent, and prices are mean reverting, such that and diversifiable, and it is not correlated with the Q Q price shocks are not independent and identically dis- market prices, such that EE ½P ¼ E½E ½P. PjPt PjPt tributed. Second, Property 1 in Sobel and Babich We model commodity processing under yield uncer- (2012) cannot be applied to our model because price is tainty as an echelon model. We define the expected a log-normal random variable, and the futures price is ^ inventory position of ethanol as zt, such that not linear in the current state. Therefore, future ^ = + e + zt ¼ ayt þ xt, and xt+1 (a )yt xt dt+1. The expected inventory cannot be written as a linear com- stochastic dynamic program in Equation (1) is modi- bination of past states of inventory and price. Notice fied to incorporate yield uncertainty as follows: that our modeling approach for yield will be Goel and Tanrisever: Financial Hedging and Procurement 1934 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society intractable under a higher number of echelons due to policy) to carry inventory in earlier periods as a hedge the “curse of dimensionality.” Nevertheless, we are against poor yield outcomes in later periods. We call able to characterize an optimal policy in a two- this effect the propagation effect, and we expect it to be echelon structure because in our model there is no amplified under a low holding cost. In the presence of incentive to carry inventory at the upper echelon due the propagation effect, we expect the myopic policy to to marginal convenience yield, which reduces the perform poorly. In addition, as the holding cost problem to a single echelon. increases, the cost of mismatch between the myopic The complication in solving Equations (5) and (6) is and optimal policies increases, decreasing the perfor- that the transition probabilities from state Pt to Ptþ1 mance of the myopic policy. This situation is called are not easy to compute, particularly when the price the look-ahead effect because it only occurs when the ^ process is at least two-dimensional with one dimension constraint zt xt is violated, as myopic policies are each for input and output commodity prices. In section not forward looking. As a result, we expect myopic 5, we discretize the price process through a binomial policies to perform better under a moderate holding lattice to calculate Equations (5) and (6) in order to cost. The numerical analysis6 presented in Table 1 cor- compute zt and bt . Nevertheless, as the time horizon roborates our intuition. In addition, the myopic policy for decision making increases, the size of the binomial performs poorly as yield uncertainty increases. lattice increases. Due to the curse of dimensionality it becomes computationally challenging to numerically 5. Numerical Analysis calculate the dynamic program. As a result, we aspire to obtain myopic policies as an approximation of the In section 5.1, we describe the stochastic price pro- optimal policy. The following theorem develops the cesses used to jointly model the input and output myopic policy for the model defined in Equation (4). commodity prices, and describe the method to estimate the price process parameters for corn and etha- THEOREM 5. (MYOPIC POLICY UNDER YIELD UNCERTAINTY). nol using the futures price data from the CME. Then, Let bt and zt be determined by simultaneously solving the in section 5.2 we discuss the discretization of the price following two equations: process and the algorithm to compute the dynamic (a) for t = 1, 2, ...,T 2 program. In section 5.3, we discuss the managerial b b i b insights generated through the sensitivity analysis of ^ st þ a h f þ ct ctþ1 CðM Þ¼ t the price process parameters and firm characteristics. t abðr þ hÞ r2 ; = ^ ; ; 5.1. Stochastic Price Process 2 oð1 btÞ¼Covðstþ1 ptþ1Þ a þ gðzt btÞ We model the input and output commodity prices as (b) for t = T 1 a mean-reverting stochastic process. In particular, we b 0 b model the logarithm of the price of a commodity as ^ st þ ct a f þ a T Cð Þ¼ t – vi Mt an Ornstein Uhlenbeck process, defined as d t ¼ 2baT jiðai vi iÞdt þ ridZi, dvo ¼ joðao vo oÞdt ZZ^ t t t 1 Mt þ rodZo, and dZi.dZo = qdt. Superscripts i and o repre- b ¼ þ T ðfo p Þw ðpÞdp/ðÞd; t 2 r2 t tþ1 tþ1 sent input and output commodities, respectively. v is 0 0 j R R ^ R the log of the price, represents the rate of mean ^ ; Mt o w 1 a k where gðzt btÞ¼ fr ðf ptþ1Þ þ ðpÞdp þ h ^ reversion, is the long-run average price, is the risk 0 t Rt 1 Mt ^ ^ r ðfo; p Þw ðpÞdpg/ðÞd, CðM Þ¼ ð þ ÞW ðM Þ premium per unit of mean reversion, denotes the t tþ1 tþ1 t 1 a tþ1 t q c o c ^ ^ = volatility in the commodity price, represents the ^ Atþ1 btft tþ1 zt ðzt xtÞ a /ðÞd, and Mt ¼ . Then, cð1 btÞ instantaneous correlation between the two stochastic ^ myopic procurement and hedging policies ^z and b, respec- processes, and dZ is the increment of a Brownian t t tively, can be obtained, such that (i) if bt 2½0; 1 then ^ ^ ; Table 1 Percentage Difference in Value Function between Myopic and zt ¼ zt and bt ¼ bt and (ii) if bt 62½0 1 then Optimal Policies ^ ; ; ^ bt ¼ maxfminfbt 1g 0g, and zt is obtained by solving the above equations. Coefficient of variation in yield Holding cost 0.04 0.06 0.08 0.10 0.12 Our objective is to explore the conditions under 0.01 0.99 3.19 7.62 10.70 14.00 which the myopic policy performs close to the opti- 0.02 0.84 0.82 3.13 5.64 11.00 mal policy. In this regard, we expect to experience 0.04 0.82 0.62 2.68 3.50 4.50 two effects, namely the propagation effect and the 0.10 0.84 0.58 2.68 3.37 4.20 look-ahead effect. Under the optimal policy, the firm 0.20 0.88 0.59 3.27 4.57 7.00 0.30 0.93 0.61 4.28 7.15 11.00 processes a higher quantity (compared to the myopic Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1935 ^ Motion associated with the stochastic process. Under four possible nodes. From every state node PtPt,we the risk-neutral measure, the futures price at time t consider the possibility of four transitions as s ^ ^ vî Di; vô Do for a contract that expires at time is defined as Pt Ptþ1, defined as Pt ¼ft t g, 2 i vi jiðs tÞ ai i jiðstÞ ðriÞ depending upon the up or down jump for the combi- ft;s ¼ exp½ te þð Þð1 e Þþ 4ji i nation of input and output prices. ð1 e2j ðs tÞÞ for the input commodity, and 2 The transition probabilities of mean-reverting pro- o vo joðs tÞ ao o joðstÞ ðroÞ – ft;s ¼ exp½ t e þð Þð1 e Þþ 4jo cesses, such as an Ornstein Uhlenbeck process, are jo s 2 ð tÞ known to be state dependent. We denote P^ as the ð1 e Þ for the output commodity. Pt We estimate the parameters of the price process by transition probability under the risk-neutral measure applying a Kalman filter technique on the futures ^ ^ þ from node Pt to Pt . As an example, P^ represents price data for corn and ethanol from the CME Pt ^ between 4/1/2005 and 12/31/2011. According to the the transition probability from node Pt Pt to an up corn futures price data, the average price for corn is node for the input commodity and a down node for 300 cents/bushel. Corn futures contracts typically the output commodity. These four probabilities at mature in the months of March, May, July, September, each node are obtained by requiring them to sum up and December, and there are about 15 such contracts to 1, and equating the risk-neutral conditional expec- traded at a time for various maturities. The average tations, variance and covariance of the discretized price for ethanol is 120 cents/gallon. Ethanol futures process to those of the original process, as described have been trading on the CME since early 2005, and in Kamrad and Ritchken (1991) and Hahn and Dyer the contracts mature every month. Table 2 illustrates (2008). Nevertheless, as the transition probabilities are the parameters of the joint stochastic price process state dependent, they may be required to censor. In for the two commodities (see Schwartz and Smith this regard, we follow the two-step approach of Hahn 2000 for details). We observe moderate levels of mean and Dyer (2008) to develop the conditional transition reversion for both commodities. In general, the mean- probabilities. The jump probabilities from the state vî vî Di þ reversion factor is difficult to estimate, but we observe variable t are defined as Pf t þ g¼Pvî and t from the low values of the standard deviation that the vî Di vô Pf t g¼Pvî . Using Bayes’ rule, Pf t þ coefficient of mean reversion is significant. The t Do vî Di þþ= þ vô Do vî Di þ= volatility of the two commodities is around 30%, with j t þ g¼P^ Pvî , Pf t þ j t g¼P^ Pt t Pt a strong correlation in prices. The parameters in P, Pfvô Dojvî þ Dig¼Pþ=Pþ, and Pfvô vî t t P^ vî t Table 2 are used in the numerical section to compute t t t Do vî Di = j t g¼P^ Pvî . The conditional transition the futures prices and conduct a sensitivity analysis Pt t on the optimal procurement and hedging policies. probabilities are given as: ffiffiffiffiffi 5.2. Optimization Algorithm 1 jiðai i vîÞp Pfvî þDig¼ 1þ t Dt To compute the optimal procurement and hedging t 2 ri policies, we discretize the price process on a binomial ji ai i vî pffiffiffiffiffi i i 1 ð Þ lattice as a function of the input and output price vari- Pfv^ D g¼ 1 t Dt t 2 ri ables. We then solve the dynamic program using The- "#pffiffiffiffiffi orem 4 to obtain the optimal policies. We discretize the 1 ðjoðao o vôÞ DtþqroÞri Pfvô þDojvî þDig¼ 1þ t pffiffiffiffiffi t t ro ri ji ai i vî D stochastic price process as a recombinant lattice, as in 2 ð þ ð tÞ tÞ Kamrad and Ritchken (1991). The state space of price "#pffiffiffiffiffi o o o o o i i o ^ 1 ðj ða v^ Þ Dtqr Þr Pt is a function of v and v .WedefinePt as the lattice vô Do vî Di t ffiffiffiffiffi ^ t t ^ Pf t þ j t g¼ 1þ p nodes, such that P P ,whereP ðvî; vôÞ.Thejump 2 roðri jiðai i vîÞ DtÞ t t t t t "#ffiffiffiffiffi t size on the lattice corresponding to the input commod-ffiffiffiffiffi p p 1 ðjoðao o vôÞ DtþqroÞri ity is denoted by Di, and it is given by Di ¼ ri Dt, Pfvô Dojvî þDig¼ 1 t pffiffiffiffiffi t t 2 ro ri ji ai i vî D where Dt is the time interval between successive ð þ ð tÞ tÞ "#pffiffiffiffiffi jumps. Similarly, the jump size on the lattice corre- 1 ðj0ðao o vôÞ DtqroÞri Do Pfvô Dojvî Dig¼ 1 t pffiffiffiffiffi : sponding to the output commoditypffiffiffiffiffi is denoted by , t t i o 2 roðri jiðai i v^ Þ D Þ and it is given by D ¼ ro Dt. The lattice starts at time t t = ^ vî ; vô t 0fromnodeP0 ð 0 0Þ and then transitions to ð7Þ

Table 2 Estimated Stochastic Price Process Parameters

Symbol ai ji ki ri ao jo ko ro q Mean 6.505 0.170 0.579 0.323 5.532 0.142 0.552 0.275 0.772 SE 0.797 0.007 0.799 0.019 0.833 0.010 0.842 0.013 0.033 Goel and Tanrisever: Financial Hedging and Procurement 1936 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society

We discretize the price process on two factors, such volatile. On the other hand, the effect of input price that we compute t2 nodes at time t. For our numerical volatility is driven by a different mechanism. As the analysis, we also truncate the price distribution from input price becomes more volatile, the effect of self- the above by the monopoly price, to ensure that the hedging due to correlated prices becomes more pro- firm remains a price taker. The outline of the algo- nounced. In particular, higher input price volatility rithm that calculates (4) is as follows: creates more opportunities to reduce the procurement Step 1: Initialization cost when the firm is not hedging due to correlation between the prices. This situation increases the bene- (a) Set up the lattice to discretize the price profits of underhedging leading to lower motivation to cess, and calculate the transition probabilities ^ hedge. Hence, as the volatility of the input price from P P to the corresponding nodes in ^ t t increases, the value of hedging decreases, as shown in P . tþ1 Figure 1c. This is a unique result of this study. In (b) Censor the transition probabilities based on addition, this effect gets further amplified with an Equation (7). c o vô jo increase in demand elasticity . (c) At each node, calculate ft;tþ1 ¼ exp½ t e 2 Effect of price correlation and holding cost: When input ao o jo ðroÞ 2jo þð Þð1 e Þþ 4jo ð1 e Þ. and output prices are positively correlated, selling in Step 2: Recursive Calculation the spot market provides a natural hedge, resulting in higher profits by reducing the expected procurement ^ ; ^ ^ (a) Calculate VTðxT PTÞ for all nodes PT PT. Set cost. In our model, if the realized output price is high, t = T 1. this results in low demand and high inventory of Q ^ ; ^ (b) Calculate E^ ^ Vtþ1ðxtþ1 PÞ to obtain zt and bt ethanol. Due to a positive correlation between input PjPt ^ from Theorem 4 for all nodes PtPt. Then cal- and output prices, it is likely that the realization of the ^ ^ culate Vtðxt; PtÞ. input price is also high. This case implies that when (c) Set t = t 1. If t > 0 then go to step 2b: other- the price of the input is high, the firm needs to pro- wise stop. cure less since it has a high inventory of ethanol. Simi- larly, if the realized price of the output commodity is A similar algorithm can be used to compute the low, then it leads to high demand and low ethanol optimal policies and the value function for the case inventory. In this case, the firm needs to procure without yield uncertainty using Theorem 1. We next more, but also faces a low input price. To summarize, develop managerial insights based on the numerical the positive correlation between input and output analysis. prices controls the procurement cost, either through a lower input price or a lower procurement quantity. 5.3. Managerial Insights Therefore, as the correlation increases, the firm gets In this section, to gain further managerial insights, we more motivated to sell in the spot market resulting in conduct a sensitivity analysis for the parameters, such a lower value of hedging, as shown in Figure 1b. On as the volatility of input and output commodity the other hand, the value of hedging increases with prices, the correlation between the prices, and the higher holding costs, as illustrated in Figure 1d. A holding cost. We consider a planning horizon of higher degree of hedging allows better operational T = 10 weeks.7 We change one parameter at a time efficiency by eliciting advanced demand information, while keeping the other parameters at their base-case resulting in less mismatch in demand and supply. An values, as detailed in Table 2. We first develop man- increase in the holding cost leads to a higher cost for a agerial insights for the deterministic yield case, and supply and demand mismatch, resulting in a higher then examine the impact of yield uncertainty. We value from hedging. Similar results can be shown for compute the value function of selling through the the penalty cost. spot market, defined as V(spot), using Equation (1) Effect of yield uncertainty: The value of the firm when b = 0. Similarly, we compute the value of the t decreases with an increase in yield uncertainty. This optimal policy, defined as V(optimal), using Equation result is consistent with the economics and finance lit- (1). Then we denote their percentage difference on erature. Nevertheless, the percentage benefit of hedg- % VðoptimalÞVðspotÞ graphs, such that Gains ¼ VðspotÞ 100. ing is non-monotonic in yield uncertainty, as evident Effect of output and input commodity price volatility: from Figure 2. In Figure 2a, the base case refers to the Hedging creates more value as the volatility of the base-case volatility of corn, and the other cases refer output commodity price increases, as shown in Fig- to the 80% and 120% volatility of corn with respect to ure 1a. This result is driven by the concavity of the the base case. Similarly, in Figure 2b, the base case revenues due to the negative correlation between the refers to the base-case volatility of ethanol, and the demand and output price, which creates a bigger other cases refer to the 90% and 110% volatility of incentive to hedge as the output price becomes more ethanol in the base case. As yield uncertainty Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1937

Figure 1 Effect of Hedging on Firm Value [Color ﬁgure can be viewed at wileyonlinelibrary.com]

12 6

10 5

8 4 6 3 4 2 2

0 1 -0.8 -0.4 0 0.4 0.8

(a) (b)

5.5 5 4.5 4 3.5 3

% Gains 2.5 2 1.5 1 10% 30% 50% 70% 90%

Figure 2 Effect of Yield Uncertainty on the % Beneﬁt of Hedging [Color ﬁgure can be viewed at wileyonlinelibrary.com]

(a) (b) increases, it increases the risk exposure of the firm, sharply. As a result, the percentage benefit of hedging but only market risk can be hedged through the con- is non-monotonic in yield uncertainty. tract. Therefore, the percentage of risk that can be hedged decreases with an increase in yield uncer- 6. Concluding Remarks tainty, decreasing the value of hedging. However, as yield uncertainty becomes very high, the percentage We consider the operations of a commodity proces- benefit of hedging increases as the value of firm, sor that is a price taker in the commodity markets. which relies purely on spot procurement, decreases In general, commodity processors operate with thin Goel and Tanrisever: Financial Hedging and Procurement 1938 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society profit margins, making it imperative to implement commodity, and (3) more expensive storage of the optimal procurement, processing, and hedging poli- output commodity than the input commodity. These cies. We formulate a multi-period model where the results are significant because they are contrary to the processor procures an input commodity in the spot existing literature on price-setter firms, where order- market to process and sell it to the downstream up-to policies have been shown to be optimal. For a retailer. The commodity processor may sell the given hedging policy, the optimal input commodity output through a spot, forward or an index-based procurement policy has a newsvendor-like structure contract. In this study, we jointly optimize procure- as a function of the spot and futures prices of the ment policies for the input commodity, and finan- input commodity. Our research also elucidates the cial hedging policies for the output commodity role of the term structure of futures prices on the opti- when demand is negatively correlated with output mal procurement policy. price. We also assume that the input and output Agricultural commodity processors also deal with commodity prices are correlated and follow a joint yield uncertainty in pursuit of matching supply with stochastic process that offers no risk-free arbitrage demand. In the presence of such uncertainty, how- opportunities. In summary, we develop an inte- ever, a myopic policy is not optimal. We model yield grated risk management model for the commodity uncertainty as stochastically proportional to the pro- processor that accounts for correlation between cessing quantity, and show that an expected base- demand and the output commodity price, and also stock policy is optimal. As the time horizon of deci- captures the correlation between input and output sion making increases, the state space of the joint prices. price process on the lattice increases exponentially, Under this integrated framework, we show that in rendering it impossible to compute the optimal pol- general, neither selling exclusively in spot nor for- icy due to the curse of dimensionality. In this con- ward markets is optimal, but selling through an text, we develop myopic policies and conclude that index price, which is a combination of spot and for- they perform reasonably well for moderate values of ward prices, is optimal. This leads to an optimal holding cost. However, their performance deterio- hedge ratio of less than 1, which is in contrast to the rates as the yield becomes more uncertain. This is classic economics literature that considers optimiz- the first paper in the operations literature that stud- ing only the output end of the supply chain and ies hedging under yield uncertainty. We find that concludes that the optimal hedge ratio is one in the yield uncertainty has a U-shaped effect on the bene- absence of yield uncertainty (Ho 1984, Rolfo 1980). fits of hedging. Our research concludes that the correlation between This research contributes to the growing literature input and output prices provides a natural hedge, at the interface of operations and finance. Our analy- resulting in a decrease in reliance on financial hedg- sis concludes that the correlation coefficient between ing. One of the key managerial insights of our the input and output prices is key in determining the research is that hedging is most beneficial when out- optimal hedging policy. In this study, we assume a put price volatility is high and input price volatility static correlation coefficient between prices. Modeling is low. a stochastic correlation coefficient as an additional Financial theory explains the value of hedging factor in the price process would generate additional through capital market imperfections, such as bank- insights. Furthermore, we show that index-based con- ruptcy costs, taxation, agency problems, and ineffi- tracts create value for a firm, and it could be further cient pricing of derivatives. In our study, we consider investigated how other financial contracts, such as k a form of friction, that is, logistical costs, t, to access swaps, options, or swing options, could be used by a the exchange market. This logistical cost is also the firm to create value. The role of capacity constraints driver of the downward sloping demand in the on optimal hedging policies could also be explored study. In particular, there are two distinct reasons further. We believe that the managerial insights why hedging creates value in our value-maximiza- developed from our analysis will be useful for pro- tion framework: (1) Logistical costs result in the non- curement and sales managers in a commodity supply linearity of the profit function in the output price, chain. leading to the optimality of hedging. (2) Hedging elicits demand information from the downstream retailer to allow efficient operational planning, eliminating Appendix A. Proofs wasteful inventory due to a mismatch in demand and supply. A.1. Proof of Lemma 2 We identify three conditions under which a myopic PROOF. The proof is by backward induction. For policy is optimal: (1) absence of yield uncertainty, (2) this purpose, we first need to show that ; ; ; linearly decreasing demand in the price of the output Jtðxt zt bt PtÞ is jointly concave in zt and bt for Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1939 t = T 1, i.e., for the last decision period. Recall we obtain: ; ; ; that JT1ðxT1 zT1 bT1 PT1Þ is defined as: @2 JT1 2 Q cr ~ 2 ¼2 o J ðx ; z ; b ; P Þ¼bE R ðb ; p Þ @ T 1 T 1 T 1 T 1 T 1 PjPT1 T 1 T 1 T bT1 ðc o þ c þ Þ2 ðsT1 þ cT1Þ þ cb fT1 AT T zT1 w : ð Þ ð Þ 2 T ðMT1Þ zT1 xT1 r xT1 2 3 a c ð1 bT1Þ þ Q hðxT1Þ þ bE VTðzT1 dT; PÞ; @2 PjPT1 JT1 \ Now it is trivial to observe that @ 2 0. Finally, bT1 ; þ the cross-partial derivative of JT1 with respect to where VTðxT PTÞ¼ðpT TÞxT ðpT þ TÞxT = and xT zT1 dT. zT1 and bT1 is obtained as: ; The first-order partial derivative of JT1ðxT1 @2 c o c z ; b ; P Þ with respect to z is shown JT1 f AT þ T þ zT1 T1 T1 T1 T1 ¼ b T1 wðM Þ: w Ψ @ @ 2 T 2 T1 below (for brevity, we will not index and , and bT1 zT1 cð1 bT1Þ omit the arguments of the function JT1ðxT1; zT1; bT1; PT1Þ). Accordingly, the Hessian matrix of JT1 is given by: Z M 2 3 @ T1 @2 @2 JT1 sT1 þ cT1 o JT1 JT1 b w 2 ¼ þ ðfT þ TÞ ðpÞdp @z @bT1@zT1 @ 4 T1 5 AB; zT1 a Z 0 @2 @2 ¼ 1 JT1 JT1 CD @ @ 2 b o w ; bT1 zT1 @b þ ðfT TÞ ðpÞdp T1 MT1 where c 0 c AT bT1fT1 T zT1 where MT1 ¼ c . Following, the c 0 c ð1 bT1Þ 1 AT bT1fT1 T zT1 A ¼2bT w ; second-order partial derivative of JT1 with respect cð Þ cð Þ 1 bT1 1 bT1 to zT1 is given by: c 0 c AT bT1fT1 T zT1 B ¼C ¼2bTw 2 cð1 b Þ @ J @M !T 1 T 1 ¼ 2b T 1 wðM Þ: @ 2 T @ T1 c o þ c þ zT1 zT1 fT1 AT T zT1 ; 2 and @ cð1 b Þ Observing MT1 ¼ 1 \ 0, we conclude T 1 @zT1 cð1 bT1Þ @2 c o c 2 JT1 b 1 w \ 2 ð fT1 AT þ T þ zT1Þ that 2 ¼2 T c ðMT1Þ 0. Similarly, cr cb @z ð1 bT1Þ D ¼2 2 T T1 o c2 3 ð1 bT1Þ the first-order partial derivative of JT1 with respect ≤ ≤ A cb f0 c z to bT1 is shown below for 0 bT1 1. w T T1 T1 T T1 : cð1 bT1Þ @J T1 ¼ 2c ð1 b Þr2 @ T1 o We have already proven that the first-order prin- bT1 Z MT1 cipal minors of the Hessian matrix is negative, i.e., cb o w @2 @2 þ ðfT1 pTÞðpT þ TÞ ðpÞdp JT1 \ JT1 \ @ 2 0 and @ 2 0. In this case, the second- Z0 z b 1 T 1 T 1 cb o w : order principal minor is the determinant of the þ ðfT1 pTÞðpT TÞ ðpÞdp MT1 matrix and is given by @2 @2 @2 2 Following, the second-order partial derivative of JT1 JT1 JT1 @ 2 @ 2 @ @ JT1 with respect to bT1 is given by: zT1 bT1 bT1 zT1 1 @2 @ ¼ 4cr2b wðM Þ; JT1 2 MT1 o o T c T1 ¼2cr þ 2cb ðf M ÞwðM Þ: ð1 bT1Þ @ 2 o T @ T1 T1 T1 bT1 bT1 which is positive. This concludes that the Hessian is Then, substituting for negative definite and JT1 is strictly jointly concave c 0 c in zT1 and bT1. Then, from concavity preservation AT bT1fT1 T zT1 MT1 ¼ under maximization, we conclude that the value cð1 bT1Þ function VT1 is also strictly concave in xT1 (see, and e.g., Porteus 2002, p. 227). @ c o c Now to complete the induction argument, let’s MT1 fT1 AT þ T þ zT1 ¼ ; V + @ 2 assume that the value function t 1 is strictly con- bT1 cð1 bT1Þ cave in xt+1, and check whether Jt is strictly jointly Goel and Tanrisever: Financial Hedging and Procurement 1940 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society

< concave in zt and bt for t T 1. Recall that @ Jt cb r2 cb = cb ; ; ¼2 ð1btÞ Covðstþ1;ptþ1Þ a utðzt btÞ¼0 Q ~ ðst þ ctÞ @ o J ðx ;z ;b ;P Þ¼ bE R ðb ;p Þ ðz x Þ bt t t t t t PjPt t t tþ1 t t a o ðA5Þ þ þ Q rðxtÞ hðxtÞ þ bE Vtþ1ðxtþ1;PÞ ; R R PjPt Mt o 1 where u ðz ; b Þ¼r ðf p þ ÞwðpÞdp þ h t t t 0 t t 1 Mt c o c + = + + o Atþ1 btft tþ1 zt since xt 1 zt dt 1 is linear in zt and bt,ifVt 1 is ðf p þ ÞwðpÞdp and M ¼ . t t 1 t cð1btÞ strictly concave in xt+1, then Vt+1 is also strictly (a) If Cov(st+1, pt+1) > 0, then from Equation (A5), jointly concave in zt and bt. Hence it is sufficient to it can be easily shown that b \ 1. show that the remaining terms of J , i.e., t Q t Ignoring l and l + , and substituting E t t 1 PjPt ½s ¼fi in Equation (A3), we obtain z as a solu- b Q ~ ; ðst þ ctÞ þ tþ1 t t Kt ¼ E Rtðbt ptþ1Þ ðzt xtÞrðxtÞ = ... PjPt a tion to Equation (2) for t 1, , T 2. Similarly, þ hðxtÞ ; we can show that Equation (3) follows from Equation (A5) for t = 1, ..., T 2. Also, we can obtain z @2 @2 T1 Kt Kt cr2 = + k are concave. Observing that @ 2 ¼ 0, @ 2 ¼2 o and bT for period T 1 by substituting r pT T @2 zt bt 1 \ 0 and Kt ¼ 0 concludes that K is concave, and = k @bt@zt t and h T pT in Equations (A3) and (A4), and \ hence Jt is strictly jointly concave in zt and bt (note ignoring the Lagrange multipliers. If bt 0, then that Vt+1 is strictly jointly concave). This completes since the objective function is concave, the myopic h the induction argument. solution is obtained by setting bt ¼ 0 and solving Equation (2). Subsequently, we now prove the opti- ≤ ≤ A.2. Proof of Theorem 1 mality of this myopic policy when 0 bt 1. PROOF. The first-order condition for zt from Equa- We structure the proof by showing that the optimal tion (1) is given by myopic policy never leads to the violation of con- s þ bha bfi þ c bc [ ¼ t t t tþ1 @J s þ c Q @V ðz d ; PÞ straint ztþ1 zt dtþ1.LetGt bðr þ hÞa , t ¼ t t þ bE tþ1 t tþ1 þ l PjPt t W1 @zt a @zt then from Equation (2) and equating Mt ¼ ðGtÞ, ; [ ¼ 0 and ðA1Þ the constraint ztþ1 zt dtþ1 can be written as the following (note that the problem is myopic by defini- @ tion for t = T 1): Vt st þ ct : : l ; ¼ h 1fxt [ 0g þ r 1fxt\0g t ðA2Þ @xt a o 1 Atþ2 cbtþ1f cð1 btþ1ÞW ðGtþ1Þctþ2 tþ1 ðA6Þ l ≥ [ c W1 : where t is the Lagrange multiplier for zt xt. Com- ð1 btÞðptþ1 ðGtÞÞ bining Equations (A1) and (A2) gives Now, let U + be the highest possible output price @ Jt st þ ct Q stþ1 þ ctþ1 + Atþ2 ¼ þ bE in the local market. At time t 2, 2c is the optimal @ PjPt zt a a monopoly price for the firm, which is always greater

: o than the price realized in competitive markets. In h 1fzt [ Atþ1cðbtf þð1btÞptþ1þtþ1Þg t what follows, given that the firm is a price taker, Atþ2 þ r:1 \ c o l þ l ¼ 0: i.e., c U þ , we show Equation (A6) is always fzt Atþ1 ðbtft þð1btÞptþ1þ tþ1Þg tþ1 t 2 satisfied. ðA3Þ The worst-case scenario, which violates the con- For ease of exposition, we assume that the dis- straint in Equation (A6), is that the RHS is large and = b counted processing cost is fixed, i.e., ct ct+1 for the LHS is small. The RHS can achieve maximum by = ... ≤ ≤ = Ψ1 = t 1, , T 2. Ignoring the constraint 0 bt 1, setting bt 0, (Gt) 0, and ptþ1 ¼ U þ ,such @z the first-order condition for b from Equation (1) is tþ1 1 t that RHS = c(U + ). In addition, ¼ðW ðG þ Þ @btþ1 t 1 given by o c ftþ1Þ ,hence,ztþ1 (i.e., the LHS of A6) is monotoni- @ cally either increasing or decreasing in b + .Ifz is Jt b Q c ; o t 1 tþ1 ¼ E ½ðAtþ1 2 Wtþ1ðbt ptþ1ÞÞðf ptþ1Þ @ PjPt t decreasing in b + , then after setting b + =1, Equation bt t 1 t 1 ðA4Þ c o c c Q @Vtþ1ðzt dtþ1;PÞ @dtþ1 (A6) becomes Atþ2 ftþ1 þ tþ2 þ ðU þ Þ.On bE ¼ 0: PjPt the other hand, if z is increasing in b + ,thenafter @ðzt dtþ1Þ @bt tþ1 t 1 1 + = cW @ setting bt 1 0, Equation (A6) becomes Atþ2 dtþ1 o We know that @ ¼cðf ptþ1Þ; substituting it c c bt t ½Gtþ1þ tþ2 þ ðU þ Þ. Finally, note that by con- in Equation (A4) results in the following myopic + o struction, U is greater than both ftþ1 þ tþ2 and condition: Ψ1 + k [Gt+1] t+2. Now it is easy to observe that, since Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1941

Atþ2 2cðU þ Þ, Equation (A6) is satisfied in both This, in conjunction with the concavity of Jt in zt,results c ≤ cases. in zt decreasing in .WhenCov(pt+1, st+1) 0, from The- = c (b) if Cov(st+1, pt+1) < 0, then at bt 1weget orem 1, zt is non-increasing in . @J t [ 0 from Equation (A5). Since b 2½0; 1 and the (b) When Cov(pt+1, st+1) > 0, from Equation (A5) @bt t and Corollary 1, we get objective function is concave, this implies that bt ¼ 1. @ 2 o Jt @ J A þ cf c þ z A þ z Note that at b ¼ 1 the derivative @ in Equation (A3) t t 1 t t 1 t t 1 t t zt @ @c ¼ c ðr þ hÞ does not exist due to the kinks in the objective func- bt 1 bt ð1 btÞ \ : tion. Therefore, the optimality condition has to satisfy and bt 1 @J t @2J 0 @z as explained in Rockafellar (1970). Observ- t [ \ G c o c t Then @b @c 0whenzt zt ¼ Atþ1 ft tþ1, @J Q þ t t st þ ct l b stþ1 ctþ1 l @2 ing that @ ¼ þ t þ E f tþ1gþ Jt G zt a PjPt a and \ 0whenz [ z . This, in conjunction with @bt@c t t b : o ; : o = ½r 1fzt \ Atþ1 cðf þ tþ1Þg h 1fzt [ Atþ1 cðf þ tþ1Þg at bt t t the concavity of Jt in bt, shows that bt increases (de- 1, now there are two possible cases: c \ G [ G creases) with when zt zt ðzt zt Þ. When Cov b i b c o + + ≤ (i) st ft ra: In this case, zt ¼ At ft (pt 1, st 1) 0, then from Theorem 1, bt is always one. c tþ1 is an optimal solution and myopic (c) When Cov(pt+1, st+1) > 0, from EquationR (A5) @2 c o c Jt Mt 0 indeed. Notice, at zt ¼ At ft tþ1 the and Corollary 1, we can show @ @ ¼ c ðf ptþ1Þ bt r 0 t R leftover inventory is zero; therefore, M Q wðpÞdp [ 0 and b \ 1 since by definition t bE ½l ¼0. t 0 PjPt tþ1 0 ðf ptþ1ÞwðpÞdp [ 0. Also, using Equation (A5) we i t R b [ b l @2 (ii) st ft ra: In this case, t > 0, such that Jt 1 0 obtain ¼c ðf p þ ÞwðpÞdp [ 0. This, in @bt@h Mt t t 1 zt ¼ xt is an optimal solution and myopic indeed. h conjunction with the concavity of Jt in bt, proves the ≤ desired result. When Cov(pt+1, st+1) 0, then bt is A.3. Proof of Lemma 3 always one. + + PROOF. Immediately follows from the concavity of (d) When Cov(pt 1, st 1) > 0, we can show that @2 Q @2 0 Q Jt b [ Jt function R in p + , and f ¼ bE ½p þ . h @ @ ¼ E ½1fz \A cðb foþð1b Þp þ Þg 0, @ @ ¼ t t 1 t PjPt t 1 zt r PjPt t tþ1 t t t tþ1 tþ1 zt h b Q \ \ E ½1fz [ A cðb foþð1b Þp þ Þg 0andb 1. A.4. Proof of Corollary 1 PjPt t tþ1 t t t tþ1 tþ1 t This, in conjunction with the concavity of Jt in zt, PROOF. From Lemma 2 we know that Equation (3) ≤ proves the desired result. When Cov(pt+1,st+1) 0, from has a unique solution. Then, if Cov(pt+1, st+1) > 0, from @ Theorem 1, z is a constraint with respect to r and h. h Equation (A5) it implies that Jt \ 0atb = 1. This con- t @bt t \ = cludes that bt 1. If Cov(pt+1, st+1) 0, from Equation A.8. Proof of Theorem 4 @ Jt = h (A5) it implies that @ ¼ 0atbt 1. PROOF. We can construct a similar proof as in Lemma bt ^ 1 to show that Vtþ1ðða þ Þyt þ xt dtþ1; PÞ is con- A.5. Proof of Theorem 2 cave in yt. Similar to Lemma 2, we can establish the joint PROOF. From Lemma 3 and Equation (1), we can concavity of the objective function in (4), which estab- easily show that Vtðxt; PtÞVtðxt; PtÞ. From Corol- lishes that there exists a unique solution for the first- lary 1, we know that b \ 1, and hence by the definition order conditions (5) and (6), and that the base-stock pol- t of optimality it follows that Vtðxt; PtÞVtðxt; PtÞ icy is optimal. h \ Vtðxt; PtÞ. h A.9. Proof of Theorem 5 A.6. Proof of Theorem 3 PROOF. From Equation (4), we approximate the deriva- ^ PROOF.SinceJt is concave in zt for a given bt and Pt, tive of the value function Vt with respect to the current 0 ^ @V s þ c the base-stock policy is optimal. If h < h /a,thenitis t t t [ \ inventory xt as @x ¼ a h1fxt 0g þ r1fxt 0g. b i t possible that st þ ha ft 0. In this case, from Thereafter, = b = Equation (A3), for ct ct+1 and bt 1, it implies that b Q l [ @^ @ ^ ; E ½ tþ1 0. This results in the possibility of con- Jt st þ ct Q Vtþ1ðxtþ1 PÞ PjPt ¼ þ bEE 1 þ straint z + ≥ x + to be binding such that the myopic @^ PjPt @ t 1 t 1 zt a xtþ1 a policy is not optimal. h i st þ ct ft þ ctþ1 ¼ þ b h ðA7Þ a a ) A.7. Proof of Proposition 1 ZZ^ Mt PROOF. (a) When Cov(pt+1, st+1) > 0, using Equation þðr þ hÞ 1 þ wðpÞ/ðÞdpd ; @2 Jt a (A3) and Corollary 1, we obtain @ @c ¼ðr þ hÞ 0 zt c o c ^ ^ = ^ Atþ1 btf tþ1zt ðztxtÞ a ð Þ t Atþ1 zt w \ \ where Mt ¼ cð1b Þ . 2 ðMtÞ 0andb 1. Note that At+ zt >0. t c ð1 btÞ t 1 Goel and Tanrisever: Financial Hedging and Procurement 1942 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society

EquatingR (A7) to zero, and defining forward price of the input commodity agreed ^ ^ ^ + CðMtÞ¼ ð1 þ ÞWðMtÞ/ðÞd, we get CðMtÞ¼ at time t to be delivered at time t 1. a = s þ abh bfi þ c bc • At time t, the firm starts with x z d t t t tþ1. Similarly, ignoring the constraint t t 1 t abðr þ hÞ units of output inventory and q units of ≤ ≤ t1 0 bt 1, we obtain: input inventory to be delivered at time t through the input forward agreement engaged @^ @ ^ ; Jt bcr2 bc Q Vðxtþ1 PÞ o t ¼ 2 ð1 btÞþ EE ðf ptþ1Þ in at time 1. @ o PjPt @ t bt xtþ1 • The firm then decides the echelon output Q s þ c bcr2 bc tþ1 tþ1 o inventory level zt by processing the existing ¼ 2 ð1 btÞþ E ½ ðf ptþ1Þ o PjPt a t input (from the forward contract) and using Z Z ^ Mt the spot market to buy additional input com- o bc ½r ðf p þ ÞwðpÞdp zt xt þ t t 1 modity if needed. In particular, ð qt1Þ Z 0 a 1 denotes the additional input purchased from o w / zt xt þ þ h ðf ptþ1Þ ðpÞdp ð Þd the spot market and ðq Þ is the ^ t t 1 a Mt excess input after processing. Note that one bcr2 bc ; = ¼ 2 o ð1 btÞ Covðstþ1 ptþ1Þ a unit of input converts to a units of output, and Z Z ^ zt xt Mt hence a gives how much input is processed. bc o w ½r ðft ptþ1Þ ðpÞdp Z 0 Before we present the revised mathematical model, 1 below we first show an important lemma that simpli- o w / þ h ðf ptþ1Þ ðpÞdp ð Þd ^ t fies the exposition of the model. Mt

R R ^ ^ ; Mt o LEMMA B1. It is suboptimal to carry unprocessed input As a result,R defining gtðzt btÞ¼ ½r 0 ðft w 1 o w / commodity, i.e., it is optimal to sell unprocessed input ptþ1Þ ðpÞdp þ h ^ ðft ptþ1Þ ðpÞdp ð Þd , and set- Mt zt xt þ @^ commodity ðqt1 Þ in the spot market at the end Jt 2 a ting @ ¼ 0, we get 2r ð1 b Þ¼Covðs þ ; p þ Þ= bt o t t 1 t 1 of each period t. ^ a þ gðzt; btÞ. Now, solving this equation together with equation (A7) = 0 gives the unconstrained PROOF. The proof follows from our no-arbitrage solutions bt and zt. Since the objective function is assumption in commodity markets. Suppose at time concave, the optimal policy is given by (i)if t, the firm owns one unit of input commodity. Now, ; ^ ^ consider that the firm does not process it and carries bt 2½0 1 then zt ¼ zt and bt ¼ bt and (ii) if 0 ^ it to the next period (by paying a holding cost of h ) b 62½0; 1 then b ¼ maxfminfb ; 1g; 0g, and ^z is t t t t with the expectation of using it in the future. Alterna- obtained by solving Equation (A7) = 0 for tively, at time t, the firm may sell the commodity in ^ ; ; = bt ¼ maxfminfbt 1g 0g. Solutions for t T 1 the spot market and buy a forward contract for the h can be obtained analogously. delivery of one unit of input commodity next period. b i The cost of this alternative strategy is ft;tþ1 st, Appendix B. Value-Neutrality of Hedging where b is the one period risk-free discount rate. Input Procurement Note that both strategies provide the firm with one In this appendix, we provide a mathematical proof unit of input commodity next period. The first one 0 b i of the value-neutrality of input hedging. In addi- costs h and the second one costs ft;tþ1 st.When tion to the original modeling setup in the manu- we take the difference between the cost of the first 0 b i script, suppose that the firm may also buy forward and second strategies, we obtain st þ h ft;tþ1, contracts for the delivery of the input commodity which is called the marginal convenience yield (Goel to reduce its exposure to input price risk. In this and Gutierrez 2011, Williams and Wright 1991), and case, the timeline of events and decisions in the this quantity should always be non-negative; other- original manuscript will change as follows (below wise, it implies arbitrage. In particular, when we only list the changes due to forward buying of 0 b i \ st þ h ft;tþ1 0, investors may simultaneously the input commodity as the rest of the events and long the physical commodity and short the contract decisions are identical to the ones in the original and lock in a risk-free profit. This situation is known manuscript): as cash-and-carry arbitrage in commodity markets (see, • At time t, the firm decides the volume of input e.g., Geman 2005, Ch. 2.5). As a result, the no-arbit- forward contracts, qt, to be purchased for rage assumption leads to the fact that carrying unpro- delivery at time t + 1, in addition to other cessed input is suboptimal. That is, it is optimal to i zt xt þ existing decision variables. Let ft;tþ1 denote the clear the excess input at time t, ðqt1 a Þ in the Goel and Tanrisever: Financial Hedging and Procurement Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society 1943 spot market. If needed, the firm can buy the same @ ; ; ; ; Jtðxt zt bt Pt qtÞ b i b Q amount in the forward market and avoid paying the ¼ f ; þ E ½stþ1I f þ g @ t tþ1 PjPt dtþ1 zt aqt ztþ1 h qt marginal convenience yield. Q þbE ½s I ¼0 PjPt tþ1 fdtþ1 ztþaqtztþ1g The no-arbitrage assumption and the resulting Lemma B1 are important to prove the value neutrality @ ; ; ; ; Jtðxt zt bt Pt qtÞ i Q ¼bf ; þ bE ½s I ^ of input hedging. Using Lemma B1, now the revised t tþ1 PjPt tþ1 fptþ1 ptþ1g @qt optimization model with forward and spot procure- b Q þ E ½stþ1I f^ g¼0 ment of the input commodity reads as follows: PjPt ptþ1 ptþ1

; ; ; ; ; ; ; ð þb foÞ Vtðxt qt1 PtÞ¼ max Jtðzt bt qt xt qt1 PtÞ ^ Atþ1ðztþaqtztþ1Þ tþ1 t t ; ; where p þ ¼ . Then, zt bt qt2AtðxtÞ t 1 cð1btÞ ð1btÞ ð Þ @ ; ; ; ; B1 Jtðxt zt bt Pt qtÞ i Q ¼bf ; þ bE ½s ðI ^ n t tþ1 PjPt tþ1 fptþ1 ptþ1g @qt ; ; ; ; ; b Q ~ ; b i Jtðzt bt qt xt qt1 PtÞ¼ E Rtðbt ptþ1Þ f ; qt PjPt t tþ1 þI ^ ¼0 fptþ1 ptþ1g þ þ zt xt zt xt @ ð ; ; ; P ; Þ st qt þst qt Jt xt zt bt t qt i Q 1 1 ¼bf ; þ bE ½s ¼0: a a t tþ1 PjPt tþ1 @qt ct þ þ ðzt xtÞrðxtÞ hðxtÞ a o Q Finally, note that the no-arbitrage assumption also þbE Vtþ1ðzt dtþ1; qt; PÞ PjPt requires that the contracts are fairly priced, i.e., @ : i Q Jtð Þ i f ; ¼ E ½stþ1.Hence,itholdsthat @ ¼bf ; þ ; ; t tþ1 PjPt qt t tþ1 where, AtðxtÞ¼fzt xt 0 bt 1g, and VTðxT b Q þ þ E ½stþ1¼0forallqt, and the level of forward ; PjPt qT1 PTÞ¼sTqT1 þðpT TÞðxTÞ ðpT þ TÞðxTÞ . 8 The first term in the objective function, buying, qt, is irrelevant to firm value. This result may Q ~ also be generalized to other financial derivatives for bE Rtðbt; ptþ1Þ, represents the expected revenue PjPt trading the input commodity. h from the sales of the output commodity. The second fi q term, t;tþ1 t, gives the cost of forward procurement. Appendix C. The Choice of Demand The third and fourth terms, s ðzt xt q Þþ and t a t1 Parameters s ðq zt xtÞþ, are the cost of spot procurement and t t1 a Economics literature shows that ethanol demand the revenue from the spot sales of the input commod- presents strong price elasticity. In particular, ct ity, respectively. The fifth term, a ðzt xtÞ, is the pro- empirical research finds that price elasticity of etha- cessing cost. The remaining terms are identical to the nol demand ranges between 0.43 and 2.92 (Elo- ones in the original paper, except that the state space beid and Tokgoz 2008, Luchansky and Monks 2009, of the cost-to-go function now also includes the for- Roberts and Schlenkera 2013). Following these ward input commodity position qt. The following results in the literature, we consider a downward lemma describes the value neutrality of input hedging: sloping linear demand function for ethanol, i.e., = c dt+1 A pt+1. We normalize the firm specific LEMMA B2. Forward procurement of input commodity market size to A = 500 units, and consider three is value-neutral. levels of price sensitivity: high (c = 160), medium (c = 120) and low (c = 80). We select these levels of PROOF. For t = 1, ..., T 2, we can write the FOC c such that the resulting price elasticities are con- for q as the following: t sistent with the empirical observations. In particu- Q lar, in our demand model, ethanol price elasticities @ ; ; ; ; @bE Vtþ1ðzt dtþ1;qt;PÞ Jtðxt zt bt Pt qtÞ i PjPt ¼bf ; þ ¼0; for the mean price ranges from 0.56 (for c = 80) @q t tþ1 @q t t to 2.57 (c = 160). Note that our model is tested on @ ; ; ; ; futures price data for corn and ethanol from the Jtðxt zt bt Pt qtÞ b i @ ¼ ft;tþ1 Chicago Mercantile Exchange between 4/1/2005 qt and 12/31/2011. In this period, the average ethanol Q @b ztþ1xtþ1 þ ztþ1xtþ1 þ EPjP fstþ1ð qtÞ þstþ1ðqt Þ g price was $2.25/gallon. þ t a a ¼0; @qt Notes observing that, in this case, the order of expectation and derivative can be interchanged and using some 1According to Van Mieghem (2003), “Mitigating risk, or algebra we obtain: hedging, involves taking counterbalancing actions so that, Goel and Tanrisever: Financial Hedging and Procurement 1944 Production and Operations Management 26(10), pp. 1924–1945, © 2017 Production and Operations Management Society loosely speaking, the future value varies less over the pos- Devalkar, S. K., R. Anupindi, A. Sinha 2016. Dynamic risk man- sible states of nature. If these counterbalancing actions agement of commodity operations: Model and analysis. involve trading financial instruments, including short-sell- Working paper, Indian School of Business, India. ing, futures, options, and other financial derivatives, we Ding, Q., L. Dong, P. Kouvelis. 2007. On the integration of pro- call this financial hedging.” In our context, we refer to duction and financial hedging decisions in global markets. Oper. Res. 55(3): 470–489. financial hedging as selling the output commodity using a forward contract or an index-based contract. Dong, L., H. Liu. 2007. Equilibrium forward contracts on non- 2Perfect hedging is defined as a hedge that completely storable commodities in the presence of market power. Oper. Res. 55(1): 128–145. eliminates price risk. 3A jobber procures gasoline from the refineries, then pro- Elobeid, A., S. Tokgoz. 2008. Removing distortions in the us ethanol market: What does it imply for the United States and Bra- cesses it with additives in accordance with state and environ- zil? Am. J. Agr. Econ. 90(4): 918–932. mental regulations to sell the gasoline to unbranded retailers. 4 Federgruen, A., A. Heching. 1999. Combined pricing and inven- For models with similar assumptions on the correlation tory control under uncertainty. Oper. Res., 47(3): 454–475. between demand and price, see Kazaz (2004) and Seco- Franken, J., J. Parcell. 2003. Cash ethanol cross-hedging opportuni- mandi and Kekre (2014). ties. J. Agric. Appl. Econ. 35(3): 509–516. 5 In the economics literature, Pindyck (2004) defines mar- Froot, K. A., D. S. Scharfstein, J. C. Stein. 1993. Risk management: ginal convenience yield as the economic cost of storage. Coordinating corporate investment and financing policies. 6In section 5, we illustrate the computational procedure. J. 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