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1 Faculty & Research Working Paper he Interaction of echnology Choice and Financial Risk Management: An Integrated Risk Management Perspective Onur BOYABALI L. Beril OKAY 2006/54/OM

2 he Interaction of echnology Choice and Financial Risk Management: An Integrated Risk Management Perspective Onur Boyaatlı * and L. Beril oktay** Septemer 2006 * PhD Student, echnology and Operations Management at INSEAD, Boulevard de Constance, Fontaineleau Cedex, France, onur.oyaatli@insead.edu ** College of Management, Georgia Institute of echnology, Atlanta, GA, USA, eril.toktay@mgt.gatech.edu A working paper in the INSEAD Working Paper Series is intended as a means wherey a faculty researcher's thoughts and findings may e communicated to interested readers. he paper should e considered preliminary in nature and may require revision. Printed at INSEAD, Fontaineleau, France. Kindly do not reproduce or circulate without permission.

3 Astract his paper analyzes the integrated operational and financial risk management portfolio of a firm that determines whether to use flexile or dedicated technology and whether to undertake financial risk management or not. he risk management value of flexile technology is due to its risk pooling enefit under demand uncertainty. he financial risk management motivation comes from the existence of deadweight costs of external financing due to capital market imperfections. Financial risk management has a fixed cost, while technology investment incurs oth fixed and variale costs. he firm's limited udget, which depends partly on a tradale asset, can e increased y orrowing from external markets, and its distriution can e altered with financial risk management. In a parsimonious model, we solve for the optimal risk management portfolio, and the related capacity, production, financial risk management and external orrowing levels, the majority of them in closed form. We characterize the optimal risk management portfolio as a function of firm size, technology and financial risk management costs, product market demand variaility and correlation) and capital market external financing costs) characteristics. Our analysis contriutes to the integrated risk management literature y characterizing the optimal risk management portfolio in terms of a more general set of operational and financial factors; providing the value and limitation of operational and financial risk management y explicitly modeling their costs and enefits; demonstrating the interactions etween the two risk management strategies; and relating our theoretical results to empirical oservations. Key Words: Risk Management, Capacity Investment, Flexiility, Financing, Operational Hedging.

4 Introduction his paper is aout integrating operational and financial risk management and characterizing the drivers of the optimal integrated risk management portfolio. he two means of risk management are motivated y the existence of different market imperfection costs and utilize different tools. On the operational side, firms are exposed to demand and supply uncertainties in product markets. hese uncertainties, which we call forms of product market imperfection, impose supply-demand mismatch costs. o manage these costs, firms rely on different types of operational flexiility that provide a etter response to product market imperfections and counteralance the effect of supply-demand mismatch costs. On the financial side, firms do not always have sufficient internal cash flows to finance their operations and depend on external capital markets to raise funds. he transaction costs in capital markets ankruptcy costs, taxes, underwriter fees, agency costs etc.), which are forms of capital market imperfection, impose deadweight costs of external financing on firms. o manage these costs, firms rely on different types of financial instruments written on tradale assets with which their cash flows are correlated. hese financial instruments engineer the internal cash flows of firms to meet their optimal investment needs and counteralance the effect of external financing costs. Despite responding to two different types of market imperfection, operational and financial risk management interact with each other: he choice of operational risk management has implications for financial risk management and vice versa. herefore, operational and financial risk management should e viewed as constituting an integrated risk management portfolio. In practice, most corporate-level risk management programs of non-financial firms focus only on financial risk management Bodnar et al. 998). At the same time, a numer of large non-financial firms are ecoming more interested in operational solutions to manage their risk exposures Business Week 998). Due to the existence of oth product and capital market imperfections in practice, using oth risk management tools and doing so in an integrated fashion is important. he academic literature on risk management has largely documented the value and effectiveness of each risk management tool in isolation. Relatively little progress has een made in understanding their interactions and the main drivers of an optimal integrated risk management portfolio. he ojective of this paper is to enhance our understanding of integrated risk management. Our main contriutions are to model and analyze an integrated risk management prolem that i) yields structural results aout the characteristics and drivers of an optimal risk management portfolio; ii) provides managerial guidelines that can e used in designing risk management programs; and iii) can e used to generate hypotheses that account for operational and product market characteristics to a greater extent than the existing empirical risk management literature.

5 o this end, we model a udget-constrained manufacturer who produces and sells two products. Product demands are random, which is the product market imperfection, and correlated. he firm chooses etween flexile and dedicated technologies that incur fixed and variale costs, and determines the capacity level of the chosen technology. Because of its risk pooling enefit, the flexile technology is the firm s operational risk management tool. he firm s limited udget partially depends on a perfectly tradale asset. he firm can relax its udget constraint y orrowing from external markets, ut orrowing incurs external financing costs that originate from capital market imperfections. Forwards written on the asset price can e used as the firm s financial risk management tool to alter the udget distriution and help counteralance the effect of external financing costs. he fixed and variale investment costs of flexile technology are higher than those of dedicated technology, and financial risk management has a fixed cost. herefore, it may e undesirale to use these tools despite their value. In this rich ut parsimonious model, we answer the following research questions:. What is the optimal risk management portfolio of the firm defined as choosing flexile versus dedicated technology, and engaging in financial risk management or not) as a function of firm size, technology and financial risk management costs, product market conditions demand variaility and correlation) and capital market conditions external financing costs)? 2. What are the fundamental drivers of the optimal risk management portfolio? 3. Are financial and operational risk management complements or sustitutes? 4. What are the consequences of the interaction etween financial and operational risk management? What is the effect of financial risk management on operational decisions? 5. Can our results e used to support or refine existing empirical research? We derive the optimal integrated risk management portfolio and the related capacity, production, financial risk management and external orrowing levels, the majority of them in closed form. Our analysis reveals that there are three fundamental drivers that explain the optimal portfolio choice: the roustness of the optimal capacity investment level to product market conditions, the level of reliance on external financing and the opportunity cost of financial risk management. hese drivers work in opposite directions for large and small firms due to differences in their orrowing needs under financial risk management. As a result, the size of the firm is highly relevant the same underlying conditions lead to different optimal portfolio choices as a function of firm size. Conversely, it may e optimal for small and large firms to choose the same optimal portfolio for 2

6 entirely different reasons. hese results generate managerial insights and guidelines for designing an integrated risk management program. Our analysis clearly illustrates the intertwined nature of operational and financial risk management strategies. We show that firms can use financial risk management for speculative purposes with flexile technology, whereas they may prefer to hedge with dedicated technology. he reason is that firms with a limited internal udget can optimally increase their asset risk exposure to cover the higher fixed cost of flexile technology and invest in capacity to generate revenue. We demonstrate that engaging in financial risk management may induce the firm to change its technology decision; flexile technology and financial risk management can e complements or sustitutes. his is a direct consequence of the difference etween each technology regarding the counteralancing value of financial risk management with respect to external financing costs. We relate our theoretical findings to empirical oservations concerning risk management practices of firms. Our results provide theoretical support for some oservations and highlight additional trade-offs in others. For example, we estalish that the value of financial risk management increases in external financing costs only for large firms and not for small firms. his is in contrast to existing understanding that this is true for any firm. We show that if firms use financial instruments only for hedging purposes, it is optimal for small firms to not undertake financial risk management; existing arguments attriute this oservation only to the fixed cost of estalishing a financial risk management program. he distinction we make etween large and small firms, and our results related to the effect of technology and product market characteristics on the risk management portfolio provide new hypotheses that can e tested empirically. We note that all of the results otained are analytical and are valid for any demand and asset price distriution with positive and ounded support. With these results, we contriute to the growing operations management literature that incorporates financial considerations in operational decision making. In the next section, we provide more detail aout how our work contriutes to the existing literature. In 3, we descrie the model and discuss the asis for our assumptions. 4 analyzes the optimal strategy of the firm, culminating in a characterization of the optimal risk management portfolio. 5 and 6 flesh out the results of the previous section to descrie the impact of various factors on the optimal portfolio choice. We analyze the value and effect of integrated decision making y comparing with the non-integrated enchmark in 7. In 8, we discuss the roustness of our results to our assumptions. 9 concludes. 3

7 2 Literature Review In this section, we review the streams of literature related to our paper and delineate our contriutions to each stream. he operations management literature has documented the risk management value of operational flexiility. Starting with the influential studies of Huchzermeier and Cohen 996), Cohen and Huchzermeier 999) and Kouvelis 999), this stream delineates the value of various operational flexiilities e.g. technology flexiility, geographical diversification, postponement) in the firm s network structure, referred as operational hedges, in managing demand-side product market imperfections Van Mieghem 2003, 2006, Aytekin and Birge 2004, Kazaz et al. 2005). We refer the reader to Boyaatlı and oktay 2004) for a recent review of papers in this stream. A numer of papers take this analysis further and study the interaction etween different operational flexiilities of firms Bish and Wang 2004, Goyal and Netessine 2005, Chod et al. 2006a, Dong et al. 2006). his stream of papers often implicitly) assumes perfect capital markets and hence there are neither deadweight costs of external financing nor any value for financial risk management. We demonstrate the effect of external financing costs and financial risk management on the value of operational risk management, and document several interactions etween operational and financial risk management. he finance literature on risk management, in turn, focuses on financial risk management e.g. forwards, options, etc.) and typically does not consider product market imperfections and operational risk management. he majority of this literature i) provides different explanations for the existence of financial risk management that are ased on different types of capital market imperfections; or ii) focuses on the optimal use of financial instruments in a variety of settings. Since the focus of these papers is financial risk management, the interactions etween the two risk management strategies are not studied. We refer the reader to Fite and Pfleiderer 995) for a review of the first stream and Brown and oft 200) for a review of the second. here are a few theoretical papers that study the firm s integrated risk management portfolio choice. In operations, Chod et al. 2006) and Ding et al. 2005) analyze the interaction etween financial risk management and different types of operational flexiility, where financial risk management is motivated y the risk aversion of the decision maker. Chod et al. 2006) analyze whether financial risk management complements or sustitutes operational flexiility. hey demonstrate that this depends on whether the optimal flexiility level increases or decreases with financial hedging. We show that financial and operational risk management can again e either complements or sustitutes under external financing, ut the driver is firm size. Ding et al. 2005) is closest to our paper in terms of its research ojective. hey study the integrated operational postponement) and financial risk management currency options) decisions of a multinational firm and delineate 4

8 the value of each risk management strategy under demand and exchange rate uncertainty. In a numerical study, they show that engaging in financial risk management alters the roustness of operational decision variales capacity) with respect to demand variaility and changes the strategic decision variales gloal supply chain structure). We demonstrate similar results analytically. In addition, we analyze the effect of external financing costs, demand correlation and firm size on the optimal risk management portfolio. Incorporating the costs of each risk management strategy enales us to also explore the limits of their use. In finance, Mello et al. 995) and Chowdry and Howe 999) model a multinational firm that has sourcing flexiility sourcing from oth domestic and foreign production facilities is possile) and that uses financial instruments to manage the exchange rate risk. hese papers demonstrate the value of sourcing flexiility in conjunction with financial risk management. he focus of these papers is mainly financial risk management, and they do not consider a detailed representation of the firm s operations. Our analysis generates a numer of insights aout integrated risk management in a more detailed model of firm operations. All of these papers assume that financial risk management is costless, in which case financial risk management is trivially included in the optimal risk management portfolio since it has positive value. In contrast, the fixed cost of financial risk management e.g. software and personnel costs) can e a deterrent in practice. Motivated y this oservation, we incorporate a positive fixed cost for engaging in financial risk management. his makes whether to engage in financial risk management or not a nontrivial question. he answer to this question goes eyond a oundary invest/do not invest decision divorced of the other decision variales: Under a udget limit and external financing costs, the effective cost of financial risk management is larger than its fixed cost ecause the firm may need to orrow an additional amount as a result of incurring this fixed cost. herefore, engaging in financial risk management has an impact on the level of other decisions variales. Similarly, the fixed cost of the technology investment has a sutle effect on the optimal portfolio. hese interactions add interesting dimensions to the optimal risk management portfolio. In contrast to the theoretical finance research, the empirical finance literature has paid more attention to operational risk management, as reviewed in Smithson and Simkins 2005). his literature either statistically or qualitatively attriutes a numer of empirical oservations to the firm s operational risk management capailities, which we discuss these oservations in detail in 5 and 6. We contriute to this stream in a numer of ways: We provide theoretical support for some empirical oservations and delineate additional trade-offs in some others; we provide alternative explanations to some oservations that are ased on the interplay etween the two risk management strategies; and we identify potential future empirical research avenues. 5

9 In summary, our major contriution is to the integrated risk management literature. We contriute to this literature y i) characterizing the optimal risk management portfolio in terms of a more general set of operational and financial factors; ii) providing the value and limitation of each risk management strategy y explicitly modelling the costs and enefits of each strategy; iii) demonstrating the interactions etween the two risk management strategies; and iv) relating our theoretical predictions to empirical oservations. Note that we have made a distinction etween papers that augment the financial risk management analysis with operational risk management versus operational decisions only. Up to this point, we focused on the former, which involves a type of flexiility that can e used for risk management and susumes a numer of operational decisions). he latter focuses only on operational decisions in analyzing financial risk management. In the latter stream, we highlight Froot et al. 993) from the finance literature since their modelling of the financial risk management motive is the same as in our paper. he authors use a concave increasing investment cost function to capture the operational dimension. hey demonstrate that financial risk management adds value y generating sufficient internal funds to finance operational investments when there exist deadweight costs of external financing. We extend their framework y formalizing the operational investments y incorporating product market characteristics, and technology and production decisions), and y imposing a cost for financial risk management. We illustrate that some of their predictions continue to hold, whereas some change due to the interplay etween financial and operational decisions. In the operations literature, Birge 2000), Chen et al. 2004), Gaur and Seshadri 2005), and Caldentey and Haugh 2005, 2006) document the value of financial risk management when the operating cash flows are correlated with a financial index. he financial risk management rationale is the risk-aversion of the decision maker in the latter three papers. Among these papers, we can link our paper to Caldentey and Haugh 2005) who motivate financial risk management y imposing a udget constraint on the firm, ut without the possiility of external financing. his can e viewed as a special case of our model: When the external financing cost is sufficiently high, the firm never orrows. he external orrowing feature of our model is an important determinant of the risk management portfolio: the reliance on external orrowing determines the technology choice and the value of financial risk management with each technology. Finally, our work is related to two other streams in operations management. he stochastic capacity investment literature analyzes the question of flexile versus dedicated technology choice with demand-side uncertain demand) and supply-side unreliale supply) product market imperfections. We refer readers to Van Mieghem 2003) for an excellent review and to omlin and Wang 6

10 2005) for a specific focus on the supply-side imperfection. As highlighted in Van Mieghem 2003), stochastic capacity models often implicitly) assume perfect capital markets. We demonstrate that under financing frictions, there exist additional trade-offs in technology choice: the level of reliance on external financing and the value of financial risk management with each technology. A second stream relaxes the perfect capital market assumption and models the firm s joint financial and operational decisions Lederer and Singhal 994, Buzacott and Zhang 2004, Baich and Soel 2004, Xu and Birge 2004 and Baich et al. 2006). he primary focus of these papers is to analyze the effect of external financing costs and the financing decision on operational decisions. hey demonstrate the value of integrated financing and operational decision making. We extend the interaction argument in these papers y considering another facet of financial decisions, financial risk management. Our analysis reveals that the effect of external financing costs are largely dependent on the value of financial risk management and that technology choice is a key determinant of the firm s reliance on external markets: the higher investment cost of flexile technology requires higher external financing levels than dedicated technology. 3 Model Description and Assumptions We consider a monopolist firm selling two products in a single selling season under demand uncertainty. he firm chooses the technology dedicated versus flexile), the capacity investment level and the production level so as to maximize expected shareholder wealth. Differing from the majority of traditional stochastic technology and capacity investment prolems, we model the firm as eing udget constrained, where the udget partially depends on a hedgeale market risk. We allow the firm to undertake financial risk management to hedge this market risk, and to orrow from external markets to augment its udget. After operating profits are realized, the firm pays ack its det; default occurs if it is unale to do so. We model the firm s decisions as a three-stage stochastic recourse prolem under financial market and demand risk. In stage 0, the firm chooses its integrated risk management portfolio. he firm decides its technology choice flexile or dedicated), whether to engage in financial risk management, and if so, its financial risk management level under demand and financial market risk. In stage, the financial market risk is resolved and the financial risk management contract if any) is exercised; these two factors determine the internal cash level of the firm. he firm then determines the level of external orrowing and makes its capacity investment using its total udget internal cash and orrowed funds). In stage 2, demand uncertainty is resolved and the firm chooses the production quantities for each product. Susequently, the firm either pays ack its det or defaults. In the remainder of this section, we define the firm s ojective and discuss 7

11 the assumptions concerning each decision epoch in detail. We discuss the roustness of our results with respect to the majority of these assumptions in 8. Assumption he firm maximizes the expected stage 2) shareholder wealth y maximizing the expected value of equity. he shareholders are assumed to e risk-neutral and the risk-free rate r f is normalized to 0. Shareholders have limited liaility. he main goal of corporations is to maximize shareholder wealth. he expected shareholder wealth is a function of the expected cash flows to equity of the firm and the required rate of return of the shareholders. By assuming the risk neutrality of shareholders, we focus on maximizing the expected equity value of the firm. he required rate of return is the risk-free rate, which is normalized to 0 y assumption. Although the shareholders are risk-neutral, the existence of external financing costs creates an aversion to the downside volatility of the internal cash level in stage : he firm may e forced to underinvest in capacity at low internal cash level realizations ecause of external financing costs. his creates a motivation for undertaking firm-level financial risk management activities Froot et al. 993). 3. Stage 0 In this stage, the firm determines its technology choice {D, F }, whether to use financial risk management, and if so, the financial risk management level H under financial market and demand uncertainty. he flexile technology F ) has a single resource that is capale of producing two products. he dedicated technology D) consists of two resources that can each produce a single product. Assumption 2 echnology has fixed F ) and variale c ) capacity investment costs. he fixed cost of the flexile technology is higher than that of the dedicated technology; F F F D. he variale capacity investment cost of the two dedicated resources are identical. Both technologies are sold immediately at the end of the selling season at a reduced price of γ F where γ is the salvage rate and 0 γ <. he firm commits to technology in this stage whose fixed cost is incurred in stage. Since flexile technology is generally more sophisticated than dedicated technology, the fixed cost of flexile technology is assumed to e higher. he stage 0 commitment of the firm to technology choice can e justified y the lead time of the acquisition if outsourced) or the development time if uilt in-house) of the technology. When the technology is resold, ecause of depreciation and liquidation costs, the fixed cost of the technology cannot e fully retrieved γ < ). 8

12 Assumption 3 he firm uses a loan commitment contract to finance its capacity investment and to cover the fixed cost of the committed technology. he terms of the contract are known at stage 0, while orrowing takes place at stage. Loan commitment is a promise to lend up to a pre-specified amount at pre-specified terms. In practice, most short-term industrial and commercial loans in the US are made under loan commitment contracts Melnik and Plaut 986). At stage 0, the firm owns the right to a loan contract that can e exercised in stage. We discuss the characteristics of the loan commitment contract in Assumption 6 of stage. Assumption 4 At stage 0, the firm has rights to a known internal stage endowment ω 0, ω ). Here, ω 0 represents the cash holdings and ω represents the asset holdings of the firm. he asset is a perfectly tradeale asset that has a known stage 0 price of α 0 and random stage price of α. he random variale α has a continuous distriution with positive support and ounded expectation α. With this assumption, in stage 0, the firm knows that the value of its endowment will e ω 0 + α ω in stage, where α is random; this is the financial market risk in our model. his representation is consistent with practice: In general, firms hold oth cash and tradale assets on their alance sheet, such as a multinational firm that has pre-determined contractual fixed payments denominated in oth domestic and foreign currency, or a gold producer that produces a certain level of gold that is exposed to gold price risk. In these examples, the asset price α represents the exchange rate and the gold price in stage, respectively. Although the cash and the asset holdings are certain, the price of the asset makes the stage value of the internal endowment random. he firm can use financial risk management tools to alter the distriution of this quantity. Assumption 5 he firm uses forward contracts written on asset price α to financially manage the market risk. here is a fixed cost F F RM of engaging in financial risk management that is incurred in stage 0 y transferring the rights of the firm s claims ω 0 and ω, in proportions β and β. Forward contracts are fairly priced. We restrict the numer of forward contracts H such that the firm does not default on its financial transaction in stage. Forward contracts are the most prevalent type of financial derivatives used y non-financial firms Bodnar et al. 995). he fixed cost of financial risk management F F RM ) includes the costs of hiring risk management professionals, and purchasing hardware and software for risk management; it is independent of the numer of forward contracts used. In a recent survey, non-financial firms report this fixed cost as the second most important reason for not implementing a financial risk management program Bodnar et al. 998). Since we focus on loan commitment contracts and 9

13 the firm can orrow from external markets only at stage, F F RM is deducted in stage 0 from the firm s stage endowment y transferring the rights of the claims ω 0 and ω with β and β proportions respectively. In other words, rights for βf F RM of the cash holdings and β)f F RM α 0 of the asset holdings are transferred in stage 0. his leaves the firm with a stage endowment of ω0 F RM, ω F RM ) =. ω 0 βf F RM, ω β α 0 F F RM ). he firm can only engage in financial risk ). Since management if these quantities are non-negative, or equivalently, if F F RM min ω0 β, α 0ω β the firm is exposed to external financing costs in stage, there is an opportunity cost associated with F F RM : he firm has lower internal cash in stage and may need to orrow more from external markets after paying for F F RM. he fair-pricing assumption ensures that the firm can only affect the distriution of its udget in stage and not its expected value y financial risk ] 0 management. We restrict the feasile set of forwards to the range α, ω F RM. Within this ωf RM [ range of forwards the firm never defaults on its financial transaction in stage. his ensures that we can use default-free prices in forward transactions. 3.2 Stage In stage, the market risk α is resolved. he value of the firm s internal endowment and the exercise of the financial contract if any) determine the firm s udget B. In this stage, the firm can raise external capital if the udget is not sufficient to finance the desired capacity investment. he firm determines the amount of external orrowing and the capacity investment level under demand uncertainty. Assumption 6 With the loan commitment contract, the firm can orrow up to credit limit E from a unit interest rate of a > r f = 0. he face value of the det e + a) is repaid out of the firm s assets in stage 2. he firm has physical assets of value P e.g. real estate) that are pledged to the creditor as collateral. he loan is secured fully collateralized), i.e. E + a) P. he physical assets are illiquid; they can only e liquidated with a lead time. he value of the physical assets P is sufficient to finance the udget-unconstrained optimal capacity investment level of the firm. he salvage value of technology γ F ) cannot e seized y the creditor among the firm s assets. Any possile costs that may e incurred in the orrowing process y the creditor e.g. fixed ankruptcy costs) are charged ex-ante to the firm in a. We assume that the loan commitment is fully collateralized y the firm s physical assets P, i.e. E + a) P, since most ank loans are secured y the company s assets Weidner 999) and modelled as such Mello and Parsons 2000). Although the loan is fully collateralized, if the firm s final cash position is not sufficient to cover the face value of the det, the firm cannot immediately liquidate the collateral assets to repay its det since the physical assets are illiquid. Under limited 0

14 shareholder liaility, this leads to default, in which case the creditor can seize these physical assets, liquidate them and use their liquidation value to recover the loan. he salvage value of technology is assumed to e non-seizale; the creditor cannot use the salvage value to recover the face value of the loan. We also assume that the creditor s transaction costs associated with default e.g. fixed ankruptcy costs) are charged to the firm ex-ante in the unit orrowing cost. A positive unit financing cost a > 0) and a credit limit less than the value of the collateralized asset E < P ) can e interpreted as the deadweight costs of external financing that arise from capital market imperfections: If the capital markets are perfect i.e. there are no transaction costs, default related costs, information asymmetries), then the contract parameters are determined such that the loan is fairly valued in terms of its underlying default exposure. Since we focus on a collateralized loan, in the asence of default-related deadweight costs, there is no risk for the creditor associated with default. Consequently, in perfect capital markets, the fair unit financing cost of the loan commitment contract would e the risk-free rate a = 0), and the credit limit would e the value of collateralized physical asset E = P ). If there are capital market imperfections, then a > 0 and E < P would e otained in a creditor-firm interaction. herefore, although we assume that they are exogenous parameters in this paper, a positive unit financing cost a > 0) and a credit limit less than the value of the collateralized asset E < P ) can e interpreted as capturing the deadweight costs of external financing that arise from capital market imperfections. his parallels the assumptions in Froot et al. 993) who take the external financing costs as exogenous and state that they can e argued to arise from deadweight costs associated with capital market imperfections. In a creditor-lender equilirium, the endogenous) contract parameters need not e identical for each technology. In 8, we discuss conditions under which our results with identical contract parameters are valid in a general equilirium setting, and refer the reader to Boyaatlı and oktay 2006) for an analysis of equilirium contract a, E ) for each technology in a creditor-firm Stackelerg game. o conclude, we note that our external financing cost structure provides a parsimonious model that is consistent with real-life practices; allows us to implicitly capture capital market imperfections and enales us to preserve tractaility. 3.3 Stage 2 In this stage, demand uncertainty is resolved. he firm then chooses the production quantities equivalently, prices) to satisfy demand optimally. If the firm is ale to repay its det from its final cash position, it does so and terminates y liquidating its physical assets. Otherwise, default occurs.

15 In this case, ecause of the limited liaility of the shareholders, the firm goes to ankruptcy. he cash on hand and the ownership of the collateralized physical assets are transferred to the creditor. he firm receives the remaining cash after the creditor covers the face value of the det from the seized assets of the firm. Assumption 7 Price-dependent demand for each product is represented y the iso-elastic inversedemand function pq i ; ξ ) = ξ i q / i for i =, 2. Here,, ) is the constant elasticity of demand, and p and q denote price and quantity, respectively. ξ i represents the idiosyncratic risk component. ξ, ξ 2 ) are correlated random variales with continuous distriutions that have positive support and ounded expectation ξ, ξ 2 ) with covariance matrix Σ, where Σ ii = σ 2 i and Σ ij = ρσ σ 2 for i j and ρ denotes the correlation coefficient. ξ, ξ 2 ) and α have independent distriutions. he marginal production costs of each product at stage 2 are 0. 4 Analysis of the Firm s Optimal Risk Management Portfolio In this section, we descrie the optimal solution for the firm s technology choice, and the levels of financial risk management, external orrowing, capacity investment and production. A realization of the random variale s is denoted y s and its expectation is denoted y s. Bold face letters represent vectors of the required size. Vectors are column vectors and denotes the transpose operator. Vector exponents are taken componentwise. xy denotes the componentwise product of vectors x and y with identical dimensions. We use the following vectors throughout the text: ξ = ξ, ξ 2 ) product market demand), K F = K F flexile capacity investment) and K D = K D, K2 D ) dedicated capacity investment). P r denotes proaility, E denotes the expectation operator, χ.) denotes the indicator function with χϖ) = if ϖ is true, x) +. = maxx, 0) and Ω 0. = Ω 0 Ω. Monotonic relations increasing, decreasing) are used in the weak sense otherwise stated. ale summarizes the decision variales. ale 6 that summarizes other notation and all proofs are provided in Appendix A. We solve the prolem y using ackward induction starting from stage 2. Stage Name Meaning Stage 0 {D, F } echnology choice, dedicated or flexile H Numer of forwards with technology Stage e Borrowing level with technology K Capacity investment level with technology Stage 2 Q Production quantity with technology ale : Decision variales y stage 2

16 4. Stage 2: Production Decision In this stage, the firm oserves the demand realization ξ and determines the production quantities Q = q, q2 ) within the existing capacity limits to maximize the stage 2 equity value. Proposition he optimal production quantity vector in stage 2 with technology {D, F } for given K and ξ is given y Q D = K D, Q F = K F ξ + ξ 2 Since the unit production cost is zero, the firm optimally utilizes the entire availale capacity. With dedicated technology, the optimal individual production quantities are equal to the availale capacity levels for each product. With flexile technology, the firm allocates the availale capacity K F etween each product in such a way that the marginal profits for each product are equal. ξ. 4.2 Stage : Capacity Choice and External Financing In this stage, the firm exercises the forward contract H if the firm has already decided to engage in financial risk management at stage 0) and oserves the asset price α. With fair pricing, the strike price of the forward is equal to α. he stage udgets with and without financial risk management are therefore B F RM α, H ). = ω F RM 0 + α ω F RM H ) + α H and B F RM α ). = ω 0 + α ω, respectively. We henceforth suppress α and H and denote the availale udget realization y B [0, ). For given B and, the firm determines the optimal capacity investment level K B) and the optimal external orrowing level e B). Proposition 2 he optimal capacity investment vector K B) and the optimal external orrowing level e B) for technology {D, F } with a given udget level B are K 0 if B Ω 0. = { B : B c K 0 + F } K if B Ω. = { B : c K + F B < c K 0 + F } K B) = K if B Ω 2. = { B : B B, c K + F E B < c K + F } K if B Ω 3. = { B : B B < c K + F E } 0 if B Ω 4. = { B : 0 B < B } e B) = c K B) + F B ) + ) χ B > B. 2) Here, χ.) is the indicator function and B is the unique udget threshold for technology {F, D} such that the firm optimally does not orrow e B) = 0) and does not invest in capacity K B) = 0) for B B. 3 )

17 he explicit expressions for the capacity vectors in the proposition are given in 28) in the proof. K 0 is the optimal capacity investment in the asence of a udget constraint the udget-unconstrained optimal capacity ). If the udget realization is high enough to cover the corresponding cost F + c K 0 B Ω 0 ), then K B) = K 0 with no orrowing. Otherwise, for each udget level B Ω 234, the firm determines to orrow or not y comparing the marginal revenue from investing in an additional unit of capacity over its availale udget with the marginal cost of that investment including the external financing cost, +a)c. For B Ω, the udget is insufficient to cover K0, and the marginal revenue of capacity is lower than its marginal cost. herefore, the firm optimally does not orrow, and only purchases the capacity level K that fully utilizes its udget B. For B Ω 23, the marginal revenue of capacity is higher than its marginal cost + a)c. herefore, the firm optimally orrows from external markets to invest in capacity. K is the optimal capacity investment with orrowing, in the asence of a credit limit the credit-unconstrained optimal capacity ). If the udget realization and the credit limit can jointly cover its cost, K is the optimal capacity investment; otherwise, the firm purchases the capacity level K that fully utilizes its udget and its credit limit. For B Ω 4, the firm must orrow to e ale to invest in technology, ut the total cost of the capacity that can e purchased with the remaining B + e F cannot e covered y the expected revenue it generates for any e. herefore, the firm optimally does not orrow and does not invest in capacity. Appendix B characterizes B and provides a closed-form expression for a suset of parameter values. he optimal external orrowing level e B) is such that the firm orrows exactly what it needs to cover its capacity investment. Since production is costless, the firm does not incur any further costs eyond this stage. Moreover, since the face value of the det is always deducted from the firm s assets, the firm cannot transfer wealth from the creditor to shareholders y orrowing more money than what is needed for its capacity investment. herefore, the firm only orrows for funding the capacity investment, which yields 2). he optimal expected stage ) equity value of the firm with a given udget level B, π B), is otained in closed form Equation 34 in Appendix A). Corollary π B) strictly increases in B for B 0, and is concave in B on [ B, ). It is not concave in B on [0, ). As we will see in 4.3., this structure has implications for the optimal financial risk management level. 4

18 4.3 Stage 0: Financial Risk Management Level and echnology Choice In this stage, the firm decides on the technology choice {D, F }, whether to engage in financial risk management FRM) and if so, the financial risk management level H, the numer of forward contracts written on the stage asset price α. he optimal expected stage 0) equity value Π W) as a function of the internal stage ) endowment W = ω 0, ω ) is Π W) = max { Λ F RM, Λ F RM, ω 0 + α ω + P }. 3) Here, Λ F RM and Λ F RM denote the expected stage 0) equity value of the etter technology with and without financial risk management FRM), respectively, where Λ F RM is calculated at the optimal risk management level H. In 3), the firm compares these equity values with ω 0+α ω +P, the expected stage 0) equity value of not investing in any technology derives H, characterizes the optimal technology choice with and without FRM, and characterizes the solution to 3). his characterization is valid for any continuous α and ξ distriution with positive support and ounded expectation Financial Risk Management he expected direct gain from the financial contract is 0 due to the fair pricing assumption. At the same time, financial risk management affects the distriution of the stage udget B F RM α, H ), which is used to finance the firm s capacity investment after paying for the fixed cost commitment. In choosing H, the goal of the firm is to engineer its udget to maximize the expected gain from the technology commitment made in stage 0. When H > 0 H < 0), the firm decreases increases) its exposure to the asset price risk α. Following Hull 2000, p.2), we refer to the first case as financial hedging, and to the second as financial speculation. We call H = ω F RM full ωf RM 0 hedging ecause it isolates the udget from the underlying risk exposure. We call H = α full speculation ecause it maximizes the firm s asset risk exposure within the feasile range of forward contracts. Proposition 3 characterizes H. Proposition 3 here exists a unique technology fixed cost threshold F such that i) If F F, then the firm fully hedges H = ωf RM ). ii) If F > F then. if ω F RM 0 + α ω F RM B, then full speculation is optimal H 2. if ω F RM 0 + α ω F RM > B, H {{ dependent. B H < ω0 F RM 5 α } { ω F RM ωf RM 0 = α ); } } and is distriution

19 he structure of π is key to these results. If π is a concave function of the availale udget B on [0, ), then full hedging is optimal. his follows y Jensen s inequality: For concave π, E [π B F RM α, H ))] π E[B F RM α, H ]) = π ω F RM 0 + α ω F RM ), the equity value under full hedging. However, π is not concave if Ω 4, i.e. if there is a udget range in which the firm would not invest in capacity in stage despite having made the technology investment in stage 0. his happens when the fixed cost of the technology investment is too high to leave sufficient funds for a profitale capacity investment. Below the fixed cost threshold F, Ω 4 =, π is concave, and full hedging is optimal. Aove this threshold, H depends on the expected value of the internal stage ) endowment ωf RM 0 + αω F RM, which is also the udget availale to the firm under full hedging. When this value is lower than B, the firm would optimally not invest in capacity if it were to fully hedge. Instead, the firm optimally chooses to increase its exposure as much as possile so as to maximize the proaility of realizing high-udget states in which it is ale to invest in capacity and generate revenue from its technology investment. his also increases the proaility of realizing low-udget states, ut the outcome in those states does not change - no capacity investment is optimal.) For ω F RM 0 + α ω F RM > B, the optimal risk management level is distriution dependent and a full characterization is not possile without making further assumptions echnology Choice We now turn to the technology selection prolem with and without financial risk management. he choice etween flexile versus dedicated technology is determined y a unit cost threshold that makes the firms indifferent etween the two technologies. Proposition 4 For given technology cost parameters F, γ ) and financing cost scheme a, E), and under the financial risk management level H for each technology, there exists a unique variale cost threshold c F c D, H ) such that when c F < c F c D, H ) it is more profitale to invest in flexile technology = F ). Without financial risk management, there is a parallel threshold c F c D, 0). hese thresholds increase in c D, F D, γ F and demand variaility σ), and they decrease in F F, γ D and the demand correlation ρ) 2. With symmetric fixed costs and salvage rates, c F c D, H ) = c F c D, 0) = c S F c D ) = c D [ E ξ + ξ2 ) E [ξ ] + E [ξ 2 ] ] + c D, 4) where the equality only holds if the product markets are deterministic σ = 0), or the product markets are perfectly positively correlated ρ = ) and ξ has a proportional ivariate distriution. 6

20 he comparative statics results developed here are used in 6 to analyze the drivers of the firm s optimal risk management portfolio. he threshold c S F c D) is independent of unit financing cost a, credit limit E, and engaging in financial risk management. Although these factors do have an effect on the equity value of each technology, the differential value of this effect is never sufficient to induce the firm to alter its technology decision. his threshold is independent of α and valid for any distriution of ξ. he threshold c S F c D) is a variant of the mix flexiility threshold in Chod et al. 2006a), and has the same structure. It is interesting to note that the same threshold structure is valid despite the existence of external financing costs and financial risk management policy in the symmetric cost case. Due to the risk pooling enefit of flexile technology, we have c S F c D) c D. Proposition 4 shows that there is no risk pooling enefit c S F c D) = c D ) only if the product market demand is deterministic, or the multiplicative demand uncertainty is perfectly positively correlated and it has a proportional ivariate distriution ρ =, σ = kσ 2 and ξ = kξ 2 for k > 0). Flexile technology can have risk pooling value even if the product markets are perfectly positively correlated. his oservation is in the spirit of Proposition 6 in Van Mieghem 998), which is ased on the pricedifferential of two products in a price-taking newsvendor setting. In our case, the value comes from the fact that for non-proportional ivariate distriutions, the optimal production quantities with the flexile technology in stage 2 are state dependent such that there is still value from production switching at different ξ realizations Optimal Portfolio Choice he cost thresholds developed in Proposition 4 reveal which technology is more profitale with and without financial risk management, ut we need several more elements to fully characterize the solution to 3). Four more cost thresholds achieve this purpose. hese thresholds are summarized in ale 2 and derived in the Appendix A. he algorithm to solve 3) is as follows: We use the variale cost thresholds derived in Proposition 4 to determine the optimal technologies yielding Λ F RM and Λ F RM. Using the fixed F RM technology cost thresholds F and F F RM, if we determine that not investing in any technology dominates either exactly one or oth of Λ F RM and Λ F RM, 3) is solved. Otherwise, we need to compare Λ F RM and Λ F RM. If the same technology is optimal in oth cases, then the fixed financial risk management cost threshold F F RM is used to determine whether FRM or no FRM is optimal with that technology and 3) is solved. If different technologies are optimal with and without FRM, then c c, H, 0) is used to determine the optimal solution. his completes the characterization of the optimal portfolio. he next three sections highlight and discuss a series of 7

21 hreshold c F c D, 0) c F c D, H ) c S F c D) F RM F F F RM F F RM c c, H, 0) Usage Comparison etween technologies without engaging in FRM Comparison etween technologies with optimal FRM Comparison etween technologies with symmetric F and γ Comparison etween investing in without FRM and not investing in any technology Comparison etween investing in with FRM and not investing in any technology Comparison etween FRM and no FRM with technology Comparison etween technology with FRM and the other technology ) without FRM ale 2: hresholds used in solving for the firm s optimal strategy. he first three were derived in Proposition 4 and the last four are derived in Propositions, 2 and 3 in the Appendix. insights that can e otained from this analysis. 5 Oservations Concerning the Optimal Risk Management Portfolio In this section, we make several oservations aout the structure of the optimal risk management portfolio and its managerial implications. We start with an oservation that illustrates the limits of the value of each risk management strategy. Corollary 2 If capital markets are perfect, F F F RM = F D F RM = 0: financial risk management has no value. If product markets are perfect, and asent a fixed cost or salvage value advantage, c F c D, H ) = c F c D, 0) = c D : flexile technology has no value. Without capital market imperfections, the firm is not exposed to deadweight costs of external financing, as discussed in Assumption 6. In this case, financial risk management does not have any value. his is consistent with the decoupling of operational and financial decisions in perfect capital markets Modigliani and Miller 958). If there is no demand uncertainty Σ = 0), the product markets are perfect, and the firm is not exposed to supply-demand mismatch costs. Asent a fixed cost or salvage value advantage, flexile technology does not have any value. Oservation 2 confirms our intuition aout the risk management role of each strategy in counteralancing the effects of costs that originate from product and capital market imperfections. Corollary 3 he firm can optimally speculate with forward contracts. Flexile technology can trigger speculative ehavior. While firms frequently use financial derivatives for hedging purposes, Bodnar et al. 998) document that some firms take speculative positions with financial derivatives. Froot et al. 993) 8

22 show that speculation may indeed e optimal when there is an external financing cost and the return on the operational investments and the risk variale are statistically correlated. hey also conclude that in the asence of such correlation, the firm optimally fully hedges. In Proposition 3, we prove that the full-hedging conclusion need not hold if there are fixed costs of technology investment: Firms with limited expected internal endowment may optimally speculate to e ale to invest in capacity. he majority of empirical papers assume that firms use financial derivatives for hedging purposes Geczy et al. 997). Oservation 3 illustrates that such an assumption can e prolematic in industries with fixed cost requirements. It is interesting to note that speculation can e triggered y investment in flexile technology. he higher investment cost of flexile technology induces the firm to speculate while it uses forward contracts for hedging purposes with dedicated technology. his illustrates the intertwined nature of the integrated risk management portfolio. Engaging in operational risk management flexile technology) may have a structural effect going from hedging to speculation) on financial risk management. Firms may limit their usage of financial risk management to hedging only, since speculation is typically not viewed as a desired strategy. Non-speculative use of financial risk management imposes a hedging constraint on the feasile set of forwards y imposing H 0, which yields the following outcome: Proposition 5 If the firm uses forward contracts for hedging purposes only, then the firm optimally may not engage in financial risk management even if it is costless F F RM = 0). he intuition of this result is similar to the full speculation case aove, otained in the case of low expected internal endowment value. he firm is etter off y leaving the exposure to asset price as high as possile this corresponds to H = 0) to e ale to invest in capacity. Empirical studies unanimously demonstrate more widespread usage of financial risk management among large firms, and this oservation is attriuted to the fixed costs of estalishing a financial risk management program Allayannis and Weston 999). Proposition 5 proposes another possile explanation: the no-speculation constraint on financial derivative usage. With this constraint, small firms that have low internal endowments) do not engage in financial risk management. In a recent empirical study, Guay and Kothari 2003) find no significant usage of financial risk management among non-financial firms, and suggest that these firms may e using operational hedges instead to manage their risks. We oserve that indeed, firms can rely only on operational hedges in an integrated risk management framework. Corollary 4 Any risk management portfolio can e optimal. Financial risk management is not a panacea. Firms can rely only on flexile technology for risk management purposes. 9

23 If financial risk management was costless, it would always e in the optimal risk management portfolio. Our analysis finds two reasons why firms may not use financial risk management: i) Its fixed cost is high. Since non-financial firms do not have as much expertise as financial firms in financial risk management, its effective fixed cost could e higher for them, which provides support for the oserved difference in usage. ii) he firm limits itself to only hedging even if it is costless. hus, not only the investment cost of financial risk management, ut also the interplay etween financial and operational decisions is important in determining the optimal risk management portfolio. he firm should evaluate financial risk management as an integral part of the firm s overall investment strategy. he next section provides guidelines aout optimal portfolio selection. 6 Characteristics of the Optimal Risk Management Portfolio In this section, we delineate the main drivers of the optimal risk management portfolio and analyze the interplay etween financial and operational risk management. In 6., we relate the optimal risk management portfolio to firm, industry, technology, product market demand variaility and correlation) and capital market external financing frictions) characteristics. We then analyze the interaction etween operational and financial risk management strategies in 6.2. For this analysis, we proxy the firm size using the level of internal stage ) endowment. In particular: Definition he firm is defined to e small large) if the firm orrows does not orrow) from external markets with flexile technology and full hedging, ω F RM 0 + α ω F RM Ω 2 F Ω0 F ). he finance literature qualitatively refers to small and large firms according to the degree to which they are affected y external financing frictions. his definition formalizes this concept in the context of our model. We parameterize the internal stage ) endowment as λω 0, λω ) and the fixed technology costs as F D = F, F F = F + δ with δ 0. For tractaility, we impose some parameter restrictions. Assumption 8 Let β = F RM F. ω 0 ω 0 +α 0 ω, γ = 0, E c K a) +)a, F F = c K +a) +)a, and F < hese assumptions ensure the following: F F RM ω 0 + α 0 ω, so that financial risk management is feasile, and undertaking financial risk management or not can e optimal. If the firm engages in financial risk management, it optimally fully hedges; this rules out cases where the optimal financial risk management level cannot e uniquely characterized. he firm is not constrained y the credit limit, so the effective financing friction is the unit financing cost a. Finally, the optimality of not investing in either technology is ruled out. 20

24 6. Comparative Statics Results We define as the value of financial risk management FRM) with technology :. = E [ π BF RM α, ω F RM ) )] E [π B F RM α ))]. 5) o investigate the main drivers of the optimal portfolio choice, we carry out comparative statics analysis on the variale cost thresholds c F c D, H ) and c F c D, 0), and on. he results elow hold locally such that Assumption 8 and the defining regions for small and large firms are not violated. Proposition 6 echnology Choice) With symmetric fixed technology costs F F = F D ), c F c D, H ) and c F c D, 0) are invariant to the unit financing cost a), the fixed costs of oth technologies F ) and the internal endowment λ) of the firm. With asymmetric fixed costs F F > F D ), c F c D, H ) and c F c D, 0) decrease in the fixed costs of oth technologies and the unit financing cost, and increase in the internal stage ) endowment of the firm. With symmetric fixed costs, the technology ordering is independent of financing cost, fixed costs and internal stage ) endowment. With asymmetric fixed costs, since flexile technology has a higher investment cost, any increase in costs fixed cost, financing cost) favors the dedicated technology; a decrease in costs such as an increase in the internal stage ) endowment), favors the flexile technology. Proposition 7 Value of FRM) he value of FRM increases in the external financing cost a) for large firms. For small firms, the value of full hedging increases decreases) in the external financing cost at low high) levels of F F RM. For large small) firms, the value of FRM increases decreases) in the fixed cost of technology F ) and the demand variaility σ), and decreases increases) in the internal stage ) endowment λ) and the demand correlation ρ). We now explain the drivers of Proposition 7 y grouping the results that have similar intuition. Since with Assumption 8, the firm optimally fully hedges with financial risk management, we refer to the firm engaging not engaging) in financial risk management as the hedged unhedged) firm. he effect of external financing cost. Financial risk management is valuale since it reduces risk exposure and hence the expected orrowing level. At the same time, it is costly, and there is an opportunity cost for engaging in FRM: the firm may even need to orrow additional funds to finance its operational investments. hese two drivers comine to determine how an increase in financing cost impacts the financial risk management decision of the firm. For large firms, the hedged firm y Definition does not orrow at all, while the unhedged firm is adversely affected 2

25 from increasing financing costs. herefore, the value of financial risk management increases in the financing cost. For small firms, this trade-off depends on the fixed cost of financial risk management. For low fixed costs, the value of financial risk management increases in financing costs; at high fixed costs, the opposite occurs. he effect of fixed technology cost and internal stage ) endowment. he proof of the proposition reveals that there is one fundamental driver that explains oth comparative statics results: the level of reliance on external financing, as summarized in ale 3. A firm s reliance on external financing increases as the fixed investment cost F increases and the internal stage ) endowment level λ decreases. By Definition, the large hedged firm does not need to orrow and the large unhedged firm orrows in some udget realizations. herefore, increasing the reliance on external financing adversely affects the unhedged firm while not affecting the hedged firm. We conclude that for large firms, the value of FRM increases as the need for external financing increases. Since the small hedged firm, y Definition, always orrows and the small unhedged firm only orrows in some udget realizations, increasing the reliance on external financing adversely affects the unhedged firm, ut it affects the hedged firm even more. We conclude that for small firms, the value of FRM decreases as the need for external financing increases. Case Borrowing level Increasing reliance on external financing Large unhedged firm Borrows in some states Increases the value of FRM since Large hedged firm Does not orrow the unhedged firm orrows more in expectation Small unhedged firm Borrows in some states Decreases the value of FRM since Small hedged firm Borrows in all states the hedged firm orrows more in expectation ale 3: Increasing the reliance on external financing has the opposite effect on the value of financial risk management for large and small firms. A firm s reliance on external financing increases as the fixed investment cost F increases, and it decreases as the internal stage ) endowment level λ increases. he effect of demand correlation and demand variaility. hese two factors have an effect on the firm only with flexile technology. he proof of the proposition reveals that there is one fundamental driver that explains these two comparative statics results: the marginal change in the optimal investment level with changes in these factors, as summarized in ale 4. A firm s optimal investment level decreases as the demand variaility decreases or the demand correlation increases. he small unhedged firm orrows only in some udget realizations, while the small fully hedged firm always orrows. As a result, the small hedged firm employs a more conservative investment policy the capacity investment level is lower at each state) than the unhedged firm since its exposure to external financing costs is higher. Consequently, a similar change in variaility or correlation alters 22

26 the small hedged firm s optimal investment policy to a lower extent than the unhedged firm s; its optimal investment level is more roust to changes in these factors. herefore, while a reduction in the optimal investment level due to a decrease in variaility or an increase in correlation) adversely affects the small hedged firm, it affects the small unhedged firm even more. We conclude that for small firms, the value of FRM increases as the optimal investment level decreases. For large firms, the opposite result holds. his follows from parallel arguments ased on the fact that the large unhedged firm needs to orrow in some udget realizations, while the large hedged firm does not. Case Borrowing level Reduction in the optimal investment level at each udget state Large unhedged firm Borrows in some states Decreases the value of FRM since the hedged firm s Large hedged firm Does not orrow optimal investment is less conservative and less roust Small unhedged firm Borrows in some states Increases the value of FRM since the hedged firm s Small hedged firm Borrows in all states optimal investment is more conservative and more roust ale 4: A reduction in the optimal investment level at each state has the opposite effect on the value of financial risk management for large and small firms. A firm s optimal investment level decreases as the demand variaility decreases or the demand correlation increases. Synthesis. ale 5 summarizes the main drivers of each optimal portfolio choice for large and small firms y comining Propositions 4, 6 and 7 for technologies with asymmetric fixed cost F F > F D ). By definition, if the variale cost thresholds increase in a parameter, flexile technology is preferred under a larger set of conditions as that parameter increases, and we say that flexile technology is favored. Similarly, if increases in a parameter, we say financial risk management is favored. While not exact, this usage captures the direction of change. For example, high demand variaility and low demand correlation favor investing in flexile technology and undertaking financial risk management for large firms. his is how ale 5 is constructed. We note that the capital intensity of an industry can e captured y keeping the internal endowment level constant and altering the fixed technology costs. With a given internal endowment level, a sufficiently high low) fixed cost implies a small large) firm according to our definition. herefore, our results aout small and large firms can e interpreted as eing relevant for capital intensive and non-capital intensive industries, respectively. he main message of ale 5 is that the size of the firm is key to optimal portfolio choice. As explained earlier, the three fundamental drivers ehind the optimal portfolio choice opportunity cost of financial risk management, level of reliance on external financing, and roustness of the optimal capacity investment level to variaility and correlation) work in opposite directions for small and large firms. herefore, different size firms may choose the same optimal portfolio for entirely different reasons. 23

27 Portfolio Choice Large Firms Small Firms High demand variaility High internal endowment F with FRM Low demand correlation Low technology fixed costs Low financing costs with low F F RM Low internal endowment Low demand variaility D with FRM High technology fixed costs High demand correlation High financing costs High financing costs with low F F RM High internal endowment High demand variaility F without FRM Low technology fixed costs Low demand correlation Low financing costs Low financing cost with high F F RM Low demand variaility Low internal endowment D without FRM High demand correlation High technology fixed costs High financing costs with high F F RM ale 5: Main Drivers of the Optimal Risk Management Portfolio with Asymmetric Fixed echnology Costs. ale 5 is for asymmetric fixed technology costs. With symmetric fixed costs, it follows from Proposition 4 that the technology ordering is independent of changes in any parameter. herefore, changes in parameter levels only affect the choice etween undertaking FRM or not. Consequently, all the conditions in ale 5 that favor flexile or dedicated technology with FRM and without FRM for a given firm size favor using FRM and not using FRM, respectively. We conclude that the technology cost characteristic is also key to the optimal portfolio structure. We now relate our theoretical findings to the associated empirical literature. he financial risk management literature relates the value of financial risk management to underlying exposure, growth opportunities and size of firms Allayannis and Weston 999). Our results demonstrate that the value of financial risk management also depends on the product market and technology characteristics, and that there are sutle differences etween large and small firms. Gay and Nam 998) say that firms with higher investment opportunities that are exposed to higher external financing frictions and lower levels of cash make greater use of financial derivatives. We show in the proof of Proposition 7) that the effect of cash ω 0 is the same as the effect of internal stage ) endowment: A lower internal stage ) endowment increases the value of hedging for small firms, ut not for large firms. herefore, our results support their argument for small firms, ut not for large firms. he financial risk management literature hypothesizes that the value of financial risk management increases as financing frictions increase y invoking the counteralancing effect of financial 24

28 risk management with respect to external financing frictions Mello and Parsons 2000). Our results support this argument for large firms, ut not for small firms. he key is how much the firm needs to orrow after undertaking financial risk management. 6.2 he Interaction of Operational and Financial Risk Management We first investigate whether flexile technology and financial risk management are sustitutes or complements in an integrated risk management framework. hey are defined to e sustitutes if the firm invests in flexile technology when the firm is not allowed to use financial risk management and switches to dedicated technology when the firm engages in financial risk management; they are called complements if the switch is from dedicated to flexile technology. Proposition 8 Flexile technology and financial risk management can e complements or sustitutes. Small large) firms tend to sustitute complement) flexile technology with financial risk management. he main driver of Proposition 8 is the value of financial risk management with each technology. Flexile technology is more expensive, so it is more exposed to external financing costs. he use of financial risk management allows large firms to secure a udget level sufficient to eliminate orrowing. hus, large firms complement flexile technology with financial risk management in their integrated risk management portfolio. Small firms need to orrow to invest in flexile technology, even using financial risk management, ut may not need to orrow for dedicated technology if they use financial risk management. In other words, the value of financial risk management is higher with dedicated technology. his explains why flexile technology and financial risk management are sustitutes for small firms. Interestingly, the empirical literature also finds mixed results on this question, aleit in other contexts. Geczy et al. 2000) document complementarity etween operational physical storage) and financial means of risk management among natural gas pipeline firms. In a multinational context, Allayannis et al. 200) find that financial and operational geographical diversification) risk management tools are sustitutes. In a different framework, Chod et al. 2006) provide another theoretical justification for these mixed empirical results y focusing on the effect of financial risk management on the optimal flexiility level of the firm. hey demonstrate that financial risk management is a complement sustitute) to operational flexiility when the optimal flexiility level increases decreases) with financial hedging. We next analyze whether the value of operational risk management defined as the expected stage 0) equity value difference etween flexile and dedicated technologies) is more or less roust 25

29 to changes in product and capital market conditions when financial risk management is undertaken. Roust strategies are preferale ecause they perform well under a wider range of parameters, and can e implemented with more confidence. Proposition 9 For large small) firms, the value of operational risk management is less more) roust to changes in product market conditions ρ, σ) and more less) roust to changes in capital market conditions a) with financial risk management than without. he proof of the proposition reveals that the roustness with respect to product market conditions is linked to the value of FRM with flexile technology. he value of operational risk management is more or less roust with respect to correlation if the value of FRM decreases or decreases in correlation, respectively. his is valid for small and large firms, respectively, as we discussed in 6.. Roustness with respect to variaility follows from a similar argument. Roustness with respect to the unit financing cost is determined y the difference etween the value of FRM with flexile and dedicated technologies: he value of operational risk management is more roust to changes in a if the value of FRM with flexile technology increases more rapidly than the value of FRM with dedicated technology in response to an increase in a. Proposition 9 again illustrates the intertwined nature of operational and financial risk management strategies: Engaging in financial risk management has the opposite impact on the roustness of the value operational risk management with respect to product and capital market conditions. 7 Value and Effect of Integrated Decision Making Sections 5 and 6 analyzed the properties of the optimal integrated risk management portfolio and its drivers. In practice, firms may not take an integrated approach to these decisions; operational and financial risk management decisions may e taken independently. In this section, we focus on the value and effect of integrated decision making. We relax the restrictions of Assumption 8 and focus on general parameter settings. If we ignore its effects on operational decisions, financial risk management does not have any value ecause forward contracts are investments with zero expected return. For this reason, we take no FRM as the non-integrated enchmark. Since the non-integrated enchmark is no FRM, the results of this section can also e interpreted as the effect of engaging in FRM on the firm s performance and optimal decisions. he effect of FRM on the optimal expected capacity investment and external orrowing level is amiguous: Proposition 0 Engaging in financial risk management can increase or decrease the optimal expected capacity investment and the optimal expected orrowing levels. 26

30 Since financing frictions negatively impact the stage capacity investment level at each udget state, and the firm uses FRM to counteralance the effect of financing frictions, one may expect that with FRM, the firm s expected orrowing level would e lower and the expected capacity investment level would e higher than without. On the other hand, if there is cost associated with engaging in financial risk management F F RM > 0), the firm has less internal endowment to invest in capacity at each udget state, and has to orrow additionally to compensate for F F RM. In the proof of Proposition 0, we illustrate that even if FRM is costless, the optimal expected capacity investment can decrease and the expected orrowing level can decrease. his is a direct consequence of the joint optimization in external orrowing and capacity levels. he fundamental driver of this result is the marginal profit of the capacity investment in the joint optimization prolem as we discussed in 4.2. Proposition 0 shows the dependence of capacity investment on financial risk management. We now analyze the effect of engaging in financial risk management on the technology choice: Corollary 5 he firm may make different technology decisions with and without financial risk management. In their numerical analysis, Ding et al. 2005) demonstrate that financial risk management can alter more strategic operational decisions gloal supply chain structure) than the capacity investment levels. Oservation 5 is in line with their conclusion. We analytically prove that the technology choice of the firm may e altered y engaging in FRM. he direction of change in technology choice is determined y the value of FRM with each technology. Proposition 8 is an example for such changes and provides the intuition with some restrictions on the parameter levels. he analysis aove illustrates the effect of integrating risk management decisions on the firm s decisions. We now analyze the value of such integration as a function of firm size. o separate the value of integration from the cost of FRM, we use F F RM = 0. Here, our definition of a large firm is the same as Definition, ut our definition of a small firm is slightly more restrictive. We refer to firms with very limited expected internal endowment value that optimally fully speculate with FRM as small firms. Since under the conditions of Assumption 8, these firms fully hedge with FRM, the new definition is consistent with Definition and corresponds to a suset of small firms in 6 that have a significantly low expected internal endowment value. Corollary 6 he value of integration is low for small firms with low cash levels ω 0 ) and large firms with high cash levels. If the firm uses financial risk management only for hedging purposes, the value of integration is higher for large firms than for small firms. he value of integration is equivalent to the value of engaging in FRM. Since large firms with high 27

31 cash levels are not significantly exposed to external financing frictions without FRM, the value of FRM, and hence the value of integration is low. In the extreme case, a cash level sufficient to finance the udget-unconstrained optimal investment level completely removes the exposure to external financing frictions and FRM has no value. For small firms with low levels of cash, the additional enefit of full speculation H = ω 0 α ) over not using FRM H = 0) is low. In the extreme case, if the small firm does not have any cash ω 0 = 0), then FRM has no value. When the firm uses financial risk management only for hedging purposes, it follows from Proposition 5 that small firms optimally do not engage in FRM. In this case, integration has no value. Large firms generally fully hedge with FRM, therefore integration has value for them. In a numerical analysis not reported here, we oserve a similar pattern without imposing the hedging constraint. 8 Roustness of Results to Model Assumptions In this section, we investigate the roustness of our results to the assumptions presented in 3. Non-identical and exogenous financing costs. We assumed a unique external financing cost structure a, E). he firm can e exposed to a different external financing cost structure a, E ) with each technology {D, F }. All the analytical results of 4 continue to hold y replacing a, E) with a, E ) where a lower unit orrowing cost is associated with a higher credit limit. he main insights of the paper do not change except that the technology with lower a and higher E is favored in the optimal risk management portfolio. Endogenous financing costs. In this paper, we focus on a partial equilirium setting where the financing costs are exogenous and identical for each technology. In a general equilirium setting, the financing cost for each technology is determined y the interaction etween the firm and a creditor. In Boyaatlı and oktay 2006), we derive the equilirium level of secured loan commitment contracts a, E ) for each technology in a creditor-firm Stackelerg game using a similar firm model. We show that the orrowing terms will e independent of technology choice when the creditor has limited information aout the firm and the technologies, there is no credile way of information transmission, and the creditor ases its assessment of default proaility on the same cash flow distriution of the firm for any technology. hese conditions are relevant for ank financing where anks rely on the credit history of the firm for credit risk estimation and do not have operational expertise. All of the results in this paper are valid in the general equilirium sense under these conditions. We refer the reader to Boyaatlı and oktay 2006) for a detailed treatment of endogenous financing costs. Unsecured loan commitment contracts. If the firm uses unsecured loan commitment contracts 28

32 P = 0), the firm only receives the salvage value of the non-pledgale technology in the default states. he limited liaility of the shareholders left-censors the stage 2 equity value distriution at 0. he expected stage ) equity value is calculated using conditional expectations with respect to default and non-default events. he proaility of default depends on the capacity investment level, external orrowing level and the risk-pooling value of the technology choice. At stage, similar to secured lending, the firm optimally orrows so as to finance the optimal capacity investment level. In a single-product price-taking newsvendor setting, Baich et al. 2006) provide conditions under which the expected stage ) equity value is unimodal though not concave) in capacity. With two products and endogenous pricing, the optimal capacity investment level is very hard to solve and ecomes intractale for flexile technology ecause of the dependence on default regions with ivariate product market uncertainty. In our paper, the effect of limited liaility is inherent in the financing cost structure a, E). When the capital market imperfection costs are default-related e.g. ankruptcy costs), if there were no limited liaility then the creditor would e sure to recoup the face value of the loan and default-related costs from the shareholders personal wealth. With such a riskless loan, the cost of the loan would e the risk-free rate a = 0) and the firm could raise sufficient funds to finance the udget unconstrained capacity level E = P ). If we allow unsecured lending in our setting, we conjecture that the optimal capacity investment level would e lower: he marginal cost of orrowing is less than +a ecause of the default, which should induce the firm to orrow more and invest more in capacity. Structural results related to financial risk management are expected to hold. How the technology choice would change is not clear ecause of the dependence on default regions. he arguments in this section are also relevant for i) partially secured lending P is positive ut not sufficient to finance the udget unconstrained capacity investment), and ii) secured lending with default-related costs deducted from the firm s seized assets y the creditor in the case of default. Positive production cost at stage 2. Let y denote the unit production cost for oth products with either technology. With y > 0, the optimal production vector at stage 2 is limited y the cash availaility of the firm in addition to the physical capacity constraints. In this case, the literature often uses a clearing-pricing strategy for tractaility that fully utilizes the physical capacity see for example, Chod and Rudi 2005). If we assume a clearing-pricing strategy, the firm optimally orrows so as to fully utilize the physical resource in stage 2 and all the results of our paper continue to hold y replacing c with c + y. If we focus on the optimal pricing policy with y > 0, the optimal production vector with flexile dedicated) technology is state dependent and has a complex form that is characterized y a tworegion six-region) partitioning of the demand space ξ, ξ 2 ) with respect to capacity constraints 3. 29

33 he optimal capacity level is lower than the y = 0 case, and accounts for the state-dependent optimal production vector. With flexile technology, the firm optimally orrows the exact amount required for the full utilization of the physical resource. With dedicated technology, the optimal orrowing level is such that the physical resources are never fully utilized. Financial capacity has a risk-pooling enefit with dedicated technology ecause the firm can allocate the financial resource to each physical capacity contingent on the demand realization. Because of this additional riskpooling enefit of dedicated technology, flexile technology is more adversely affected from y > 0 compared to y = 0. With y > 0, the majority of the insights and the structural results otained with y = 0 remain valid. he results concerning the product market characteristics ρ, σ) are among the few exceptions. Similar to flexile technology, the value of dedicated technology decreases in ρ and increases in σ. his is a direct consequence of the declining risk-pooling value of the financial capacity. he optimal technology choice as a function of product market conditions is not clear in this setting. Seizale salvage value of technology. We assume that the creditor cannot seize the salvage value of the technology in case of default. If the salvage value of the technology is offered as an additional collateral, then the creditor can seize the technology. With exogenous financing costs, seizale technology does not have any impact on the results of this paper. With endogenous financing costs and immediate liquidation of technology, collateralizing the technology reduces the default risk and hence external financing costs in equilirium. Different salvage values of the technologies have a significant impact on the technology choice in equilirium as we discuss in Boyaatlı and oktay 2006). Fixed cost of technology is incurred at stage 0. If the firm incurs the fixed cost of technology at the time of commitment at stage 0), then this fixed cost is deducted from the firm s internal stage endowment ω 0, ω ) in the same way as F F RM. With this assumption, the firm always optimally fully hedges with financial risk management; hence Oservation 3 and Proposition 5 do not hold. All the other results remain valid. he same conclusions hold in the asence of technology fixed costs F F = F D = 0). 9 Conclusions his paper analyzes the integrated operational and financial risk management portfolio of a firm that determines whether to use flexile or dedicated technology and whether to undertake financial risk management or not. he risk management value of flexile technology is due to its risk pooling enefit under demand uncertainty. he financial risk management motivation comes from the existence of deadweight costs of external financing. Financial risk management has a fixed cost, 30

34 while technology investment incurs oth fixed and variale costs. he firm s limited udget, which depends partly on a tradale asset, can e increased y orrowing from external markets, and its distriution can e altered via financial risk management. In a parsimonious model, we solve for the optimal risk management portfolio, and the related capacity, production, financial risk management and external orrowing levels, the majority of them in closed form. We characterize the optimal risk management portfolio as a function of firm size, technology and financial risk management costs, product market demand variaility and correlation) and capital market external financing costs) characteristics. We find that three fundamental drivers explain the optimal portfolio choice: the roustness of the optimal capacity investment with respect to product market characteristics, the level of reliance on external financing and the opportunity cost of financial risk management. Our results provide managerial insights aout the design of integrated operational and financial risk management programs. A firm that operates in highly variale or highly negatively correlated product markets should use flexile technology with financial risk management if the firm has sufficiently high internal endowment large firm); and without financial risk management if the firm has limited internal endowment small firm). For large firms with low high) external financing costs, flexile technology with financial risk management dedicated technology without financial risk management) is the est risk management portfolio. For small firms, the insights related to technology choice under high and low external financing costs continue to hold ut the firm should only use financial risk management if the fixed cost of financial risk management is sufficiently low. Our analysis clearly shows the intertwined nature of operational and financial risk management strategies and illustrates their sutle interactions. For example, operational and financial risk management can e complements or sustitutes depending on the firm size. Flexile technology and financial risk management tend to e sustitutes for small firms and complements for large firms. he fundamental driver of this result is the difference in the value of financial risk management with each technology. We also show that the firm s use of financial instruments for speculative reasons can e triggered y choosing the higher cost flexile technology. Our analysis extends the modelling framework of Froot et al. 993) y formalizing operational investments and imposing a cost for financial risk management. With our more detailed operational model, some of their findings do not continue to hold. For example, firms can optimally use financial risk management for speculative purposes even if the returns from operational investments are independent from the financially hedgale risk variale. he driver of this result is the fixed cost of technology. In addition, we show that firms may choose not to use financial risk management due to its cost when resources are limited. he effective cost of financial risk management is larger 3

35 than its fixed cost ecause of the existence of operational investments: After incurring the fixed cost of financial risk management, the firm may need to orrow additional funds to finance its operational investments, which imposes an opportunity cost on the firm. hese results enhance our understanding of the effect of operational factors in risk management and underline the importance of integrated decision making. his paper rings constructs and assumptions motivated y the finance literature into a classical operations management prolem. In turn, we provide theoretical support for some oservations made in the empirical finance literature and highlight additional trade-offs in some others. For example, we estalish that the value of financial risk management increases in external financing costs only for large firms and not for small firms. his is in contrast to the existing understanding that this is true for any firm. here is evidence that large firms use financial instruments more frequently than small firms. his oservation is attriuted to the fixed cost of estalishing a financial risk management program. Our analysis proposes another explanation that is ased on the hedging constraint sometimes imposed in practice: If firms are allowed to use financial instruments for hedging purposes only, it is optimal for small firms to not undertake financial risk management even if it is costless. Our paper opens new empirical avenues. he existing literature on risk management typically does not capture operational aspects such as characteristics of different technologies and product market characteristics. As demonstrated y our analysis, these can have a significant effect on the risk management portfolio and generally have opposite effects for large and small firms. he distinction we make etween large and small firms or equivalently, etween capital intensive and non-capital intensive industries), and our results related to the effect of technology and product market characteristics on the risk management portfolio provide new hypotheses that can e tested empirically. For example, we expect to see that large firms engage in financial risk management less frequently than small firms in highly positively correlated markets. We also expect to see a positive relation etween fixed technology costs and the frequency of engaging in financial risk management for large firms and a negative relation for small firms. In 8, we discussed the implication of relaxing some of our assumptions. Other interesting research directions remain. For example, this paper focuses on a monopolistic firm. In an integrated risk management framework, strategic risk management has not received much attention. Goyal and Netessine 2005) analyze the value of flexile technology under product market competition. It would e interesting to incorporate financial risk management decisions of the firm in this competitive setting. he financially hedged firm may invest in more costly flexile technology whereas the non-hedged competitor may not ecause of external financing frictions. Financial risk management 32

36 will certainly have a non-trivial impact on the equilirium of the game. Dong et al. 2006) take a step in this direction y modeling operational flexiility and financial risk management decisions of a gloal firm facing a local competitor that can only respond y setting its production quantity. We assume an exogenous external financing cost structure. echnology characteristics can affect the external financing costs in equilirium; this occurs if the lender has information aout the firm s technology options and the aility to assess their operational and collateral value. In this case and with loan commitment contracts, the financing cost structure would depend on the firm s likelihood of orrowing and the default risk conditional on the orrowing level. Flexile technology has higher costs, and requires more external orrowing than dedicated technology; ut the riskpooling value of flexile technology decreases the default risk. he different collateral values of each technology ring another facet to this interaction. It is interesting to analyze which effect dominates under what conditions. he roader question is whether firms should use flexile versus dedicated technology in imperfect capital markets. We analyze these issues in a companion paper Boyaatlı and oktay 2006). Notes With the exception of sensitivity results with respect to demand variaility and correlation: hese results require formalization of demand variaility and correlation via specific distriutional or structural using stochastic orderings) assumptions. 2 o capture the effect of demand correlation and variaility, we use different measures that are commonly used in the literature. hroughout the paper, y an increase in demand variaility, we refer to any one of the following cases: i) ξ has a symmetric ivariate lognormal distriution and σ monotonically increases, ii) ξ with independent marginal distriutions is replaced with ξ with independent marginal distriutions such that ξ = ξ and ξ i is stochastically more variale than ξ i for i =, 2, or iii) ξ with σ = 0 is replaced with ξ with σ 0. By an increase in demand correlation, we refer to any one of the following cases: i) ξ has a ivariate lognormal distriution and ρ monotonically increases, ii) ξ is replaced with ξ which dominates ξ according to the concordance ordering, or iii) ξ with ρ is replaced with ξ with ρ =. he details of the analysis can e found in the proof. 3 he proofs for the stage 2 optimal production vector for each technology with y > 0 are availale upon request. 33

37 Acknowledgement his research was partly supported y the INSEAD Booz-Allen-Hamilton research fund. We thank Paul Kleindorfer and the participants of the BCIM Risk Management Mini-Conference in Washington University, St. Louis, MO, 2005 for their insightful comments. References Allayannis, G., J. Ihrig, J., J. Weston Exchange-rate hedging: Financial vs operational strategies. American Economic Review Papers&Proceedings Allayannis, G. J. Weston he use of foreign currency derivatives and industry structure. G. Brown, D.H. Chew, eds. Corporate Risk: Strategies and Solutions. Risk Books Aytekin, U., J. Birge Optimal investment and production across markets with stochastic exchange rates. Working Paper, he University of Chicago Graduate School of Business. Baich, V., M.J. Soel Pre-IPO operational and financial decisions. Management Science Baich, V., G. Aydin, P. Brunet, J. Keppo, R. Saigal Risk, financing and the optimal numer of suppliers. Working paper, University of Michigan. Bish, E., Q. Wang Optimal investment strategies for flexile resources, considering pricing and correlated demands. Operations Research Birge, J. R Option methods for incorporating risk into linear capacity planning models. Manufacturing and Service Operations Management Bodnar, G. M., G. Hyat, R.C. Marston Wharton survey of derivative usage y US non-financial firms. Financial Management Bodnar, G. M., G. Hyat, R.C. Marston Wharton survey of financial risk management y US non-financial firms. Financial Management Boyaatlı O., L.B. oktay Operational Hedging: A Review with Discussion. Working Paper 2004/2/M, INSEAD, Fontaineleau, France. Boyaatlı O., L.B. oktay Capacity investment in imperfect markets: he interaction of operational and financial decisions. Working Paper, INSEAD. Brown, G., K.B. oft How firms should hedge. he Review of Financial Studies

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41 A Appendix A Name Meaning ω 0, ω ) cash and asset holdings of the firm, called the firm s endowment β α 0 proportion of F F RM deducted from cash holdings of the firm stage 0 price of tradale asset F, c ) fixed and variale capacity costs of technology γ F F RM B a, E) r f = 0) P α ξ = ξ, ξ 2 ) Σ ρ salvage rate of fixed cost of technology fixed cost of financial risk management FRM) stage udget interest rate and credit limit of the loan contract risk-free rate value of collateral physical asset stage price of tradale asset multiplicative demand intercept in product markets covariance matrix of ξ coefficient of correlation in ξ σ standard deviation of ξ and ξ 2 ) Γ Π π Λ F RM F RM Λ expected Π optimal stage 2 operating profits optimal stage 2 equity value optimal expected stage ) equity value expected stage 0) equity value of etter technology with FRM stage 0) equity value of etter technology without FRM optimal expected stage 0) equity value Value of financial risk management with technology ale 6: Summary of Notation Proof of Proposition : We start y formulating the stage 2 optimization prolem. Let ) Γ K, e, B, ξ denote the optimal stage 2 operating profit as a function of the state vector K, e, B, ξ). Since we assume production is costless, this profit is equal to the maximum sales revenue that can e otained with the existing capacity. In stage, the firm will have oserved the udget realization B and orrowed e to invest in capacity level K. he remaining cash holdings of B + e c K F, non-negative y construction, will have een invested into a cash account with return r f = 0). wo outcomes are possile in stage 2: If the firm s final cash position operating profits and cash account holdings) is sufficient to cover the face value of the loan, i.e. Γ K, e, B, ξ) + B + e c K F ) e + a), then the firm does not default; otherwise, it does. If the firm does 38

42 not default, it repays the face value of its loan and liquidates the non-pledged technology and the physical assets, generating γ F and P, respectively. If the firm defaults, the cash on hand and the ownership of the collateralized physical asset are transferred to the ank. he firm receives the salvage value of the technology γ F and the cash RK, e, B, ξ) remaining after the face value of the loan is deducted from its seized assets. We write RK, e, B, ξ) = P + Γ K, e, B, ξ) + B + e c K F ) e + a), 6) where we invoke the assumptions that any additional fees in the default state e.g. ankruptcy fee) are orne y the creditor as out-of-pocket expenditures, and that the loan is fully-collateralized y the physical asset. Since the shareholders are risk neutral and the risk-free rate is 0, the stage 2 equity value can e written as the sum of the individual components cash flows, regardless of when they are realized: ) Γ K, e, Π K B, ξ) + B + e c K F ) if no default, e, B, ξ = e + a) + γ F + P 7) γ F + RK, e, B, ξ) if default Inspecting 7) reveals that the equity value can simply e written as ) Π K, e, B, ξ = Γ K, e, B, ξ) + B + e c K F ) + γ F e + a) + P 8) regardless of whether the firm defaults or not. Otaining this unique functional form is essential in preserving tractaility and in deriving closed-form expressions for the firm s capacity, technology and financial risk management decisions for a suset of parameter levels. he production decision only affects the operating profit Γ in 8), so optimizing the stage 2 equity value is equivalent to the following optimization prolem: max Q pq; ξ) = max ξ Q +. 9) Q Θ Q Θ Here, pq; ξ) = pq ; ξ ), pq 2 ; ξ ). 2 ), Θ F = {Q : Q 0,. Q K F } and Θ D = {Q : Q 0, Q K D } are the feasiility sets for production quantity levels for each technology. Let fq). = ξ Q + and Q denote the optimal production vector that solves 9) for technology {F, D}. It is easy to estalish that fq) is strictly concave in Q = q, q 2 ). Since the constraints are linear, KK conditions are necessary and sufficient for optimality and Q is unique. Since f q i = + /) ξ q / i > 0, and with, ), lim qi 0 f + q i, the non-negativity constraints will e non-inding and the capacity constraint will e inding at optimality. With the dedicated technology, this yields Q D = K D and ) Γ D K D, e D, B, ξ = fq D) = ξ + K D. 39

43 With the flexile technology, according to the KK conditions, Q F solves After some algera, we otain Q F = K F ξ + ξ 2 ξ and f q q = f KF q 2. F qf Γ F K F, e F, ) B, ξ = fq F) = ξ + K + F ξ 2 ) + [ ξ + ] ξ ξ 2 = + ) ξ 2 K + F. Defining N F. = ξ + ) ξ 2 the optimal equity value Π : Π K, e, and N D. = ξ and sustituting Γ in 8) yields the expression for ) B, ξ = N + K + B + e c K F ) + γ F e + a) + P 0) Proof of Proposition 2: We start y formulating the stage optimization prolem. he optimal expected stage ) equity value of the firm, π B), is given as follows: { π B) max Ψ B), B } γ )F + P if B + E > F = ) B γ )F + P if B + E F where Ψ B) = max K, e s.t. )] B + e c K + F ) B + e c K F ) + E [Π K, e, B, ξ e c K + F B e E 2) K 0, e 0. We start with explaining the formulation of the optimization prolem 2). he firm has availale udget B and orrows e from the creditor. Out of this sum B + e, the firm invests c K + F in capacity and places the remainder B + e c K F ) into the cash account. he return from the cash account and the operating profits from the capacity investment are included in the )] expected value of the equity in stage 2, E [Π K, e, B, ξ. Using 8), the ojective function can e rewritten as B + P γ )F + Γ K, e, B, ξ) c K ae. Here, the first three terms are equal to the equity value of the firm if the firm does nothing does not orrow and does not invest). Note that since the firm has already committed to technology, the fixed cost F is incurred even if K = 0. he last three terms are the net profit derived from orrowing and investing in capacity. he first constraint ensures that the amount of external orrowing is greater than the difference etween the cost of the investment and the availale udget, otherwise the investment is not feasile. 40

44 he second constraint states that the external orrowing is less than the credit limit E) of the firm. Equation ) states the firm will either choose a positive capacity level in stage or do nothing not orrow and not invest in capacity). he former will e the case when the optimal capacity investment level otained in 2) is positive, and this solution dominates doing nothing; with π B) = Ψ B). In the latter case, the equity value of the firm is B + P γ )F, with K B) = 0 and e B) = 0. his is the optimal solution if i) the udget plus the credit limit is insufficient or only sufficient) to cover the fixed cost of investment B + E F ), so the firm liquidates the physical asset and salvages the technology; or if ii) the udget plus credit limit is sufficient to cover the fixed cost, ut the firm optimally chooses not to invest in capacity B + P γ )F > Ψ B) when B + E F ). Note that if K = 0 in the optimal solution of 2), the formulation in 2) forces the firm to suoptimally) orrow E B, ut the optimal ojective function value is then dominated y B + P γ )F, the value of doing nothing, so the joint formulation in 5) and 6) yields the correct optimal solution. Since a > 0, the firm optimally does not orrow if it does not invest in capacity e = 0 if K = 0) and only orrows exactly enough to cover the capacity investment when this investment B) + level is positive e = c K + F if K > 0). Sustituting Π from 8) and Γ from Proposition in 2), we otain the equivalent formulation Ψ B) = max K B B) + c K γ )F a c K + F + E [N ] + K + P s.t. c K + F B E 3) K 0. Let gk ) denote the ojective function in 3) and K p B) e the optimal solution of 3). he corresponding optimal orrowing e p B) is equal to c K p B) B) +. + F For B > F, the function gk ) has a kink and is not differentiale at K = B F c. We rewrite 3) as a comination of two su-prolems i = 0, with Ψ B) max i Ψ i = B) if B > F Ψ B) 4) if B F such that Ψ i B) = max K B c K γ )F a i c K + F B ) + E [N ] + K + P s.t. Z i L c K + F B Z i U 5) K 0, 4

45 where a 0 = 0, a = a and ZL 0 =, Z L = 0, Z0 U = 0, Z U = E. Suprolem 0 ) is the restriction of the prolem to the no orrowing orrowing) regions. Let g i K ) denote the ojective function and K p i B) e the optimal solution of su-prolem i. We have g 0 K ) if c K + F B gk ) = g K ) if c K + F > B. he remainder of the proof has the following structure:. We show that g i K ) is strictly concave and solve each su-prolem i for K p i B). 2. We show that gk ) is strictly concave. It follows that K p B) = K p i B) arg max i Ψ i where i = B) if B > F if B F We derive Ψ B) y using K p i B). 3. We compare Ψ B) with B γ )F + P, the value of not investing in capacity, and derive K B) and e B).. Solution for K p i B).a. Flexile echnology: Let A =. [ ) ] E [N F ] = E ξ + ξ2. he first and second order conditions in 5) are g i = c F a i c F + + /) A K / K F 2 g i = + /) A K/ ) F. K 2 F F, Since <, we have lim 2 K / ) KF 0 + F and 2 K / ) KF 2 F > 0 K KF 2 F 0. With <, it follows that 2 g i < 0 for K KF 2 F 0 and the function g i K F ) is strictly concave for i = 0,. Since the constraints in 5) are linear, first-order KK conditions are necessary and sufficient for optimality for each su-prolem i and K p i F B) is unique. From KK conditions if i has a non-empty feasile region then the optimal solution is either the solution of = 0, K p i F B) A+ = ) c F +a )), or is a oundary solution. Since B > F i F for i = 0 g i K F from 4) and B > F F E for i = from ), the non-negativity constraint is never inding in 5). Since lim KF 0 + gi K F, K F = 0 is never optimal. If Zi L + B F F c F > 0 and gi K F < 0 at this point, then K p i F B) = Zi L + B F F c F, i.e., the optimal solution occurs at the lower ound of 42

46 the financing constraint. If g i K F > 0 at K F = Zi U + B F F c F > 0, then K p i F B) = Zi U + B F F c F, i.e., the optimal solution occurs at the upper ound of the financing constraint. o summarize, K p i F B) for i = 0, is characterized y K p 0 F B) = K p F B) = K 0 F K F K F K F K F ). A+ = ) c F. = B FF c F ). = B FF. = c F ) A+ ) c F +a) ). = E+ B FF c F if c F K 0 F + F F B 0 if c F K 0 F + F F B > 0, ) if c F K F + F F B 0 if 0 < c F K F + F F B E if c F K F + F F B > E. Here, K 0 F is the udget-unconstrained optimal capacity investment and K F is the credit-unconstrained optimal capacity investment... Dedicated echnology: We otain 2 g i K D )2 2 g i K j = D )2 + /) ξ j K j D )/ ) < 0, 2 g i [ 2 KD 2 g i ] 2 )2 KD = K2 D + /) ξ j K j D )/ ) 0 > 0 for i = 0, and j =, 2. herefore, the Hessian matrix D 2 g i K D ) is negative definite for K D 0 and g i K D ) is strictly concave. Since the constraints in 5) are linear, first-order KK conditions are necessary and sufficient for optimality in each su-prolem i and K p i D B) is unique. If K p i D B) is an optimal solution to 5), then there exist λ i = λ i, λi 2 ) and µi = µ i, µi 2 ) that satisfy j 6) c D K p i D B) + F D B Z i U, 7) c D K p i D B) + F D B Z i L, 8) K p i D B) 0, 9) + a i )c D + + /) ξ K p i D B) / c D λ i λ i 2) + µ i = 0, 20) λ i [Z i U c D K p i D B) F D + B] = 0, 2) λ i 2[ Z i L + c D K p i D B) + F D B] = 0, 22) µ i K p i D B) = 0 23) with λ i 0 and µ i 0 for i = 0,. Oserve that lim K j for j =, 2, so it is never D 0+ K j D optimal to invest in only one of the resources. Since we will compare Ψ D B) with B γ D )F D +P 43 g i

47 the value of not investing in either resource) in Step 3, we can focus on K p i D B) > 0 here. his implies µ i = 0 for 23) to e satisfied. Case : c D K p i D B) + F D B < Z i U and c D K p i D B) + F D B > Z i L In this case λ i = 0, and 20) yields K p i D B) = K i D. = ) ) + c D + a i ξ. ) For 7), 8) and 9) to e satisfied, and the solution K p i D B) = K i D to e valid, we need ZL i < c D K i D +F D B < ZU i. Here, K0 D is the udget-unconstrained optimal capacity investment and K D is the credit-unconstrained optimal capacity investment. Case 2: c D K p i D B) + F D B = Z i U In this case 8) holds as a strict inequality, so λ i 2 = 0 for 22) to e satisfied. equality as KD 2 = Zi U + B F F c D KD c D, and comining this with 20) yields Z i U + B F D Z i U + B F D K p i D B) = c D ) ξ ) ξ + ξ, 2 c D Rewriting the ) ξ )) 2 ξ + ξ. 24) 2 he condition λ i 0 should e satisfied at optimality. After some algera, this condition implies that 24) is optimal if B c D K i D + F D Z i U. Case 3: c D K p i D B) + F D B = Z i L his case is only relevant for i = since ZL 0 =. In this case, 7) holds as a strict inequality, so λ = 0 for 2) to e satisfied. Rewriting the equality as K2 D = Z L + B F F c D KD c D, and comining with 20) yields K p D B) = Z L + B F D c D ) ξ ) ξ + ξ, 2 Z L + B F D c D ) ξ )) 2 ξ + ξ. 25) 2 he condition λ 2 0 should e satisfied at optimality. After some algera, this condition implies that 25) is optimal if B c D K D + F D Z L. Comining cases, 2 and 3, K p i D B) for i = 0, is characterized y K p 0 D B) = K p D B) = K 0 D = + ) ) c D ξ if c D K 0 D + F D B 0 ) ) ) )) 26) K D = B FD ξ c D, B FD ξ ξ 2 +ξ c D if c 2 ξ D K 0 +ξ D + F D B > 0, 2 ) ) ) )) K D = B FD ξ c D, B FD ξ ξ 2 +ξ c D if c 2 ξ D K +ξ D + F D B 0 2 ) K D = + ) c D +a) ξ if 0 < c D K D + F D B E K ) ) ) )) D = E+B FD ξ c D, E+B FD ξ 2 c D if c D K D + F D B > E. ξ +ξ 2 44 ξ +ξ 2

48 2. Solution for K p B) and Ψ B): o show that gk ) is strictly concave, we need to show that K I, KII 0 and λ 0, ), gλk I + λ)k II ) λgk I ) λ)gk II ) > 0. 27) Since g i K ) is strictly concave, we only need to focus on K I, KII such that c K I +F B and c K II + F > B. We have two cases to consider. First, if c λk I + λ)kii ) + F B then after some algera, the left-hand side of 27) ecomes E [N ] λk I + λ)k II ) + Since x + λe [N ] K I + λ)e [N ] K II + + λ)ac K II + F B). is strictly concave for x 0 and c K II + F B is positive y definition, the aove equation is strictly greater than 0. Second, if c λk I algera, the left-hand side of 27) ecomes + λ)kii ) + F > B then after some E [N ] λk I + λ)k II ) + λe [N ] K I + λ)e [N ] K II + λa c K I + F B). Since x + is strictly concave for x 0 and c K I +F B is negative y definition, the equation aove is strictly greater than 0. Since 27) is satisfied for oth cases, gk ) is strictly concave. It follows that K p B) = K p i B) where i = arg max i Ψ i B) if B > F if B F is the unique maximizer of g. Comining 6) and 26), the unique optimal solution to prolem 3) and the corresponding optimal amount of orrowing are given y K p B) = e p B) = K 0 if c K 0 + F B K if c K + F B < c K 0 + F K if c K + F E B < c K + F K if B < c K + F E, c K p B) + F B ) + 28) 45

49 where K 0 D = K D = K D = K D = KF 0 = K F = KF = K F = )) ξ + )) ξ2 +, c D B ) FD ξ ) B FD c D ξ + ξ, c D 2 )) ξ + )) ξ2 +, c D + a) c D + a) E ) + B FD ξ ) E + B FD ξ + ξ, c D 2 A + c F ) B FF c F c D )) ) A + c F + a) E + B FF c F ) ). c D ) ξ )) 2 ξ + ξ 2 ) ξ )) 2 ξ + ξ 2 We sustitute 28) in 3) and find B γ )F + c K 0 +) + P if c K 0 + F B ) + Ψ B) M B F c = + γ F + P if c K + F B < c K 0 + F B F ) + a) + c K +a) +) + γ F + P if c K + F E B < c K + F E + a) + M E+ B F + γ F + P if B < c K + F E. [ ) ] where M F = E ξ + ξ2 c ) + and M D = relevant and is defined) only for B > F E. 3. Solution for K B) and e B): ξ + ξ 2 ). It follows from ) that Ψ B) is o complete the characterization of K B) and e B), we compare Ψ B) with B γ )F +P the value of the not orrowing and not investing in capacity) for B > F E and estalish that the two functions intersect at most once on B F E, ); and find K B) and e B). For B > F E, we define G B). = Ψ B) B γ )F + P ), the difference etween the equity values in 29) and not orrowing and not investing in capacity. It is easy to verify that, lim B Bk + Ψ B) = lim B Bk Ψ B) for B k > F E therefore, Ψ B) and, in turn, G B) 29) 46

50 are continuous functions of B. We have 0 if c K 0 + F B ) G B) M c B = + ) B F c if c K + F B < c K 0 + F a if c K + F E B < c K + F ) + ) E+ B F c if B < c K + F E. M c 30) For c K + F B < c K 0 + F, M + B c ) F and for B < c K + F E, c ) > M + c ) K 0 ) = 0, 3) M + c ) E + B ) F > M + c c ) K ) = a. 32) It follows that lim B Bk + B G B) = lim B Bk G B) is differentiale for B > F E and c K 0 + F. For c K 0 + F B, B G B) on the domain of G.). herefore B G B) 0 with equality holding only for B G B) = B γ )F + c K 0 + ) + P B γ )F + P ) = c K 0 > 0. 33) + ) We showed that G B) strictly increases for B F E, c K 0 + F ) and is positive for B [c K 0 + F, ). Let B denote the udget level at which the two equity value curves intersect, i.e. G B ) = 0. For F E, we have lim B F E) + G B) = ae < 0. Since G B) strictly increases in B, it follows that for F E, there exists a unique B > F E such that G B ) = 0. For F < E, the domain of G B) is [0, ). For notational convenience, we let. B = 0 if the two curves do not intersect on this domain G B) > 0 for B 0). Since G B) strictly increases in B, it follows that for F < E, B, if it exists on [0, ), is unique. For B B we have K B) = 0 and e B) = 0. Comining this with 28) gives the desired result. Proof of Corollary : he expected stage ) equity value of the firm with a given udget level B follows directly from Proposition 2: B γ )F + c K 0 +) + P if B Ω 0 M B F + γ F + P if B Ω π B) = B F ) + a) + c K +a) +) + γ F + P if B Ω 2 ) + c E + a) + M E+ B F c ) + + γ F + P if B Ω 3 B γ )F + P if B Ω 4 34) 47

51 [ ) ] where M F = E ξ + ξ2 and M D = We calculate M c ξ + ξ 2 )., 0) if B Ω 0 ) ) + /) B F c, M + /) B F ) if B Ω π B) B, 2 π B) ) c +/) B = 2 + a, 0) if B Ω 2 ) ) + /) E+ B F c, M + /)E + B F ) if B Ω 3 M c c +/), 0) if B Ω 4 at the points where π B) is differentiale. It is easy to verify that lim B Bk + B π B) = lim B Bk B π B) for B k Ω 023, and π B) is differentiale everywhere in its domain except at B. Since π B) is a continuous function of B it follows that π B) is strictly increasing in B. We have 2 B π 2 B) 0 for each Ω i and π B) is piecewise concave. From 3) we otain B π B) > for B Ω and from 32) we have B π B) > + a for B Ω 3. Since π B) is only kinked at B it follows that π B) is concave in B for B B, ut not gloally concave. Proof of Proposition 3: Since ξ and α are independent, he optimal risk management level H is given y H = argmax H E [π B F RM α, H ))] 35) s.t. ωf RM H ω 0α F RM E ξ,α [π B F RM α, H ))] = E α [E ξ [π B F RM α, H ))]] = E α [π B F RM α, H ))]. herefore we can write the expectation in 35) over α. Let r α.) and R α.) denote the density and distriution function of α, respectively. Since B F RM α, H ) = ω F RM 0 + α ω F RM H ) + α H, for each H the unique distriution function of B F RM H ) is ) B R BF RM H ) B) ω F RM 0 α H = R α ω F RM H B ω F RM 0 + α H. 36) It follows that H determines the range and the proaility distriution of the availale udget in stage. Since we do not impose any specific assumption on the type of the distriution of α, we will use general structural properties of the optimization prolem 35) to solve for H. In particular, we will focus on the functional form of π B) since the expected stage 0) value of the equity is the expectation of this function with respect to the udget random variale. We first provide the following lemma that we will use throughout the proof. he proof is relegated to Appendix C. 48

52 Lemma here exist unique fixed cost threshold F such that B = 0 iff F F, and B > 0 iff F > F. We now conclude the proof y analyzing each case in Proposition 3. Case i), F F : It follows from Lemma that B = 0. Since B = 0, from Corollary we have that π B) is concave for B 0. From Jensen s inequality, for H E [π B F RM α, H ))] π E[B F RM α, H )]) = π ω F RM 0 + α ω F RM ) ) ωf RM [ 0 α, ω F RM Case ii), F > F : ]. his implies that H = ωf RM. = π BF RM α, ω F RM From Lemma, we have B > 0 and we cannot guarantee the concavity of π for the whole range of B. herefore Jensen s inequality is not sufficient to find H. In this case, H is either a solution to the first order condition H E[π ] = 0, or occurs at a oundary, i.e. H { ωf RM 0 α, ω F RM }. o write the first-order condition, we utilize the following lemma proven in Appendix C: Lemma 2 For any argument κ of π, the expectation and the derivative operators can e interchanged, i.e. [ κ E[π ] = E κ π ]. Let α 0. = c K 0 +F ω0 F RM H α, α. ω F RM H = c K +F ω0 F RM H α, ω F RM H α 2. = c K +F E ω0 F RM H α, α B. B ω F RM H = ω0 F RM H α. ω F RM H From Lemma 2 letting κ = H ), we can write the first-order condition )) 37) H E[π ] y using the expression for π B) in 34) of Corollary and the equivalence in 36). he integration ranges correspond to the regions Ω i in 34) of Corollary. E [ ] π = α x) r α x) dx H maxα 0,0) 38) + maxα 0,0) M + ω F RM c ) 0 + xω F RM ) H ) + α H F α x) r α x) dx c maxα,0) maxα,0) maxα 2,0,αB ) α x) + a) r α x) dx maxα 2,0,α B ) max0,α B ) max0,α B ) 0 M + ω F RM c ) 0 + xω F RM ) H ) + α H + E F α x) r α x) dx c α x) r α x) dx 49

53 Both the limits of integration and the integrants in 38) are functions of H. Since we do not impose any distriutional assumptions on α it is not always possile to find a closed-form solution for H. We have α 0 > α > α2 y definition. For ωf RM 0 + α ω F RM B, α B α. herefore, for ω F RM 0 + α ω F RM B, we either have α 0 > α > α2 t > α B α > 0 or α 0 > α > αb α > 0 > α 2. Similar to 3) and 32) we estalish α 0 α M + ω F RM c ) 0 + xω F RM H ) + α H F c ) α x) r α x) dx < α 0 α α x) r α x) dx, maxα 2,α B ) α B It follows that M + ω F RM c ) 0 + xω F RM H ) + α H F π H < 0 c < α x) r α x) dx + a maxα 2,α B ) α B α he first term is equal to 0 and the second term is negative, therefore π H his concludes the proof for part ) of this case. α B ) α x) r α x) dx α x) + a) r α x) dx. α x) r α x) dx. 39) < 0 and H = ωf RM 0 α. If ω0 F RM + α ω F RM > B [ ], then H either satisfies E π H H = 0 or occurs at a oundary ωf RM 0 { α, ω F RM } depending on the distriutions of α and ξ. From Jensen s inequality, ω F RM B dominates H ω0 F RM α ecause y 36) and Corollary, π B F RM α, H )) is concave over B its domain for H {{ ω0 F RM α. It follows that H B H < } ω0 F RM α { ω F RM } }. Proof of Proposition 4: We first prove the existence of c F c D, H ). Notice from 35) that the optimal financial risk management level H depends on c 4. For each financial risk management level H, the expected stage 0) equity value E [π c, B F RM α, H ))] is a continuous function of c. It follows that the expected stage 0) equity value at the optimal risk management level E [π c, B F RM α, H c )))] is also a continuous function of c ecause it is the upper envelope of continuous functions). For a finite c D > 0, E [π D c D, B F RM α, H D c D)))] is also finite. It is easy to prove that lim E [π F c F, B F RM α, H c F F c F )))] = ω0 F RM + α ω F RM γ F )F F + P, lim E [π F c F, B F RM α, HF c F )))]. c F 0 4 Since from Proposition 3 we cannot guarantee the uniqueness of H, H c ) is a correspondence. 50

54 Since the equity value is continuous in c F, if E [π D c D, B F RM α, H D c D)))] > ω F RM 0 + α ω F RM γ F )F F + P, then there exists a c F such that the equity values with oth technologies coincide. If E [π D c D, B F RM α, H D c D)))] ω F RM 0 +α ω F RM γ F )F F +P then the threshold does not exist and the flexile technology is always preferred over the dedicated technology. his concludes the proof for existence of c F c D, H ). he existence of c F c D, 0) can e proven in the same manner y sustituting B F RM.) with B F RM.) and H c ) with 0. o prove the uniqueness of c F c D, H ) and c F c D, 0) we first provide the following lemma and relegate the proof to Appendix C: Lemma 3 In the optimal set of financial risk management levels, for a fixed level of H, the expected stage 0) value of the equity with technology strictly decreases in the unit capacity investment cost c E[π c, B F RM α, H))] < 0). From Lemma 3 it follows that the expected stage 0) equity value with flexile technology is strictly decreasing in c F for any relevant) financial risk management level H F. his implies the uniqueness of c F c D, H ). he uniqueness of c F c D, 0) follows from Lemma 3 using the identity B F RM α ) = B F RM α, H) for H = 0 and F F RM = 0. For the comparative statics results with respect to demand variaility and correlation we first provide the following [ two lemmas and relegate ) ] their proofs to Appendix C. Recall from Corollary that M F ξ) = E ξ + ξ2. Lemma 4 M F ξ) M F ξ ) for ξ following ways: that is otained from ξ with an increase in σ in one of the i) ξ is otained y an increase in σ where ξ has a symmetric ivariate lognormal distriution, ii) ξ and ξ have independent marginal distriutions, equal means ξ = ξ ), and ξ i v ξ i ξ i is stochastically more variale than ξ i ) for i =, 2 or the variaility ordering holds for only one of the marginals and the other marginal is identical, iii) ξ is random σ 0) while ξ is deterministic σ = 0). Lemma 5 M F ξ) M F ξ ) for ξ following ways: that is otained from ξ with an increase in ρ in one of the i) ξ is otained y an increase in ρ where ξ has a symmetric ivariate lognormal distriution, ii) ξ dominates ξ according to the concordance ordering ξ c ξ), iii) ξ is perfectly positively correlated ρ = ) and ξ is less than perfectly positively correlated ρ < ). 5

55 In Lemma 4 and Lemma 5, case i imposes distriutional assumptions on ξ to analyze the effect of σ and ρ, respectively. Case ii of each lemma analyzes different stochastic orderings to capture the effect of product market conditions. Variaility ordering is often used in the literature to analyze the effect of increasing variaility. Concordance ordering ξ c ξ, as stated in Corett and Rajaram 2005, p. 3), essentially means that ξ, ξ 2 ) move together more closely than ξ, ξ 2 ). Case iii focuses on limiting cases. o estalish the comparative statics results, we provide the following lemma and relegate the proof to Appendix C: Lemma 6 In the optimal set of financial risk management levels, for a fixed level of H, the expected stage 0) value of the equity with technology i) strictly decreases in the fixed cost of technology and strictly increases in the salvage rate F E[π F, B F RM α, H))] < 0 and γ E[π γ, B F RM α, H))] > 0), ii) decreases in unit financing cost a E[π a, B F RM α, H))] 0), and the equality only holds for H such that ω F RM 0 + α H c K + F, iii) increases in credit limit E E[π E, B F RM α, H))] 0), and the equality only holds for H such that ω F RM 0 + α H c K + F E, iv) increases in demand variaility σ E[π E, B F RM α, H))] 0), v) decreases in demand correlation ρ E[π E, B F RM α, H))] 0). Since the expected stage 0) equity value is a continuous function of parameters a, E, F, γ, ρ, σ for a given financial risk management level H, the expected stage 0) equity value at the optimal risk management level which also depends on these parameters) is also continuous in these parameters. herefore the monotonic relations stated in Lemma 6 are also satisfied in the weak sense not strict inequality) at the optimal financial risk management level without assuming differentiaility ecause the expected stage 0) equity value might not e differentiale at the points where the optimal financial risk management level changes). he comparative static results for c F c D, H ) follow from Lemma 6. he comparative static results for c F c D, 0) also follow from Lemma 6 using the identity B F RM α ) = B F RM α, H) for H = 0 and F F RM = 0. With symmetric fixed costs and salvage rates, we estalish the functional form of c S F c D) with the following Lemma and relegate the proof to Appendix C: Lemma 7 When the fixed costs and the salvage rates of the two technologies are symmetric, at c F = c S F c D) expected stage ) equity values, expected stage 0) equity values at an aritrary financial 52

56 risk management level H and the optimal financial risk management actions are the same for oth technologies, i.e.π F c F, B) = π Dc D, B) for B 0, E[πF c S cf =c S F c F c D), B F RM α, H))] = D) E[π D c D, B F RM α, H))] and H F cs F c D)) = H D c D). It follows from Lemma 7 that c S F c D) is the unique threshold with financial risk management in the symmetric case c F c D, H ) = c S F c D)). Using the identity B F RM α ) = B F RM α, H) for H = 0 and F F RM = 0, it follows from Lemma 7 that c S F c D) is also the unique threshold without financial risk management in the symmetric case c F c D, 0) = c S F c D)). We now prove the relation c S F c D) c D. It is sufficient to show [ ) ] E ξ + ξ2 E [ξ ] + E [ξ 2 ]. From Hardy et al. 988, p.33,46) if d 0, ) and X and Y are non-negative random variales then the following is true: E /d [ X + Y ) d] E /d [X d ] + E /d [Y d ] 40) where the equality only holds when X and Y are effectively proportional, i.e. X = λy. In the expression for c S F c D) we have d = 0, ) and ξ > 0 therefore we can use this inequality. Replacing X with ξ and Y with ξ2 gives the desired result. Notice that c S F c D) = c D only if ξ = kξ 2 for k > 0. his is only possile if either ξ is deterministic or it is perfectly positively correlated and has a proportional ivariate distriution. Proof of Corollary 2: If the capital markets are perfect we have E = P c K 0 + F and a = 0 as we discussed in Assumption 6). Since we have Ω 234 =, it follows from Proposition 2 that the firm invests in the udget-unconstrained capacity investment level for any udget realization, K B) = K 0, and orrows to finance this capacity level, e B) = [c K 0 +F B] +. We otain E [π B F RM α ))] = E [π B F RM α, H))] FF RM =0 = ω 0 + α ω γ )F + c K 0 + ) + P, and it follows from Proposition 2 that F F RM = 0 for {D, F }. If the product markets are perfect Σ = 0), then with symmetric fixed costs and salvage rates, it follows from Proposition 4 that c F c D, H ) = c F c D, 0) = c D. Proof of Corollary 3: he proof of the first argument follows from Proposition 3. For the second argument, we provide a numerical example where the firm optimally fully speculates with flexile technology and fully hedges with dedicated technology. We focus on the case with F F RM = 0 such that financial risk management is costless. he horizontal line in Figure denotes the value of not investing any technology; hence the firm optimally chooses flexile technology with full speculation in this example. 53

57 Figure : Optimal Speculation is triggered y flexile technology investment: Dedicated technology with full hedging H D = ω = 4) is dominated y flexile technology with full speculation H F = w 0 α = 0.6. Proof of Proposition 5: With a hedging constraint, the range of forward contracts is [0, ω F RM ] in 35). Sustituting F F RM = 0 in 38) of Proposition 3, similar to 39), we otain π H follows that H = 0. Proof of Corollary 4: < 0. It It follows from Proposition 4 that for symmetric fixed costs and salvage rates of technologies and for F F RM = 0, the optimal risk management portfolio is flexile dedicated) technology with financial risk management if c F < c S F c D) c F > c S F c D)). From the proof of Proposition 2, for β = ω 0 ω 0 +α 0 ω we can have a sufficiently large feasile F F RM such that engaging in financial risk management is not profitale. In this case, the optimal risk management portfolio is flexile dedicated) technology with financial risk management if c F < c S F c D) c F > c S F c D)). Proof of Proposition 6: he invariance of c F c D, H ) and c F c D, 0) to the unit financing cost, the fixed cost of oth technologies and the internal endowment of the firm follows from the definition of c S F c D) in Proposition 4. For F = F D < F F = F + δ with δ > 0, we otain c F c D, H ) < c S F c D) and c F c D, 0) < c S F c D) from Proposition 4. We first provide the proof of the results with respect to technology fixed costs. Comparative statics with respect to the internal endowment follow from a similar argument. We define S F RM c F ). = E[π F c F, F + δ, B F RM α ))] E[π D c D, F, B F RM α ))] where S F RM c F c D, 0)) = 0.4) From the implicit function theorem we have F c F c D, 0) = F S F RM c F S F RM ) cf c D,0). From Lemma 2, we can interchange derivative and expectation operators, and using Lemma 3 with B F RM α ) = B F RM α, H) for H = 0 and F F RM = 0, we otain [ ] S F RM πf B F RM α )) c F = E cf c D,0) c F 54 c F c D,0) < 0.

58 Similarly we have S F RM F = cf c D,0) + Ω F [ E [ πf B F RM α )) F M F + ) [ Ω D c F c D, 0)) + ] [ πd B F RM α ))) E F B F δ ) + M D c + D ]]. c F c D,0) Since c F c D, 0) < c S F and δ > 0 it follows that c F K i F cf c D,0) + F + δ > c D K i D + F for i = 0,. his implies that Ω 0 F Ω0 D and Ω2 F Ω2 D. We otain F RM S F = + ) dr B F RM B) 42) cf c D,0) Ω 0 F Ω 0 D + M ) ) F + B F δ + dr B F Ω F Ω 0 D c F c D, 0) c F c D, 0) RM B) ] ) B F From 3), we have we have M D c + D = Ω 2 F + a) + ) dr B F RM B) Ω 0 D [ ] + a) + + ) M ) D B F dr B F RM B) Ω 2 F Ω D Ω 2 F c + D Ω 2 D + a) + + a)) dr B F RM B). dr B F RM B) F S F RM cf c D,0) < 0 for B Ω F Ω 0 D and B Ω 2 F Ω D. From Lemma 7, M F c S F c D)) +. Since c F c D, 0) < c S F c D) and δ > 0, we otain F S F RM cf c D,0) < 0 for B Ω F Ω D. In conclusion, we have c F S F RM cf c D,0) < 0 and F S F RM cf 0. It c D,0) follows from the implicit function theorem that F c F c D, 0) 0 where the equality holds only for ω 0 > c F K 0 F cf c D,0) + F + δ. o prove the result for c F c D, H ), we define S F RM c F ), the counterpart of 4) y replacing B F RM α ) with B F RM α, H ). We have H F c F ) = HD = ωf RM for c F = c F c D, H ). We estalish c F S F RM cf < 0 using c D,H ) c F HF = 0. he rest of the proof follows in a cf c D,H ) similar manner using the facts that = 0 and that with full-hedging ωf RM 0 +α ω F RM F H cf c D,H ) is realized in only one of the regions in 42). In conclusion, it follows from the implicit function theorem that F c F c D, H ) 0 where the equality holds only for ω F RM 0 + α ω F RM Ω 0 F Ω 0 D or ω F RM 0 + α ω F RM Ω 2 F Ω 2 D. o prove the results with respect to the unit financing cost for c F c D, 0), we follow the same 55

59 steps y replacing F with a in S F RM c F ). We otain S F RM a = B cf K F + F F ) cf c D,0) Ω 2 cf cd,0)) dr B F RM B) 43) F \Ω2 D + c D K D + F D c F K F + F F ) ) Ω 2 F Ω 2 cf dr c D,0) B F RM B) D + E dr B F RM B) + Ω 3 F \Ω23 D Ω 3 F Ω 2 D B + c F K F + F F ) cf c D,0) E ) dr B F RM B). he first term and the last integrands are negative y the definition of the regions. From aove comparative static with respect to fixed cost) we have c D K D + F D < c F K F cf c D,0) + F F. a S F RM cf c D,0) his implies ω 0 > c F K F cf c D,0) + F F. 0. We conclude a c F c D, 0) 0 where the equality holds for he result for c F c D, H ) can e proven in a similar fashion. It follows that a c F c D, H ) 0 where the equality holds if ω0 F RM + α ω F RM Ω 0 ) Ω 3 F Ω 3 D. Proof of Proposition 7: follow from similar arguments. We define Υ ϕ. = ϕ = E F We only prove the results for small firms. Results related to large firms [ π BF RM α, ω F RM ) )] E [π B F RM α ))] ϕ ϕ as the derivative of the value of full hedging with respect to the argument ϕ. For small firms, we have E [ π BF RM α, ω F RM ) )] = ω0 F RM + α ω F RM F ) + a) + c K +a) +) + γ F + P. We analyze each comparative static result separately. Fixed cost of technology. We otain 44) Υ F = + a) dr B F RM B) Ω 0 Ω + )M c B F c + a) dr B F RM B). ) dr B F RM B) Ω 2 ) B F c It is easy to show that + ) M c < + a for B Ω. It follows that ΥF < 0. Initial endowment. After parameterizing the initial endowment, we otain β λ = ω 0 ω 0 +α 0 ω λω 0 λω 0 +α 0 λω = = β. We have ω0 F RM, ω F RM ) =. λω 0 βf F RM, λω β α 0 F F RM ) and for small firms, it fol- ))] = ω 0 + α ω ) + a). After parameterizing the initial endowment, lows that E[π B F RM α,ω F RM λ 56

60 we define α 0. = c K 0 +F ω λ 0 ω, α. = Υ λ = ω 0 + α ω ) + a) maxα 0,0) maxα,0) maxα,0) 0 M + /) c c K +F ω λ 0 ω. We otain ω 0 + xω ) r α x) dx maxα 0,0) λω0 + xω ) F c ω 0 + xω ) + a) r α x) dx ) ω0 + xω ) r α x) dx Notice that negative terms aove are the expected value of the following function ω 0 + α ω if α α 0 ) fα ) = M +/) λω0 +α ω ) F c c ω 0 + α ω ) if α 0 > α α ω 0 + α ω ) + a) if α < α with respect to the asset price distriution α. It is easy to prove that ω 0 +α ω )+a) fα ) for α 0 with strict inequality for some α. It follows that E[ω 0 +α ω )+a)] = ω 0 +α ω )+a) > E[fα )] and we otain Υ λ > 0. o analyze the effect of cash holdings ω 0 ) on the value of financial risk management, we only parameterize the cash holdings as λ ω 0, ω ) and set β = 0 such that F F RM is only deducted from the value of asset holdings ω. It follows that ω F RM 0 = λ ω 0 and ω F RM = ω F F RM α 0. Υ λ > 0 follows from the similar lines with Υ λ > 0. Demand variaility and correlation. We only provide the proof for demand variaility. he proof for demand correlation is along the similar lines. It is sufficient to focus on flexile technology ecause dedicated technology is not affected from changes in σ and ρ. We otain Υ σ = M F σ Ω F cf M F + ) ) M F σ It is easy to show B FF + a B FF c F ) + c F and it follows that Υ σ 0. Ω 0 F M F σ ) + dr B F RM B) Ω 2 F > Unit financing cost. We otain c F M F + ) ) dr B F RM B) M F σ Υ a = ω 0 + α ω β + β)α )F F RM c K F α 0 cf M F + ) ) dr B). B F RM + a ) cf M F + ) +a for B Ω F. From Lemma 4, we have σ M F 0 Ω 2 B c K F ) dr B F RM B). It follows that for ω 0 > c K + F, when the non-hedged firm does not orrow at all, we have Υ a < 0. We focus on the case where the firm orrows at some udget states without financial risk 57

61 management ω 0 < c K + F ). For F F RM = 0, we have Υ a = Ω 2 c K + F B) dr B F RM B) c K + F B) where B = ω 0 + α ω. Notice that the first term is the expected value of the function f B) c K = + F B if B c K + F 0 if B > c K + F with respect to the udget distriution. Since f B) is a convex function, Υ a 0 for F F RM = 0 follows from Jensen s inequality. For F F RM > 0, we have Υ a = c K + F B) dr B F RM B) c K + F + β + β)α )F F RM B). α 0 Ω 2 We oserve that the first term is strictly less than c K + F ω 0. For F F RM F 0 F RM = α 0 ω, we otain c β)+ β K + F + β + β)α α 0 )F F RM B c K + F ω 0 and it follows α that Υ a < 0. Notice that F F RM α 0ω β) is the feasility condition; hence such F F RM exists. We calculate F F RM Υ a = β + β)α ) < 0. Since Υ a strictly decreases in F F RM, Υ a 0 for α 0 F F RM = 0 and Υ a < 0 for F 0 F RM, we conclude that there exists a unique F F RM such that Υ a < 0 for F F RM > F F RM and Υ a 0 for F F RM F F RM. Proof of Proposition 8: We focus on the case where it is profitale for the firm to engage in financial risk management. o prove the proposition, we use the ordering etween c F c D, H ) and c F c D, 0). If c F c D, 0) < c F c D, H ) c F c D, 0) > c F c D, H )) then flexile technology and financial risk management are complements sustitutes) ecause engaging in financial risk management enales the firm to invest in flexile dedicated) technology at some technology cost levels where dedicated flexile) technology was more profitale without financial risk management. From Proposition 4, we otain c F c D, H ) < c S F c D) and c F c D, 0) < c S F c D). From Assumption 8, we have H D c D) = H F c F c D, H )) = ω F RM. From Lemma 3, it follows that c F c D, H ) c F c D, 0) if and only if E [ π D B F RM α, ω F RM )) ] E [ π F c F c D, 0), B F RM α, ω F RM ))) ]. 45) Recall that c, F ) is the value of financial risk management with technology {D, F } at given cost parameters c, F ) as defined in 5). Inequality 45) holds if and only if F c F c D, 0), F F ) D c D, F D ). We will use the relation etween F c F c D, 0), F F ) and D c D, F D ) to prove the proposition. We provide the following lemma and relegate the proof to Appendix C. Lemma 8 For F < F and E > c K + F, i) If ω F RM 0 + α ω F RM Ω 0 Ω2 ) then F 0 F < 0); ii) If ω F RM 0 + α ω F RM Ω 0 Ω2 ) then c 0 c > 0). 58

62 For large firms ω F RM 0 + α ω F RM Ω 0 F ), we otain from Lemma 8, cs F c D) > c F c D, 0) and F F F D that D c D, F D ) = F c S F c D ), F D ) F c F c D, 0), F D ) F c F c D, 0), F F ). From the proof of Lemma 8, the inequalities aove are strict for sufficiently low ω 0. We conclude that c F c D, H ) c F c D, 0) and large firms tend to use flexile technology and financial risk management as complements. For small firms ω0 F RM + α ω F RM Ω 2 F ), we otain D c D, F D ) = F c S F c D ), F D ) > F c F c D, 0), F D ) > F c F c D, 0), F F ). We conclude that c F c D, H ) < c F c D, 0) and small firms tend to sustitute flexile technology with financial risk management. Proof of Proposition 9: We only prove the results for small firms. Results related to large firms follow from similar arguments. Recall that in the proof of Proposition 6 we defined S F RM = E [ π F BF RM α, ω F RM ) )] E [ π D BF RM α, ω F RM ) )] S F RM = E [π F B F RM α ))] E [π D B F RM α ))] as the value of operational risk management with and without financial risk management respectively. he value of operational risk management is more roust to a change in ϕ {a, ρ, σ} with financial risk management then without if S F RM ϕ < S ϕ F RM o analyze the roustness of the value of operational risk management, we focus on the cases where operational risk management has a value, i.e. flexile technology is preferred over dedicated technology with and without financial risk management. Recall from the proof of Proposition 6 that we have c F c D, H ) < c S F c D) and c F c D, 0) < c S F c D) in this setting. relevant unit investment cost pair c F, c D ) we have c F condition separately.. herefore, for any < c S F c D). We now analyze each market Roustness with respect to capital market condition a). Since c F < c S F c D), it follows from 43) that a SF RM 0 and a S F RM 0. herefore, it is sufficient to show a SF RM asf RM to prove the result of lower roustness. It follows from 44) that this condition is equivalent to a F a D. We otain F a D a = ω 0 + α ω sα )F F RM c F K F F F [ω 0 + α ω sα )F F RM c D K D F D ]χb Ω 2 D) + 59 Ω 2 D Ω 2 F B c F K F F F ) dr B F RM B) B c D K D F D ) dr B F RM B).

63 where sα ) = β + β)α α 0, B = ω 0 + α ω sα )F F RM and χ.) is the indicator function. We have the indicator function ecause a small firm that always orrows with financial risk management with flexile technology) need not to orrow with financial risk management with dedicated technology. We now show that a F a D y focusing on two cases. Case i : B Ω 2 D ) We otain F a D = c F K F + F F a B) dr B F RM B) Ω 2 F \Ω2 D + c F K F + F F c D K D F D ) dr B F RM B) c F K F + F F c D K D F D ). Ω 2 D Ω 2 F Since for B Ω 2 F \Ω2 D we have B c D K D + F D, it follows that a F < a D. Case ii : B Ω 0 D + F a Ω 2 F D a = ) We otain c F K F + F F B) dr B F RM B) Ω 2 F \Ω2 D Ω 2 D c F K F + F F c D K D F D ) dr B F RM B) c F K F + F F ω 0 α ω + sα )F F RM ). Since we have ω 0 + α ω sα )F F RM > c D K D + F D, it follows that a F < a D. his concludes the proof for the roustness result with respect to capital market condition. Roustness with respect to product market conditions ρ, σ). We only provide the proof for ρ. From Lemma 6, we have ρ SF RM 0 and ρ S F RM 0. herefore, it is sufficient to show ρ SF RM ρ SF RM to prove the result of higher roustness for small firms. It follows from 44) that this condition is equivalent to a F 0. he result follows from Proposition 7. Proof of Proposition 0: o demonstrate the amiguous effect of financial risk management on expected stage 0) capacity investment level, it is sufficient to provide examples for each case of E[ K B F RMα ))] E[ K B F RMα, H ))]. We consider F F = F D = 0 which implies from Proposition 3 that the firm optimally fully hedges with oth technologies Ω 4 = ). Let F F RM = 0 such that financial risk management is costless. Without loss of generality we consider c F < c F which implies from Proposition 4 that = F with or without financial risk management. Let E e sufficiently large E c K a) +)a is sufficient as follows from Lemma 9 in Appendix B) such that the firm does not orrow up to the credit limit Ω 3 F = ). With these parameter restrictions, we otain E[K FB F RM α ))] = E[K FB F RM α, ω ))] = Ω 0 F K 0 F dr B F RM B) + K F dr B F RM B) + K F Ω F Ω 2 K 0 F if ω 0 + α ω Ω 0 F K F if ω 0 + α ω Ω F K F if ω 0 + α ω Ω 2 F. 60 dr B F RM B),

64 We have K 0 F > K F, and K0 F > K F K F for B Ω F with equality only holding for the lower ound of the region Ω F. For ω 0 Ω 0 F and hence ω 0 + α ω Ω 0 F ), E[K F B F RMα ))] = E[K F B F RMα, ω ))]. For ω 0 Ω 2 F and ω 0+α ω Ω 0 F, E[K F B F RMα ))] < E[K F B F RMα, ω ))]. For ω 0 + α ω Ω 2 F and hence ω 0 Ω 2 F ), E[K F B F RMα ))] > E[K F B F RMα, ω ))]. If we relax our assumption on E, we otain E[e F B F RM α ))] = [c F K F B] dr B F RM B) + E[e F B F RM α, ω ))] = It follows that for ω 0 + α ω Ω 3 F Ω 2 F Ω 3 F 0 if ω 0 + α ω Ω 0 F c F K F ω 0 α ω if ω 0 + α ω Ω 2 F E if ω 0 + α ω Ω 3 F. E dr B F RM B), we have E[e F B F RMα ))] < E[e F B F RMα, ω ))] and for ω 0 + α ω Ω 0 F we have E[e F B F RMα ))] > E[e F B F RMα, ω ))]. Proof of Corollary 5: he proof follows from Proposition 8. Proof of Corollary 6: From Proposition 3, it follows that small firms, as we define in 7, optimally fully speculates H = ω 0 α. For ω 0 = 0, the firm optimally does not engage in financial risk management. he low value of integration follows from a continuity argument and the ounded derivative of expected stage 0) equity value with respect to ω 0. management does not have any value if ω 0 c K 0 + F, i.e. For large firms, financial risk the cash level is sufficient to finance the udget-unconstrained optimal capacity investment level. Low value of financial risk management at high cash levels follow from similar arguments with small firms. When the firm uses financial risk management only for hedging purposes, it follows from Proposition 5 that small firms optimally do not engage in financial risk management. herefore, the value of integration is zero for small firms. Large firms tend to use financial risk management for full-hedging purposes. For ω 0 < c K 0 +F, financial risk management has positive value; hence the value of integration is higher for large firms than small firms. his concludes the proof. Proposition If the firm does not engage in financial risk management, there exists a unique F RM technology fixed cost threshold F F < F F RM not investing in technology. < c K 0 +) γ ) for technology {D, F } such that when, investing in technology without financial risk management is more profitale than If the firm engages in financial risk management, only one of the following cases holds, depending on the level of the fixed cost F F RM : i) here exists a unique technology fixed cost threshold F F RM technology {D, F } such that when F 6 c K 0 +) β+ β)α F α F RM 0 γ for < F F RM, investing in technology is more

65 profitale than not investing in technology; this case occurs at sufficiently low levels of F F RM. ii) Not investing in technology is more profitale for F 0. Proof of Proposition : We first prove the first part of the proposition. From Lemma 6 in the proof of Proposition 4 using H = 0 and F F RM = 0), E [π B F RM α ), F )] is strictly decreasing in F. We define L B). = π B) B + P ), the difference etween the equity values of investing in technology and not investing in technology at each state B. It is easy to verify that for F 0. = 0, π B) > B + P for B 0. It follows that E [ π F 0, B F RMα )) ] > ω 0 + α ω + P. For F > c K 0 +) γ ) we have L B) < 0 for B 0. It follows that E [ π F, B F RMα )) ] < ω 0 + α ω + P. Since E [π F, B F RM α ))] is strictly decreasing in F, there exists a unique F RM F < c K 0 +) γ ). he second part of the proposition follows from a similar argument. We otain E [π B F RM α, H ) ))] ω 0 + α ω β + β)α α 0 F F RM γ )F + c K 0 +) + P, where the latter is the expected stage 0) equity value with udget-unconstrained optimal capacity investment. It follows that for F > F = c K 0 +) β+ β)α F α F RM 0 γ, not investing in technology is more profitale. wo cases may arise with respect to the level of F F RM. When F F RM is sufficiently low, for F 0 = 0 we have E [ π F 0, B F RMα, H ))] > ω 0 +α ω +P. In this case case i), a unique F F RM < F exists since E [π F, B F RM α, H ))] is strictly decreasing in F. For a sufficiently high level of F F RM and appropriate allocation scheme β that makes such a F F RM feasile), not investing in technology is more profitale for F technology is more profitale for F 0. = 0. In this case case ii), F F RM does not exist and not investing in Proposition 2 Only one of the following cases holds for technology : i) here exists a unique financial risk management fixed cost threshold F F RM such that when F F RM < F F RM, it is more profitale to engage in financial risk management than not; ii) For any feasile F F RM, engaging in financial risk management is more profitale than not. Proof of Proposition 2: he proof follows from showing that E [π B F RM α, H ))] strictly decreases in F F RM. From Lemma 2, we can interchange the derivative and expectation operators 62

66 and using the Leiniz rule we otain [ ] π B F RM α, H)) E = β β x) r α x) dx 46) F F RM maxα 0,0) α 0 maxα 0,0) ) M + /) Ux) β + β x) r α x) dx c c α maxα,0) maxα,0) β β x) + a) r α x) dx maxα 2,0,αB ) α 0 ) M + /) Ux) + E β β x) r α x) dx c c α 0 maxα 2,0,α B ) max0,α B ) max0,α B ) 0 β β α 0 x) r α x) dx for any feasile H, where Ux) = ω F RM 0 + xω F RM H) + α H F. Since all terms are negative, it follows that E [π B F RM α, H ))] is strictly decreasing in F F RM. For F F RM = 0, we have E [π B F RM α, H ))] E [π B F RM α ))] from the optimality of H. he existence of ) F F RM min ω0 β, α 0ω β depends on the allocation scheme β. If β is such that a sufficiently large level of F F RM is feasile, then since E [π B F RM α, H ))] is strictly decreasing in F F RM, there exists a unique F F RM case i). Otherwise, since E [π B F RM α, H ))] is preferred for F F RM = 0, case ii holds. ω We show that β such that case i holds. Let β = 0 ω 0 +α 0 ω. It follows that the condition F F RM ) min ω0 β, α 0ω β is equivalent to F F RM ω 0 +α 0 ω. We otain lim FF RM ω 0 +α 0 ω B F RM α, H) = 0; therefore E [π B F RM α, H ))] F F RM ω 0 +α 0 ω F F RM exists. < E [π B F RM α ))]. It follows that a unique Proposition 3 For technology {D, F } there exists a unique variale cost threshold c c, H, 0) such that investing in technology with financial risk management is more profitale than investing in the other technology ) without financial risk management. Proof of Proposition 3: he proof follows as in Proposition 4, and is omitted. 63

67 B Appendix B. Characterization of B Recall from Proposition 2 that B is the udget threshold elow which the firm does not orrow or invest. From the proof of Proposition 2, for F E, B > F E is the unique solution to G B ) = 0 where G B). = Ψ B) B γ )F + P ), the difference etween the equity values in 29) and not orrowing and not investing in capacity. For F < E, B, if it exists on [0, ), is unique. For notational convenience, we let B. = 0 if the two curves do not intersect on the domain of G.) for F < E. From 29) for B F we otain lim K 0 + K Ψ. It follows that the firm always optimally invests in capacity if internal udget B is sufficient to cover the fixed cost of the technology. We conclude that F E < B < F. Since Ψ B) can take four different forms we have four different cases to analyze. Case : c K 0 + F B From 33), G B) > 0 in this range, so it is not possile to have c K 0 + F B. Case 2: c K + F B < c K 0 + F G B) = M B F ) c M B F ) c c B F ) K 0 + γ F + P B γ )F + P ) ) + F B = + B F ) > 0. herefore, it is not possile to have c K + F B < c K 0 + F. Case 3: c K + F E B < c K + F G B ) = B F ) + a) + c K + a) + γ F + P + ) B γ )F + P ) = 0 B = F c K + a). + )a For B to e feasile in Case 3, B 0 and B c K + F E should hold. herefore, if F c K +a) +)a and E c K a) +)a c K + F E B < c K + F. Case 4: c K + F E > B then B is feasile. Otherwise, it is not possile to have In this case, we can derive a sufficient condition for non-existence of intersection. We otain G B) = E + a) + M E + B F ) c E + a) + M E + B F ) c E + a) + ) + a + ) B F ). c E + B F ) + F B K 64 ) + γ F + P B γ )F + P )

68 herefore if F < E+a) a, then G B) > 0 and it is not possile to have c K + F E > B. Otherwise, B is a solution of a non-integer polynomial of degree + and it is not possile to find closed-form expression in the whole range of parameters. he following lemma summarizes the analysis and provides a closed-form expression for B for a suset of parameter levels. Lemma 9 Let E e such that E c K a) +)a. If F c K +a) +)a then B = 0 and Ω 34 =. If F > c K +a) +)a then B = F c K +a) +)a and Ω 3 =. We also provide the following lemma which we will occasionally use in the comparative statics analysis throughout the paper. Lemma 0 he udget threshold B is increasing in c, F, a and decreasing in E. Proof We only provide the proof for the result related to a. he other results can e shown in a similar fashion. Let B a i ), i = 0, define the threshold levels for an aritrary a 0 < a. We want to show that B a 0 ) B a ). Notice that not only the functional form of G B) in any region ut also the udget levels defining the regions in 29) depend on a. We otain G B) a 0 if c K 0 + F B 0 if c K = + F B < c K 0 + F B c K F if c K + F E B < c K + F E if F E < B < c K + F E. at the points where G B) is differentiale in a. It follows that a G B) 0 for any B where the function is differentiale. Since G B) is a continuous function of B for any a, we conclude that G B) is decreasing in a. his implies G B, a 0 ) G B, a ) for B > F E. At this point, two different cases may arise regarding the definition of B a 0 ). If B a 0 ) is the solution of G B, a 0 ) = 0, then we have G B a 0 ), a ) G B a 0 ), a 0 ) = 0. Since G B) is increasing in B from 30), it follows that B a ) B a 0 ). If B a 0 ) = 0 ecause G B, a 0 ) > 0 for B 0, then from 30) either we have B a ) = 0, G B, a ) > 0 for B 0) or B a ) is a solution to G B, a ) = 0. In either case, we have B a ) B a 0 ). 65

69 C Appendix C. Proofs of Supporting Lemmas Proof of Lemma : From Appendix B, we calculate 0 if c K 0 + F B G B) F = M c M c ) + ) B F c + if c K + F B < c K 0 + F a if c K + F E B < c K + F ) + ) E+ B F c + if F E < B < c K + F E. From 3), 32) and the continuity of G B), it follows that G B) strictly decreases in F B < c K 0 + F. Recall from Proposition 2 or Appendix B) that either B G B) = 0 or B = 0 if G B) > 0 for B 0). We first prove the necessity of the second argument. Let F for is a solution to e the fixed cost that satisfies G B F ), F ) = 0 with B F ) = 0. In other words, F is the fixed cost of technology that makes the two equity values intersect at B = 0. From Appendix B, it follows that for F = 0, G B) > 0 for B 0. For F E, we have lim B F E) + G B) < 0, and two curves intersect at B > F E. Since G B) is strictly decreasing in F, such an F < E always exists. Let F 0 F e an aritrary fixed cost. We have G B F ), F 0 ) < G B F ), F ) = 0 since B < c K 0 +F follows from Appendix B) and G strictly decreases in F. From 30) we have G B) is strictly increasing in B so it follows that B F 0 ) > B F ) = 0. We now prove the necessity of the first argument. Let F < F e an aritrary fixed cost. Since B F ) = 0 and G B) strictly decreases in F, we have G B, F ) > 0 for B 0. his implies that B F ) = 0 for F < F. he uniqueness of F follows from the fact that G B) is strictly decreasing in F and the uniqueness of B. he proof for sufficiency follows easily using a contrapositive argument. Proof of Lemma 2: he expectation and differentiation operators can e interchanged if the function under expectation is integrale and satisfies the Lipschitz condition of order one Glasserman 994, p.245). he function π α ) satisfies the Lipschitz condition of order one if π α ) π α ) α α Y π α, α ) > 0 for some Y π with E[Y π ] <. 47) ) ) π α is ounded. Note that α π = B π α B = ) ω H ). From Corollary, we know that π is differentiale in α everywhere except Clearly, condition 47) is satisfied if B π at α B as defined in 38). If B Ω we have π B M + c ) K ) 66 + a),

70 and for B Ω 3 since B 0 and E > F from )) we have π B M + ) E c ) F Y c where + a < Y <. It follows that π α Y ω H ) < for α 0 except α B. Since π is continuous in α and the first derivative is ounded at the differentiale points of π, the non-differentiaility at α B does not violate 47). Since π α ) is integrale, the interchange of the derivative and expectation is justified. Proof of Lemma 3: From Lemma 2, we can interchange the derivative and the expectation operators and using the Leiniz rule we otain E[π B F RM α, H))] = K 0 dr c BF RM H) B) 48) H Ω 0 Ω Ω 2 Ω 3 + )M c B F c K + a) dr BF RM H) B) + )M c ) + dr BF RM H) B) E + B ) + F dr c BF RM H) B). It follows that c E[π B F RM α, H))] 0 with equality holding only for H = ω F RM and ω0 F RM + α ω F RM Ω 4 i.e. Ω023 = ). From Proposition 3, we know that in this case ωf RM 0 = α, so we can ignore H = ω F RM. In other words, in the relevant set of B F RM α, H) we have c E[π B F RM α, H))] < 0. Proof of Lemma 4: Case i): he proof follows from Lemma 3 of Chod et al. 2006) y sustituting τ = and noting that ρ and σ in that paper correspond to parameters of the underlying ivariate normal distriution ln ξ) of ξ. In our paper, ρ and σ are the parameters of ξ in the covariance matrix Σ. Case ii): We only prove the more general case where oth of the marginal distriutions of ξ pairwise stochastically more variale than the marginal distriutions of ξ. he proof for the case where one of the marginals is identical for ξ i and ξ i is a special case of this proof. For ξ i 0, ξ i 0 and ξ i = ξ i it follows from Ross 983, p.27) that ξ i v ξ i if and only if E[hξ i )] E[hξ i )] for all convex functions h.). With independent marginal distriutions of ξ we have [ ) ] ) E ξ + ξ2 = x + x 2 f x )f 2 x 2 )dx dx 2 = gx ; x 2 )f x )dx 0 0 k + x 2 o conclude the proof, we need to show that gk; x 2 ) is convex in k and where f i.) is the marginal distriution of ξ i and gk; x 2 ) = ) are f 2 x 2 )dx 2 for k 0. k + x 2 ) is convex

71 in x 2. o prove oth of the desired convexity results, it is sufficient to show that g k, x 2 ) is convex in k. We otain 2 g k = ) k + x 2 for k 0 and x 2 0. his concludes the proof. ) k 2 Case iii): Follows from 40) in the proof of Proposition 4. Proof of Lemma 5: x 2 k + x 2 ) 0 Case i): he proof follows from Lemma 4 of Chod et al. 2006) y sustituting τ = and noting that ρ and σ in that paper correspond to parameters of the underlying ivariate normal distriution ln ξ) of ξ. In our paper, ρ and σ are the parameters of ξ in the covariance matrix Σ. Case ii): he proof of this case is adapted from Corett and Rajaram 2005). If ξ c ξ, it follows from Muller and Scarsini 2000, p.0) that ξ sm ξ ξ dominates ξ in the sense of supermodular order). From the definition of supermodular stochastic ordering, it is sufficient to ) show that gξ, ξ 2 ) = ξ + ξ2 is supermodular. From Muller and Scarcini 2003), it follows that g is supermodular if and only if all mixed derivatives are non-negative, i.e. 2 ξ ξ 2 g 0 for ξ 0. We otain his concludes the proof. 2 g ) = ) ξ + ξ 2 2 ξ ξ 2 ) 2 0. ξ ξ 2 Case iii): Follows from 40) in the proof of Proposition 4. Proof of Lemma 6: Case i): As in Lemma 3, we otain E[π B F RM α, H))] = F Ω 0 Ω Ω 2 Ω 3 Ω 4 γ ) dr BF RM H) B) 49) + )M c B F c ) γ + a) dr BF RM H) B) + )M c E + B ) F c γ ) dr BF RM H) B). γ dr BF RM H) B) γ dr BF RM H) B) Since γ < y definition, it follows from 3) and 32) that the second and the fourth terms are negative. his implies that F E[π B F RM α, H))] < 0. We have 68 γ E[π B F RM α, H))] = F

72 and it follows that the expected stage 0) equity value is strictly increasing in the salvage rate for F > 0. Case ii): We otain E[π B F RM α, H))] E = Ω 3 + a) + M ) + E + B F c c ) dr BF RM H) B). It follows that E E[π B F RM α, H))] 0 with equality holding for H such that ω F RM 0 + α H c K + F E; or H = ω F RM and ω0 F RM + α ω F RM < B. From Proposition 3 we know that in the latter case H ωf RM 0 = α and we can ignore this case in the relevant set of financial risk management levels. Case iii): We otain E[π B F RM α, H))] a = Ω 2 B c K F ) dr BF RM H) B) E Ω 3 dr BF RM H) B). It follows that a E[π B F RM α, H))] 0 with equality holding for ω F RM 0 + α H c K + F ; or H = ω F RM and ω F RM 0 + α ω F RM < B. From Proposition 3 we know that in the latter case H = ωf RM 0 α and we can ignore this case in the relevant set of financial risk management levels. Case iv): he expected stage 0) equity value with dedicated technology is independent of σ. herefore, we focus only on flexile technology. We otain E[π F B F RM α, H))] σ = Ω 0 F Ω F Ω 2 F Ω 3 F M F σ M F σ M F σ M F σ c F M F + ) ) dr BF RM H) B) 50) B FF c F ) + dr BF RM H) B) cf M F + ) ) dr BF RM H) B) + a E + B F F c F ) + dr BF RM H) B). From Lemma 4, we have σ M F 0 with respect to our definitions of demand variaility. It follows that σ E[π B F RM α, H))] 0. Case v): he proof of the comparative static result with respect to ρ is similar to σ and is omitted. Proof of Lemma 7: It is easy to verify that we have c F K j F = c D K j c S D F c D ) for j = 0,. Since F F = F D and γ F = γ D from 29) we have Ψ F B) = Ψ D B) which implies B F = B D. It follows that the regions in ) overlap, i.e. Ω i F Ωi D for i = 0,.., 4. Since the udget distriution 69

73 B F RM H) is independent of cost parameters, the expected stage 0) equity values are the same at the threshold level. Moreover, from 35), it follows that H F cs F c D)) = H D c D) ecause oth of them are solutions to the same optimization prolem. Proof of Lemma 8: Recall from the proof of Proposition 7 we have Υ ϕ. = ϕ = E [ π BF RM α, ω F RM ) )] E [π B F RM α ))] ϕ ϕ For ϕ = c ϕ = F ), we calculate the derivative from Lemma 3 Lemma 6) y letting Ω 34 ecause of our assumptions on F and E). In 48) of Lemma 6, for B Ω we have γ < + )M B F c c ) γ < + a γ. = For ω0 F RM + α ω F RM Ω 0, it follows that E[π B F RM α, ω F RM ))] F = γ ) E[π B F RM α ))] F, and we otain Υ F 0 where the equality holds for ω 0 > c K 0 + F. For ω F RM 0 + α ω F RM Ω 2, and we otain Υ F E[π B F RM α, ω F RM ))] F = + a γ ) < E[π B F RM α ))] F, < 0. his concludes the proof for part i). Similarly, in 49) of Lemma 6, for B Ω we have K 0 > + )M B F c c For ω0 F RM + α ω F RM Ω 0, it follows that ) + > K + a). E[π B F RM α, ω F RM ))] c = K 0 E[π B F RM α ))] c and we otain Υ c 0 where the equality holds for ω 0 > c K 0 + F. For ω F RM 0 + α ω F RM Ω 2, E[π B F RM α, ω F RM ))] c = K + a) > E[π B F RM α ))] c and we otain Υ c > 0. his concludes the proof for part ii). 70

74 References Chod, J., N. Rudi, J.A. Van Mieghem Mix, time and volume flexiility: Valuation and corporate diversification. Working Paper, Kellogg School of Management, Northwestern University. Corett, C.J., K. Rajaram A generalization of the inventory pooling effect to non-normal dependent demand. Working Paper, Anderson Graduate School of Management, University of California, Los Angeles. Glasserman, P Perturation analysis of production networks, D. Yao, eds. Stochastic modelling and analysis of manufacturing systems, Sringer-Verlag, NY. Hardy, G., J.E. Littlewood, G. Polya Inequalities. Camridge Press, NY. Muller, A., M. Scarsini Some remarks on the supermodular order. Journal of Multivariate Analysis Muller, A., M. Scarsini Archimedean Copulae and Positive Dependence. Working Paper, ICER. Ross, S.M Stochastic Processes. Wiley Series in Proaility and Mathematical Statistics. 7

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