Joint Production and Financing Decisions: Modeling and Analysis


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1 Joint Production and Financing Decisions: Modeling and Analysis Xiaodong Xu John R. Birge Deartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208, USA The University of Chicago Graduate School of Business, Chicago, Illinois 60637, USA October 4, 2004 Abstract This aer develos models to make roduction and financing decisions simultaneously in the resence of demand uncertainty and market imerfections. While the Modigliani and Miller roositions demonstrate that a firm s investment and financing decisions can be made indeendently in a erfect caital market, our models illustrate how a firm s roduction decisions are affected by the existence of financial constraints. We analyze the interactions between a firm s roduction and financing decisions as a tradeoff between the taxes benefits of debt and financial distress costs. Our numerical examles illustrate that a traditional allequity manufacturing comany can imrove its erformance significantly by making real and financial decisions together. The results illustrate greater firm value sensitivity to roduction decisions than to financing decisions and that lowmargin roducers face significant risk in not coordinating roduction and financing decisions. Key Words: Production Decisions; Financial Constraints; Caital structure; Debt Caacity This work was suorted in art by the National Science Foundation under Grant DMI
2 2 1 Introduction How much should a firm roduce and what should be its otimal caital structure? While these two decisions are often indeendent, the first one made by a roduction manager or oerating officer and the second one under the resonsibility of a financial officer, the firm s roduction/investment decisions and caital leverage are closely related. The oerations management literature, however, tends to focus on the areas of caacity exansion, inventory control, or suly chain management without considering the effects of financial constraints or caital structure on the firm s oerating decisions; on the other hand, while financial economists have long considered the caital structure of a comany, they usually assume that the investment or roduction decisions are exogenously determined. Recently, a growing trend in the oerations and financial economic communities aims to unite these two views of the firm by analyzing interactions between roduction and financing decisions. These studies follow two main directions: a focus in the roductmarket literature on how caital structure influences a firm s incentive to comete in the roduct market, and an emhasis in the market imerfection literature concentrating on the effect of financial leverage on the cost of roduction, and, hence, the comany s outut decisions. Three major theories aear in the roductmarket category. According to the strategic commitment theory of Brander and Lewis (1986), a comany s decision to use debt works as a commitment to more aggressive behavior in the roduct market. Because debt alters the ownershi of residual cashflows, this threat becomes credible, which induces a firm s unlevered rival to reduce outut. A second theory is the long urse or redation argument formalized by Bolton and Scharfstein (1990), wherein rivals may increase outut to drive down rices and cause the highlyleveraged firm to exit the industry. Finally, in the industrial equilibrium model of Williams (1995), a firm s financial contract is jointly determined by various industry characteristics, such as the number of firms, roject riskiness, and technologies. Williams model suorts the coexistence of rofitable
3 3 firms that are large, make more fixedcaital investment, and use more debt, with smaller firms that are less rofitable and use less debt. The seminal work by Modigliani and Miller (1958) shows that a firm s investment and financial decisions can be made searately within a erfect caital market. Due to market imerfections, such as taxes, agency costs, and asymmetric information, however, the choice of a firm s caital structure may in fact be closely related to its roduction decisions (see Harris and Raviv (1991) for a general review on the theory of caital structure). Discussions of market imerfection effects mainly focus on the tradeoffs among tax benefits, financial distress costs, and the effect of ersonal tax. Modigliani and Miller s (1963) traditional tradeoff model indicates that the tax benefit of debt is the tax saving that results from deducting interest from taxable oerating income; the rime costs are those associated with financial distress. Miller (1977) oints out that the traditional aroach ignores ersonal tax because the interest income of bondholders is taxed at the ersonal income level while equity holdings are subject to cororate tax and then ersonal tax according to realized caital gains. Hite (1977) made an early study of the interdeendence of real and financial decisions taking a marketimerfection aroach. He shows that an increase in financial leverage leads to an increase in the otimal outut level of the firm in a situation with riskless debt (i.e., no bankrutcy). In the DeAngelo and Masulis (1980) statereference model, nondebtrelated tax shields, such as dereciation deductions, investment tax credits, and asymmetric cororate tax code, are introduced into Miller s framework. Dotan and Ravid (1985) treat debt as an indeendent decision variable, include ossible bankrutcy, and exlicitly model a firm s investment as a dereciationrelated tax shield. They show that high roduction caacity should be financed by less debt, while increasing debt usage reduces otimal caital investments, thus, reversing Hite s conclusion. Our aer follows a similar treatment to Dotan and Ravid s, but we model future demand uncertainty instead of rice
4 4 uncertainty to lace our model in the context of oerations models for firms that set rices and determine roduction volume before demand is realized. Several other studies in the literature examine interactions between caital structure and investment decisions in contingent claims models. Brennan and Schwartz (1984) consider firm valuation in a setting in which bond covenants restrict financial olicy and influence investment olicy. Mello and Parsons (1992) comare the oerating decisions of a mine under allequity financing to those when the mine is artially debt financed and maximizes leveraged equity value. In Mauer and Triantis (1994) analysis, the firm has the flexibility to shut down or reoen its roduction facility in resonse to rice fluctuations. Although these analysis are valuable in understanding the deendent mechanism between financing and investment decisions, they usually include strong assumtions on the firm s roduction or financing flexibility, limiting their alicability. For examle, Mauer and Triantis assume the firm continuously roduces an infinitely divisible commodity at the rate of one unit er year if the roduction facility is in oeration, while, in reality, comanies make varying quantitative roduction decisions according to realized demands. Recently, several studies in the oerations management community have addressed the interface of roduction and financial decisions. Among these analysis, Lederer and Singhal (1994) consider joint financing and technology choices when making manufacturing investments and show that considerable value can be added to investments through financing decisions. Birge and Zhang (1999) seek to use otion theory to introduce risk into inventory management. In another examle, Birge (2000) adats contingent claim ricing methods to incororate risk into a caacity lanning model. Babich and Sobel (2004) examine the relationshi between oerational decisions and timing of IPO of a startu firm. The aer by Buzacott and Zhang (2004) also considers the connection between roduction and finance for a comany with financial constraints. These studies do not, however, consider otimal caital structure or discuss debt caacity.
5 5 This aer addresses the shortcoming of revious analyses by considering a firm s caital structure in a model which characterizes roduction inut and financial olicy as endogenous decision variables; we maintain analytical tractability by limiting decisions to a single eriod and obtain results on the relationshi between roduction decisions and caital structures choices. While we sacrifice fidelity to the dynamics of investment over time, our models consideration of couled investment decisions and revenue realizations rovides insight into ractical management decisions. We find that financial constraints lay an imortant role in determining the firm s roduction decisions because the internal cash osition of the constrained comany may not be able to suort its otimal outut level. Our research indicates that the constrained firm can imrove its rofit by issuing debt in the financial market. We further show that the firm s otimal roduction decision is negatively related to its financial decisions and that misidentifying the comany s otimal leverage ratio or outut decreases firm value; hence, the firm s roduction and financial decisions should be made jointly. We also demonstrate that the firm s otimal debt level is less than its debt caacity, i.e., it is not otimal to borrow u to the debt limit. Our numerical examles illustrate that a firm s investment and financing decisions are both sensitive to critical oerating arameters, such as rofit margin, demand volatility, cororate tax rate, and bankrutcy recovery rate. These sensitivity analyses rovide useful managerial insights for different tyes of comanies. The results illustrate, for examle, greater firm value sensitivity to roduction decisions than to financing decisions and that lowmargin roducers face significant risk in not coordinating roduction and financing. The aer is organized as follows. In Section 2 we show that the firm s roduction decision does not deend on its caital structure in a erfect caital market. We also find that, without market imerfection, the value of the comany is indeendent of its caital structure as long as its otimal roduction level can be fully suorted. Our debt valuation method is resented in
6 6 Section 3. With the assumtion of a riskneutral equivalent measure, we show the existence of debt caacity and illustrate the relationshi between credit risk and face value of debt. To fully understand the interactions between investment and caital structure decisions, Section 4 models debt leverage ratio as an indeendent variable in the resence of market imerfections. Finally, Section 5 contains results and analysis of a numerical examle. 2 The simle model This section first shows the effects of financial constraints on a firm s roduction decision, articularly that financial constraints can reduce the value of the firm. The firm can, therefore, imrove its erformance by raising debt from financial market. We then oint out that, without market imerfection, the value of the comany is indeendent of its caital structure as long as its otimal roduction level can be fully suorted. To analyze the relationshis between roduction and financing decisions, we first emloy a simle model, which is essentially a classical news vendor roblem with financial constraints k. The assumtions of the model are as follows: the firm is in a quantity cometitive industry, makes a single tye of roduct, and only oerates for one eriod within an equivalent riskneutral world. The stochastic demand s, realized at the end of the oerating eriod, has a riskneutral equivalent cumulative distribution function F and density function f. We also assume F is continuous, differentiable, and strictly increasing. At the beginning of the eriod, the comany roduces x units of roduct at a constant cost of c dollars er unit so that x is used as the caacity constraint on roduction. The firm then sells min(x, s) units of roduct at a fixed rice c(1+r f ), where r f is the riskfree interest rate, and then liquidates the remaining inventory. To simlify the roblem, we assume the firm roduces erishable or fashion goods with no salvage value. The riskneutral equivalence assumtion reresents a transformation from a nominal robabil
7 7 ity distribution that is usually considered in studies of the news vendor model and extensions. By transforming to a riskneutral measure, the otimal roduction decision for the allequity firm considers market risk and is equivalent to an evaluation with the caital assetricing model (CAPM). Singhal (1998) derives conditions for the allequity news vendor decision under the riskneutral measure or CAPM in a general case. Birge and Zhang (2000) give an exlicit solution to the allequity news vendor decision under riskneutral equivalence and lognormal demand distribution and show that this solution equals the standard news vendor solution under the original demand distribution times the market remium for the risk of the roduct s overall market. To find the otimal roduction decision, x, we have maximize ( x s df (s) + x 0 subject to 0 cx k. x ) df (s) cx(1 + r f ) (1) Let ˆx be the solution to F (x) = c(1 + r f ), then the otimal roduction olicy for the financially constrained comany is ˆx = min (k/c, ˆx). If k/c < ˆx, the financial constraint is redundant and has no effect on the firm s roduction decision; however, if internal cash is not sufficient to suort otimal roduction, i.e., k/c > ˆx, the firm cannot achieve its otimal outut level and incurs a loss in market value. Model (1) illustrates that the firm s financial situation may affect its roduction by caing its initial investment. Deending on the tightness of the constraints, the effects of the financial constraints can be significant. In reality, many comanies face financial constraints and raise funds to suort roduction in multile ways. In our case, we assume the firm can issue a single homogeneous class of zerocouon discount bond with rice D and interest rate r. Let s b = D(1 + r) be the amount of demand for which the endoferiod revenues are just sufficient to cover romised ayments to bondholders. If demand falls below s b, the comany is forced into bankrutcy where the ayoff to debtholders is s; otherwise, the comany ays debt in full. Under the riskneutral equivalence assumtion, the
8 8 investors are indifferent to the risk. The ayoff of the riskfree account and the zero couon bond are identical; hence, the interest rate of the bond is given by the following equation: D(1 + r f ) = s b 0 s df (s) + D(1 + r) s b df (s). The following model then gives the otimal roduction decision including financial considerations: maximize subject to ( x ) s df (s) + x df (s) cx(1 + r f ) 0 x ( s b 0 cx k + (1 + r f ) 1 s df (s) + D(1 + r) where k reresents the initial cash osition of the comany. 0 s b df (s) If the firm s otimal investment level cx is less than its initial cash osition k, the above model is simly the traditional news vendor roblem. From now on, we assume the firm s internal cash osition is not sufficient to suort its otimal investment, i.e., cx > k. Because of interest cost, the financial constraint in Model (2) is tight. Denote V as the value of the firm. Substituting (cx k)(1 + r f ) = terms, (2) becomes, s b 0 s df (s) + D(1 + r) maximize V (x, L) = x s b ), (2) df (s) into the objective function and rearranging s b (s L) df (s) + (x L) x df (s), subject to nonnegative constraints on x and L resectively, where L = D(1 + r) is the face value 1 of debt. Substituting dx/dl = df (s) into dv/dl = (x 1) df (s), we have c(1 + r f ) s b x ( ) dv/dl = df (s) 1 df (s). (3) c(1 + r f ) s b x From (3), we have that the caital structure of the comany has no influence on the roduction decision in a erfect market (as also follows from the ModiglianiMiller (MM) theory). Proosition 2.1 In a erfect market, the firm s roduction decision is indeendent of its caital structure whenever its otimal investment level can be fully financed. The otimal roduction and
9 9 financial decisions have the following roerties: (1) the single eriod roduction olicy is to roduce u to level x without considering financial ( ) c(1 + constraints, i.e., x = F 1 rf ) ; (2) the otimal amount of debt raised is max (0, cx k). 3 Debt Valuation and Cororate Debt Caacity The conclusions in Proosition 2.1 only hold in erfect caital markets. This section shows the effects of bankrutcy costs on debt ricing. If the firm decides to finance art of its investment by selling cororate debt at time zero, the endoferiod market value of the debt is uncertain because it deends on the market demand for the firm s roduct. If oerating income is insufficient to reay the debt, debtholders take ownershi of the firm, ay bankrutcy costs, and acquire the residual value of the comany. Bankrutcy costs include administrative exenses, such as fees aid to lawyers, trustees, auctioneers, and accountants, and indirect costs due to financial distress. Similar to Leland (1994), our aer takes a roortional form with bankrutcy cost reresented as (1 α) s s < s b, where s b = L/ is the bankrutcy oint in terms of sales and 0 < α < 1 reresents the asset recovery rate after bankrutcy. If bankrutcy occurs, a fraction 1 α of the oerating income reresents the loss due to bankrutcy costs. From the analysis above and our bankrutcy form, the endoferiod ayoff to debtholders is: D(1 + r(d)) if s s b, Y D (x, D) = α s if s b > s, where r(d) is the nominal interest charged by debtholders for lending D; this rate deends on the risk characteristics of the market, such as the demand distribution, the rofit margin, and the amount of debt. Because of market uncertainty and bankrutcy costs, the debtholders actual income may be less than the firm s romised ayment. Following Dotan and Ravid (1985), we
10 10 again assume an equivalent riskneutral measure of future demand, so that we can analyze otimal decisions as if the firm is in a riskneutral world; hence, the interest rate aid to bondholder must guarantee that the exected ayment equals the return obtained at the riskfree rate, i.e., E(Y D ) = D(1 + r f ). The exlicit form is D(1 + r f ) = D(1 + r) s b f(s)ds + α s b 0 sf(s)ds. (4) We also assume that, in the single eriod model, no additional debt can be issued; thus, Equation (4) effectively revents stockholders from transferring wealth away from debtholders and guarantees that bondholders rights are not violated. The debtraising ability of a comany deends on the willingness of debtholders to extend credit. Given the existence of bankrutcy costs, the lenders require an exected rate of return to comensate for the risk of default and other associated costs. After the debt amount reaches a certain level, the risk remium increment cannot rovide a market equilibrium return because of the comany s rofitability characteristics. Since other investment oortunities are available in the caital market, lenders invest no more than the maximum level. Cororate debt caacity is defined as this maximum amount that a firm can borrow. Unless the firm has already reached its debt caacity, the firm can borrow more by romising to ay more to lenders. While a firm may attemt to increase its debt usage by increasing the romised future reayment, debt caacity can be reached where increasing the romised reayment does not increase the market value of debt. In other words, once the comany reaches its debt caacity, it can borrow no more regardless of how much it romises to ay. Let D denote cororate debt caacity and L = D(1 + r( D)) reresent the amount the firm romises its debt holders to reach D. The existence of debt caacity means that d2 D dl 2 L= L< 0 and dd dl L= L= 0; hence, debt caacity can be determined by setting dd/dl = 0, and solving for D and L. Conditioning on the demand distribution function, the following roosition shows the existence of debt caacity D.
11 11 Proosition 3.1 If the roduct market demand distribution function satisfies sb f (s b ) f(s b ) where s b solves dd = 0, then there exists a finite debt caacity for the firm. dl < 2 α 1 α, Proosition (3.1) indicates that once the comany reaches its debt caacity, the market value of debt D is a decreasing function of L. An intuitive exlanation for this roerty is that, before the firm reach its debt caacity, the debtholders can ass on the entire bankrutcy cost to stockholders by charging an aroriate risk remium. Once the debt level asses D, the return from this firm becomes less than the market equilibrium return; therefore, the romised high interest from equity holders is not sufficient to comensate for the debtholders loss due to overinvestment. Note also that at the oint where the debt caacity is reached, the marginal contribution of an additional unit of romised ayment is balanced by the marginal bankrutcy cost. The interest rate charged by debtholders and the resent market value of debt, must satisfy Equation (4). The following Proosition 3.2 characterizes the relationshi between r(d) and D. Proosition 3.2 The interest rate charged by the debtholders, r(d), is a monotone increasing function of the resent value of the debt D. 4 Investment & Caital Structure in an Imerfect Market To analyze the effects of market imerfections, such as tax and bankrutcy costs, on the firm s roduction and financial decisions, we assume cororate rofits are taxed at a constant rate τ and the debt ayments are fully deductible in comuting taxable cororate income. If the comany s internal cash is not sufficient to suort its otimal roduction level x, the firm can borrow at an interest rate r(d), which is a function of the face value of debt borrowed. The firm s taxable income is max[0, s cx rd], where s is the realization of the demand. Since the comany only oerates for a single eriod, we assume gains are taxed at a constant rate τ, while all tax losses are
12 12 not allowed for tax carrybacks or carryforwards. The face value of the debt must be fully aid at the end of the eriod, excet in the case of bankrutcy when, as before, all assets are sold and roceeds are distributed to creditors minus a roortional bankrutcy cost. The cash flow to the equity holder is then: x τ(x cx rd) D(1 + r) if x s, Y E (x, D) = s τ(s cx rd) D(1 + r) if s s < x, s D(1 + r) if s b s < s, where x is the roduction caacity, s = cx + rd, is the amount of demand for which accounting income equals zero. If the realized demand s is greater than the breakeven oint s, the comany s oerating income is taxed at a constant rate τ. For s b = D(1 + r), the bankrutcy oint, if demand falls below s b, the comany is forced into bankrutcy and a bankrutcy cost (1 α)s is charged. The debtholders cash flow, Y D, is the same as in Section 3. Note that the equityholders cannot transfer wealth from the debtholders by the above constraint, maximizing the value of the firm or the value of the equity yields the same results. We can comute the exected future value of the firm at the end of the eriod by combining the exected values for debtholders and equityholders together and deducting the initial investment, i.e., V = E(Y E ) + E(Y D ) cx(1 + r f ). The model which maximizes the firm s value is then: maximize V (x, D) = x x + + (x τ(x cx rd)) f(s) ds s (s τ(s cx rd)) f(s) ds s s b s b sf(s)ds + α subject to D(1 + r f ) = D(1 + r)[1 F (s b )] + α 0 D cx. 0 sf(s)ds cx(1 + r f ), s b 0 sf(s)ds, (5) Model (5) rovides an initial exlanation for why we need to make roduction and financial decisions at the same time. The exected value of the firm deends on both decisions x and D. The
13 13 following comutation gives the otimal decisions the comany should take and the relationshi between these two decision variables. Setting derivatives to zero, we first obtain: 0 = V x = (1 τ) x f(s)ds + cτ s f(s)ds c(1 + r f ). (6) The interretation of equation (6) is that the exected rofit of additional caacity (1 τ)[1 F (x)] + cτ[1 F (s )], i.e., the sum of the marginal after tax revenue lus the marginal tax benefit, equals the marginal cost c(1 + r f ) at time T. We find that the level of debt affects caacity choice because of its imact on the breakeven level and, hence, on the robability that debt tax benefits will in fact be used. We can show that, s D = 1 ( r + D r ) 0, which indicates that, as more D debt is taken on, s increases and, consequently, the exected marginal roduction cost increases. Equation (6) indicates that the otimal roduction decision should be achieved at the oint where marginal roduction cost equals marginal rofit. Since we assume constant marginal cost, the otimal decision is determined by the sum of the marginal after tax revenue and the marginal tax benefit. Both of these are decreasing functions of the roduction decision x because an additional caacity unit results in a higher chance of salvage loss and a higher breakeven sales level. Notice that the effect of caital structure is illustrated by the marginal tax benefit cτ[1 F (s )]. Since s = cx + rd, a higher debt level results in a smaller tax benefit. For the overall effect of debt on firm value, we consider: ( V D = τ r + D d r ) ( f(s)ds (1 α) 1 + r + D d r ) s b f(s b ). dd s dd Let L = D(1 + r), we can rewrite V D 0 = V ( d L D = τ[1 F (s )] dd 1 less exlicitly and obtain the first order condition: ) (1 α)s b f(s b ) d L dd. (7) An exlanation for this equation is that, at otimality, the change in the exected bankrutcy cost is balanced by the exected tax shield benefit of an additional unit of debt. In the extreme
14 14 situation of no bankrutcy cost, i.e., α = 1, Equation (7) cannot be satisfied; the firm s otimal decisions are then achieved at an extreme oint; the demand amounts corresonding to the bankrutcy and breakeven oints, cx(1 + r), become identical. Because debt rovides a taxshield for oerating income, the firm s otimal caital structure is alldebt financing. Equations (6) and (7) indicate that the firm s roduction decision and caital structure olicy must be made jointly since the breakeven oint s, which is a function of the investment decision, x, and the financing decision, D, aears in both equations. Comaring (3) and (6) we find that the otimal roduction decision of an allequity comany is a secial case of a levered comany with the allequity financing constraint. For the concet of otimal caital structure to be meaningful, it is necessary to show that the firm s otimal debt level, D, is less than its debt caacity D. The following roosition gives this result. Proosition 4.1 The debt level corresonding to the firm s otimal caital structure is less than its debt caacity. Proosition 4.2 For a firm oerating in a market with taxes and bankrutcy cost, the roduction and financing decisions are interdeendent; (a) the otimal roduction decision is a decreasing function of financial leverage; (b) the simultaneous otimal roduction and financial decisions yield otimal debt D with D > D E for D E the otimal financial decision corresonding to the otimal roduction level x E of an allequity comany. Proosition (4.2) shows that the roduction and financial decisions of the firm are insearable, indicating that the roduction decisions of an allequity firm are different from that of a levered firm. Increasing the debt level results in a higher breakeven demand realization that decreases the tax shield. Consequently, the otimal roduction level decreases due to the increase in marginal
15 15 roduction cost. On the other hand, it is not otimal for the comany to oerate as an allequity comany because of the taxshield benefit. A small increase in debt level leads to an increase in the cost of caital, and hence, a decrease in roduction. The otimal decisions are achieved at the oint where the marginal benefit of the tax shield, which occurs over the firm s ositive income states of nature, lus the exected marginal rofit of additional caacity, are equal to the marginal roduction cost. The following roosition discusses the conditions under which the firm has an otimal roduction and financial solution. Proosition 4.3 A solution (x, D ) of Equations (6) and (7) that satisfies f (s b ) > 0 and 1 α > τ is an otimal solution to (5). 2 α The following roosition gives the effects of cororate tax rate changes on the firm s otimal roduction and debt decisions. Proosition 4.4 If the conditions in Proosition 4.3 are satisfied, the otimal roduction decision, x, is negatively correlated with the cororate tax rate, τ; while the otimal debt decision, D, is ositive correlated with τ. 5 Analysis of the Model This section resents a comarative statics analysis of the interactions between roduction and caital structure decisions. We first examine the relationshi between these two decisions by conducting a sensitivity analysis for different combinations of oerating and environmental variables. We then investigate the effects of misidentifying otimal roduction and financial leverage decisions on the rofit of the comany. Our goal is to determine under what conditions making simultaneous oerational and financial decisions is most critical.
16 16 We consider misidentification of roduction and financing decisions to determine the imact of not following the otimal olicies. We find that the value of the comany is a convex function of debt usage and roduction level. Deviating from otimal roduction and financial decisions can incur significant losses to the comany; therefore, traditional oerations management models, such as caacity lanning, inventory management, or suly chain management models, that seldom consider the effects of caital structure, tend to undervalue the comany or roject. Our results also suggest that misidentifying roduction levels generally has a more significant imact on firm value than misidentifying debt levels but that lowmargin firms may suffer significant value loss from misidentifying caital structure. We choose arameters that are roughly consistent with that of a commodity manufacturing comany. For the base case arameters, we assume the selling rice,, of one unit is $ 1, and the roduction cost, exressed as a fraction of, is $ The industrial average tax rate, τ, and risk free interest rate, r f, are initially set at 35 % and 5 % resectively. We also assume the market demand of the roduct aroximately follows a lognormal distribution. We suose the current market demand is 1000 units, the exected market growth rate and volatility are 10 % and 40 % er year resectively. We also assume the base case debt recovery rate is 30 %. 5.1 Sensitivity analysis of otimal roduction and financial decisions Our analysis of the interactions between oerating decisions and financing olicies mainly concentrates on roduction outut level, x, and market leverage ratio, D/V. The market value of the comany, V, equals the discounted value of exected future cash flow minus the initial investment. We find that the firm s otimal leverage ratio is negatively related to its otimal roduction decisions, which is consistent with our revious analysis in Section 4. Figure 1 shows financial leverage as a function of roduction cost c for different market demand
17 17 Panel A Panel B sigma : 0.4 sigma : 0.5 sigma : sigma : 0.4 sigma : 0.5 sigma : Debt/Market Value Ratio Outut Level Production Cost Production Cost Figure 1: Relation between leverage ratio and roduction decision uncertainty levels. Clearly as the firm s marginal roduction cost increases, its otimal leverage ratio increases and otimal roduction level decreases. This observation suggests that a lowmargin comany should take a conservative roduction decision and an aggressive financial decision, while a highmargin comany should take aggressive roduction decisions and conservative debt olicy. Because lower margin means higher roduction cost, the firm s roduction outut level decreases with decreasing margin. For a comany facing uncertain demand, a lower roduction level decreases the risk of the future cash flow; therefore, the debt holder charges a smaller risk remium to comensate for bankrutcy risk. To take advantage of a lower cost of debt, the firm refers to use more debt in its caital structure. For different demand volatility levels, Figure 1 illustrates that both a firm s investment and financial leverage decisions are negatively correlated with market uncertainty. A rise in the demand volatility causes the robability of bankrutcy to increase; the rice of debt therefore rises. As debt becomes more costly, the firm lowers its investment level, hence, decreasing the risk of the cash flow. A negative relationshi, therefore, should exist between demand volatility and the firm s outut. Another observation from Figure 1 is that the otimal leverage ratio increases as the market uncertainty diminishes. It suggests that a firm with less volatile cash flow is likely to have a smaller chance of bankrutcy. This attern confirms the hyothesis of the tradeoff theory that firms with less variable earnings have more leverage, as also observed in the emirical work by
18 18 Fama and French (2002). 5.2 Effects of Missecifying Debt & Production Decisions on Firm Valuation The results in Section 4 indicate that roduction/investment and financing decisions should be made simultaneously to maximize firm value. Financial leverage decisions can affect investment decisions because debt financing rovides a deductible tax shield for oerating income and may incur financial distress costs in the case of bankrutcy, thus altering cash flow. We consider the following questions: How much can the firm increase value by making roduction and financing decision simultaneously? Is this increase significant? The following analysis exlores these questions by examining the effects of missecifing caital structure and roduction outut level on the value of the comany. To analyze the significance of misidentification, we comare the net rofit of missecified decisions with that corresonding to otimal investment and caital structure decisions. By changing the value of the roduction cost, demand volatility, bankrutcy recovery rate, and cororate tax rate, we observe the sensitivity of the misidentification effects. For simlicity of comarison, we define the normalized net income, I(x, l)/i(x, l ), as the ratio between the net income associated with a certain roduction and debt leverage decision air (x, l), and that corresonding to the otimal decision air (x, l ). Clearly, the normalized net income ranges from 0 to 1 with maximize value achieved at the otimal decision oint. To identify the missecification effects of the two contributor factors searately, we analyze the sensitivity of value on the roduction decision and the debt decision resectively. We first analyze the effects of debt misidentification on the value of the comany by keeing the roduction decision at an otimal level. Figure 2 lots the normalized firm value as a function of book leverage ratio for different oerating arameters. The sensitivities of the missecification
19 19 Panel A Panel B 1 1 Normalized Firm Value c=0.6 c=0.7 c=0.8 Normalized Firm Value sigma=0.4 sigma=0.5 sigma=0.6 Normalized Firm Value Leverage Ratio 1 alha=0 alha=0.3 alha=0.6 Panel C Leverage Ratio Normalized Firm Value Leverage Ratio tau=0.25 tau=0.35 tau=0.45 Panel D Leverage Ratio Figure 2: Effects of caital structure missecification on the normalized net income effects of roduction cost, c, demand volatility, σ, bankrutcy recovery rate, α, and cororate income tax rate, τ, are illustrated by Panels A, B, C, and D resectively, controlling all other arameters to be the same as in the base case. Observe that the firm s value is a convex function of financial leverage, suggesting the existence of an otimal caital structure as we observed analytically. An underleveraged firm can increase its rofit by raising more debt, taking the benefits of the tax shield. Once the debt usage crosses the otimal leverage ratio, the cost incurred by financial distress cannot be balanced by the tax benefit; so, the comany s value begins to decrease. As illustrated by Panel A, if the roduction cost is 80 % of the selling rice, an allequity financed comany can increase its normalized value from 0.9 to 1 by raising debt usage to its otimal leverage ratio, 0.68; however, if the debt leverage rises above the otimal caital structure, more debt issue actually reduces firm value. In this case, one hundred ercent debt financing only earns 82 % of the value with otimal caital structure. The above analysis indicates that missecifing debt leverage can incur significant losses to the comany.
20 20 An interesting observation from Figure 2 is that the effect of overleverage is more severe than underleverage. The overleveraged firm faces a higher chance of bankrutcy, which has two effects on the firm value: first, raising financial distress cost; second, decreasing the rofit margin, resulting in lower outut. Both effects contribute negatively to the erformance of the comany. Although an underleveraged firm does not fully take advantage of otential debt service deductibility, the overleveraged comany faces more severe financial distress loss; therefore, for the same ratio of leverage deviation, the effect of overleverage is more significant than underleverage. With longer term debt available for lower issuing cost, this observation suggests that firms should have lower leverage ratios than given here to rotect against ossible future financial distress, which is also consistent with emirical results. Figure 2 also indicates that the effects of missecifing leverage ratios are sensitive to changes in c, σ, α, and τ. For examle, Panel A indicates that the missecification effect becomes more significant as roduction cost increases. When c is 60 % of the selling rice, the firm faces a 3 % and 7 % value loss by taking an allequity or alldebt financing caital structure resectively. If the roduction cost accounts for 80 % of the rice, the value ga between the allequity financed and the otimally leveraged comany rises to 10 %, while the alldebt financed comany faces an 18 % value loss. For a comany with high margins, the contribution of the financing decision to the value of the comany is small; the highmargin firm s manager does not need to ay significant attention to the caital structure of the comany (relative to the need for careful management of oerational decisions). Increasing the roduction cost lowers the firm s rofitability, hence, increasing the robability of bankrutcy. Both effects make the tax shield lay an more imortant role in asset return; therefore, decreasing the rofit margin increases the significance of debt missecification. Another observation from Figure 2 is that, for an underleveraged comany, a decrease in rofit margin, market volatility, or an increase in bankrutcy recovery rate or tax rate leads to
21 21 a larger value loss; for an overleveraged comany, an increase in market volatility, or a decrease in bankrutcy rate or cororate tax rate results in higher loss. These observations suggest that, to avoid misidentification effects, a comany with stable cash flow, low margin, a large volume of fixed assets, and high tax rate, refers a high debt leverage ratio; a firm with uncertain cash flow, rofitable roduct, small fixed asset, and lower cororate tax rate should tend to use conservative debt olicy. This observation is consistent with Graham and Harvey (2001) who find that tax advantage is more imortant for large, regulated, and dividendaying comanies and that those comanies usually have high tax rates and large tax incentives to use debt. The above analysis suggests that otimal caital structure decision may increase a firm s asset return and vice versa. Clearly, the comany s initial roduction decision also lays an imortant role in determining the cash flow of the comany. Corresonding to the revious discussion, Figure 3 dislays the firm value as a function of roortional roduction levels for different roduction costs, c, bankrutcy recovery rates, α, and cororate tax rates, τ. To facilitate comarison, the roduction levels, x, are standardized with resect to the otimal roduction levels, x, which together with the otimal debt financing decision maximize the value of the comany. To identify the effect of roduction decisions, we hold the debt decision at the otimal leverage ratio. We summarize our observations from Figure 3 concisely. All four anels illustrate that the value of the comany is a concave function of the roduction level, which suggests that missecifying the roduction decision incurs a loss. Notice also that the firm s value function is much more sensitive to the change of the roduction level comared to the revious analysis for the debt decision. In other words, missecifying the otimal roduction decision incurs a higher loss than missecifying otimal debt leverage. For examle, Panel A illustrates that, when the roduction cost equals $ 0.60, if the comany s investment decision decreases to 50 % of the otimal roduction level, the firm s value is just 64 % of that of the otimal decision; while the maximum loss incurred
22 22 Normalized Firm Value c=0.6 c=0.7 c=0.8 Panel A Production Level Panel C Normalized Firm Value Panel B sigma=0.4 sigma=0.5 sigma= Production Level Panel D 1 1 Normalized Firm Value alha=0 alha=0.3 alha=0.6 Normalized Firm Value tau=0.25 tau=0.35 tau= Production Level Production Level Figure 3: Effects of outut level missecification on the normalized net income by debt missecification is only 7 % of the otimal rofit. The same attern is reeated in the other three anels. Another observation is that the firm s value is more sensitive to oerating arameters: roduction cost or market demand volatility, comared to financial arameters, such as bankrutcy recovery rate and cororate tax rate. 6 Conclusions In this article, we develoed models to determine roduction and financing decisions simultaneously assuming conditions for riskneutral equivalent distributions. We analyzed the interactions between these sets of decisions and showed when the interactions are most significant. While MM theory demonstrates that, in a erfect caital market, the value of a firm does not deend on its caital structure, and consequently that investment and financial decisions can be made searately, our research rovides a characterization of how financial constraints may still affect a firm s roduction decisions by caing its maximum outut level. Comanies with internal financial constraints can imrove erformance (in terms of maximum value or equity return) by aroriately considering
23 23 debt (and the resulting tax shield) along with roduction quantities. We find how market imerfections can negatively affect a firm s roduction decision and its value. We also find that otimal roduction decisions are negatively correlated with the otimal debttomarketvalue leverage ratio. An increase in this factor, which contributes to higher net margins, leads to greater roduction levels and lower marketvalue leverage ratios. Our numerical results illustrate that both roduction and financial decisions are sensitive to changing oerating and environmental variables, indicating that there may exist large differences in outut and caital structure even in the same industries due to differences in those exogenous factors. Our sensitivity analyses also demonstrate the imortance of joint roduction and financial decisions; we observe that missecification losses are most severe for roduction decisions (relative to financing decisions) and, in general, for lowmargin comared to highmargin firms. The models resented here can be extended in a number of ways. To consider the effects of a firm s growth oortunities and the stochastic roerties of the roduct and financial markets, one ossible direction is to model the interactions in a multile eriod environment. Since we only consider the interface of financial and roduction decisions inside a comany while ignoring the effects of cometition in the industry, another ossible aroach might be to look at otimal financing for a comany to comete with other firms in the industry and to show how to coordinate financing and oerations with both ustream suliers and downstream buyers. These considerations remain challenges for future research. Aendix PROOF of PROPOSITION 3.1 Define L = D(1 + r) as the face value of debt, also let G(D, L) = L f(s)ds + α L/ L/ 0 sf(s) D(1 + r f ). (8)
24 24 Differentiating (8) with resective to D and L yields G D = (1 + r f ), and G L = f(s)ds (1 α) L L/ f(l/). Substituting the above artial derivatives into dd dl = G L / G D gives ( ) dd dl = 1 f(s)ds (1 α) L 1 + r f L/ f(l/). (9) To show that the secondorder condition is satisfied, we find all the second order derivative terms of G are zero excet 2 G L 2 derivative terms and (9) into = 2 α L(1 α) f(l/) 2 f (L/). Substituting the second order ( d 2 D dl 2 = 2 ( G 2 L 2 + G L D + ) 2 G dd D L dl + 2 G D 2 ( ) ) dd 2 / G dl D, (10) ( ) and reorganizing items yields d2 D 1 L(1 α) dl 2 = (2 α)f(l/) + f (L/). Since (1 + r f ) s b f (s b ) f(s b < 2 α ) 1 α, we have d2 D < 0 which shows the existence of the firm s debt caacity. d L 2 PROOF of PROPOSITION 3.2 To rove dr > 0, let G(D, r(d)) = D(1 + r) f(s)ds + dd s b s b α sf(s) D(1 + r f ). Taking artial derivatives with resective to D and r, we obtain: 0 G D = (1 + r) f(s)ds D(1 + r)2 (1 α) s b f(s b ) (1 + r f ), and G r = D f(s)ds D2 (1 + r)(1 α) f(s b ). s b Substitute the above artial derivatives into dr dd = G D / G, we have r dr dd = D 1 + r f f(s)ds D2 (1 + r)(1 α) s b To show dr dd > 0, we only need 1 + r f > (1 + r) 1 α 0 and 1 + r f > (1 + r) s b f(s b ) f(s)ds D(1 + r)2 (1 α) s b 1 + r D. (11) f(s)ds from Equation (4); it follows that dr dd > 0. f(s b ). Notice that
25 25 PROOF of PROPOSITION 4.1 Let D be the debt level corresonding to the otimal caital structure satisfying V/ D = 0. V/ D is given by (7). Reorganizing terms, we have: τγ[1 F (s 1)] (1 α)s b 1f(s b 1) = 0, (12) where γ = dl 1/dD 1 dl 1 /dd, L 1 = D [1 + r(d )], while s b 1 = D [1 + r(d )] and s 1 = cx + r(d )D are the bankrutcy oint and breakeven oint associated with D. From (9), the comany s debt caacity D solves where s b 2 = D[1 + r( D)] 1 F (s b 2) (1 α)s b 2f(s b 2) = 0, (13) is the bankrutcy oint corresonding to debt caacity D. From Proosition 3.2, we know d r dd > 0. Therefore, dl dd 1 = r + D d r dd > 0 leads to 0 < τγ < τ < 1. Comaring (12) and (13), it is easy to show that s 1 < sb 2. Since sb 1 < s 1, we have s b 1 < sb 2. Notice that r(d) is a monotone increasing function of D, so that D < D, i.e., the otimal caital structure involves less debt financing than the comany s debt caacity. PROOF of PROPOSITION 4.2 (a) Let x(d) denote the otimal value of roduction decision x for a given debt level D. Total differentiation of x (D) with resect to D yields dx dd = V 2 2 x D / V 2. From (6), we have x 2 V x 2 = (1 τ)f(x) c2 τ f(s ) < 0, (14) and 2 V x D = cτf(s ) s D < 0, (15) since s D dx dx > 0. Substituting (14) and (15) into, it is clear that dd dd < 0. (b) Assume x E to be the otimal investment level of an allequity comany. The corresonding otimal debt amount D E is derived for this certain investment level x E by Equation (7). Let x, D be the otimal decisions by solving (6) and (7) simultaneously. From (a), the firm determines
26 26 an otimal x < x E since D 0. To satisfy the first order conditions given by Equation (6), the firm must increase its debt level, i.e., D > D E. PROOF of PROPOSITION 4.3 From Equation (14), we have 2 V x 2 < 0; therefore, to demonstrate that the second order condition is satisfied, it is enough to show H = 2 V x 2 2 V D 2 2 V x D 0. Taking crossderivatives with resect to x, D and D, x yields: 2 V x D = 2 V D x = cτ ( ) dl dd 1 f(s ) < 0, 2 V D x > where L = D(1 + r) is the face value of debt. The second order derivative with resect to D is given by: 2 V D 2 = d2 L dd 2 ( τ ) (1 α)s b f(s b ) τ[1 F (s )] ) 2 f(s ) 1 α ( dl dd 1 To show H > 0, it is enough to show that d 2 L ( ) dd 2 τ[1 F (s )] (1 α)s b f(s b ) < 1 α From Equation (9), we know ( dl dd ) 2 ( f(s b ) + s b f (s b )). (16) ( ) dl 2 ( f(s b ) + s b f (s )) b. (17) dd dl dd = L/ 1 + r f f(s)ds (1 α) L. f(l/) We also have d 2 L dd 2 = (1 + r f ) 2 (2 α)f(l/) + L(1 α)f (L/) 2 ( ) 3. f(s)ds (1 α) L L/ f(l/) Substituting dl dd, d2 L dd 2, and 1 F (s ) > 1 F (s b ) > (1 α)s b f(s b ) (from Proosition 4.1 and Equation (9) ) into Equation (17), and reorganizing terms yields [ 1 τ 2 α 1 α ] f(s b ) s b f(s)ds + s b f 2 (s b ) + (1 τ)s b f (s b ) s b f(s)ds > 0. (18)
27 27 By assumtion, Condition (18) is satisfied; hence, V (x, D) is a concave function of (x, D) so that (x, D ) is an otimal decision. PROOF of PROPOSITION 4.4 From the assumtion and Equation 14, we have 2 V 2 x < 0 and H > 0, hence the first order conditions aly for an otimal decision (x, y ). Taking total [ 2 V 2 V x D D τ 2 V 2 ] V x τ D 2 / H, and differentials of x and D with resect to τ yields dx dτ = dd [ 2 dτ = V 2 V x τ D x 2 V 2 V x 2 D τ by ] / H. From Equation (6), the otimal roduction level is given F (x ) = (1 τ) c[1 + r f τ(1 F (s ))] (1 τ) < c(1 + r f ), which indicates that the otimal roduction decision in an imerfect market is always less than that of the decision in a erfect market. From Equation (6), we can show that 2 V x τ = x f(s)ds + c f(s)ds < 0 s since [1 F (x )] c(1 + r f ) 0. Taking crossderivative with resect to D and τ yields: 2 ( V D τ = [1 F (s )] r + D r ) > 0. D From Equation (15) and (16), we have 2 V 2 < 0, and 2 D dx dd < 0 and dτ dτ > 0. V x D = 2 V D x < 0; hence, it is clear that References Babich, V., M. J., Sobel PreIPO oerational and financial decisions. Management Science Birge, J. R Otion methods for incororating risk into linear caacity lanning models. Manufacturing and Service Oerations Management Birge, J. R., R. Q. Zhang Riskneutral otion ricing methods for adjusting constrained cash flows. Engineering Economist
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