Equity valuation, production, and financial planning: A stochastic programming approach

Equity Valuation, Production, and Financial Planning:A Stochastic Programming Approach

Xiaodong Xu,1 John R. Birge2

1 Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208

2 University of Chicago Graduate School of Business, Chicago, Illinois 60637

Received 8 October 2004; revised 26 October 2005; accepted 13 December 2005DOI 10.1002/nav.20182

Published online 7 August 2006 in Wiley InterScience (www.interscience.wiley.com).

Abstract: Most of the operations management literature assumes that a firm can always finance production decisions at an optimallevel or borrow at a constant interest rate; however, operational decisions are constrained by limited capital and often critically dependon external financing. This paper proposes an integrated corporate planning model, which extends the forecasting-based discountdividend pricing method into an optimization-based valuation framework to make production and financial decisions simultaneouslyfor a firm facing marker uncertainty. We also develop an efficient algorithm to solve the resulting integer stochastic programmingmodel with nonlinear constraints. Compared with traditional valuation and planning models, our method yields higher equityvaluations, indicating that valuation without considering contingent decisions is inherently inaccurate. © 2006 Wiley Periodicals,Inc. Naval Research Logistics 53: 641–655, 2006

Keywords: production planning; financial planning; stochastic programming; debt pricing; capital structure

1. INTRODUCTION

The operations management literature tends to focus onthe areas of capacity expansion, inventory control, and sup-ply chain management without considering the effects offinancial constraints or capital structure on a firm’s operat-ing decisions. In contrast, while financial economists havelong considered the capital structure of a company, theyusually assume that investment or production decisions areexogenously determined. The separation between operationaland financial decisions in both literatures simplifies com-plex problems and can be justified by the seminal work ofModigliani and Miller [30], which proves that a firm’s invest-ment and financial decisions can be made separately in aperfect capital market; however, due to the existence of taxes,agency costs, and asymmetric information, the capital mar-ket is not perfect in reality (see [19] for a general reviewon the theory of capital structure). Furthermore, many firms’growth potential is constrained by limited internal capital andcritically depends on bank loans, equity issues, or venture

Supported in part by the National Science Foundation under GrantDMI-0100462. The second author also gratefully acknowledgesfinancial support from The University of Chicago Graduate Schoolof Business.

Correspondence to: X. Xu ([email protected]); J. R. Birge([email protected])

capital investments. In this practical context, a firm’s opera-tional decisions may in fact be closely related to its financialchoices.

In this paper, we propose an integrated corporate planningmodel for making production and financial decisions simul-taneously to maximize the value of the firm within a dynamicuncertain environment. While an extensive optimization lit-erature in the areas of production and financial managementhas developed over the past several decades, a general frame-work linking production planning, capital structure decisions,market demand prediction, and the interactions among thosedecisions is not yet available. Indeed, the financial manage-ment literature has in general developed in isolation fromresearch on production planning, while the operations man-agement literature has not typically considered the effects offinancial constraints or capital structure on a firm’s operatingdecisions.

Production planning addresses decisions on the acquisi-tion, utilization, and allocation of resources to meet customerrequirements in the most efficient way. Typical decisionsinclude purchasing parts and supply from vendors, set-ting inventory levels, deciding production lot sizes, andscheduling personnel. Optimization models can provide deci-sion support in this context. General references that reviewsuch models in production planning include Graves, thoseby Rinnooy Kan, and Zipkin [18] and Silver, Pyke, and

© 2006 Wiley Periodicals, Inc.

642 Naval Research Logistics, Vol. 53 (2006)

Peterson [37]. In our case, to model and analyze the inter-actions between production and financing decisions, wefocus on high-level decisions of production and inventoryquantities.

Applications of optimization in corporate financial plan-ning include currency hedging for multi-national corpo-rations; asset allocation for pension plans and insurancecompanies; risk management for large public corporations,etc. Many articles in this literature have illustrated thatstochastic programming models are flexible tools for describ-ing such financial optimization problems under uncertainty.Mulvey and Vladimirou [32], for example, propose a multi-period stochastic network model for the purpose of assetallocation; Cariño et al. [8] formulate the asset/liability man-agement problem of a Japanese insurance company as amulti-period stochastic linear program; Seshadri et al. [35]develop a methodology for strategic asset-liability manage-ment, which has been applied to the Federal Home Loan Bankof New York. Although these planning systems have achievedsome success, most existing models are either restricted todeterministic environments or focus on a single function ofthe company while ignoring the interactions among differentplanning units.

Several researchers in the operations management commu-nity have begun to address the interface between productionand financial decisions. Among them, Kogut and Kulatilaka[25] model the coordination of geographically dispersed sub-sidiaries as an option dependent upon the real exchange rate.Other examples include Huchzermeier and Cohen [22], whodevelop a firm valuation model through the exercise of pro-duction and supply chain network options, and Birge [4], whoadapts contingent claim pricing methods to incorporate riskinto capacity planning. For recent reviews of the literature,please also see Cohen and Huchzermeier [9], Kouvelis [26],and Kleindorfer and Wu [24].

More recently, Babich and Sobel [2] considered capacityexpansion and financial decisions to maximize the expectedpresent value of a firm’s IPO income. Buzacott and Zhang[7] incorporate financial capacity into production decisionsusing asset-based constraints on the available working capitalin a leader–follower game. Gaur and Seshadri [17] addressthe problem of hedging inventory risk for a seasonal productwhen its demand is correlated with the price of a financialasset. Ding, Dong, and Kouvelis [12] study the integratedoperational and financial hedging decisions faced by a globalcompany. These papers, however, either concentrate on asingle-period analysis or ignore the effects of bankruptcy ona firm’s financing and production decisions.

A growing trend in the financial economic communitiesalso aims to unite the firm’s investment and finance deci-sions. Fazzari, Hubbard, and Petersen [16], for example,argue and find that if a firm faces financial constraints, itsinvestment will be more sensitive to its cash flow. Following

their research, many studies focus on the effects of exter-nal financial constraints on the firm’s investment decisions.The product-market literature includes other examples, suchas Brander and Lewis [6], Bolton and Scharfstein [5], andWilliams [38], who analyze how capital structure influencesa firm’s incentive to compete in the product market. In addi-tion to this line of research, the market imperfection literatureconcentrates on the effect of financial leverage on the cost offinancing. Leland [27] and Anderson and Sundaresan [1],for example, examine a firm’s optimal debt financing pol-icy with financial distress costs, while Hennessy and Whited[21] develop a dynamic trade-off model including financeand real investment choice in the presence of tax, bankruptcy,and flotation costs.

Although these analyses are valuable in understandingthe dependent mechanism between financing and invest-ment decisions, they usually include strong assumptions onthe firm’s production and financing flexibility, limiting theirapplication. For example, Anderson and Sundaresan [1]model the value of a firm as a stochastic process, whileHennessy and Whited [21] assume that the firm can borrowat the risk-free rate.

In a previous paper [39], we explored financing and oper-ational decision interactions for a single period. We showedthat integrating production and financial decisions can indeedbe significant in this model when taxes and bankruptcy costsare included. We also showed that low-margin producerscan especially increase firm value through financing andproduction decision integration. This setting, however, didnot consider the possibility of additional borrowing, equityinvestments, or the value of inventory beyond the singleperiod. Our current model aims to determine how these addi-tional factors affect management decisions in a multi-periodsetting.

In addition to coordinating production and financial deci-sions in an integrated planning framework, our paper hasa non-traditional, from the operations perspective, criterionfor measuring the performance of the firm: maximizing theequity value instead of maximizing profit or minimizing costas is used in most of the traditional operations managementliterature. We consider firms whose management teams oper-ate on behalf of the shareholders and take actions to maximizethe expected cash flow to equity holders. This operating cri-terion has been widely accepted as an objective of the firmin the finance and economics literature and is, in fact, thefiduciary responsibility of a corporate board.

This paper also integrates operational decisions in finan-cial valuation by proposing an optimization-based valuationframework, which follows a synthesized pricing approachas in the accounting literature (see, e.g., [34, 29]). The tra-ditional valuation methods in the accounting or financialliterature usually assume that the firm faces forecasted cashflows or the value of the equity follows a given stochastic

Naval Research Logistics DOI 10.1002/nav

Xu and Birge: Equity Valuation and Financial Planning 643

process, leaving the company without managerial flexi-bility. In reality, however, firm valuation is inherently aninterdisciplinary concept, involving skills that span account-ing, finance, economics, marketing, and corporate strategy.The company should have the right to alter productionscale, adjust capital structure, and even abandon opera-tions in disadvantageous situations; hence, effective deci-sion making can create value. Valuation without contingentdecision making is then inherently inaccurate. Our paperextends the traditional passive-forecasting-based valuationtechnique, particularly the discount dividend model, into anactive-optimization-based valuation framework by applyingstochastic programming methodology.

The remainder of this paper is organized as follows. Inthe next section we give a brief review of the accounting firmvaluation models and show how we can introduce these meth-ods into operations management and how we can improvethe accuracy of these pricing methods. In Section 3, wefirst describe the integrated corporate planning (ICP) prob-lem in detail. A multi-stage stochastic programming model isdeveloped to analyze the interactions between production andfinancial decisions. We also identify some of the challengesinvolved in developing a solution strategy. In Section 4, weanalyze properties of the ICP model and propose an efficientalgorithm. In Section 5, a numerical example and computa-tional results are discussed. Finally, we conclude our work inSection 6.

2. BASIC MODEL OF FIRM VALUATION

The accounting literature discusses two broad approachesto estimate shareholder value. The first is direct valuation,which can be expressed in terms of projected future payoffs.The most popular and simplest model for equity valuationis the dividend discount model (DDM), which forecasts thedividends for equity holders and gives the value of equity asthe present value of expected dividends. The second methodtakes a relative valuation approach, which obtains the firmvalue by applying a multiple from a comparable firm to a tar-get firm’s value driver, such as earnings, sales, cash flows, orbook values. We integrate these two techniques into a unifiedframework in which a multi-period valuation model is usedto calculate the value of the cash flow during a finite plan-ning horizon, while the relative comparative method is usedto estimate the terminal value of the firm beyond the planninghorizon.

Without managerial flexibility, the DDM model only pas-sively considers the firm’s future cash flows. In reality, thecompany can alter production scale, adjust capital struc-ture, and even cease operation. A methodology that does notaccount for firm differences in this range of choices maynot capture essential aspects that determine value. In the

following section, we introduce an active valuation frame-work as an integrated corporate planning model to allowconsideration of such managerial flexibility. We then applystochastic programming methodology to the modified DDMmodel to improve pricing accuracy and take advantage of thevalue of information.

3. PROBLEM DESCRIPTIONAND FORMULATION

The purpose of this section is to develop an integratedcorporate planning model for a company facing financialconstraints. We consider a discrete-time, finite-horizon, par-tial equilibrium model, in which the firm takes output price,risk-free interest rate, and the market demand evolution pro-cess as given. The objective is to maximize the expecteddiscounted value of net cash flow to shareholders subject toperiod-by-period constraints that model resource evolution.

The company is assumed to produce a single product forsimplicity. The unit production cost is c and the selling price isp, where both c and p are constant through time. The stochas-tic demand, realized at the end of period t , has a risk-neutralequivalent cumulative distribution Ft(q).1 At the beginningof every operating period t ∈ {0, 1, . . . , H −1}, the companyobserves an inventory level x and cash level u. If it is optimalto continue operations, a production decision y is made. Ifoptimal to default on debt, the firm is liquidated; all growthoptions of the firm and its cash flow producing ability are lost.

Due to internal financial constraints, the company may notbe able to reach an optimal investment level. We assumethe firm has four potential sources of funds: internal sav-ings, current cash flow, single-period debt, and externalequity. Whenever the firm’s desired investment exceeds inter-nally generated cash flow, the company can obtain externalfunds, up to debt capacity, at a premium. We assume thatall agents have full information so that, in particular, debtholders know the firm’s operating decisions and set the debtpremium as the expected cost (under the risk-neutral mea-sure) of default. Our model could incorporate some forms of

1 The existence of a risk-neutral equivalent distribution, or equiva-lent martingale measure, follows from assuming that market doesnot allow arbitrage (see, for example, [20]). This distribution isunique under a complete market assumption, while multiple dis-tributions may exist in incomplete markets. Our model uses oneof these distributions that is consistent with shareholders’ utility.The use of this general approach for project evaluation appears in[11, 10] and elsewhere. Birge [4] describes the process of obtainingthis measure when the demand distribution has known correlationwith the overall market and can be priced with the Capital Asset Pric-ing Model (CAPM) of Sharpe [36] and Lintner [28]. For lognormaldemand distributions, the transformation simplifies to shifting themean of the log of demand by the risk premium implied by theCAPM.



information asymmetry, but our analysis focuses on symmet-ric information to maintain a consistent value-maximizingobjective.

At every decision epoch, the firm’s manager first selects aninvestment or default policy to maximize the value of share-holders’ claims. As in [27] and [14], we assume default istriggered by the decision of shareholders to cease raisingadditional equity to meet the debt payment. If the managerdecides to default on the debt obligation, either it is notoptimal for the equity holders to continue operation or thecompany’s financing ability cannot meet the interest paymentplus the par value of debt at the end of an operating period;the company is then forced into bankruptcy with inventorysold at a fraction of the market price.

If the firm decides to finance part of its investment byselling corporate debt at the beginning of period t , the end-of-period payoff to the debt-holders, YD , is uncertain anddepends on the market demand and the firm’s operationaldecisions. If equity value falls below the face value of debt,debt holders will force the company to bankruptcy, which,as we have assumed, implies liquidation of the firm’s assets.We also assume the liquidation situation incurs a bankruptcycost. Similar to Leland [27], our paper takes a propor-tional bankruptcy form with bankruptcy cost represented by(1 − α)pq ∀ q < qb, where qb = D(1 + r)/p is the bank-ruptcy point in terms of sales, D denotes the present valueof debt, and 0 ≤ α ≤ 1 represents the asset recovery rateafter bankruptcy. If bankruptcy occurs, a fraction 1 − α ofthe operating income represents the loss due to bankruptcycosts; therefore, the end-of-period debt holders’ payoff is

YD(D) ={D(1 + r) if q ≥ qb,αpq if q < qb,

where r is the nominal interest rate charged by debt holdersfor lending D (and, therefore, r depends on D).

Because of market uncertainty and bankruptcy costs, thedebt holders’ actual income may be less than the firm’spromised payment. Following Dotan and Ravid [13], weassume the debt can be priced with a risk-neutral equivalentdistribution, so that we can analyze optimal decisions as if thefirm was in a risk-neutral world; hence, the interest rate paidto debt holders must guarantee that the expected paymentsunder the risk-neutral equivalent distribution equals the returnobtained at the risk-free rate rf , i.e., E(YD) = D(1 + rf ).The inexplicit form for nominal interest rate calculation is

(1 + rf )D = (1 + r)D

∫ ∞

qb

dFt (q) + α

∫ qb

0pqdFt(q).

(1)

The firm’s operating profit during a certain period equals(p− c)z− rD −K , i.e., sales profit less interest expense and

fixed operating cost, where again r is the interest rate chargedby debt holders for debt level D. Note that the realization ofthe sales, z, should be less than the initial inventory x plus theoutput level y of the current period, i.e., z = min[x + y, q],where q is the realization of the random demand. For simplic-ity, we assume there is no backordering. We also assume thefirm is taxed at a flat rate of τ ∈ (0, 1) on taxable corporateincome i = max[0, (p − c)z − rD −K]. The debt paymentsare assumed fully deductible with no tax loss offset or carryforward provisions. The firm then decides how much residualincome should be paid out as dividend. If the default decisionis taken, the company is liquidated immediately; otherwise,the operation continues until the end of period H .

3.1. Chronology

To briefly summarize the model, the firm’s decisions atstage t are as follows (where the notation is summarized inthe Appendix):

1. Observe current period initial inventory level xs,t andcash position us,t = u+s,t − u−s,t ;

2. Find an optimal production decision ys,t and optimalfinancing decisions {es,t , ds,t } to maximize the valueof the equity V s,t (xs,t , us,t );

3. If V s,t (xs,t , us,t ) ≥ 0, continue operating the com-pany, and

(a) borrow an amount ds,t , issue new equity es,t+ ,

or pay dividends es,t− ; the firm’s net dividend

payout is es,t = es,t− − e

s,t+ ;

(b) pay the fixed operating cost K and variablecost cys,t to produce ys,t units of goods;

(c) observe the market demand qs,t and satisfydemand at level zs,t = min[xs,t + ys,t , qs,t ]to obtain revenue pzs,t − cys,t ;

(d) pay (1 + rs,t )ds,t to the debt holders forprincipal and interest;

(e) realize operating profit is,t = (p − c)zs,t −rs,t ds,t − K , and pay out τ max[is,t , 0] ascorporate tax;

else, if V s,t (xs,t , us,t ) < 0, stop operations, and goto bankruptcy (liquidating all assets and payingproportional costs).

If the company does not go to bankruptcy by the endof the last period, the comparative multiple method intro-duced in Section 2 is applied to calculate the terminal valueV s,H (xs,H , us,H ).

3.2. Integrated Corporate Planning Formulation

We depict future uncertainty in the form of an event tree.Every node in the scenario tree at time t ∈ {0, 1, . . . , H −



1} represents a realization of the uncertainty with a positiveprobability. Branches do not intersect because the bankruptcyprocesses of the company are path-dependent. We call a pathin the event tree between time 0 and time t a scenario s attime t , referred to by {s, t}. We denote St as the set of allpossible scenarios at time t and the probability of the sthoutcome in period t as ps,t

r . In this model, we assume thatthese probabilities correspond to a risk-neutral distribution.

In practice, they may, for example, be approximations of alognormal distribution for demand that includes an adjust-ment for the risk premium in this market. Let Ns,t be thedecision node associated with scenario {s, t}. Each node, Ns,t

∀ t ∈ [1, H − 1], has an ancestor Ns−,t−1 and one or moredescendants Ns+,t+1 ∈ Ds,t . With the notation and proce-dures introduced above, the ICP problem can be formulatedas follows:

MaxH∑

t=0

∑s∈St

e−rf tps,tr (e

s,t− − e

s,t+ ) + e−rT

∑s∈ST

ps,Tr θ(xs,T , us,T

+ ) (2)

s.t. xs−,t−1 + ys−,t−1 ≥ xs,t + zs,t ∀ t ∈ [1, H ], s ∈ St , (3)

us−,t−1+ + e

s−,t−1+ − e

s−,t−1− − rs−,t−1ds−,t−1 − KIs−,t−1 − cys−,t−1

+ pzs,t − τ is,t ≥ us,t+ − u

s,t− ∀ t ∈ [1, H ], s ∈ St , (4)

is,t − [(p − c)zs,t − rs−,t−1ds−,t−1 − KIs−,t−1

] ≥ 0 ∀ t ∈ [1, H ], s ∈ St , (5)

us,t+ + e

s,t+ − e

s,t− + ds,t − cys,t − KIs,t ≥ 0 ∀ t ∈ [0, H − 1], s ∈ St , (6)

MIs,t − [ds,t + e

s,t− + e

s,t+ + xs,t + ys,t + zs,t + is,t + u

s,t+

] ≥ 0 ∀ t ∈ [0, H ], s ∈ St , (7)

M(1 − I s,t ) − us,t− ≥ 0 ∀ t ∈ [0, H ], s ∈ St , (8)

I s−,t−1 − I s,t ≥ 0 ∀ t ∈ [1, H ], s ∈ St , (9)

zs,t ≤ qs,t ∀ t ∈ [1, H ], s ∈ St , (10)

ds−,t−1(1 + rf ) = ds−,t−1(1 + rs−,t−1)∑

Ns,t∈Ds− ,t−1

ps,tr I s,t

+ αp∑

Ns,t∈Ds− ,t−1

ps,tr qs,t [1 − I s,t ] ∀ t ∈ [1, H ], s ∈ St , (11)

ys,t ≥ 0 xs,t ≥ 0 UE ≥ es,t+ ≥ 0 e

s,t− ≥ 0, I s,t ∈ {0, 1},

us,t+ ≥ 0 u

s,t− ≥ 0 is,t ≥ 0 zs,t ≥ 0 Ud ≥ ds,t ≥ 0 rs,t ≥ 0 ∀ t ∈ [1, H ], s ∈ St .

all variables non-anticipative and t integral.

The first term in the objective function represents thepresent value of the cash flow from time 0 to H , which isequal to the dividend payout net of equity issuing proceeds,where rf is the risk-free rate. The last item, θ(xs,T , us,T

+ ) =u

s,T+ + βxs,T + ρzs,T , is the terminal value of the firm if the

default decision has not been taken by the end of the planninghorizon, β is the salvage factor of the inventory, and ρ is theequity-sales multiple. Taking sales as a value driver is simplyused here for illustration. Any value driver may be used.

The inventory transfer constraints (3) link inventorybetween successive periods in each scenario, i.e., the inven-

tory level of a node at stage t is determined by its parent node’sinventory xs−,t−1 plus the surplus of production ys−,t−1 oversales zs,t at the beginning of stage t . The greater or equalrelations are employed because we assume the salvage valueof the product is zero once the company goes to bankruptcy;hence, equality will not hold at bankruptcy nodes.

The cash transfer constraints are given by (4), whichdescribes the relationship among net operation income, cap-ital investment, and cash increments, i.e., the net operatingincome in each period is the sum of capital investment and thedividend payment e

s−,t−1+ − e

s−,t−1− , where the net operating



income of a scenario at stage t is equal to the sales revenuepzs,t minus production cost cys,t , interest cost rs−,t−1ds−,t−1,and tax cost τ is,t . The capital investment consists of two parts:the working capital investment c(ys−,t−1 − zs,t ) and the cashincrement u

s,t+ − u

s−,t−1− . Because both u

s,t+ and u

s−,t−1− are

nonnegative variables, us,t− > 0 indicates that the value-to-go

of the equity is negative; in this case, it is not optimal for theequity holders to invest further in the firm; the best policy isto abandon the firm and let it fall into bankruptcy.

The tax constraints (5) give the amount of taxable income.If sales income exceeds the sum of the production cost, fixedoperating cost, and interest cost, the company incurs a profitand is subject to a tax obligation. Note that we assume debtpayments are fully tax deductible, so that debt provides a taxshield that should be deducted from the operating income.If the company operates at a break-even or loss position, thetaxable income is assumed to be zero.

The first set of resource constraints forms the productionconstraints. The capital required to support the productionoperation comes from three categories: the internal cash u

s,t+ ,

the net equity issuing income es,t+ − e

s,t− , assumed only avail-

able to the incumbent shareholder, and the debt financingproceeds ds,t . Constraints (6) ensure that the total amount ofcash income from these financial instruments can cover theproduction and fixed operating costs.

Bankruptcy constraints (7) and (8) imply that, after thecompany goes into bankruptcy, the contribution of the futurecash flow to the wealth of the equity holders is zero. Ifthe company does not go to bankruptcy at a certain sce-nario, Constraint (7) will have no effect on the decisionsof the company since we assume M ↑ +∞; otherwise, ifu

s,t− < 0, Constraints (7) and (8) will stop the operation of the

firm. Constraints (9) then indicate that, once the firm goesinto bankruptcy, it cannot resume operations again.

Constraints (10) specify that realized sales should not begreater than the market demand, qs,t . The nonlinear inter-est equilibrium constraints (11) are a discretized version ofEq. (12), which indicate that the interest rates charged by thedebt holders are functions of the market demand distributionand the firm’s operational decisions. Since the amounts ofdebt crucially depend on the interest costs, the firm’s financ-ing decisions are then related to its production decisions. Thecompany’s output levels are also contingent on its financingability; the firm’s production and financial decisions are thenmade simultaneously.

All the variables in the above model are subject to non-negativity constraints (and integrality constraints for thebankruptcy decision variables). We also assume that thereis an upper bound on the dividend payout and the debt issue.The decisions are implicitly non-anticipative since our treeconstruction forces each variable for {s, t} only to depend onthe history of the process at t . We also state that t is assumedto be integral without loss of generality.

4. MODEL PROPERTIES ANDALGORITHMIC STRATEGY

In this section, we detail an algorithm for solving the inte-grated corporate planning model (2)–(11). We first analyzeproperties of the ICP model and show that, for each rationaloperating strategy, there exists a unique equilibrium interestrate for every decision node. We then propose a two-stagealgorithm to find an optimal solution by taking advantage ofthe rational-operations property. This approach significantlydecreases the computational complexity to find an optimalsolution.

4.1. Properties of the IntegratedCorporate Planning Model

Compared with traditional stochastic programming prob-lems, a key difficulty in solving model (2)–(11) is that theinterest rate parameter vector, r ≡ {rs,t |Ns,t ∈ N t , t ∈[0, T − 1]}, is a decision vector that depends on other deci-sion variables as indicated by Eq. (11). An intuitive strategyfor dealing with this complex nonlinear equality is to removethe nonlinear constraints (11) from the ICP model, iterativelyupdating the interest rate parameters by substituting the solu-tions of the previous iteration into Eq. (11), and stoppingwhen an equilibrium is achieved. Our following analysis,however, indicates that, for each rational tree (defined later inthis section), there exists a unique equilibrium r, maximizingthe objective value of model (2)–(11); furthermore, closed-form solutions for r can be given. Hence, we need not use theiterative strategy.

Lemma 4.1 gives properties of the equity value V s,t

as inventory level, cash position, and demand realizationschange.

LEMMA 4.1: Let V s,t be the equity value of the com-pany under scenario realization s ∈ S t at stage t ∈ [0, H ];then, V s,t is nondecreasing in (i) inventory level xs,t ; (ii) cashposition us,t ; and (iii) demand realization qs,t .

PROOF: From the ICP model, it is clear that increasingxs,t , qs,t , or us,t = u

s,t+ − u

s,t− expands the feasible region

of the corresponding subproblem starting from node Ns,t ;therefore, the associated objective value is nondecreasing inxs,t , u

s,t+ , and qs,t . �

Denote Ds,t as the set of descendants of scenario s− at timet − 1. Also, let Ns,t , Ns ′,t be two sibling scenarios belongingto Ds,t . Lemma 4.2 states that if Ns,t is the node with higherdemand realization, then it is not optimal to operate the firmunder the lower demand while shutting it down in the statewith higher demand.

LEMMA 4.2: ∀ Ns,t , Ns ′,t ∈ Ds−,t−1, t ∈ [2, . . . , H ], ifqs,t ≥ qs ′,t , then I s,t ≥ I s ′,t .



PROOF: Let X = Xs,t , s ∈ S t , t ∈ [0, . . . , H ] be theoptimal solution of the ICP model, and Xs,t , Xs ′,t be thesubproblem solution associated with scenario s and s ′,respectively. Denote V s,t as the equity value at node Ns,t .Suppose I s,t< I s ′,t is an optimal strategy; we must haveV s ′,t ≥ 0 ≥ V s,t since the company will stop operating onlyif its equity value falls below zero. From (iii) of Lemma 4.1,we know V s,t ≥ V s ′,t since the realized demand at node Ns,t

is higher than that of node Ns,t , and all the other decisionsand status variables up to time t − 1 are exactly the samefor both nodes; therefore, if V s ′,t ≥ 0, keeping the companyoperating at node Ns,t increases the value of the firm’s equity,which contradicts the assumption that I s,t< I s ′,t is the optimalstrategy. Hence, I s,t ≥ I s ′,t if qs,t ≥ qs ′,t . �

To facilitate our analysis and exploit the special structureof the ICP model, we use the following definitions.

DEFINITION 4.3: Let Ns ′,t+1, Ns ′′,t+1 ∈ Ds,t , qs ′,t+1 ≥qs ′′,t+1, t ∈ [0, H − 1],

(i) A scenario tree, T , is called a stopping tree if ithas an associated stopping policy I ∈ I ≡ {I s,t ∈{0, 1}|Ns,t ∈ N t , t ∈ [0, H ]}.

(ii) A stopping tree, TR , is rational if both I s,t ≥ I s ′,t+1

and I s ′,t+1 ≥ I s ′′,t+1 hold.(iii) A tree, TE , is in equilibrium if it is rational and

satisfies the debt pricing constraints,

(1 + rf )ds,t = (1 + rs,t )ds,t∑

Ns′ ,t+1∈Ds,t

ps ′,t+1r I s ′,t+1

+ αp∑

Ns′ ,t+1∈Ds,t

qs ′,t+1ps ′,t+1r [1 − I s ′,t+1]. (12)

Definition 4.3 indicates that, if a stopping tree is not arational tree, then the equity value given by that operatingstrategy is dominated by another strategy. The equilibriumdefinition further specifies that, at every decision node, theoptimal decisions satisfy the debt pricing Eq. (12). For everystage t ∈ [0, H − 1], the rational tree realization TR pre-specifies the stopping region. If demand realization qt is lessthan the lowest operating demand qt

b specified by TR , then theinitial cash position of that scenario is negative, i.e., u′(qt ) <

0, ∀ qt < qtb.

To show that there exists an equilibrium interest rate vectorr for each rational tree TR , we start with analysis of the prop-erties of the optimal policy of the relaxed ICP model (2)–(10)without the debt pricing constraints (11).

LEMMA 4.4: Let V s,t (u, x) be the value-to-go function atstage t ∈ [0, . . . , H − 1] of a rational realization TR , withinitial cash position u and inventory level x; then,

(i) given inventory level x and production decision y,the sales volume is z = min[q, x + y];

(ii) if sb = max{s|Ns,t ∈ Ds−,t−1, I s,t = 1}, thenusb ,t = 0;

(iii) given values of the production and dividend deci-sions, the debt level is

d = max[0, cy + e + K − u].PROOF: (i) To show that z = min[q, x + y] is optimal,

we first show that V s,t is nondecreasing in z. Fromthe cash transition constraints u′ = u − e + pz −cy−rd −K −τ i, it is clear that u′ and e are nonde-creasing in z. Note also that V t(u + c�, x − �) ≥V t(u, x), and that V s,t is nondecreasing in z. Sincez ≤ q and z ≤ x + y are the only constraintson z, the optimal sales decision is z = min[q,x + y].

(ii) Suppose u′ = usb ,t = 0 is another optimal policy.Since the company can raise money from the finan-cial market at the beginning of each stage as longas the firm does not default on its debt payment, thevalue of Vt(u, x) can be increased by the follow-ing strategy: (1) if d ≥ u′, decreasing debt usageby u′ will increase Vt(u, x) by ru′; (2) if d = 0,increasing the dividend payout by u′ will boost thevalue of Vt(u, x) by (1 − ρ)u′; (3) if 0 < d < u′,the company can combine strategies (1) and (2),the increased value will be rd + (1 − ρ)(u′ − d).Since all the above strategies achieve higher valuesof Vt(u, x) and satisfy the rational policy require-ment given by TR , u′ > 0 could not be an optimalpolicy; hence, us,t = 0.

(iii) Let d ′ = d be another optimal debt policy. Ifd ′ < d , either the company takes negative debt,d ′ < 0, or the debt level is not sufficient to sup-port the production and dividend decisions. Hence,d ′< d is infeasible. If d ′> d , from (ii) the companyincurs an additional interest cost, r(d ′ − d), whichdecreases the value of Vt(u, x); hence, d ′ = d.

�Lemma 4.4 indicates that a property of optimal decisions

of the ICP model is that the initial cash position correspond-ing to the scenario with the lowest demand realization is zero.Recognizing this fact, Lemma 4.5 shows that, for the opti-mal decisions of the relaxed ICP problem (2)–(10), the debtinterest rate is negatively correlated with the debt amount foreach decision node.

LEMMA 4.5: Let rs,t be the interest parameter of nodeNs,t for a rational realizationTR and letds,t be the correspond-ing optimal debt decision; then ds,t is a monotone decreasingfunction of rs,t .



PROOF: We first show that zsb ,t = qsb ,t . Note that q < qsb ,t

is the bankruptcy region pre-specified by the rational policyTR corresponding to Vt(u, x). From part (i) of Lemma 4.4, wemust have z = q ≤ (x + y); otherwise, the market demandstate corresponding to q ′ = x + y ≤ qsb ,t does not includebankruptcy, which contradicts the assumption that q < qsb ,t

is the bankruptcy region; hence, zsb ,t = qsb ,t .From Lemma 4.4, we know (i) cy = u + d − e − K , (ii)

usb ,t = 0. Substituting i = max[0, (p − c)qb − rd − K] andy = (u + d − e − k)/c into usb ,t = 0, we have

d =

(p − τ(p − c))qb + τK

1 + (1 − τ)rif i > 0 (or r < rb),

pqb

1 + ro.w.,

(13)

where rb = (p−c)qb−K

cqb+K. We now show that r < rb is equiv-

alent to i > 0. If i > 0, substituting Eq. (13) into rd <

(p−c)qb−K and reorganizing terms, we have d < cqb+K ,i.e., r < rb. On the other hand, r ≥ rb leads to negative tax-able income, i.e., i = 0. Substituting y = (u + d − e − k)/c

into usb ,t = 0 yields (1+ r)d = pqb, which is Eq. (13) wheni = 0. From Eq. (13), ds,t is a monotone decreasing functionof rs,t . �

To show that there exists a unique equilibrium interest vec-tor r, we only need to show that the debt–interest relationshipspecified by Eq. 12 is positively correlated.

LEMMA 4.6: Let (rs,t , ds,t ), ∀Ns,t ∈ N t , t ∈ [0, H−1] bea debt–interest pair associated with an equilibrium tree real-ization, then ds,t is a monotone increasing convex functionof rs,t .

PROOF: Let Ns ′,t+1 ∈ Ds,t be the descendant nodes ofNs,t ; also let I s ′,t+1 ∈ {0, 1} be the associated rational oper-ating decision of node Ns ′,t+1. We know at equilibrium thatrs,t and ds,t satisfy

(1 + rf )ds,t = (1 + rs,t )ds,t∑

Ns′ ,t+1∈Ds,t

ps ′,t+1r I s ′,t+1

+ αp∑

Ns′ ,t+1∈Ds,t

qs ′,t+1pt+1,s ′r

[1 − I t+1,s ′]

, (14)

where qs ′,t+1 and P s ′,t+1r are the demand realization and

probability density associated with scenario s ′. For a givenrational operating strategy T , I s ′,t+1 is constant. For sim-plicity, let a = ∑

Ns′ ,t+1∈Ds,t P s ′,t+1r I s ′,t+1 and b be the

second term of the right-hand side of Eq. (14). Notice that1 + rf − (1 + rs,t )a > 0, so that dds,t

drs,t = ab[1+rf −(1+rs,t )a]2 ≥ 0

and d2ds,t

d2rs,t = −2a2b[1+rf −(1+rs,t )a]3 ≤ 0; therefore, ds,t is convex

in rs,t . �

PROPOSITION 4.7: There exists a unique equilibrium foreach rational operating strategy TR ∈ TR .

PROOF: From Lemmas 4.5 and 4.6, there exists a uniqueequilibrium for every Ns,t ∈ N t , ∀ t ∈ [0, . . . , H ] such thatEqs. (4.3) are satisfied. �

Lemmas 4.5 and 4.6 not only show the unique existenceof the equilibrium interest rates rE , but also give a method tocalculate these parameters, which solve Eqs. (12) and (13).Note that the equity value of the rational stopping tree dom-inates that of the non-rational stopping tree; we only needto consider rational trees instead of all stopping trees to findthe optimal decisions, which significantly decreases the com-putational burden. This is formally stated in the followinglemma.

PROPOSITION 4.8: Let T ∗ ≡ arg maxT : T ∈TSVE(u, x) be

a stopping tree that maximizes the value of the equity; thenT ∗ ∈ TE , where TE := {TE} is the set of equilibrium scenariotrees.

PROOF: To show T ∗ ∈ TE , we only need to show thatV ∗ is not achievable if T /∈ TE . From Definition 4.3 thereare three cases: Eq. (12) does not hold, Lemma 4.2 is notsatisfied, or both. In the first case, the ICP problem is infea-sible. In the second case, we know the optimal value cannotbe achieved from Lemma 4.2; therefore, T ∗ ∈ T . �

4.2. Algorithm and Example

We now detail a two-stage algorithm for the ICP model,which identifies the set of rational scenario trees, {T̃R}, in thefirst stage, and then runs over {T̃R} to find optimal decisionsduring the second stage.

STEP 1: Identify a rational scenario realization set TR

such that I s,t ≥ I s ′,t+1 and I s ′,t+1 ≥ I s ′′,t+1 hold for everyTR ∈ TR , where Ns ′,t+1, Ns ′′,t+1 ∈ Dt ,s , qs ′,t+1, ≥ qs ′′,t+1,t ∈ [0, H − 1]. Let J = ||TR||, denote the j th element of TR

by Tj

R . Set j = 0.

STEP 2: Let j = j + 1. If j > J , go to step (3); Set theinterest rate parameter set rj from the solutions of Eqs. (13)and (14). Solve the ICP problem with rational stoppingrealization T

j

R ; denote V (Tj

R , rj ) and X(Tj

R , rj ) as the cor-responding objective value and decision, respectively.

STEP 3: Identify the index of the optimal rational sce-nario tree as j ∗ = arg maxj∈[1,J ]V (T

j

R , rj ); optimal decisions

corresponding to the optimal value V (Tj∗R , rj ) are X(T

j∗R , rj ).

The following proposition demonstrates the efficiency ofour integer decomposition strategy by comparison with the



computational complexity of the worst case for the branch-and-bound method.

PROPOSITION 4.9: Let m and n be the number of stop-ping trees and rational scenario trees of the ICP model,

respectively; then n < kH(H+1)

2 � 2kH+1−1

k−1 = m if H andk are large, where k is the number of children of each node,and [0, H ] is the length of the planning horizon.

PROOF: Let {TS} and {TR} be the sets of stopping treesand rational trees, respectively. From Definition 4.3 we knowthe size of the rational tree is n ≡ ||{TR}|| <

∏H−1t=0 kt+1 =

kH(H+1)

2 ; therefore, n < kH(H+1)

2 � 2kH+1−1

k−1 = m, where m ≡||{TS}|| = 2

kH+1−1k−1 is the total number of stopping trees. �

We illustrate the above valuation/planning procedure byconsidering an ICP model for a hypothetical firm. In ourexample, the firm’s planning horizon is three periods. Theequity-sales multiple is used to calculate its future cash flowsat Time 3. The risk-free interest rate is assumed to be 0.05. Wealso assume the market demand evolution follows a geomet-ric diffusion process with a growth rate of 0.1 and volatilityof 0.5 per year.2 The current market demand level is set to100 units per year. The unit commodity production cost is$0.40 and the selling price is $1. Both initial cash positionsand inventory levels are set to zero. The salvage value ofinventory at the end of the planning horizon is assumed tobe 50% of its production cost. We also let the terminal valuemultiples be 1, i.e., the value of future cash flows to equityholders equals the firm’s sales revenue during the secondstage. The base case bankruptcy recovery rate is 60% and thefixed operating cost is $40 per stage.

For simplicity, we assume there are three scenarios for eachdecision node; hence, there are 13 binary integer variablesin this example. The total number of stopping trees is 213,while the total number of rational trees is only 85, indicatingsignificant decreases in computational complexity even insuch a small-scale situation. For each rational stopping tree,the interest rate parameter set solves Eqs. (13) and (14). Weuse the ILOG CPLEX solver (Version 8.0, 2002) to find the

2 This represents the risk-neutral equivalent demand process. Theoriginal demand process would have had a growth rate risk premiumequal to the product of the market risk premium to volatility ratio,the correlation of the demand to the market return, and the demandvolatility. If the natural demand process had a correlation of 0.5 withthe market and the market risk premium was 0.1 with volatility of0.2, then the premium for this product demand would have been(0.1/0.2)× (0.5)× (0.5) = 0.125. The natural demand growth ratewould then have been 0.225 in that case. Other values would alsobe consistent with the risk-neutral distribution given here. Since weonly used the risk-neutral equivalent form, we did not specify theoriginal demand distribution.

optimal decisions for each rational tree and choose the treewith highest equity value as the optimal operating strategy.

Figure 1 illustrates the results of the optimal decisions.The optimal operating strategy for the equity holder at Stage2 is to abandon the firm if the demand realization is poor,corresponding to N2,3; otherwise, it is optimal to continueoperation. At the sixth scenario of Stage 3, the value of theequity is not sufficient to cover the debt payment; there-fore, the debt holders force the firm to bankruptcy if theuncertain realization is N3,6; otherwise, the firm can alwayspay back the debt plus interest in full. Note that in the ICPmodel, rather than subjectively fixing the borrowing rate, weuse the decision-adjusted interest rates for different marketrealizations. The interest rates associated with the three deci-sion nodes N1,1, N2,1, and N2,3 are 0.19, 0.05, and 0.18,respectively.

Given the optimal operating strategy and the debt interestcosts, we solve the ICP model and find the optimal produc-tion and financial decisions. Note that the optimal interestrate charged at node N2,1 is the risk-free rate. The low interestrate should give the equity holders incentive to take an all-debt financing strategy; however, the firm actually financesthe production mainly by a new equity issue, 102.87, whichis almost twice the amount of debt. The firm takes thisaction because the non-default operating strategy (i.e., risk-free interest cost) actually restricts the firm from aggressivedebt policy, which might eventually lead to default in thecase of low demand realizations; also note that, in this exam-ple, the optimal market leverage ratio is not a fixed targetbut rather a dynamic one that changes over time as marketdemand situations change.

Figure 1. A decisions tree of a 3 stage 3 scenario ICP model.



5. NUMERICAL RESULTS

In this section, we present numerical results to comparealternative models. To evaluate the performance of the ICPmodel described in Section 3, we compare against a fixedinterest (FI) model, which removes the debt pricing con-straints in the ICP model and assumes that the company canalways borrow at a fixed rate. We also consider a mean value(MV) model in which all random variables are replaced bytheir means. A brief discussion on the sensitivity analysis ofthe models is given by changing the demand and financialmarket environmental factors. To evaluate the effects of theplanning horizon on the firm’s valuation and capital struc-ture choices, we compare performance both in single- and inmultiple-period settings. We conclude that a longer planninghorizon increases firm valuation and leads to lower leverageratios.

As discussed in the previous section, the ICP model outper-forms the static FI model. The positive value of the stochasticsolution over the MV model also suggests that we shouldadopt a contingent decision framework in the discount divi-dend model. Our results indicate that the financial and marketdemand factors have significant effects on production deci-sions and equity valuation. The joint financing and marketdemand effects suggest the need for integrated corporateplanning.

The base case numerical example is identical to the pre-vious example in Section 4 except that we let the volatilityof the underlying demand process be 0.3 per year. We alsochange the fixed operating cost to $10 per operating period.The base case terminal value multiples are also set to zero;the company effectively operates as a two-stage project.

We first explore how operating environmental factor varia-tions affect a firm’s decisions and valuation. Figure 2 displaysthe performance of the ICP, FI, and MV models as func-tions of production cost c, demand volatility σ , bankruptcyrecovery rate α, and terminal valuation multiple ρ (equity-sales ratio). Note that the equity price given by the ICPmodel is always greater than or equal to the results of the FImodel because the ICP model identifies the best productionand financial strategy while the FI model specifies a fixeddebt interest cost that the production decisions must satisfy.The constant interest rate assumption restricts productionflexibility and decreases the feasible region for decisions.

Substituting the decisions of the traditional DDM into theFI model gives the MV solution. Figure 2 shows that thevalues given by the MV model are always dominated bythe other two models, which indicates that the DDM modelcan be improved significantly by incorporating more scenariorealizations. Specifically, the difference between the objec-tive value of the FI model and the MV model is called thevalue of the stochastic solution, which is always nonnegative(see [3]). Another explanation for the improvement is that

DDM only uses first-order information to calculate the valueof the equity. Such a simplification of future uncertainty leadsto inferior decisions and lower equity valuation.

Figure 2A also shows that the equity value is negativelycorrelated with production cost c; the differences amongthe three valuation models also decrease as production costincreases. A rise in production cost increases the marginalproduction cost, which leads to a lower production level. Thegap between the ICP and FI models becomes smaller as costsincrease because the lower output levels due to higher costsare associated with lower debt interest rates.

Figure 2B illustrates that equity value is a decreasing func-tion of demand volatility. The firm’s future cash flows becomeriskier as volatility increases, which decreases profit mar-gin and leads to higher debt cost; hence, equity value hasa downward trend. Another observation from Fig. 2B is thatthe higher the volatility, the better the performance of the ICPmodel compared with the FI and MV models. The ICP modelallows the firm to stop operating under poor conditions, whilein the FI model, the firm must meet the solvency requirementunder all scenarios, limiting the high-end possibilities for thefirm.

For different bankruptcy recovery rates and terminal valu-ation multiples, Fig. 2 shows that, as expected, equity valueis positively correlated with these two factors. An increase inthe bankruptcy recovery rate clearly decreases debt costs andincreases profit margin, leading to higher equity value.

Figure 2D illustrates that the firm’s value and decisionsare sensitive to the growth factor, indicating that the demandtrend plays an important role in decision making. This resultis intuitive since the value of a firm is determined not onlyby the income during its planning horizon but also by thefuture cash flow beyond that period. If the firm has a stronggrowth trend, it might not be optimal to shut down under poordemand realizations during early stages, since the value of thefuture cash flows could exceed the losses incurred during theinitial planing horizon. When the growth multiple reaches acertain level, the optimal policy for the company is to continueoperations under all scenarios. In this case, the ICP modelbecomes identical to the FI model. Hence, the equity valuesgiven by the ICP and FI model converge as the multiple valueincreases.

Figure 3 indicates that production cost, demand volatility,bankruptcy recovery rate, and the terminal valuation multipleplay important roles in capital structure decisions. Figure 3Ashows that the financial leverage ratio is positively correlatedwith production cost. This observation suggests that a low-margin company should take aggressive financial decisions,while a high-margin firm should follow a conservative debtpolicy. Because lower margin means higher production cost,the firm’s production output level decreases with decreasingmargin. For a company facing uncertain demand, a lower pro-duction level decreases the risk of future cash flow; therefore,



Figure 2. Equity value as a function of production cost, demand volatility, bankruptcy recovery rate, and terminal value multiple for threedifferent planning models.

the debt holder charges a small risk premium to compen-sate for bankruptcy risk. To take advantage of a low cost ofdebt, the firm prefers to use more debt in its capital structure.Another explanation is that high-margin firms usually expectlarge future investments. To balance current and expectedfinancing costs, high-margin firms tend to take conservativefinancing policies to raise low-cost debt in the future.

These observations are consistent with the pecking-ordertheory (see, e.g., [33]) that suggests a negative relationbetween profitability and leverage ratio. This relation hasalso been observed empirically by Fama and French [15].In Xu and Birge [40], we also found empirical evidence ofa strong negative relation between pre-tax operating mar-gin and market leverage for low pre-tax operating-marginfirms, but that paper also presents demand distribution con-ditions under which margins for high-margin firms exhibit a

positive relation with market leverage. The paper then showsthat a weak positive relation exists empirically between pre-tax operating margin and market leverage for high-marginfirms. This finding is then consistent with the trade-off theory(see, e.g., [31]) of capital structure.

Another observation from Fig. 3A is that the financialleverage ratio is negatively related to demand volatility, whichis again consistent with the trade-off model, in which firmswith more volatile earnings and net cash flows have lessleverage and lower dividend returns. More volatile earningsimply lower expected tax rates and high expected bankruptcycosts, which push firms toward less leverage and lower div-idend payouts. We also observe that the higher the demandvolatility, the more significant the effect of production coston capital structure. As production cost increases, the valueof equity decreases while the debt usage increases, leading



Figure 3. The market leverage ratio as a function of production cost and terminal value multiple for varying market volatility and bankruptcyrecovery rates.

to high leverage ratios. On the other hand, a rise in marketvolatility drives up interest cost and the percentage of debtfinancing declines. Although an increase in market volatilityhas a negative effect on the equity valuation, the drop in debtusage is even sharper; hence, the effect of production costincreases on capital structure becomes more significant forfirms facing higher demand uncertainty.

The effects of bankruptcy recovery rate and terminal valu-ation multiples on capital structure are illustrated by Fig. 3B.The optimal debt leverage ratio is a decreasing function ofboth factors. Since a lower bankruptcy recovery rate reducesthe debt holders’ cash flow in the case of default on the firm’sdebt payment, this yields higher debt cost; hence, the firmis reluctant to raise the debt level when the interest com-mitment increases the likelihood of bankruptcy. On the otherhand, a higher recovery rate lowers the borrowing cost, whileraising production and increasing debt. Our findings also

support the trade-off theory implication here that firms usedebt conservatively when the expected financial distress costsare high.

In our optimization-based valuation framework, we applythe multiple method to calculate the terminal value, i.e., thevalue for the cash flows subsequent to the horizon year. Thevalue driver is a summary statistic for the value of the futurecash flows; therefore, the higher the value of the multiple, thestronger the potential growth trend of the company. Figure 3Bindicates that leverage ratio is negatively correlated with thevalue of the multiple. These observations agree with the prac-tice that firms with strong growth ability prefer lower leverageratios. An intuitive explanation is that firms with high valuemultiples are expected to have large growth rates and profitmargins, implying larger equity value and lower leverage.

In [39], we considered joint production and financial deci-sions in a single-period model that did not allow future



Figure 4. Equity value and market leverage ratio as a function of production cost for two-stage and single-stage valuation models.

investment. To explore the differences between the single-and multiple-period planning models, Fig. 4 shows the equityand market leverage ratio as a function of production cost forthe single-stage and two-stage cases. The two-stage planningmodel yields higher equity valuation than the single-stagecase. The main reason is that a longer planning horizon allowsfirms to observe realizations of future uncertainty before tak-ing contingent actions. The manager can base decisions onnew information, which provides an option for the firm tohedge market risk by adjusting the investment level or ceasingoperations in undesirable situations. In the multistage setting,the firm can also take advantage of debt or equity financingby, for example, waiting for future favorable conditions toexpand financing.

Another observation from Fig. 4 is that the debt-to-market-leverage ratio is higher in the single-stage model than inthe multistage model. At first, this difference might appearcounter intuitive since the multistage model actually allows

greater overall debt capacity with larger investment andproduction alternatives than the single-stage model. Thoseexpanded opportunities, however, produce higher equity val-uation as explained earlier. This increase overcomes the debtincrease and leads to lower leverage in the multistage case.

6. CONCLUSIONS

In this paper, we develop an integrated corporate planningmodel to make production and financial decisions simulta-neously for a company facing demand uncertainty. Financialand production decisions are linked in this model because thefirm’s operational decisions depend critically on its financ-ing ability to support optimal production and the operationaldecisions affect the firms’ financing costs and choices. Wemodel the corporate planning problem with a multistagestochastic program. At each decision node the managers



make operational and financing decisions: stop or continueoperations; determine amounts of loans, dividend payout ornew equity issues; and set levels for product output. A difficultpart of this problem is that the debt interest rate is a nonlin-ear function of the operating decisions, while we also needthe debt interest rate as an input parameters to find optimaloperating decisions.

To find an optimal solution of the ICP model with nonlin-ear financial constraints and binary integer variables, we firstidentify the rational integer realization sets, which signifi-cantly decreases computational complexity from a standardimplementation. We then show that, for each set of rationalrealizations, there is a unique equilibrium interest rate sat-isfying the nonlinear financial constraint for each decisionsnode. From solving the model under these conditions, oursensitivity analysis indicates that the decisions of the ICPmodel outperform the traditional production planning modeland the discount dividend model.

Our main conclusions are: (a) production and financialdecisions should be made simultaneously in an integratedinteractive framework; (b) the ICP framework enables a firmto coordinate production and financial decisions simultane-ously and extends the passive pricing method into an activevaluation framework; (c) the ICP model can consider debtcosts as endogenous decision variables instead of exoge-nous parameters; (d) compared with a single-period staticmodel, the multistage setting yields higher equity valuationand lower leverage ratios.

APPENDIX (Notation)

V s,t (xs,t , us,t ) : equity value function under scenario s at period t

Parameters

M: constant positive parameter, M ↑ ∞;qs,t : market demand under scenario s at period t ;rs,t : single-period debt interest cost under scenario s at period t .

Status variables

xs,t : inventory level at the beginning of period t under scenario s;us,t : cash position at the beginning of period t under scenario s, us,t =

us,t+ − u

s,t− .

Decision variables

ys,t : production decision at the beginning of period t under scenario s;ds,t : debt issued by the company at the beginning of period t under scenario

s;es,t− : dividend payed at period t under scenario s;

es,t+ : stock issued at period t under scenario s;

zs,t : realized sales of product at period t under scenario s, zs,t =min(xs,t , qs,t );

is,t : taxable operating income at period t under scenario s,

is,t = max[(p − c)zs,t − rs− ,t−1ds− ,t−1 − KIs− ,t−1, 0];

I s,t : operating indicator variable, I s,t ={1 if us,t ≥ 0,0 if us,t < 0.

REFERENCES

[1] R.W. Anderson and S. Sundaresan, Design and valuation ofdebt contracts, Rev Finan Stud 9 (1996), 37–68.

[2] V. Babich and M.J. Sobel, Pre-IPO operational and financialdecisions, Manage Sci 50 (2004), 935–948.

[3] J.R. Birge, The value of the stochastic solution in stochasticlinear programs with fixed recourse, Math Program 24 (1982),314–325.

[4] J.R. Birge, Option methods for incorporating risk into lin-ear capacity planning models, Manufact Serv Oper Manage2 (2000), 19–31.

[5] P. Bolton and D.S. Scharfstein, A theory of predation basedon agency problems in financial contracting, Am Econ Rev 80(1990), 93–106.

[6] J.A. Brander and T.R. Lewis, Oligopoly and financial structure:The limited liability effect, Am Econ Rev 76 (1986), 956–970.

[7] J.A. Buzacott and R.Q. Zhang, Inventory management withasset-based financing, Manage Sci 24 (2004), 1274–1292.

[8] D.R. Cariño, T. Kent, D.H. Meyers, C. Stacy, M. Sylvanus,A.L. Turner, K. Watanabe, and W.T. Ziemba, The Russell–Yasuda Kasai model: An asset liability model for a Japaneseinsurance company using multistage stochastic programming,Interfaces 24 (1994), 29–49.

[9] M. Cohen and A. Huchzermeier, “Global supply chain man-agement: a survey of research and applications,” Quantitativemodels for supply chain management, S. Tayur, R. Ganeshan,and M. Magazine, (Editors), Kluwer Academic, Boston.

[10] G.M. Constantinides, Market risk adjustment in project valu-ation, J Finance 33 (1978), 603–616.

[11] J.C. Cox and S.A. Ross, The valuation of options for alterna-tive stochastic processes, J Finan Econ 3 (1976), 145–166.

[12] Q. Ding, L. Dong, and P. Kouvelis, On the integration ofproduction and financial hedging decisions in global mar-kets, Working paper, Olin School of Business, WashingtonUniversity, St. Louis, MO, 2004.

[13] A. Dotan and S.A. Ravid, On the interaction of real and finan-cial decisions of the firm under uncertainty, J Finance 40(1985), 501–517.

[14] D. Duffie and D. Lando, Term structures of credit spreadswith incomplete accounting information, Econometrica 69(2001), 633–664.

[15] E.F. Fama and K.R. French, Testing trade-off and peckingorder predictions about dividends and debt, Rev Finan Stud15 (2002), 1–33.

[16] S.M. Fazzari, R.G. Hubbard, and B.C. Petersen, Financingconstraints and corporate investment, Brookings Papers EconActiv 1 (1988), 141–195.

[17] V. Gaur and S. Seshadri, Hedging inventory risk throughmarket instruments, Manufact Serv Oper Manage 7 (2005),103–120.



[18] S.C. Graves, A.H.G. Rinnooy Kan, and P.H. Zipkin, Hand-books in operations research and management science, Vol-ume 4, Logistics of production and inventory, Elsevier Science,Amsterdam, 1993.

[19] M. Harris and A. Raviv, The theory of capital structure,J Finance 46 (1991), 297–355.

[20] M.J. Harrison and D.M. Kreps, Martingales and arbitragein multi-period securities markets, J Econ Theor 20 (1979),381–408.

[21] C.A. Hennessy and T.M. Whited, Debt dynamics, J Finance60 (2005), 1129–1165.

[22] A. Huchzermeier and M.A. Cohen, Valuing operational flexi-bility under exchange rate risk, Oper Res 44 (1996), 100–113.

[23] ILOG, CPLEX Version 8.0, 2002.[24] P.R. Kleindorfer and D.J. Wu, Integrating long- and short-term

contracting via business-to-business exchanges for capital-intensive industries, Manage Sci 49 (2003), 1597–1615.

[25] B. Kogut and N. Kulatilaka, Operating flexibility, global man-ufacturing, and the option value of multinational network,Manage Sci 40 (1994), 123–139.

[26] P. Kouvelis, “Global sourcing strategies under exchange rateuncertainty,” Quantitative models for supply chain manage-ment, S. Tayur, R. Ganeshan, and M. Magazine (Editors),Kluwer Academic, Boston, 1999.

[27] H.E. Leland, Corporate debt value, bond covenants, andoptimal capital structure, J Finance 49 (1994), 1213–1252.

[28] J. Lintner, The valuation of risk assets and the selection ofrisky investments in stock portfolios and capital budgets, RevEcon Statist 47 (1965), 13–37.

[29] J. Liu, D. Nissim, and J. Thomas, Equity valuation usingmultiples, J Account Res 40 (2002), 135–172.

[30] F. Modigliani and M.H. Miller, The cost of capital, corporationfinance, and the theory of investment, Am Econ Rev 48 (1958),261–297.

[31] F. Modigliani and M.H. Miller, Corporate-income taxes andthe cost of capital—A correction, Am Econ Rev 53 (1963),433–443.

[32] J.M. Mulvey and H. Vladimirou, Stochastic network pro-gramming for financial planning problems, Manage Sci 38(1992), 1642–1664.

[33] S.C. Myers, The capital structure puzzle, J Finance 39 (1984),579–592.

[34] S.H. Penman, A synthesis of equity valuation techniques andthe terminal value calculation for the dividend discount model,Rev Account Stud 2 (1998), 303–323.

[35] S. Seshadri, A. Khanna, F. Harche, and R. Wyle, A method forstrategic asset-liability management with an application to theFederal Home Loan Bank of New York, Oper Res 47 (1998),345–360.

[36] W.F. Sharpe, Capital asset prices: A theory of market equilib-rium under conditions of risk, J Finance 19 (1964), 425–442.

[37] E.A. Silver, D.F. Pyke, and R. Peterson, Inventory manage-ment and production planning and scheduling, 3rd ed., Wiley,New York, 1998.

[38] J.T. Williams, Financial and industrial structure with agency,Rev Finan Stud 8 (1995), 431–474.

[39] X. Xu and J.R. Birge, Joint production and financing deci-sions: Modeling and analysis, Working paper, NorthwesternUniversity, Evanston, IL, 2004.

[40] X. Xu and J.R. Birge, Operational decisions, capital structure,and managerial compensation, Working paper, NorthwesternUniversity, Evanston, IL, 2004.


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Equity valuation, production, and financial planning: A stochastic programming approach