A lattice claims model for capital budgeting - IEEE Xplore

IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 42, NO. 2, MAY 1995

140

A Lattice Claims Model for Capital Budgeting Bardia Kamrad

Abstruct- Techniques of Contingent Claims Analysis (CCA), extend current capital budgeting practices in two specific ways. First, by explicitly accounting for project uncertainty and second, by quantifying the flexibility value afforded due to the presence of real options. When applied appropriately, CCA techniques can provide a powerful and robust valuation approach and are particularly useful in providing insight to key strategic factors that affect project value. These advantages, however, come at some expense as most applications of CCA to project valuation result in complex partial differential equations which cannot be solved for simple analytic formulas. This, combined with the intricate mathematical structure of these methods often make it difficult for an intuitive grasping and may result in implementation problems. The purpose of this article is to provide a prefatory perspective on the use of CCA techniques as applied to engineering, production, mining, and manufacturing projects. To that end, by using efficient numerical techniques this article formulates a simple and unified CCA framework for valuing a large class of projects that contain real options. The approach is straightforward, readily implementable, and computationally efficient. The framework presented in this paper also provides an important introduction to the use of CCA methods and the quantification of flexibility value in the management of operations.

I. INTRODUCTION

C

ONVENTIONAL discounted cash flow (DCF) techniques are used by a majority of firms in performing valuations for the purposes of capital budgeting. These methods allow investment projects to be evaluated according to the traditional hurdles of positive NPV (or IRR 2 cost of capital), and provide an alternative valuation approach to the payback and accounting rate of return methods. Despite their manifest popularity, these techniques have been criticized primarily due to their structural inadequacy to properly account for the Real and Strategic options associated with a project. That these options provide operating flexibility and have value has, of course, long been recognized [26], [14], [20]. Widely acknowledged too, is that omission of this component of value from the analysis may induce a systematic undervaluation of some projects, resulting in rejection of opportunities which should not have otherwise been rejected, at least because of their strategic value in creating other future opportunities. These omissions, however, reflect the deficiency of the particular evaluation technique adopted and not the quantitative approach itself. For instance, to employ DCF methods it is often assumed that the future cash flows from a project follow an expected pattern which Manuscript received February 23, 1994; revised September 1994. Reviews of this manuscript were arranged by Editor J. Evans. The author is with the School of Business, Georgetown University, Washington, DC 20057 USA. E E E Log Number 9410772.

can be treated as given at the outset. This tacit convention presupposes a static approach to the evaluation problem. It ignores the fact that managers respond to market conditions by adjusting operating strategies which impact future cash flows. The flexibility afforded by having the (real) option to adapt operating decisions also affects the risk structure within a project. Consequently, estimation of future cash flows and risk-adjusted discount rates as typically obtained from an asset pricing model, (e.g., CAPM), become even more complicated.' Moreover, the irreversible nature of many investment projects together with the compound feature of future and follow up investment opportunities also make it difficult to obtain reliable valuations. As conventional methods, typically treat projects on a stand-alone basis. In most projects real options arise due to uncertainty in costs, benefits, and the opportunity to favorably alter the project's course contingent on future information. The operating flexibility furnished by these options can, in some circumstances, result in significant risk truncation. Examples of these options typically include: Differing initial investment outlays (option to wait), altering input rates, changing production rates (volume flexibility), expanding or contracting capacity (capacity flexibility), varying the product mix (product flexibility), switching technologies (dedicated versus flexible systems and option to switch), temporarily shutting down, and abandonment (for or without salvage value). On the other hand, the strategic options associated with a project result from its interdependencies with future and follow up investments. Strategic consideration of these interdependencies may justify acceptance of negative NPV projects on the basis of their potential in creating subsequent opportunities. Therefore, in some situations, by opting not to commit to a current project, other future investment opportunities may be blocked-ff or at least delayed+learly, the value of retaining options to other future projects is deeply embedded into the value of these strategic options. As we have already remarked, omission of such options from the analysis may result in underestimation of project value. Yet, this is not to imply that conventional methods systematically undervalue all classes of projects, simply because not all projects contain real or strategic options. In fact, for applications in which project cash flows can be estimated with reasonable accuracy and where the scope of future managerial actions is limited, conventional DCF methods provide an extremely useful valuation approach. It is when uncertainty and strategic reactions to market conditions are of considerable 'Cash flow estimates, risk analysis and strategy considerations have been cited as the most common concerns in the analysis of investment decisions and in implementation of capital budgeting methods [2].

0018-9391/95$04.00 0 1995 IEEE

KAMRAD: A LATI’ICE CLAIMS MODEL FOR CAPITAL BUDGETING

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consequence that the application of these methods may result a shutdown option, and in valuing the option to defer the in suboptimal decisions. initial investment, respectively. In the later case, they show Though, arguably decision tree analysis (DTA) and sim- that the value resulting from such an alternative can be ulation methods can be used to account for the dynamics highly significant for irreversible projects. That is, the class of strategic reactions to the market, problems remain since of projects for which the initial investment expenditures are in using DTA future cash flows must be forecasted and essentially sunk costs because they are mostly nonrecoverable. scenario dependent discount rates would have to be identi- For these projects, the value of the option to defer initial fied. Decision tree methods are in principle the appropriate investment (a reversible decision) may be highly significant valuation mechanism by considering both the hierarchical and since the arrival of new information may influence the timing, sequential nature of investment decisions under uncertainty. desirability or perhaps the level of expenditures: Of course, An obstacle to their use, however, concerns the problem of the same line of reasoning can also be extended to the analysis determining the appropriate discount rate in working back of production and supply contracts frequently encountered in through the tree. Simulation methods, on the other hand, the manufacturing, electronic, chemical, or mining industries. can also handle complex problems under uncertainty. Their That is, the decision to enter a contract or defer commitment. distinct advantage lies in their ability to account for a large The flexibility (value) derived from delaying construction number of input variables and that the approach can be activities in projects that produce no cash flows until comimplemented with relative ease.* A potential drawback in the pleted (e.g., aircraft production) is considered by Majd and use of simulation methods, however, concerns interpretation Pindyck [19], while Paddock, Siegel, and Smith [29], value of the output results. Specifically, the actual translation of option-like features in developing offshore petroleum leases. simulated output results (i.e., the probability distribution of the Morck, Schwartz, and Stangeland [24], consider a production NPV) into meaningful managerial strategies may be difficult valuation model (for timber) when price and inventories are since the resulting (NPV) risk profiles may not easily be stochastic. Their example is a general solution to the classical deciphered into clearcut managerial decisions [ 181, [27], [6]. “duration” problem of the optimal control of a long term Perhaps a more suitable approach would be to simulate the renewable resource. Andreou [ 11, provides a model for valuing project’s cash flows from which the appropriate discount flexible plant capacity when market demand conditions are rate can be derived to determine a single-valued expected uncertain. Trigeorgis and Mason [33], and Kensinger [l5], NPV, suggesting a more lucid decision regarding the p r ~ j e c t . ~also value flexibility from an options perspective. Kulatilaka Simulation techniques should be considered as a powerful [16], develops a model for valuing flexibility in flexible support mechanism, rather than the approach to the analysis manufacturing systems (FMS), while Kulatilaka and Marks of investment decisions. [17], consider the strategic value of flexibility in a world with A suggested valuation methodology to overcome these limi- incomplete contracting. Issues pertaining to irreversibility and tations has emerged from the theory of financial option pricing. the investment option are addressed extensively by Pindyck These techniques, commonly referred to as contingent claims [31], [32]. Trigeorgis [34], considers the impact of interaction analysis (CCA), extend current capital budgeting methods among real options within a project. by explicitly accounting for uncertainty and the operating To employ CCA methods directly, it is often assumed that flexibility afforded due to the presence of real and strategic risks are well defined by allowing project cash flows to be options within a project. functionally dependent on the uncertain prices of a traded For certain classes of projects (e.g., investments in produc- ~ecurity.~ Accordingly, most applications of CCA to valuation tion facilities, refineries, chemical plants, production contracts, of real investments concern the producers of natural resource supply agreements and mining ventures), the application of based commodities for which either trading markets (e.g., CCA methods for the purposes evaluation is attractive since futures markets) exist or, at least, the data for quantifying their in addition to quantifying the flexibility value the methodology risk is obtainable. Moreover, since CCA methods preclude also provides optimal guidelines for managing the project. prediction of prices into the future, customarily prices are The advantages of a CCA approach to the analysis of in- assumed to follow well defined (continuous time) stochastic vestment decisions and capital budgeting problems have been processes. well documented in recent literature. For instance, Pindyck Within this setting, frequently techniques of continuous time [301 considers the optimal exploitation of an exhaustible arbitrage are employed to obtain project value. In particular, if resource under uncertainty; Myers and Majd [25] examine a trading strategy can be constructed in the financial markets the value of the abandonment (for salvage) option. Production such that all the cash inflows and outflows from the resulting flexibility in mining ventures with multiple options to open, portfolio continuously duplicate the inflows and outflows close, and to subsequently abandon the project has been associated with the project, then the present value of the project considered by Brennan and Schwartz [8]. McDonald and should equal the current value of the replicating portfolio. Siegel [21], [221, in valuing investments when the firm retains 2Existence of powerful simulation packages and the state of art in computing clearly facilitate ease of implementation. 3Brealy and Myers [ 6 ] ,suggest that some of the difficulties associated with simulation studies may be. avoided by presenting the distribution of intemal rates of retum instead.

4Pindyck [32], provides an in-depth review of the economic implications of irreversibility and the option-like characteristics inherent in irreversible projects. 5Constantinides [9], McDonald and Siegel [22], and Kamrad and Ritchken [13], also provide a CCA valuation scheme wher the underlying asset may not be a traded commodity.

142


This arbitrage-free approach to valuation is advantageous since by eliminating risks through a replicating strategy the methodology does not require as input risk-adjusted discount rates or preference functions. In the absence of arbitrage opportunities, the risk free rate is an “implied” discount rate for the project.6 A further advantage is that the approach also eliminates the necessity for estimating the expected rate of change in the underlying price process. When a replicating portfolio cannot be constructed, the market equilibrium approach becomes necessary usually at the expense of increased complexity in the valuation process; it is often difficult to estimate the expected rate of change in the underlying price variable (and therefore of the cash flow process) since they are typically nonstationary. Yet, regardless of the approach taken the value obtained from a CCA model has the advantage of being consistent with an equilibrium price structure. Therefore, CCA methods are particularly useful in providing insight to key strategic factors that affect project value, since any unusual features or irregularities (in value) encountered during simulated runs are likely due to specific options or strategies considered and not because of inconsistencies in the structure of the model. A general limitation in the use of these elegant methods, however, is the scope of their applicability. The approach cannot be adopted to value all classes of projects. Their best applications arise in situations where the major sources of uncertainty associated with future cash flows can be traced back to one or two primary sources. Furthermore, project owners are assumed to be sufficiently small so that their actions cannot influence prices. Perhaps a more pressing concern in the use of this powerful approach results from the mathematical complexity inherent in the structure of these models which often make it difficult for an intuitive grasping. Specifically, in most applications of CCA, the resulting project value is driven by a second degree partial differential equation (PDE), subjected to various constraints and boundary conditions. Frequently, however, these PDE’s do not yield simple analytic solutions and numerical techniques must be invoked to obtain results. One class of numerical solution procedures commonly referred to as a lattice accounts for a discrete time approximation to the stochastic evolution of the underlying source of variability (i.e., the price process).’ While the importance of lattice methods for valuing financial options and securities has been well established [lo], [5], [12] their application for performing valuations and analysis of real options remains under utilized. In that regard, a contribution of this paper is the development of a simple and unified lattice framework for evaluating projects that use a CCA approach. Toward this objective, we will demonstrate that in the case of real options, analysis and valuations may be performed simply by superimposing the various option-like (control) features of the problem on the approximating lattice. A recursive procedure can then be developed to efficiently yield results. This 6The arbitrage-free approach was initially conceptualized in valuing stock options [3], [23]. Another common approach, for instance, is to approximate the resulting PDE with a system of difference equations.

approach is appealing, at least intuitively, since managerial reactions to the market are often exercised at discrete epochs over time. In addition, the lattice portrays decision tree like characteristics that are familiar to most analysts. However, unlike decision tree (DT) methods, the resulting values do not depend on subjective inputs of probabilities or preferences. Moreover, the approach can be used to effectively quantify the flexibility value inherent in most risky projects. This article formulates a simple and unified lattice framework for valuing a fairly large class of risky projects that use a contingent claims approach. The framework used is justified by assuming the existence of futures markets, thus allowing the valuation process to be consistent with the pricing of securities having identical risks and rewards in the economy. The general type of models presented here provide an important introduction to the use of CCA methods and the quantification of flexibility value frequently encountered in the management of operations. In addition, the conceptual framework illustrated in this paper can be easily applied to a large variety of continuous time stochastic processes for which lattice approximations exist. This paper is organized as follows. Section I1 describes the approximating lattice where the assumptions and notation are also defined. Model development and valuation applications are in Section III. In particular, in this section we develop and consider lattice based models with specific applications to manufacturing and mining valuation problems, respectively. For each application, we take into account the fact that managerial actions influence cash flows by focusing on a stochastic dynamic program defined on a lattice representing the price of the underlying stochastic variable. For each application category, stylized numerical examples provide insight to the use of the models considered. Closing remarks are provided in Section IV. 11. W E APPROXIMATING LATTICE In the economics and finance literature, it is common practice to assume that the price process evolves stochastically over time according to an Ito differential equation. That is, the sample path of the price random variable, X ( t ) has a rate of change d X ( t ) described by

d X ( t ) = lim[X(t + s) - X ( t ) ] s-0

s >0

which can be represented by an It0 process

d X ( t ) = a ( X ,t ) d t

+ c ( X ,t ) dW(t)

where X ( 0 ) = 20. In (l), a(.) and c(.)represent the instantaneous drift (the expected rate of change in X at time t) and volatility (the standard deviation of the change in X) terms respectively, while dW(t) defines an increment in the standard Wiener process W ( t ) . When a ( X ,t) = a X ( t ) and a(X,t) = a X ( t ) the above Ito process is referred to as a Geometric Brownian Motion. In our analysis we assume that the change in price can be described by a Geometric Brownian Motion. This implies that the conditional distribution of X ( t ) given X(t - s), s > 0,is lognormal.

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KAMRAD A LAmICE CLAIMS MODEL FOR CAPITAL BUDGETING

Therefore, the price variable, X ( t ) remains nonnegative with probability one which is an appealing assumption for prices to follow. This assumption also proves useful, in the absence of a forecast, for modelling positive variables that on an average basis increase at a constant exponential rate. Traded commodity and security prices are an excellent example of this property. Efficient discrete time lattice models can be constructed to approximate the evolution of a Geometric Brownian Motion. Here, we consider a trinomial lattice approximation where the convergence of the approximating process into that of the continuous counterpart is ensured by suitably controlling the parameters of the trinomial lattice.* To this end, let T define the time interval corresponding to the project's maturity, and n as an equi-distant divider of [0, 71 such that h = 7/72. Suppose that in each subinterval of duration h, the current price, X ( t ) can either increase in value by an amount u (size of an up jump) with probability p u or maintain its current value (a horizontal jump) with probability p ~or ,decrease in value by an amount d (size of a down jump) with probability p ~ where , p u p~ p~ = 1.0. That is,

+ +

X(t).u X(t) X(t).d

b

X ( t >b b

w.p. W.P. W.P.

pu PH PO.

Such a trinomial lattice has been considered by Kamrad and Ritchken [12], for valuing financial options. Here, we adopt their trinomial lattice specification mainly due to its efficiency, accuracy and ease of implementation. To ensure convergence in their model, the magnitude of an up jump, u = exp(Xa&), with ud = 1 for computational con~enience.~ The stretch parameter, A 2 1 ensures the occurrence of a horizontal jump event with positive probability (i.e., p~ > 0), where for valuation purposes the jump probability terms are 1 pu=-+ 2X2

(cw-a2/2)JiE 2Aa

Fig. 1

sense, the expected return rate cw is often very difficult to estimate. As we have already remarked, this component is typically nonstationary and therefore, preferences and aversions toward risk enter the valuation. In situations where the underlying source of variability reflects the price of a traded security, and when it is possible to construct an arbitrage free replicating strategy, the drift parameter a in (2) can be replaced by the riskless rate of return T , which simplifies the valuation problem considerably. Since we use a discrete time approximation process, let X ; j represent the time t; price given that j jumps (up, horizontal, or down)havebeenmade, withj E ( - i , - i + l , ~ ~ ~ , ~ - 5l , z ) , ~ n. Here, j represents a given state of the process as generated 1, . . , i - 1,i} by the approximating lattice, while { 4,-i defines the set of potential realizations for j at time ti. Given the initial price X o , and the constraint u d = 1, we can then write

+

(24

xij = X&)j

1

pH=l-L

X2

(2b)

1 (LY-a2/2)& Po=-(2c) 2x2 2Xa In addition, by choosing X = 1, which corresponds to p~ = 0, the possibility of a horizontal jump is removed and the above probability expressions collapse to the binomial model of Cox, Ross, and Rubinstein Note too, that in the above expressions (2), the probabilities p u and p~ depend on the constant drift and volatility parameters, LY and a, respectively. While the existence of historical data provides an opportunity for estimation of a in a statistical *It is also possible to approximate the process using a binomial model. For a binomial lattice approximation, see [lo], [ll]. 'For specific details of this trinomial lattice model, see [12]. lo An issue that may be raised concerns the order of the multinomial lattice used to approximate prices. The computational advantage of models that include a horizontal jump is well documented in the literature [4], [12]. In addition to its intuitive appeal, in that over a small interval of time prices may stay the same, the trinomial model considered here also contains the binomial model as a special case, and more fully exploits the market information. Regardless of the lattice model selected, the forthcoming analysis remains the same.

j E (4,. . . , i), i = 1 , 2 , . . . ,n

(3)

which defines the price at every node of the lattice. Having defined the state space with corresponding transition probabilities completes our trinomial lattice specification. Fig. 1, displays a three period (n = 3) trinomial lattice process, where at any time t;,( 2 i 1 ) potential price realizations (or nodes) are evident, i = 0, 1, 2, . . ,n." In the following sections we consider specific examples in which the above approximating lattice procedure has been used in providing solutions.

+

111. MODELDEVELOPMENT AND APPLICATIONS In this section we consider two examples by using the approximating procedure described previously. Our first example illustrates the problem of valuing a production agreement in which the risk of fluctuating raw material prices is fully endured by the producer. For this illustration, we assume a single raw material of significant importance whose price evolves according to a Geometric Brownian Motion. The model we develop incorporates capacity constraints both in "In our analysis, the interval [ t c - l ,t , ] is chosen to be equal to h = = 1 , 2 , . . . ,n.

7/12, i

144

inventory holding cost for Mi units, and a calibration cost of S(N;-l -+Ni).S(Ni-1--+ N ; ) may be thought of as the dollar charge for changing the level of production from one period to the other. This involves the usual set up, calibration and material handling costs. In some situations, S ( . ) may be presented in a functional form. The cost for producing N; units over the production interval [ti,ti+l] is to be incurred at time t;+l,i = 0,1,2,.-.,n - 1. The valuation model to be presented presupposes the existence of a futures market in the input commodity. Accordingly, in our analysis the local trend in the price, a, can be replaced by T , the riskless rate of return. This simplifies computations significantly both in terms of obtaining the lattice probability terms (see (2)) and that of having an appropriate discount rate, which is T . To develop a recursive lattice valuation procedure, let Fij(MilNi-1) be the maximum total expected value of the venture at time ti given the price is X i j . that Ni-lunits were produced in period [ti-l,ti], thus resulting in an inventory A. A Manufacturing Production Model level of Mi units at time ti with an optimal policy to be In this application, we consider the problem of valuing supN i ) represent the total cost followed thereafter. Let Fij(Ni-1, ply agreements commonly encountered in the manufacturing from node ( i , j ) when the current production quantity Ni-1, industries. These contracts often require the manufacturer to is the resulting inventory level at time ti is Mi, while Ni units supply fixed quantities of a product, component or subcomare scheduled for production in the (next) period [ti,ti+l]. The ponent according to a preestablished delivery time and unit time ti inventory and production policy (Mi, N i ) , generates price. Here, a production agreement reflecting a supply of A units of a product at time T = t,, and at a negotiated price a net profit of magnitude, of P dollars per unit is considered. The production quantity over a fixed production period of duration h = [ti,ti+l] is PHE+l,j(Mi+llNi) + PDFi+l,j-l(Mi+lINi)}} Ni 5 Q , i = O , l , . . . , n - 1. Finished products are then accumulated in inventory and shipped to the customer at time (6) 7 = t,. Let Mi 5 I, denote the on hand level of inventory. Q and I represent the maximum production and inventory over the time interval 7 - ti. In the above expression, the first term, Fij(Ni-1, Ni) = capacities, respectively. Define C(N i ) as the cost of producing Ni units and let b represent the unit holding cost of the item.” [ C ( N i - l ) bM; aNiX;j S(N;-1+ N;)] represents the Each unit of the finished product is assumed to require “a” cost of operations from node (i, j ) , while the latter term is the units of the input raw material whose price is determined in total expected value at time ti+l when an optimal policy is the competitive markets. At time ti,i # n, the firm opts to followed with f ( N ; ) as the set of feasible selections for Ni. produce Ni units, when the level of inventory on hand is That is (see (7) at bottom of page) represents the cash out flow at time ti together with the value in the next period. For given by Mi and where ease of exposition, let F’+l(Mi+lINi) represent the second i = 1,2,.. .n - 1 (4) term in (6), or the expected total value at time ti+l. In this 0 5 Mi = Mi-1 Ni-1 5 I light, (6) and (7) can be used to obtain a recursive valuation and where procedure for every node ( z , j ) with j E { - 2 , . . . , 0, . . , i} Nn-l Mn-l= A. ( 5 ) and i = 0, 1, . . . , n- 1. This valuation equation is given by

the production quantity as well as the finished good level of inventory. Specifically, our CCA model provides a value maximizing production and inventory policy that satisfies a known demand schedule for a product to be delivered in the future. In the second example, we redirect our analysis to account for production problems in mining ventures. Here, in contrast to the first model, we assume uncertainty in the output price. As before, a Geometric Brownian Motion describes the evolution of prices over time. The model developed accounts for an optimal extraction policy based on an initial estimate of a finite amount of a nonrenewable resource. For this example, production capacity is also accounted for where the extracted resource is assumed to be sold immediately in the market place. These illustrations provide an easily understandable introduction to the lattice CCA techniques which are a powerful valuation methodology in their own right.

+

+

+

+

Suppose no initial inventory is held SO that MO = 0, and that the raw material is obtained under a JIT purchasing and delivery system. The cost incurred at time ti includes the raw material purchase cost for production of Ni units, an 12Usually C ( N , ) is an increasing concave function over the production interval. In our analysis no specific requirements have been placed on the functional form of C ( N , ) .

s.t.

+

+

145

KAMRAD: A LAmICE CLAIMS MODEL FOR CAPITAL BUDGETING

OINisQ.

OPrUlAL PRODUCTION AND

once the cash receipts (boundary conditions) are specified. Hence, the maximum total value at time T = t, for every node (n,j ) with j E { -n, . . . ,n } accounts for profits obtained from delivery of A units of the product. Therefore, F n j ( M n J N n - 1 ) = AP -

+

{C(Nn-l) S(Nn-1 + 0 ) ) (9)

s.t.

Mn-l

+ N,-1

TABLE I

(8c)

=A

M, = O .

(9a) (9b)

Here, S(N,-l + 0) may be considered as a shutdown and delivery cost. Equations (8) and (9) provide a backward dynamic programming recursion for valuing an optimally managed project. Note too, when the risk of fluctuating input prices is completely hedgeable, and when all other costs are deterministic the riskless rate serves as the appropriate discount rate. In most situations, however, one must also account for the “benefit” resulting from having acquired the taw resource. This benefit may be thought of as a premium yield resulting from the ability to profit from temporary local shortages in the raw material resource. Practically, all commodities can be assumed to provide this benefit which is commonly referred to as the “convenience yield.” If the convenience yield is assumed to be a constant proportion of the spot price, then the current drift rate T has to be adjusted to (T - c) to account for the convenience yield, c.13 Thus in our analysis the appropriate drift adjustment is T’ = T - c, which replaces a in (2) if in fact the convenience yield were to be accounted for. Example I : We illustrate the above model by considering a manufactured item that uses copper as raw material. The price of copper being the primary source of uncertainty. Industrial applications of copper are vast and vary anywhere from production of electrical or mechanical components, to missile guidance wiring, to conductors, pipes, heating radiators, etc. In this stylized example, we assume a contractually known demand of A = 20 units of a product to be delivered in one year, hence r = 1 year. Taking n = 6 as an equi-width partition of T specifies the duration of the production period h, although we choose n = 6 for ease of exposition only.14 The current price for copper X O = $30.00 per unit. Each finished item is sold at a contractually fixed price of P = $350.00. The production cost function is C ( N ) = 10 + 2.5N2. Finished products are accumulated in inventory where a holding charge of b = $2.00 per unit applies. The inventory capacity is, I = 15 units, while the production capacity per period is Q = 7 units. The periodic calibration (or the production switching) cost, S(Ni-1 -+ N i ) = 1.5(Ni Ni-1)2, also applies. I3Brennan [7] and Brennan and Schwartz [8] provide detailed insight to the notion of convenience yield. I4The partition size can be refined by choosing larger values for n, which results in a more accurate estimation of prices. This can be done easily without altering the decision times which correspond to the beginning of each production period.

a

INVENTORYSCHEDULE

F, = $3089.59 M,* N:

0

0

1

4 5 6

10 15 20

0

F, =$3165.12 N,? 2

The price volatility, 0 = 35% per annum. The annual riskless rate of return and the constant convenience yield are T = 10% and c = 5%, respectively. We also assume that each unit of the finished item requires, a = 5 units of the raw material. For the lattice valuation procedure defined earlier, the stretch parameter is set to A = 1.22474 resulting in approximately equal probability values (pu = .33, p~ = .33, and p o = .34.) Note that in obtaining these lattice probability values (2) were applied with ( T - c) = .05 replacing a. This is a direct consequence of risk diversification by creating a riskless hedge via the futures market in copper, implying that for valuation purposes the price process can be taken as

d X ( t ) = ( T - c ) X ( t )dt

+ o X ( t )d W ( t ) .

Using (8) and (9), the time to value of the contract F,, together with the optimal production and inventory policy ( M r ,N r ) is shown in Table I. The last column entry in Table I accounts for a situation in which the producer has an option to satisfy demand gradually. That is, finished items are shipped off immediately at the end of a production period so that finished good inventory costs do not accrue. The shipments are assumed costless as they are typically accounted for in the contract’s unit sales price.15 Note that given our case parameters, this option provides a 2.5% increase in the value of the contract, and that the optimal production quantity also changes. The value of the contract as a function of current spot prices is show in Fig. 2, where depending on the level of initial prices the resulting contract value and the optimal policies also change.

B. A Mining Production Model In this section we consider a mining venture valuation model where the extracted resource is assumed to be a traded commodity such as gold, silver, platinum, or copper. Presupposing a finite and known amount of the nonrenewable resource, the problem is to establish an optimal extraction schedule corresponding to the stochastic selling price of the commodity in a fashion so as to maximize the current value of the mine. We assume that the production and extraction costs are deterministic and that production capacity is limited to a maximum of Q units per period of Operation. In light of the previous section’s notation, let Ni define the quantity to be ’’When shipping costs are not explicitly accounted for, the benefits of such an option should be weighed against the rising shipping costs as a result of increased shipping frequency.


146

raw resource were excavated, thereby leaving Mi units of the resource available for extraction, with an optimal policy for follow-up thereafter. Consider node ( i , j ) as shown below, and let Fij(Ni-1, Ni) represent the total expected value of the policy at time ti when the price is Xij, where Ni-1 units of the raw material were extracted during the pervious period while N; units are to be produced in the (next) period, [ti,ti+l].Hence, the value of the mine together with the cash flow received at time ti+l can be depicted by (see (10) at bottom of page). Based on (lo), we can now develop a recursion valuation equation resulting in a policy that maximizes the value of the mine. That is, the maximum total expected value at node ( i , j ) with i = 0, 1, . . . , n- 1 and j E {-i,. . . , i} is obtained recursively by

4 6Ooo

--

8 Moo

--

3 0

-a

s

ZOO0

Fij (Mi1Ni-l)

--

= max {eprh{

F;j (Ni-1, N ; }

f(N%)

- S(Ni-1 + N i ) } lo00

s.t.

--

05NiIQ

O