# The opportunity cost of using excess capacity.

A common problem in capital budgeting arises when a new project
proposal calls for the use of existing, but currently idle, equipment or
facilities. The issue is whether there is any opportunity cost to
diverting these facilities to the new project. For example, the firm
will incur an opportunity cost if diversion of capacity to the new
project precludes its future use by other projects or causes production
facilities to be replaced sooner than they otherwise would have been.(1)
Although the new project may be worthwhile on a stand-alone basis, the
firm should recognize that its adoption may impose externalities on
existing projects. Any such effects should be measured and included in
the calculation of the new project's net present value.

Methods for estimating this opportunity cost are available, but they focus exclusively on the firm's furture investment and make very strong assumptions about the predictability of that investment. In doing so, they fail to recognize the option elements of the problem, and in some circumstances this may result in seriously misleading estimates.(2) Our aim in this paper is to analyze these option elements explicity. We show that when uncertainty disappears, the option elements become unimportant, and the true opportunity cost collapses to the solution given by existing methods. However, as uncertainty grows, these option elements come to dominate the problem, and existing methods can substantially understimate or overestimate the true opportunity cost.

The essence of our option framework is as follows: Capacity in place gives the firm an option to produce. Thus, if a firm diverts capacity from Product A to Product B, it forgoes the option to produce A immediately, but it acquires an option to replace the diverted capacity. The true opportunity cost of using the excess capacity is the change in the value of the firm's options. By focusing on the state-contingent nature of optimal future decisions, this framework recognizes that the opportunity cost is not necessarily equal to the present value of specific investment or production decisions.

Section I describes the problem and presents the real options analysis in general terms. Existing approaches to measuring opportunity cost are then contrasted with the options approach. Section II presents a specific numerical example in which the cost of diverting excess capacity is calculated for different values of such variables as investment cost, the interest rate, product price uncertainty and unit production cost. We show that the ture opportunity cost can vary widely under these conditions in ways that are not captured by existing approaches. Section III briefly summarizes our results and offers conclusions.

I. The Opportunity Cost of Diverting Capacity

A. The Problem

Consider a firm with a unit of capacity, such as a plant, which is currently devoted to production of Product A. Each year, the firm observes the prevailing market price for A, compares that with its unit operating cost, and produces A if it is profitable to do so. The existing plant has a remaining life of N years. By investing E, the firm can build a new plant with an economic life of L years, which will become operational immediately.(3)

Suppose that the current price of A is such that the firm has just decided not to produce this year and thus, to let the plant stand as excess capacity until at least next year. Suppose further that a new opportunity suddenly arises: by investing in some renovations, the firm could refit the plant to produce an alternative product, B. For simplicity, we assume that once the renovation is complete, it is prohibitively expensive to switch back to production of A and that the salvage value of the plant is zero throughout its useful life.(4)

On a stand-alone basis, the NPV of producing B is simply the present value of profits from B for the N years remaining in the life of the plant, minus the cost of renovating the plant, plus any change in value of the options associated with this first unit of B-capacity.(5) However, this stand-alone NPV is reduced by the opportunity cost of diverting the plant away from Product A.

B. The Real Options Framework

Following Pindyck [11], we can view plants as being installed sequentially in order of decreasing value to the firm.[6] This allows us to focus on the plant that currently produces A. Since we are only concerned with measuring the opportunity cost of diverting the marginal unit of capacity, we can ignore the effect on firm value of any other units.(7)

When a firm installs a plant, it acquires two types of options: a strip of production options, one for each year of the plant's useful life, and a replacement option, which is equivalent to a future investment option. The production options are European; for example, the option to produce in Year 3 matures in that year when the firm learns the price of A. The exercise price of each production option is the unit operating cost times the level of production. The firm will exercise its production option in Year 3 if that year's unit price of A exceeds unit cost. In addition to its production options, a firm that has a plant in place has the option to replace that plant at the end of its useful life.(8) The replacement option, which is a future investment option, is American since it can be exercised either at that time or later.

If the firm does not have a plant in place in a given year, it has no production options, but it does have an immediate investment option. This option is American, since it can be exercised either now or later, and the exercise price is E, the cost of building a new plant. The investment option is a compound option, or an option on options. If the firm exercise its investment option, building a new plant, it acquires in turn the strip of production options that allow it to choose whether or not to produce in each year of the plant's life. By building a plant, the firm also acquires an American option to replace that plant, beginning L years after the plant is built.

In this framework, the opportunity cost of diverting excess capacity to a new product is the change in the value of the firm's options that is caused by diverting the capacity. If the firm does not divert capacity, it has the option to produce A any year between now and the end of the plant's useful life. If the plant will wear out in Year N, the firm also has the option to replace it in Year N (or later). On the other hand, if the firm does divert capacity, it acquires an option to invest immediately in a new plant to produce A. By diverting capacity now, the firm loses its options to produce A and its option to replace a unit of A-capacity in Year N. However, it gains the option to acquire the unit of A-capacity immediately.(9) Thus the opportunity cost, C, of diverting capacity from A to B can be represented as:

[Mathematical Expression Omitted]

where [P.sub.t] is the current value of the option to produce A in [Year.sub.t], and [I.sub.O] and [I.sub.N] are the current values of the options to invest in a unit of A-capacity effective immediately and effective in Year N, respectively.

C. Existing Approaches to Opportunity Cost Measurement

The most plausible existing method for calculating this opporunity cost calls for charging the new project with the equivalent annual cost (EAC) of a plant for each year that diversion of capacity is estimated to cause a capacity shortage for the original product.(10) With a plant life of L years, for example, the EAC is the annual payment on an L-year annuity whose present value is equal to the plant's initial investment outlay. If diverting capacity to B is expected to cause a shortage of A-capacity beginning in Year S, opportunity cost under this method is the present value of these EAC charges from Year S to Year N-1:(11)

[Mathematical Expression Omitted]

One could interpret EAC as the annual rental payment a lessor would need to charge in order to recoup the initial cost of the plant plus interest over the plant's useful life. If a rental market actually existed, a company that diverted its plant to product B could restore the lost capacity to produce A by renting a similar plant. It would then face the decision of whether or not to build a replacement plant in Year N, just as it would have if the capacity had never been diverted.

An alternative interpretation is that charging the EAC for (N - S) years is equivalent to moving up not only the first plant replacement from Year N to Year S, but all future replacement cycles as well.(12) Thus, the opportunity cost is the present value of moving up the plant investment from Year N to Year S, again from Year N + L to Year S + L, again from Year N + 2L to Year S + 2L, etc.

This latter interpretation highlights the fact that the EAC method relies on stringent assumptions about the timing of future investment expenditures. By assuming that all future investment will take place with certainty at particular dates, it ignores the option component of the investment decision. In Section II below, we show that these option components become unimportant only when uncertainty disappers and the timing of future investments can be accurately predicted today. Even if the EAC method is implemented with a risk-adjusted discount rate, it cannot capture this option component, since risk-adjusted discounting at a constant rate is ill-equipped to handle situations in which decisions will be postponed until more uncertainty is resolved.(13) In contrast, an option framework recognizes that the investment options need not be exercised at the first possible moment.

II. Valuation of the Firm's Production and Investment Options

In this section, we construct examples in which the firm's production and investment options can be valued explicity. The economic environment assumed in these examples is intentionally simplified and may not capture all features of an actual decision-making situation. Nevertheless, the examples do serve to illustrate the basic determinants of the opportunity cost of using excess capacity. In Section II.A. below, we illustrate our solution technique with two simplified, are discussed in Section II.B.

A. Examples With Only One Investment Cycle

1. The Economic Horizon Ends When the Existing Plant Wears Out

If the economic horizon ends in Year N, the replacement option for the existing plant, [I.sub.N], is worthless. In that case, the opportunity cost of diverting capacity can be illustrated with a standard option diagram, as shown in Exhibit 1. V represents the value of the options to produce A between now and Year N. If the firm diverts capacity to B, it obtains an option, I, to invest in A-capacity at a cost E. Since the economic horizon is too short to allow any replacement decisions, I is simply a call option on V with exercise price E, and, from Equation (1), the opportunity cost of diverting capacity is equal to V - I.

Note:

V = Value of production options from now to time N. E = Cost of building a new plant. I = Value of option to invest in a new plant immediately (I is a call option on V with exercise price E). C = Opportunity cost of using excess capacity = V - I.

Given V = V*, distance RS = V* - E; distance PR = E; distance PQ=C= V - I.

The diagram illustrates two different upper bounds on the opportunity cost. First, the value of I is at least V - E, so the opportunity cost is at most E. This is also true in more general cases to which the standard diagram does not apply, because no matter how valuable the forgone production options are, the firm could always replace them immediately by investing the amount E. After that, it could make all future decisions as it would have done anyway. Exhibit 1 also indicates that V is an upper bound on the opportunity cost, since the value of the investment option is always nonnegative. Even in more general settings, the opportunity cost can never exceed the value of the lost production options, since the firm can always postpone any investment until it would have occurred anyway.

The option diagram also illustrates the circumstances in which either of the two upper limits is likely to be binding. As V grows large, the firm becomes more likely to invest immediately to restore its lost production options, and the opportunity cost approaches E. For this special case in which the economic horizon ends in Year N, this is analogous to the EAC solution. That is because the plant would never need to be replaced if capacity were not diverted, so the opportunity cost is the value of any additional investment expenditures the firm makes if it does divert capacity. When V is small, on the other hand, the firm is unlikely to invest, preferring instead to simply forgo the production options.

2. The Economic Horizon Coincides With the Useful Life of a New Plant

In our second simplified example, we assume that a new plant has a useful life of six years, and that the economic horizon also ends six years from now. Since a new plant becomes operational immediately (i.e., in Year 0) the product price revealed in Year 5 is the last one before the horizon date. The plant's capacity is 1 million units of A with a unit operating cost of $10.5(14) The current price of Product A is $10 and in each future year this price will move up (down) by a multiple of 1.10 (1/1.10). This gives rise to the "tree" of possible future prices illustrated in Exhibit 2. We assume that the "risk-neutral probability" of either a price increase or decrease in a given year is equal to 0.5.(15)

The current plant will wear out after four years (N=4), and a new plant with a useful life of six years can be built then, or at any later date until the end of the horizon, for an investment expenditure of $3 million (E = $3 million). The riskless interest rate, r, is five percent per year. For this one-cycle example, all acitivity ceases at the end of six years, so any plants after the current year will cease production before their useful lives are finished.

If a firm has capacity in place, it will exercise its production option in any given year if the price of A exceeds $10.5, the unit operating cost. The current value of that year's production option is simply the present value of its "expected" cash flow, where state-contingent cash flows incorporate the optimal exercise decision and expected values are computed using the risk-neutral probabilities. Since the current price of A is $10, for example, Year O's production option is worthless, but the current value of Year 1's production option is given by:

[P.sub.1]=.5(11 - 10.5)(1 million) + .5(0)/1.05=$.238 million. (3)

Note that the production option will be exercised if the price moves up, but not if the price moves down. Similarly, the value of Year 2's production option is:

[P.sub.2]=[(.5).sub.2](12.1 - 10.5)(1 million) + 2(.5)(.5)(0) + [(.5).sub.2](0)/1.05 = $.363 million. (4)

Valuing the investment options is more complex, since two questions must be answered. The first is whether investing in a plant at any given time yields a positive NPV. Because of the short economic horizon, any plant built now or in the future, in this example, will not be replaced, so the first question can be answered by subtracting the $3 million cost of the plant from the value of the production options gained by building it. Valuing the production options for Years 0 to 5 with the risk-neutral probabilities as described above, the NPV of investing in a new plant immediately is $-0.788 million.(16) Thus, in this example, it will not be worthwhile for the firm to invest immediately.

Even if the NPV of investing immediately were positive, however, the firm would still have to consider the second question, raised by McDonald and Siegel [7], which is whether it is better to wait before investing. The value of waiting is equal to the "expected" present value of waiting until Year 1, observing the product price at that time and then following the optimal investment strategy contingent on that price. By waiting, the firm forgoes the production option in Year 0, but it also gains information about the future price of A. This information is valuable, because it enables the firm to avoid investing in capacity in situations where future production is unlikely to be profitable.

The optimal strategy at each price node, which consists of either investing (I) or waiting (W) further, is in turn found by working backward through the price tree from the economic horizon to that node, evaluating all future production options as of the time of that node, and then comparing the net present values of investing at that time versus the discounted expected value of waiting one more year. The optimal strategy for a firm that reaches a given node without a unit of capacity already in place for the next period is shown in parentheses below the relevant price in Exhibit 2. In the case at hand, the optimal strategy is to wait in Year 0, wait again in Year 1, and then invest in Year 2 only if the price is $12.1. In Year 1, the NPV of investing if the price is $11 is $1.112 million, but it is even more profitable ($1.694 million) to wait another year, because the firm can then avoid committing capital until there is a greater likelihood that the future path of prices will be favorable.

The opportunity cost of diverting the plant to production of B is given by (1) as:

[Mathematical Expression Omitted]

Using the valuation technique described above, the sum of the forgone production options is 1.066, the current value of the option to invest in Year 4 is 0.301, and the value of the option to invest immediately or later is worth 0.816, so the opportunity cost of diverting capacity is $.551 million.

Note that the opportunity cost measure given by Equation (5) is substantially below either of the two lower bounds described in the first part of Section II.A. It is less than E (which equals three in this case), because the firm would not choose to replace the diverted capacity immediately, and even when it does, it saves the replacement expenditure that might otherwise have been make in Year 4. Likewise, the true opportunity cost is less than the sum of the forgone production options (1.066 in this case), because, if the path of prices is sufficiently favorable, the firm will find it economical to invest in Year 2 and recover some of these lost options.

The EAC method is not applicable in this example, since it is predicated on an infinite economic horizon. However, the difference in present value of investment outlays, depending on whether capacity is diverted or not, would represent a single-replacement-cycle analogue to the EAC method. In the example at hand, the firm would never replace the diverted capacity immediately, since the nearest production option is out of the money. If the price of A rises, however, there is a state of nature in Year 1 in which the firm would want to produce A. Thus, we could calculate the opportunity cost of diverting capacity as the present value of moving up the $3 million investment from Year 4 to Year 1, or C = 3 [(1/1.05) - [(1/1.05).sup.4]] = $.389 million. This technique underestimates the true opportunity cost for this example, because it assumes that future replacement expenditures are certain, rather than statecontingent. In particular, it assumes that the firm would necessarily replace the existing plant in Year 4, whereas in fact this replacement is highly uncertain.

B. Results From the General Example

In general, the opportunity cost measure produced by the real options approach will vary with all of the standard determinants of option values: the exercise prices of the production and investment options, the interest rate, the degree of uncertainty in product prices, and the time periods over which the options can be exercised. This, in turn, can produce wide variations in opportunity cost both in absolute terms and relative to the measures provided by the EAC technique.

To get some idea of the sensitivity of the true opportunity cost to variations in its underlying determinants, we performed a series of simulations. For the most part, our base case assumes the same parameter values as in the example above: a unit operating cost of $10.5, and a plant production capacity of one million units of A per year; a time-to-replacement for the existing plant of two years; an investment cost of $3 million, and a useful plant life of six years. We did, however, assume an economic horizon of 50 years to get a better idea of the effect of a whole sequence of future replacement decisions. In an attempt to represent the effect of uncertainty less crudely than in the example above, we also assumed that product price follows a lognormal process with a mean of zero and a standard deviation of ten percent per year. To better approximate this distribution, we estimated all option values with a grid consisting of ten steps per year.(17) As a result of these changes, the results in the simulations are of the same order of magnitude, but not strictly comparable with the results in Section II.A.

There are some factors, such as the cost of investment and the interest rate, that affect opportunity cost as measured by either the real options or the EAC methods, but they do so in different ways. Moreover, since the EAC method focuses entirely on investment expenditures, there are several additional factors that have a significant impact on the valuation of options but do not enter into the calculation of the EAC (e.g., output price, standard deviation of output price and production cost).

1. Factors Affecting Both Option and EAC Estimates of Opportunity Cost

Exhibit 3 shows how opportunity cost varies with the cost of building a new plant. If the investment cost is zero, the diverted capacity is replaced costlessly and the opportunity cost of diverting capacity is zero for both the real options method and the EAC method. For both mehtods, the opportunity cost rises with the cost of a new plant, but it does so much more gradually under the options method. As investment cost increases, the firm becomes less and less likely to exercise its investment option before Year 4 and thus changes in investment cost become less significant. Since the firm can always simply choose not to replace the capacity to produce A, the value of the lost production options represents an upper bound on the opportunity cost. The EAC method, however, assumes that the firm always invests at the earliest opportunity, whether or not that is the best action for the firm and whatever the cost of building a plant. Thus, the EAC estimate increases at a constant rate with the cost of investment and quickly overstates the true opportunity cost. (18)

The effects of changes in the riskless rate of interest are shown in Exhibit 4. The opportunity cost calculated using the options method decreases slightly as the interest rate increases, but it is relatively insensitive to changes in the rate. The cost calculated using the EAC method increases with the riskless rate and always overestimates the true opportunity cost. Under the EAC method, increasing the interest rate increases the cost of moving forward the sequence of replacement investments. Under the real options method, by contrast, an increase in the interest rate also reduces the present value of any future production profits. Although it may be more expensive to move the sequence of investments forward, it may also be less compelling to do so.

Exhibit 5 shows the relationship between opportunity cost and time to replacement for two different plant lives, six years and ten years. In general, the longer the time to initial replacement, the higher is the opportunity cost, because the firm is giving up a longer strip of production options when it does divert capacity. For a given time to initial replacement, though, the opportunity cost is lower the longer the useful life is of a newly built plant. This is because the same initial investment buys more production options, so the firm is more likely to invest. Costs estimated with the EAC method also increase with the time to replacement since there are more periods to the annuity. Again, the method overstates the true value of the opportunity cost because it ignores the option components of the problem.

2. Factors That Affect Option But Not EAC Estimates of Opportunity Cost

Exhibit 6 shows what happens to the opportunity cost of using excess capacity when all other parameters are as in our base case, but the standard deviation of the price process (that is, the size of the yearly up and down moves) is allowed to vary. When the yearly variation in price is very small, the opportunity cost goes to zero. This is because the production options are out of the money at the start, and there insufficient price variation for them to be in the money any time in the next four years. The opportunity cost then rises as price variability rises, because this increases the value of both the production and investment options. However, [I.sub.0], the value of the option to invest immediately, is more sensitive to changes in standard deviation than [I.sub.4], the value of the option to invest beginning in Year 4. At high standard deviation levels, as Exhibit 6 illustrates, increases in [I.sub.0] can dominate the increases in [I.sub.4] and in the value of the production options, and the opportunity cost can turn down.

Because the initial production option is out of the money in our base-case example, the option and EAC measures of opportunity cost do not coincide, even when uncertainty disappears. If the initial production option were in the money, however, and the standard deviation of the price process were zero, a firm that diverted capacity would always want to invest immediately so as to capture all of the valuable options to produce A. It would thus move up the first replacement investment from Year 4 to Year 0, and it would move up all future replacements as well. If the economic horizon were infinite, then, the option and EAC methods should produce identical opportunity cost measures.

This is illustrated approximately in Exhibit 7, in which all other base-case parameters are kept the same but the unit operating cost is reduced to six, putting the initial production option into the money. At low standard deviations, the firm is virtually certain to invest in Year 0 if capacity is diverted, versus Year 4 if it is not, in Year 6 if capacity is diverted, versus Year 10 if it is not, etc. Thus, the option calculation nearly coincides with the EAC calculation. (19) As the standard deviation increases, however, the true opportunity cost rises above the EAC measure. This is because, while the firm might still invest in Year 0 when it diverts capacity, there is a greater likelihood that it would choose not to replace the capacity in Year 4 had it not diverted in the first place.

Exhibit 8 shows the variation in opportunity cost for different levels of unit production cost. At low production costs, such that the initial production option is deep in the money, the options calculation coincides with the EAC calculation, except for the small discrepancy caused by the finite horizon. As production cost increases initially from these levels, the true opportunity cost rises above the EAC. As in Exhibit 7, this is because the firm will still invest in Year 0 if it does divert capacity, but it becomes increasingly likely not to invest in Year 4 if it does not divert capacity now. For higher levels of production cost, the firm no longer invests immediately if it does divert capacity, and the opportunity cost turns down, sharply at first and then more gradually. The EAC measure, by contrast, makes a single, downward jump when production cost reaches 10. Beyond that level, production in Year 0 is no longer profitable, so the EAC is charged only in Years 1 to 3, rather than 0 to 3. However, the EAC measure does not reflect the fact that, as unit cost reaches high levels, neither investment nor production are likely to occur in the future, so that the true opportunity cost becomes negligible.

Finally, the effect of lengthening the economic horizon is shown in Exhibit 9. If the horizon is only one year, the opportunity cost is zero, because the first production option is worthless, and if the firm diverts capacity it loses nothing. As the horizon lengthens, the opportunity cost initially rises, but then begins to ratchet up and down. This is because the life of the last plant built may be prematurely terminated, and the degree of premature termination differs under the diversion strategy or the alternative strategy with the length of the horizon. However, as the economic horizon becomes very long, this ratchet effect is dampened, and the opportunity cost converges to a value of approximately $.526 million. While the ture opportunity cost varies somewhat for different horizons here, the EAC technique always produces an overestimate, because it does not account for the uncertainty inherent in the firm's future production and investment decisions.

It is clear from these examples that the true opportunity cost of diverting excess capacity to another use can vary widely, even in the context of a simplified economic environment. What general statements can be made, then, that might serve as a guide to managerial decision-making? First, there are a number of plausible circumstances in which there will be no opportunity cost at all. This occurs either when the production options the firm sacrifices by diverting capacity have little value or when they can be cheaply replaced. Second, as shown above, two upper bounds on the opportunity cost can be identified in any situation. The first is the cost of replacing the diverted capacity immediately, and the second is the value of the forgone production options. It is particularly important to recognize both of these upper bounds, since one of them may be considerably different from the other. In all of the examples above, the first upper bound the cost of immediately replacing the diverted capacity (plant cost is $3 million in our example), is a gross overestimate of the true opportunity cost. The decision-maker would be better advised in such cases to try to estimate the value of the lost production options.

III Summary and Conclusions

Existing methods for estimating the opportunity cost of diverting a unit of excess capacity to some other use focus on the cost of replacing that unit. However, these approaches ignore the option elements of the problem, and as a result they may produce estimates that are seriously distorted. By contrast, a real options perspective includes changes in the values of both production and investment options that are affected by diverting capacity. In actual practice, it may be difficult to estimate precisely the value of all of the production and investment options in Equation (1). Nevertheless, the analysis above can help to identify situations in which the true opportunity cost is likely to be high or low, and it can alert the decision-maker to conditions under which existing measurement techniques may give misleading results.

(1)An example occurs in the classic "Super Project" case in Butters, et al [3], in which a General Foods Corporation proposal for a new instant dessert product, "Super", entails the use of facilities that were previously installed for the existing product, Jell-O. Company managers in the case argue over whether the Super project should be charged an opportunity cost for using these Jell-O facilities.

(2)The option components of real assets have been the subject of a growing literature, which has recently been surveyed by Sick [12]. See also Kensinger [5], Trigeorgis and Mason [15], and Bjerksund and Ekern [1].

(3)A lag between investment and production could also be introduced without changing the essential nature of results. The absence of such a lag facilities comparison of our opportunity cost measure with that produced by the equivalent annual cost method, as described further in Section II.B.

(4)If production could be switched back to A, additional options would be created that would tend to reduce the opportunity cost of diverting capacity. Similarly, a positive salvage value for the plant would create an abandonment option, as in Myers and Majd [10]. However, including these additional option, as in Myers and Majd [10]. However, including these additional option elements would not add significant insights into the problem of opportunity cost, but would substantially complicate the analysis, since the various options may intertact. See Trigeorgis [13], [14] for further analysis of such interactions.

(5)In an options framework, for example, stand-alone NPV would include the value forgone by obtaining a unit of B-capacity immediately, since the value forgone by obtaining a unit of B-capacity immediately, since this kills the option to wait until a future year before obtaining that unit. see McDonald and Siegel [7].

(6)The total number of plants the firm can install profitably is thus limited, which rules our the possibility of infinite investment (perhaps because the firm operates under increasing long-run average costs).

(7)To see this, let the current plant be unit J. Since capacity is installed sequentially in order of decreasing value, investment in unit J is at least as attractive as investment in unit J + 1. Thus, the decision of whether or not to invest in unit J + 1 is unaffected by the decision to divert, because if the firm chooses to invest in unit J + 1, it will do so whether or not it has diverted capacity to product B. For example, if the firm has diverted capacity, and if it still wishes to invest in unit J + 1, it will also be optimal to invest in unit J. Thus, we can ignore unit J + 1 in our analysis.

(8)For simplicity, it is assumed that plants wear out all at once at the end of their useful lives and that replacement plants are identical to plants in place. Thus, an option to replace the plant before the end of its useful life would have no value.

(9)As explained in footnote 7, opportunity cost is calculated using only the investment and production options for the current plant, since that is the only capacity unit affected by the decision to divert.

(10)The EAC method is described in Brealey and Myers [2]. Two other common methods make rather implausible assumptions about useful plant life. One sets opportunity cost equal to the replacement cost of the facilities used, but this implicitly assumes that a new plant will last only as long as the existing plant (to Year N). A second method calculates the change in the present value of investment when plant replacement is moved up from Year N to Year S. This implicitly assumes that the new plant will last as long as the replacement of the existing plant would have (to Year N + L). Otherwise, the future sequence of plant replacements would depend on whether capacity is diverted today.

(11)Because we assume no lag between investment and production, we compute the EAC annuity payment using the begining-of-period timing convention and charge the EAC from Years S to N - 1. However, the results would be identical if we calculated EAC using the end-of-period convention and made the EAC charges from Years S + 1 to N.

(12)The cost of an unending series of plants with the first built at t = S is equivalent to paying the EAC every year forever, beginning in Year S. The cost of an uneneding series of plants with the first built at t = N is equivalent to paying the EAC in perpetuity, beginning in Year N. The difference in the present values of these two streams, which is the present value of an (N - S)-year annuity with payments equal to the EAC, thus represents the cost of moving all future replacement cycles ahead by (N - S) years.

(13)See Brealey and Myers [2, pp. 204, 205].

(16)Specifically, the current values of the production options for Years 3 to 5, respectively, are (in $ millions) .465, .542, and .604.

(17)The option values were estimated using a binomial lattice technique, as described in Cox, Ross, and Rubinstein [4]. With ten steps per year and an annual standard deviation of 0.10, the size of each upward move is [e.sup.(.1 square root of .1)]- 1 = 0.032, or 3.2%, and the size of each downward move is [e.sup.(-.1 square root of .1)]- 1 = -0.031, or -3.1%.

(18)Specifically, using the beginning-of-the-year payment convention, EAC is calculated as EAC = rE[(1 + r).sup.5]/[[(1 + r).sup.66 - 1], and assuming that investment is moved up from Year 14 to Year 1, this EAC is charged for Years 1, 2 and 3, so C = EAC/(1+R) + EAC/[(1+R).sup.2] + EAC/[(11r).sup.3].

(19)As above, the EAC is $.563 for E = 3, and charging this amount for Year 0 to Year 3 gives an opportunity cost measure of 2.096. In the options calculation with zero standard deviation, investment is moved up from Year 4 to Year 0, from Year 10 to Year 6, and so on. However, because of the 50-year horizon, the last investment occurs in Year 48 under the diversion strategy, versus Year 46 if the firm does not divert capacity. Since there is one more investment under the diversion strategy, the 50-year analogue to the EAC calculation is slightly higher than the infinite-horizon EAC calculation. The exact present value of the difference in the investment expenditure streams is 2.183.

References

[1.] P. Bjerksund and S. Ekern, "Mananging Investment Opportunities Under Price Uncertainty: From 'Last Chance' to 'Wait and See' Strategies," Financial Management (Autumn (1990), pp. 65-83.

[2.] R.A. Brealey and S.C. Myers, Principles of Corporate Finance, New York, McGraw-Hill, 4th edition, 1991.

[3.] J.K. Butters, W.E. Fruhan, Jr., D.W. Mullins, Jr., and T.R. Piper, Case problems in Finance, Homewood, Il, Irwin, 9th edition, 1987.

[4.] J.C. Cox, S.A. Ross, and M. Rubinstein, "Option Pricing: A Simplified Approach," Journal of Financial Economics (September 1979), pp. 229-263.

[5.] J.W. Kensinger, "Adding the Value of Active Management into the Capital Budgeting Equation," Midland Corporate Finance Journal (Spring 1987), pp. 31-42.

[6.] R.L. McDonanald and D.R. Siegel, "Option Pricing When the Underlying Asset Earns a Below-Equilibrium Rate of Return: A Note," Journal of Finance (March 1984), pp. 261-265.

[7.] R.L. McDonald and D.R. Siegel, "The Value of Waiting to Invest," Quarterly Journal of Economics (November 1986), pp. 707-727.

[8.] R.C. Merton, "An Intertemporal Capital Asset Pricing Model," Econometrica (September 1973), pp. 867-887.

[9.] S.C. Myers, "Financial Theory and Financial Strategy," Midland Corporate Finance Journal (Spring 1987), pp. 6-13.

[10.] S.C. Myers and S. Majd, "Abandonment Value and Project Life," Advances in Futures and Options Research (1990), pp. 1-21.

[11.] R.S. Pindyck, "Irreversible Investment, Capacity Choice and the Value of the Firm," American Economic Review (December 1988), pp. 969-985.

[12.] G. Sick, Capital Budgeting With Real Options, Monograph 1989-3, Salomon Brothers Center for the Study of Financial Institutions Monograph Series in Finance and Economics, New York, New York University, 1989.

[13.] L. Trigeorgis, "Option Interactions and the Valuation of Investments with Multiple Real Options," Boston University Working Paper, 1990.

[14.] L. Trigeorgis, "A Log-Transformed Binomial Numerical Analysis Method for Valuing Complex Multi-Option Investments," Journal of Financial and Quantitative Analysis (September 1991), pp. 309-326.

[15.] L. Trigeorgis and S.P. Mason, "Valuing Managerial Flexibility," Midland Corporate Finance Journal (Spring 1987), pp. 14-21.

Methods for estimating this opportunity cost are available, but they focus exclusively on the firm's furture investment and make very strong assumptions about the predictability of that investment. In doing so, they fail to recognize the option elements of the problem, and in some circumstances this may result in seriously misleading estimates.(2) Our aim in this paper is to analyze these option elements explicity. We show that when uncertainty disappears, the option elements become unimportant, and the true opportunity cost collapses to the solution given by existing methods. However, as uncertainty grows, these option elements come to dominate the problem, and existing methods can substantially understimate or overestimate the true opportunity cost.

The essence of our option framework is as follows: Capacity in place gives the firm an option to produce. Thus, if a firm diverts capacity from Product A to Product B, it forgoes the option to produce A immediately, but it acquires an option to replace the diverted capacity. The true opportunity cost of using the excess capacity is the change in the value of the firm's options. By focusing on the state-contingent nature of optimal future decisions, this framework recognizes that the opportunity cost is not necessarily equal to the present value of specific investment or production decisions.

Section I describes the problem and presents the real options analysis in general terms. Existing approaches to measuring opportunity cost are then contrasted with the options approach. Section II presents a specific numerical example in which the cost of diverting excess capacity is calculated for different values of such variables as investment cost, the interest rate, product price uncertainty and unit production cost. We show that the ture opportunity cost can vary widely under these conditions in ways that are not captured by existing approaches. Section III briefly summarizes our results and offers conclusions.

I. The Opportunity Cost of Diverting Capacity

A. The Problem

Consider a firm with a unit of capacity, such as a plant, which is currently devoted to production of Product A. Each year, the firm observes the prevailing market price for A, compares that with its unit operating cost, and produces A if it is profitable to do so. The existing plant has a remaining life of N years. By investing E, the firm can build a new plant with an economic life of L years, which will become operational immediately.(3)

Suppose that the current price of A is such that the firm has just decided not to produce this year and thus, to let the plant stand as excess capacity until at least next year. Suppose further that a new opportunity suddenly arises: by investing in some renovations, the firm could refit the plant to produce an alternative product, B. For simplicity, we assume that once the renovation is complete, it is prohibitively expensive to switch back to production of A and that the salvage value of the plant is zero throughout its useful life.(4)

On a stand-alone basis, the NPV of producing B is simply the present value of profits from B for the N years remaining in the life of the plant, minus the cost of renovating the plant, plus any change in value of the options associated with this first unit of B-capacity.(5) However, this stand-alone NPV is reduced by the opportunity cost of diverting the plant away from Product A.

B. The Real Options Framework

Following Pindyck [11], we can view plants as being installed sequentially in order of decreasing value to the firm.[6] This allows us to focus on the plant that currently produces A. Since we are only concerned with measuring the opportunity cost of diverting the marginal unit of capacity, we can ignore the effect on firm value of any other units.(7)

When a firm installs a plant, it acquires two types of options: a strip of production options, one for each year of the plant's useful life, and a replacement option, which is equivalent to a future investment option. The production options are European; for example, the option to produce in Year 3 matures in that year when the firm learns the price of A. The exercise price of each production option is the unit operating cost times the level of production. The firm will exercise its production option in Year 3 if that year's unit price of A exceeds unit cost. In addition to its production options, a firm that has a plant in place has the option to replace that plant at the end of its useful life.(8) The replacement option, which is a future investment option, is American since it can be exercised either at that time or later.

If the firm does not have a plant in place in a given year, it has no production options, but it does have an immediate investment option. This option is American, since it can be exercised either now or later, and the exercise price is E, the cost of building a new plant. The investment option is a compound option, or an option on options. If the firm exercise its investment option, building a new plant, it acquires in turn the strip of production options that allow it to choose whether or not to produce in each year of the plant's life. By building a plant, the firm also acquires an American option to replace that plant, beginning L years after the plant is built.

In this framework, the opportunity cost of diverting excess capacity to a new product is the change in the value of the firm's options that is caused by diverting the capacity. If the firm does not divert capacity, it has the option to produce A any year between now and the end of the plant's useful life. If the plant will wear out in Year N, the firm also has the option to replace it in Year N (or later). On the other hand, if the firm does divert capacity, it acquires an option to invest immediately in a new plant to produce A. By diverting capacity now, the firm loses its options to produce A and its option to replace a unit of A-capacity in Year N. However, it gains the option to acquire the unit of A-capacity immediately.(9) Thus the opportunity cost, C, of diverting capacity from A to B can be represented as:

[Mathematical Expression Omitted]

where [P.sub.t] is the current value of the option to produce A in [Year.sub.t], and [I.sub.O] and [I.sub.N] are the current values of the options to invest in a unit of A-capacity effective immediately and effective in Year N, respectively.

C. Existing Approaches to Opportunity Cost Measurement

The most plausible existing method for calculating this opporunity cost calls for charging the new project with the equivalent annual cost (EAC) of a plant for each year that diversion of capacity is estimated to cause a capacity shortage for the original product.(10) With a plant life of L years, for example, the EAC is the annual payment on an L-year annuity whose present value is equal to the plant's initial investment outlay. If diverting capacity to B is expected to cause a shortage of A-capacity beginning in Year S, opportunity cost under this method is the present value of these EAC charges from Year S to Year N-1:(11)

[Mathematical Expression Omitted]

One could interpret EAC as the annual rental payment a lessor would need to charge in order to recoup the initial cost of the plant plus interest over the plant's useful life. If a rental market actually existed, a company that diverted its plant to product B could restore the lost capacity to produce A by renting a similar plant. It would then face the decision of whether or not to build a replacement plant in Year N, just as it would have if the capacity had never been diverted.

An alternative interpretation is that charging the EAC for (N - S) years is equivalent to moving up not only the first plant replacement from Year N to Year S, but all future replacement cycles as well.(12) Thus, the opportunity cost is the present value of moving up the plant investment from Year N to Year S, again from Year N + L to Year S + L, again from Year N + 2L to Year S + 2L, etc.

This latter interpretation highlights the fact that the EAC method relies on stringent assumptions about the timing of future investment expenditures. By assuming that all future investment will take place with certainty at particular dates, it ignores the option component of the investment decision. In Section II below, we show that these option components become unimportant only when uncertainty disappers and the timing of future investments can be accurately predicted today. Even if the EAC method is implemented with a risk-adjusted discount rate, it cannot capture this option component, since risk-adjusted discounting at a constant rate is ill-equipped to handle situations in which decisions will be postponed until more uncertainty is resolved.(13) In contrast, an option framework recognizes that the investment options need not be exercised at the first possible moment.

II. Valuation of the Firm's Production and Investment Options

In this section, we construct examples in which the firm's production and investment options can be valued explicity. The economic environment assumed in these examples is intentionally simplified and may not capture all features of an actual decision-making situation. Nevertheless, the examples do serve to illustrate the basic determinants of the opportunity cost of using excess capacity. In Section II.A. below, we illustrate our solution technique with two simplified, are discussed in Section II.B.

A. Examples With Only One Investment Cycle

1. The Economic Horizon Ends When the Existing Plant Wears Out

If the economic horizon ends in Year N, the replacement option for the existing plant, [I.sub.N], is worthless. In that case, the opportunity cost of diverting capacity can be illustrated with a standard option diagram, as shown in Exhibit 1. V represents the value of the options to produce A between now and Year N. If the firm diverts capacity to B, it obtains an option, I, to invest in A-capacity at a cost E. Since the economic horizon is too short to allow any replacement decisions, I is simply a call option on V with exercise price E, and, from Equation (1), the opportunity cost of diverting capacity is equal to V - I.

Note:

V = Value of production options from now to time N. E = Cost of building a new plant. I = Value of option to invest in a new plant immediately (I is a call option on V with exercise price E). C = Opportunity cost of using excess capacity = V - I.

Given V = V*, distance RS = V* - E; distance PR = E; distance PQ=C= V - I.

The diagram illustrates two different upper bounds on the opportunity cost. First, the value of I is at least V - E, so the opportunity cost is at most E. This is also true in more general cases to which the standard diagram does not apply, because no matter how valuable the forgone production options are, the firm could always replace them immediately by investing the amount E. After that, it could make all future decisions as it would have done anyway. Exhibit 1 also indicates that V is an upper bound on the opportunity cost, since the value of the investment option is always nonnegative. Even in more general settings, the opportunity cost can never exceed the value of the lost production options, since the firm can always postpone any investment until it would have occurred anyway.

The option diagram also illustrates the circumstances in which either of the two upper limits is likely to be binding. As V grows large, the firm becomes more likely to invest immediately to restore its lost production options, and the opportunity cost approaches E. For this special case in which the economic horizon ends in Year N, this is analogous to the EAC solution. That is because the plant would never need to be replaced if capacity were not diverted, so the opportunity cost is the value of any additional investment expenditures the firm makes if it does divert capacity. When V is small, on the other hand, the firm is unlikely to invest, preferring instead to simply forgo the production options.

2. The Economic Horizon Coincides With the Useful Life of a New Plant

In our second simplified example, we assume that a new plant has a useful life of six years, and that the economic horizon also ends six years from now. Since a new plant becomes operational immediately (i.e., in Year 0) the product price revealed in Year 5 is the last one before the horizon date. The plant's capacity is 1 million units of A with a unit operating cost of $10.5(14) The current price of Product A is $10 and in each future year this price will move up (down) by a multiple of 1.10 (1/1.10). This gives rise to the "tree" of possible future prices illustrated in Exhibit 2. We assume that the "risk-neutral probability" of either a price increase or decrease in a given year is equal to 0.5.(15)

The current plant will wear out after four years (N=4), and a new plant with a useful life of six years can be built then, or at any later date until the end of the horizon, for an investment expenditure of $3 million (E = $3 million). The riskless interest rate, r, is five percent per year. For this one-cycle example, all acitivity ceases at the end of six years, so any plants after the current year will cease production before their useful lives are finished.

If a firm has capacity in place, it will exercise its production option in any given year if the price of A exceeds $10.5, the unit operating cost. The current value of that year's production option is simply the present value of its "expected" cash flow, where state-contingent cash flows incorporate the optimal exercise decision and expected values are computed using the risk-neutral probabilities. Since the current price of A is $10, for example, Year O's production option is worthless, but the current value of Year 1's production option is given by:

[P.sub.1]=.5(11 - 10.5)(1 million) + .5(0)/1.05=$.238 million. (3)

Note that the production option will be exercised if the price moves up, but not if the price moves down. Similarly, the value of Year 2's production option is:

[P.sub.2]=[(.5).sub.2](12.1 - 10.5)(1 million) + 2(.5)(.5)(0) + [(.5).sub.2](0)/1.05 = $.363 million. (4)

Valuing the investment options is more complex, since two questions must be answered. The first is whether investing in a plant at any given time yields a positive NPV. Because of the short economic horizon, any plant built now or in the future, in this example, will not be replaced, so the first question can be answered by subtracting the $3 million cost of the plant from the value of the production options gained by building it. Valuing the production options for Years 0 to 5 with the risk-neutral probabilities as described above, the NPV of investing in a new plant immediately is $-0.788 million.(16) Thus, in this example, it will not be worthwhile for the firm to invest immediately.

Even if the NPV of investing immediately were positive, however, the firm would still have to consider the second question, raised by McDonald and Siegel [7], which is whether it is better to wait before investing. The value of waiting is equal to the "expected" present value of waiting until Year 1, observing the product price at that time and then following the optimal investment strategy contingent on that price. By waiting, the firm forgoes the production option in Year 0, but it also gains information about the future price of A. This information is valuable, because it enables the firm to avoid investing in capacity in situations where future production is unlikely to be profitable.

The optimal strategy at each price node, which consists of either investing (I) or waiting (W) further, is in turn found by working backward through the price tree from the economic horizon to that node, evaluating all future production options as of the time of that node, and then comparing the net present values of investing at that time versus the discounted expected value of waiting one more year. The optimal strategy for a firm that reaches a given node without a unit of capacity already in place for the next period is shown in parentheses below the relevant price in Exhibit 2. In the case at hand, the optimal strategy is to wait in Year 0, wait again in Year 1, and then invest in Year 2 only if the price is $12.1. In Year 1, the NPV of investing if the price is $11 is $1.112 million, but it is even more profitable ($1.694 million) to wait another year, because the firm can then avoid committing capital until there is a greater likelihood that the future path of prices will be favorable.

The opportunity cost of diverting the plant to production of B is given by (1) as:

[Mathematical Expression Omitted]

Using the valuation technique described above, the sum of the forgone production options is 1.066, the current value of the option to invest in Year 4 is 0.301, and the value of the option to invest immediately or later is worth 0.816, so the opportunity cost of diverting capacity is $.551 million.

Note that the opportunity cost measure given by Equation (5) is substantially below either of the two lower bounds described in the first part of Section II.A. It is less than E (which equals three in this case), because the firm would not choose to replace the diverted capacity immediately, and even when it does, it saves the replacement expenditure that might otherwise have been make in Year 4. Likewise, the true opportunity cost is less than the sum of the forgone production options (1.066 in this case), because, if the path of prices is sufficiently favorable, the firm will find it economical to invest in Year 2 and recover some of these lost options.

The EAC method is not applicable in this example, since it is predicated on an infinite economic horizon. However, the difference in present value of investment outlays, depending on whether capacity is diverted or not, would represent a single-replacement-cycle analogue to the EAC method. In the example at hand, the firm would never replace the diverted capacity immediately, since the nearest production option is out of the money. If the price of A rises, however, there is a state of nature in Year 1 in which the firm would want to produce A. Thus, we could calculate the opportunity cost of diverting capacity as the present value of moving up the $3 million investment from Year 4 to Year 1, or C = 3 [(1/1.05) - [(1/1.05).sup.4]] = $.389 million. This technique underestimates the true opportunity cost for this example, because it assumes that future replacement expenditures are certain, rather than statecontingent. In particular, it assumes that the firm would necessarily replace the existing plant in Year 4, whereas in fact this replacement is highly uncertain.

B. Results From the General Example

In general, the opportunity cost measure produced by the real options approach will vary with all of the standard determinants of option values: the exercise prices of the production and investment options, the interest rate, the degree of uncertainty in product prices, and the time periods over which the options can be exercised. This, in turn, can produce wide variations in opportunity cost both in absolute terms and relative to the measures provided by the EAC technique.

To get some idea of the sensitivity of the true opportunity cost to variations in its underlying determinants, we performed a series of simulations. For the most part, our base case assumes the same parameter values as in the example above: a unit operating cost of $10.5, and a plant production capacity of one million units of A per year; a time-to-replacement for the existing plant of two years; an investment cost of $3 million, and a useful plant life of six years. We did, however, assume an economic horizon of 50 years to get a better idea of the effect of a whole sequence of future replacement decisions. In an attempt to represent the effect of uncertainty less crudely than in the example above, we also assumed that product price follows a lognormal process with a mean of zero and a standard deviation of ten percent per year. To better approximate this distribution, we estimated all option values with a grid consisting of ten steps per year.(17) As a result of these changes, the results in the simulations are of the same order of magnitude, but not strictly comparable with the results in Section II.A.

There are some factors, such as the cost of investment and the interest rate, that affect opportunity cost as measured by either the real options or the EAC methods, but they do so in different ways. Moreover, since the EAC method focuses entirely on investment expenditures, there are several additional factors that have a significant impact on the valuation of options but do not enter into the calculation of the EAC (e.g., output price, standard deviation of output price and production cost).

1. Factors Affecting Both Option and EAC Estimates of Opportunity Cost

Exhibit 3 shows how opportunity cost varies with the cost of building a new plant. If the investment cost is zero, the diverted capacity is replaced costlessly and the opportunity cost of diverting capacity is zero for both the real options method and the EAC method. For both mehtods, the opportunity cost rises with the cost of a new plant, but it does so much more gradually under the options method. As investment cost increases, the firm becomes less and less likely to exercise its investment option before Year 4 and thus changes in investment cost become less significant. Since the firm can always simply choose not to replace the capacity to produce A, the value of the lost production options represents an upper bound on the opportunity cost. The EAC method, however, assumes that the firm always invests at the earliest opportunity, whether or not that is the best action for the firm and whatever the cost of building a plant. Thus, the EAC estimate increases at a constant rate with the cost of investment and quickly overstates the true opportunity cost. (18)

The effects of changes in the riskless rate of interest are shown in Exhibit 4. The opportunity cost calculated using the options method decreases slightly as the interest rate increases, but it is relatively insensitive to changes in the rate. The cost calculated using the EAC method increases with the riskless rate and always overestimates the true opportunity cost. Under the EAC method, increasing the interest rate increases the cost of moving forward the sequence of replacement investments. Under the real options method, by contrast, an increase in the interest rate also reduces the present value of any future production profits. Although it may be more expensive to move the sequence of investments forward, it may also be less compelling to do so.

Exhibit 5 shows the relationship between opportunity cost and time to replacement for two different plant lives, six years and ten years. In general, the longer the time to initial replacement, the higher is the opportunity cost, because the firm is giving up a longer strip of production options when it does divert capacity. For a given time to initial replacement, though, the opportunity cost is lower the longer the useful life is of a newly built plant. This is because the same initial investment buys more production options, so the firm is more likely to invest. Costs estimated with the EAC method also increase with the time to replacement since there are more periods to the annuity. Again, the method overstates the true value of the opportunity cost because it ignores the option components of the problem.

2. Factors That Affect Option But Not EAC Estimates of Opportunity Cost

Exhibit 6 shows what happens to the opportunity cost of using excess capacity when all other parameters are as in our base case, but the standard deviation of the price process (that is, the size of the yearly up and down moves) is allowed to vary. When the yearly variation in price is very small, the opportunity cost goes to zero. This is because the production options are out of the money at the start, and there insufficient price variation for them to be in the money any time in the next four years. The opportunity cost then rises as price variability rises, because this increases the value of both the production and investment options. However, [I.sub.0], the value of the option to invest immediately, is more sensitive to changes in standard deviation than [I.sub.4], the value of the option to invest beginning in Year 4. At high standard deviation levels, as Exhibit 6 illustrates, increases in [I.sub.0] can dominate the increases in [I.sub.4] and in the value of the production options, and the opportunity cost can turn down.

Because the initial production option is out of the money in our base-case example, the option and EAC measures of opportunity cost do not coincide, even when uncertainty disappears. If the initial production option were in the money, however, and the standard deviation of the price process were zero, a firm that diverted capacity would always want to invest immediately so as to capture all of the valuable options to produce A. It would thus move up the first replacement investment from Year 4 to Year 0, and it would move up all future replacements as well. If the economic horizon were infinite, then, the option and EAC methods should produce identical opportunity cost measures.

This is illustrated approximately in Exhibit 7, in which all other base-case parameters are kept the same but the unit operating cost is reduced to six, putting the initial production option into the money. At low standard deviations, the firm is virtually certain to invest in Year 0 if capacity is diverted, versus Year 4 if it is not, in Year 6 if capacity is diverted, versus Year 10 if it is not, etc. Thus, the option calculation nearly coincides with the EAC calculation. (19) As the standard deviation increases, however, the true opportunity cost rises above the EAC measure. This is because, while the firm might still invest in Year 0 when it diverts capacity, there is a greater likelihood that it would choose not to replace the capacity in Year 4 had it not diverted in the first place.

Exhibit 8 shows the variation in opportunity cost for different levels of unit production cost. At low production costs, such that the initial production option is deep in the money, the options calculation coincides with the EAC calculation, except for the small discrepancy caused by the finite horizon. As production cost increases initially from these levels, the true opportunity cost rises above the EAC. As in Exhibit 7, this is because the firm will still invest in Year 0 if it does divert capacity, but it becomes increasingly likely not to invest in Year 4 if it does not divert capacity now. For higher levels of production cost, the firm no longer invests immediately if it does divert capacity, and the opportunity cost turns down, sharply at first and then more gradually. The EAC measure, by contrast, makes a single, downward jump when production cost reaches 10. Beyond that level, production in Year 0 is no longer profitable, so the EAC is charged only in Years 1 to 3, rather than 0 to 3. However, the EAC measure does not reflect the fact that, as unit cost reaches high levels, neither investment nor production are likely to occur in the future, so that the true opportunity cost becomes negligible.

Finally, the effect of lengthening the economic horizon is shown in Exhibit 9. If the horizon is only one year, the opportunity cost is zero, because the first production option is worthless, and if the firm diverts capacity it loses nothing. As the horizon lengthens, the opportunity cost initially rises, but then begins to ratchet up and down. This is because the life of the last plant built may be prematurely terminated, and the degree of premature termination differs under the diversion strategy or the alternative strategy with the length of the horizon. However, as the economic horizon becomes very long, this ratchet effect is dampened, and the opportunity cost converges to a value of approximately $.526 million. While the ture opportunity cost varies somewhat for different horizons here, the EAC technique always produces an overestimate, because it does not account for the uncertainty inherent in the firm's future production and investment decisions.

It is clear from these examples that the true opportunity cost of diverting excess capacity to another use can vary widely, even in the context of a simplified economic environment. What general statements can be made, then, that might serve as a guide to managerial decision-making? First, there are a number of plausible circumstances in which there will be no opportunity cost at all. This occurs either when the production options the firm sacrifices by diverting capacity have little value or when they can be cheaply replaced. Second, as shown above, two upper bounds on the opportunity cost can be identified in any situation. The first is the cost of replacing the diverted capacity immediately, and the second is the value of the forgone production options. It is particularly important to recognize both of these upper bounds, since one of them may be considerably different from the other. In all of the examples above, the first upper bound the cost of immediately replacing the diverted capacity (plant cost is $3 million in our example), is a gross overestimate of the true opportunity cost. The decision-maker would be better advised in such cases to try to estimate the value of the lost production options.

III Summary and Conclusions

Existing methods for estimating the opportunity cost of diverting a unit of excess capacity to some other use focus on the cost of replacing that unit. However, these approaches ignore the option elements of the problem, and as a result they may produce estimates that are seriously distorted. By contrast, a real options perspective includes changes in the values of both production and investment options that are affected by diverting capacity. In actual practice, it may be difficult to estimate precisely the value of all of the production and investment options in Equation (1). Nevertheless, the analysis above can help to identify situations in which the true opportunity cost is likely to be high or low, and it can alert the decision-maker to conditions under which existing measurement techniques may give misleading results.

(1)An example occurs in the classic "Super Project" case in Butters, et al [3], in which a General Foods Corporation proposal for a new instant dessert product, "Super", entails the use of facilities that were previously installed for the existing product, Jell-O. Company managers in the case argue over whether the Super project should be charged an opportunity cost for using these Jell-O facilities.

(2)The option components of real assets have been the subject of a growing literature, which has recently been surveyed by Sick [12]. See also Kensinger [5], Trigeorgis and Mason [15], and Bjerksund and Ekern [1].

(3)A lag between investment and production could also be introduced without changing the essential nature of results. The absence of such a lag facilities comparison of our opportunity cost measure with that produced by the equivalent annual cost method, as described further in Section II.B.

(4)If production could be switched back to A, additional options would be created that would tend to reduce the opportunity cost of diverting capacity. Similarly, a positive salvage value for the plant would create an abandonment option, as in Myers and Majd [10]. However, including these additional option, as in Myers and Majd [10]. However, including these additional option elements would not add significant insights into the problem of opportunity cost, but would substantially complicate the analysis, since the various options may intertact. See Trigeorgis [13], [14] for further analysis of such interactions.

(5)In an options framework, for example, stand-alone NPV would include the value forgone by obtaining a unit of B-capacity immediately, since the value forgone by obtaining a unit of B-capacity immediately, since this kills the option to wait until a future year before obtaining that unit. see McDonald and Siegel [7].

(6)The total number of plants the firm can install profitably is thus limited, which rules our the possibility of infinite investment (perhaps because the firm operates under increasing long-run average costs).

(7)To see this, let the current plant be unit J. Since capacity is installed sequentially in order of decreasing value, investment in unit J is at least as attractive as investment in unit J + 1. Thus, the decision of whether or not to invest in unit J + 1 is unaffected by the decision to divert, because if the firm chooses to invest in unit J + 1, it will do so whether or not it has diverted capacity to product B. For example, if the firm has diverted capacity, and if it still wishes to invest in unit J + 1, it will also be optimal to invest in unit J. Thus, we can ignore unit J + 1 in our analysis.

(8)For simplicity, it is assumed that plants wear out all at once at the end of their useful lives and that replacement plants are identical to plants in place. Thus, an option to replace the plant before the end of its useful life would have no value.

(9)As explained in footnote 7, opportunity cost is calculated using only the investment and production options for the current plant, since that is the only capacity unit affected by the decision to divert.

(10)The EAC method is described in Brealey and Myers [2]. Two other common methods make rather implausible assumptions about useful plant life. One sets opportunity cost equal to the replacement cost of the facilities used, but this implicitly assumes that a new plant will last only as long as the existing plant (to Year N). A second method calculates the change in the present value of investment when plant replacement is moved up from Year N to Year S. This implicitly assumes that the new plant will last as long as the replacement of the existing plant would have (to Year N + L). Otherwise, the future sequence of plant replacements would depend on whether capacity is diverted today.

(11)Because we assume no lag between investment and production, we compute the EAC annuity payment using the begining-of-period timing convention and charge the EAC from Years S to N - 1. However, the results would be identical if we calculated EAC using the end-of-period convention and made the EAC charges from Years S + 1 to N.

(12)The cost of an unending series of plants with the first built at t = S is equivalent to paying the EAC every year forever, beginning in Year S. The cost of an uneneding series of plants with the first built at t = N is equivalent to paying the EAC in perpetuity, beginning in Year N. The difference in the present values of these two streams, which is the present value of an (N - S)-year annuity with payments equal to the EAC, thus represents the cost of moving all future replacement cycles ahead by (N - S) years.

(13)See Brealey and Myers [2, pp. 204, 205].

(16)Specifically, the current values of the production options for Years 3 to 5, respectively, are (in $ millions) .465, .542, and .604.

(17)The option values were estimated using a binomial lattice technique, as described in Cox, Ross, and Rubinstein [4]. With ten steps per year and an annual standard deviation of 0.10, the size of each upward move is [e.sup.(.1 square root of .1)]- 1 = 0.032, or 3.2%, and the size of each downward move is [e.sup.(-.1 square root of .1)]- 1 = -0.031, or -3.1%.

(18)Specifically, using the beginning-of-the-year payment convention, EAC is calculated as EAC = rE[(1 + r).sup.5]/[[(1 + r).sup.66 - 1], and assuming that investment is moved up from Year 14 to Year 1, this EAC is charged for Years 1, 2 and 3, so C = EAC/(1+R) + EAC/[(1+R).sup.2] + EAC/[(11r).sup.3].

(19)As above, the EAC is $.563 for E = 3, and charging this amount for Year 0 to Year 3 gives an opportunity cost measure of 2.096. In the options calculation with zero standard deviation, investment is moved up from Year 4 to Year 0, from Year 10 to Year 6, and so on. However, because of the 50-year horizon, the last investment occurs in Year 48 under the diversion strategy, versus Year 46 if the firm does not divert capacity. Since there is one more investment under the diversion strategy, the 50-year analogue to the EAC calculation is slightly higher than the infinite-horizon EAC calculation. The exact present value of the difference in the investment expenditure streams is 2.183.

References

[1.] P. Bjerksund and S. Ekern, "Mananging Investment Opportunities Under Price Uncertainty: From 'Last Chance' to 'Wait and See' Strategies," Financial Management (Autumn (1990), pp. 65-83.

[2.] R.A. Brealey and S.C. Myers, Principles of Corporate Finance, New York, McGraw-Hill, 4th edition, 1991.

[3.] J.K. Butters, W.E. Fruhan, Jr., D.W. Mullins, Jr., and T.R. Piper, Case problems in Finance, Homewood, Il, Irwin, 9th edition, 1987.

[4.] J.C. Cox, S.A. Ross, and M. Rubinstein, "Option Pricing: A Simplified Approach," Journal of Financial Economics (September 1979), pp. 229-263.

[5.] J.W. Kensinger, "Adding the Value of Active Management into the Capital Budgeting Equation," Midland Corporate Finance Journal (Spring 1987), pp. 31-42.

[6.] R.L. McDonanald and D.R. Siegel, "Option Pricing When the Underlying Asset Earns a Below-Equilibrium Rate of Return: A Note," Journal of Finance (March 1984), pp. 261-265.

[7.] R.L. McDonald and D.R. Siegel, "The Value of Waiting to Invest," Quarterly Journal of Economics (November 1986), pp. 707-727.

[8.] R.C. Merton, "An Intertemporal Capital Asset Pricing Model," Econometrica (September 1973), pp. 867-887.

[9.] S.C. Myers, "Financial Theory and Financial Strategy," Midland Corporate Finance Journal (Spring 1987), pp. 6-13.

[10.] S.C. Myers and S. Majd, "Abandonment Value and Project Life," Advances in Futures and Options Research (1990), pp. 1-21.

[11.] R.S. Pindyck, "Irreversible Investment, Capacity Choice and the Value of the Firm," American Economic Review (December 1988), pp. 969-985.

[12.] G. Sick, Capital Budgeting With Real Options, Monograph 1989-3, Salomon Brothers Center for the Study of Financial Institutions Monograph Series in Finance and Economics, New York, New York University, 1989.

[13.] L. Trigeorgis, "Option Interactions and the Valuation of Investments with Multiple Real Options," Boston University Working Paper, 1990.

[14.] L. Trigeorgis, "A Log-Transformed Binomial Numerical Analysis Method for Valuing Complex Multi-Option Investments," Journal of Financial and Quantitative Analysis (September 1991), pp. 309-326.

[15.] L. Trigeorgis and S.P. Mason, "Valuing Managerial Flexibility," Midland Corporate Finance Journal (Spring 1987), pp. 14-21.

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Title Annotation: | Issues in Corporate Investments |
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Author: | McLaughlin, Robyn; Taggart, Robert A., Jr. |

Publication: | Financial Management |

Date: | Jun 22, 1992 |

Words: | 6793 |

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