intematkmat journal of

production economics

ELSEVIER Int. J. Production Economics 38 (1995) 281-291

An options view of investments in expansion-flexible manufacturing systems

Ram L. &mar*

Department of MIS and Operations Management, The Belk College of Business Administration, The University of North Carolina at Charlotte, Charlotte, NC 28223, USA

Received January 1994; revised August 1994

Abstract

Traditional financial investment evaluation methods are often unsuitable for evaluating investments in flexible manufacturing systems. We present a novel method of valuing expansion flexibility, based on options theory, and illustrate its use through a detailed example. This method captures additional characteristics of the problem that are not addressed by traditional methods, and can be used for comparing investment alternatives as well as justifying them. The managerial implications of this method are analyzed.

1. Introduction

Manufacturing organizations in the US have found that competition, often on a global scale, has significantly impacted manufacturing practices. There is increasing emphasis on flexibility in order to meet changing customer needs. Flexibility could be of different types. Examples of flexibility include the ability to vary production volume and/or prod- uct mix, faster response times, etc. Flexibility can be short-term or long-term [l]. A detailed discussion of different types of flexibility from economic, or- ganizational, and manufacturing perspectives can be found in [2]. These include machine, material handling, operation, process, product, routing, vol- ume, expansion, program, production, and market llexibilities. A major factor in achieving flexibility is

* E-mail: rlkumar@unccvm.uncc.edu.

the use of advanced manufacturing technology that could range from relatively simple numerically controlled machines to more complicated com- puter integrated manufacturing systems [3]. Investment justification is a serious problem that often impedes the introduction and use of flexible manufacturing systems (FMS) [4, 51. The problem arises because of the nature of benefits resulting from FMS. These benefits are often rela- tively intangible and difficult to quantify in the form of cost savings or revenue generation oppor- tunities.

A variety of techniques have been suggested for justification of investments in flexible manufac- turing systems [S-14]. These techniques can be classified into economic techniques, analytical techniques, and strategic techniques [3]. Strategic justification is often qualitative. However, there is a need to support qualitative arguments with quantitative financial information in order to en-

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282 R.L. Kumnr/Int. J. Production Economics 38 (1995) 281-291

hance the quality of decision making [6, 15, 161. Different types of flexibility may be important to organizations under different circumstances. Fo- cusing on specific types of flexibility can therefore result in tractable models that provide useful in- sights [7].

This research provides a method of quantifying the value of expansion flexibility. This method is based on options theory in finance. Managers can use this method along with qualitative arguments in investment justification of expansion-flexible manufacturing systems. The following sections dis- cuss the concept of expansion flexibility in greater detail. We start by presenting the theory of options as it pertains to FMS investments in Section 2 and introduce the concepts of primary and secondary investments in FMS. Section 3 reviews the econ- omics literature that relates to option theory and investment justification. A view of primary and secondary investments as risky assets is presented in Section 4. A model for valuing options to exchange risky assets is discussed in Section 5. Section 6 presents an example to illustrate the insights provided by this model. Section 7 discusses the managerial implications of this model. Summary and conclusions are presented in Section 8.

2. An options view of FMS investments

Investment in expansion-flexible FMS provides the investor with an opportunity to make second- ary investments at a later date. These secondary investments could yield additional profits. There is, usually, a time period within which the investment must be made. Factors like financial year, budget, or competitive pressures determine the time period for decision making. This is analogous to an American call option [17]. A person who buys an option acquires the right to buy an underlying asset (stocks, bonds, or commodities) at a particular price (called the strike price) at some point in the future, up to an expiry date. For example, one might purchase an option (say for $20) to buy 1000 units of IBM stock at $100 per unit anytime within the next year. The $20 paid to acquire the option is referred

to as the option premium or the option price. The price of $100 per unit is fixed at the time of purchas- ing the option and is called the strike price of the option. However, the option holder is not obliged to buy the underlying asset (the option to buy need not be exercised). The price of the under- lying asset (the price of IBM stock) varies stochasti- tally. Let the market price of the underlying asset at some point of time be higher than the strike price. In this case, the option holder stands to make a profit by buying at the strike price (by exercising the option to buy) and selling at the market price. Alternatively, the market prize of the underlying asset could be lower than the strike price. In this case, the option holder is not likely to exercise his option to buy at the strike price. The sum of $20 paid for the option would be lost. The holder of an option is likely to exercise the option only if a profit is to be made. Not exercising the option would result in a loss equal to the purchase price of the option. The value of an option can be calculated, based on the work of Black and Scholes [IS], and depends among other things on the strike price.

Investments in FMS can be viewed as consisting of primary and secondary investments. The pri- mary investment is the initial price of the system, and the secondary investment represents the addi- tional investment required at a later date to make use of the flexibility provided by the system. For example, a primary investment may be made in a modular manufacturing system and a secondary investment may be required for capacity increase. From the options standpoint, the primary invest- ment corresponds to the option price or premium and the secondary investment corresponds to the strike price. The investor acquires a right (but not an obligation) to make a secondary investment (for example, to increase capacity) by making a primary investment. However, the value of the secondary investment is not known exactly at the time of primary investment (unlike the strike price), and is a stochastic variable. Hence tradi- tional option pricing models cannot be directly applied to investments in flexible manufacturing systems. Sections 4 and 5 discuss an alternative option valuation method that is applicable to FMS.

R.L. KumarJInt. J. Production Economics 38 (1995) 281-291 283

3. Related research

Kester [19] analyzes the relevance of options pricing for investment analysis. The economics lit- erature contains a recent stream of research focus- ing on irreversible, uncertain investments that uses option pricing theory [20-241. Irreversible invest- ments represent situations where the investments represent sunk costs. It is not possible to sell the asset acquired by investing and recover the invest- ment. This is often true of large capital investments that represent sunk costs. Investments in many types of FMS are irreversible (partially or wholly) in this sense. A high degree of uncertainty is usually associated with the costs and/or benefits of FMS investments.

This stream of research has highlighted the inad- equacy of net present value (NPV) as a method of investment evaluation in many situations. Invest- ments often result in additional, delayed, oppor- tunities for revenue generation, and management has the option of stopping, delaying, or otherwise altering the originally planned pattern of invest- ment. The net present value method does not con- sider these characteristics of investments. Majd and Pindyck [23] discuss the limitations of NPV in the context of management’s ability to alter the pattern of investment. Pindyck [21, 241 shows how the value of installed capacity and the value of options for growth together make up the value of a firm. Triantis and Hodder [25], and He and Pindyck [20] use options theory to assign a value to a firm’s manufacturing flexibility.

Triantis and Hodder [25] examine the situation where a firm is considering investment in produc- tion equipment that can be used to costlessly switch between k different products. They propose an op- tions theory based approach to valuing such invest- ments and develop closed-form expressions for two products with no switching costs. Though the analysis focuses on product-flexible systems, they discuss extensions to include process-flexible or input-flexible systems. Consideration of switching costs is possible but complicates the analysis signif- icantly.

He and Pindyck [20] examine two types of re- lated problems in the context of flexible manufac- turing. They focus primarily on output or product

flexibility. First, they consider the problem of opti- mal capacity choice faced by a firm with stochastic demand. Second, they consider the technology choice problem for a firm that can choose between single and multiple output producing capital. The technology choice and capacity choice problems are related. A firm must choose a technology (flex- ible or nonflexible) and then select capacity. The analogous problem of input flexibility is also exam- ined. Decision rules that maximize the firm’s mar- ket value are derived based on assumptions about the firm’s profit function.

Several methods of valuing different types of flexibility that do not directly use options theory have been proposed. A discussion of some ap- proaches can be found in [2, 33. Kulatilaka [26] develops a stochastic dynamic programming based model for valuation of input-flexible capacity. Fine and Freund [27] develop a two-period quadratic programming based approach to studying the value of flexibility. They develop a decision rule for technology choice that does not necessarily maxi- mize firm value.

Pruett and Park [11] support the notion that organizations typically undertake phased invest- ments in flexible manufacturing systems. Results of earlier phases help to resolve the uncertainties in subsequent stages. They propose a method for re- solving uncertainty in a stepwise fashion and dem- onstrate its use in an investment scenario involving process flexibility. Ramasesh and Jayakumar [ 123 propose an aggregate framework for integrating different types of flexibilities and quantifying their effect on investment decisions. This model allows examination of the interrelationships between dif- ferent types of flexibility. Their primary focus is on volume and product flexibilities, though expansion flexibility can also be analyzed using the proposed model.

The literature on valuation of flexibility seems emphasize product, volume and process flexibility. A uniform measure of measuring flexibility has not emerged. Different investment scenarios are likely to attach importance to different types of flexibility.

This paper focuses on expansion flexibility. This type of flexibility refers to the ease with which the capacity can be increased or the maximum possible production can be altered. Expansion flexibility

284 R.L. Kumarllnt. J. Production Economics 38 (1995) 281-291

provides an infrastructure for growth. It is a long- term flexibility [ 11. The research on valuing expan- sion flexibility is limited. Carter [40] suggests measuring the overall effort and cost needed to add a given amount of capacity. Browne et al. [28] suggest a measure of how large a system can become. Evaluation and comparison of the degree of flexibility provided by different manufacturing systems is possible using these methods. However, these methods do not directly aid investment decisions. Sinha and Wei [ 131 propose a stochastic linear programming based approach for assessing the economic benefits of investments in flexible manufacturing systems. The proposed model aims at maximizing the NPV while consider- ing flexibilities of the system. Measures of the value of product and expansion flexibilities are provided.

This research considers a situation where there are several investment alternatives providing differ- ing degrees of expansion flexibility. Our objective is to propose a methodology that can be used to value expansion flexibility while evaluating investment alternatives. This method can be used in conjunc- tion with other methods for evaluating investments in flexible manufacturing systems

4. Primary and secondary investments, and risky assets

Capital expenditure is usually incurred before production begins in the case of dedicated produc- tion centers [8]. Such dedicated production centers usually lack expansion flexibility. In the case of flexible manufacturing systems, capital expendi- tures required are often substantially higher than dedicated production centers and there is consider- able uncertainty associated with investment deci- sions [29]. Phased investment is likely [ll, 13, 141. We assume that a primary investment is followed by a study of the resulting benefits before a second- ary investment decision in an expanded project is made. Investment justification for the flexible sys- tem is at the primary investment stage, and it is often difficult to assess the costs and benefits of secondary investment exactly at this stage. Also, it is often possible to get a clear picture of the benefits

of secondary investments only after the primary investment has been made. Hence, secondary in- vestments and resulting benefits are stochastic vari- ables at the time of making primary investments, or justifying investments in FMS. However, it is important to take secondary investments and re- sulting benefits into account during evaluation of primary investment alternatives. Investment in flexible equipment often provides opportunities for additional profit generation at a later date with relatively low additional investment. For the pur- poses of this paper, we use the term equipment in a broad sense to refer to different types of equip- ment that contribute to manufacturing capacity. This may include a wide range of facilities. Exam- ples include modular flexible manufacturing cells [28,30], a high degree of planned automation that facilitates easy expansion of capacity [31], system interfaces that are designed to facilitate integration and capacity enhancement [32], or even design of buildings that are conducive to expansion. The terms flexibility and expansion flexibility are used interchangeably in this paper. We assume that a one-time, irreversible investment decision is made in what we term a primary project. If the equipment selected has a high degree of expansion flexibility, then it is relatively easy to undertake expansion decisions at a later date. This potential opportunity that we will refer to as the secondary project has the following characteristics:

(a) There may be costs associated with the sec- ondary project (for example, equipment, tooling, additional labor, materials) which cannot be deter- mined deterministically, at the time of undertaking investments in the primary project. It is also pos- sible that the costs of the secondary project will become clear only on undertaking the primary pro- ject. On incurring these secondary costs, the organ- ization acquires an asset (expanded facilities) that is capable of producing benefits. This asset is termed a risky asset since its cost (secondary costs) is only an estimate at the time of primary investment (or even at the time of commencing secondary invest- ment). Secondary costs could vary considerably during the secondary investment (depending on market conditions, inflation, and a variety of other factors). The uncertainty is greater in situations where the secondary investment represents a

R.L. Kumar/ht. J. Production Economics 38 (1995) 28I-291 285

development project or involves construction or Table 1

manufacture with a large cycle time. Common capital and operating costs of FMS

(b) The benefits from undertaking a secondary investment represent another risky asset in the sense that they cannot be specified deterministically at the time of making initial investments. The sour- ces of uncertainty include economic factors, com- petition, and changes in technology.

We can consider the primary investment as being analogous to purchasing an option to exchange one risky asset (secondary investment) for another risky asset (benefits accruing from the secondary investment) within some time period from the in- itial investment. Management has the right not to make the secondary investment if it does not con- sider it worthwhile. In this case the management’s risk is limited to the primary investment (the option price). Primary and secondary investments in the context of flexible manufacturing systems are dis- cussed in greater detail in Section 5.

Capital expenditures

Machinery

Hardware Software

Installation

Operating expenditures

Labor Maintenance Utilities

Consumables (e.g., coolant)

Training Taxes and Insurance

benefits. This framework lends itself to valuation using a valuation model for options to exchange risky assets originally proposed by Margrabe 1351, in the context of financial arrangements such as an investment adviser’s performance incentive fee.

This view of primary and secondary investments is in some ways similar to research and develop- ment investments. However, there may be some differences. The primary investment in a research and development project is often a pilot project that may cost much less than the secondary project. In the case of FMS investments, the primary pro- ject could be more expensive than the secondary project, and substantially more expensive than in- vestments in production or transfer lines [29, 331. Also, organizations undertaking R&D projects of- ten deal with a portfolio of projects. They recognize that some projects may fail, but the portfolio of projects is likely to yield long-run benefits for the organization. Investments in flexible manufactur- ing systems are often not part of a portfolio of investments though the need for viewing these in- vestments as part of an automation portfolio has been recognized [34].

Let Si and s, denote the expected values of secondary investment and returns from secondary investment at the time of making the primary in- vestment. si represents the expected present value of all expenditures associated with the secondary pro- ject. This would include capital as well as operating costs. Table 1 provides a list of some common capital and operating costs associated with FMS. Machinery includes various components of FMS including machining centers, fixtures, robots, auto- mated material handling facilities, tools and cool- ant systems. Hardware includes computer systems and controllers that are not integrated into the machinery. Sloggy [36] and Maleki [33] provide detailed discussions of capital and operating costs associated with FMS.

5. Valuing an option to exchange risky assets

Investment in expansion-flexible manufacturing systems has been characterized as a two-stage in- vestment process, with the primary investment be- ing analogous to purchasing an option to exchange risky secondary investment for associated risky

s, represents the expected present value of all benefits associated with the secondary project. This would include increased sales, reduction in working capital, labor savings, reduced space requirements, and better quality. Sloggy t-361, Kaplan [S], and Maleki [33] provide detailed discussions and examples of benefits resulting from FMS invest- ments. In addition, expansion flexibility provides certain unique benefits. These include more effec- tive absorption of new technologies due to phased implementation, ability to dispose of parts of the system in the event of negative market events (if the system is made up of standard modules that can be dismantled and sold), and phased cash out- flows.

286 R.L. Kurnar/int. J. Production Economics 38 (I 9951 2%i-291

Let value (si, s,, t) denote the value of an option to exchange an asset with an expected value of Si for an asset with an expected return of s,, at time t. Margrabe [35] derives equations for the value of an option to exchange risky assets. These equations are valid for a scenario where the option can be exercised at any time between and including t and T. In the context of primary and secondary invest- ments, t represents the time at which the primary investment decision is made, and T represents the deadline for making the secondary investment deci- sion. The following equations result:

VdUe(Si, Sr, t) = S,N(d,) - SiN(d,),

d 1

= h(s,/sJ + fv2(T - t)

VJE-t ’

d2 = d, - vd%.

IV(.) is the cumulative standard normal density function, and V’ = vf + v,’ - 2viv,pi,. vi and v, de- note the standard deviations of the rate of change of si and s, respectively (standard deviation of the percentage change in si and s, over unit time) and Prr is the correlation coefficient between si and s,. A high correlation implies that a high degree of secondary investment results in a high degree of secondary revenues. The significance of these para-

meters in the context of FMS and methods of a priori estimation are discussed in the following section. Section 6 also explores the interesting insights pro- vided by these equations regarding flexibility.

6. An illustrative example

The objective of this section is to present a nu- merical example to illustrate the option valuation process and to highlight some insights provided by this model. Consider an investment scenario where the expected value of the secondary investment (si) is $200000, and the expected value of the returns from secondary investment (s,) is $300000. In prac- tice, si would be determined from estimates of the capital and operating costs of the second stage project. Probabilities can be associated with differ- ent cash flow streams and the expected value can be calculated. s, can be calculated in a similar fashion. Accurate a priori estimation of si and s, would depend on the ability to identify and quantify capi- tal and operating costs, and benefits of the second- ary project. These costs and benefits were discussed in Section 5.

Assume that the variance of the rate of change of secondary investment (Vi) is 0.9, and that T - t = 1

year. Tables 2-4 present values of flexibility for

Table 2 Option values for pi, = 0.2

Y 0.10 0.20 0.40 0.60 0.80 1.00 dption value 143.97 143.17 145.57 149.55 158.30 171.60

Table 3 Option values for pi, = 0.5

“r 0.10 0.20 0.40 0.60 0.80 1.00 Option value 140.79 137.59 135.24 136.82 141.57 148.55

Table 4

Option values for pi, = 0.9

Y

hption 0.10 0.20 0.40 0.60 0.80 1.00

value 137.59 130.5 119.11 110.26 107.31 110.25

R.L. Kumnr/Int. J. Production Economics 38 (1995) 281-291 287

different values of pir and v,. We thus have values of flexibility for eighteen different combinations of Pir, Vi, and v,. Note that interchanging the values of Vi and v, for each of these eighteen cases would result in the same value of flexibility. Tables 2-4 therefore present results of (1) for 36 different combinations of parameter values. Option values in these tables are in thousands of dollars.

These calculations require a priori estimation of

Pir, Vi, and v,. Estimation of Pir would require a careful analysis of the investment situation. The key question to ask in estimating pir is does in- creased spending mean increased benefits or rev- enues. The following guidelines can be used to determine if the correlation is high, medium, or low:

(a) If demand is not a constraint, and production resulting from the second stage expansion project will result in increased revenues, then the correla- tion is high. Another important consideration could be the pricing policy. If the market can ab- sorb cost-plus pricing, then pir is likely to be high since increased costs would result in increased rev- enues.

(b) If the secondary project adds capacity that is only likely to be used partly, then pi* could be low or medium depending on capacity utilization. Such situations could arise because of lumped capacity increases dictated by the technology. If pricing has to be market-based and increased costs cannot be passed on to the customer, then pir is likely to be low. A medium value of pir would be possible in situations where increased costs can be partially passed on to the consumer.

A priori estimation of vi and v, is also possible. Standard deviations of the percentage change in si or s, can be estimated using one of the following methods:

(i) It may be possible to arrive at a probability for a percentage change in si or s, over a period of time. For example, it may be possible to say that there is 90% chance of a 20% change in si or s, over the next year. This may be based on a careful study of the investment situation at hand, and compari- son with similar projects, if any. One could make assumptions about the probability distribution (for example, normal) of the rate of change of si or s, and use the probability statement to determine the standard deviation of the rate of change of Si or s,.

100 --in ~~~ 0.1 0.12 --0.c ox 0.8 1 SD of Rate of Change of Costs/Revenues

Fig. 1. Option value of secondary project as a function of project risk (vi or VJ and correlation between secondary invest-

ments and secondary returns (pi,).

(ii) If several people are involved in the project and can make estimates of percentage changes in Si or s, over a period of time (say a year), this data could be used for determining the standard devi- ation.

Fig. 1 presents a graph of the option value of the secondary project for different values of v, given that vi is 0.9. An identical graph would result from fixing v, and varying vi over the same range of values. The graph of the option value of the second- ary project is undulating and option values would increase or decrease with increase in v, depending on the position on the surface. The following sec- tions analyze the implications of varying the values of the model parameters.

6.1. The efSect of increased variance of costs and/or benejits on valuation ofjlexibility

The option value of the secondary project may increase or decrease with increase in variance of costs and benefits. In the context of expansion- flexible FMS, sources of variance in secondary project costs include equipment costs, labor costs, taxes, interest rates, etc. Sources of revenue uncer- tainty include demand, competition, new products, etc. The increase or decrease is determined by the relative values of variances of the rate of change of

288 R.L. KumarJInt. J. Production Economics 38 (1995) 281-291

investment and returns, as well as the correlation between investment and revenues. This is indicated clearly by the curves shown in Fig. 1, which de- crease and then increase. In other words, there may be some situations in which more risky or unpro- ven FMS technology may be preferred, and others in which less risky or familiar technology may be the way to go. This result differs from conventional thinking about risky investments that would imply that lower variances are always preferred. It also differs from results in the traditional option valu- ation literature using Black-Scholes formula [ 1 S] which deal with deterministic strike prices, and indicate that option value increases with increased uncertainty. The pattern of variation of the value of flexible investments is similar to that of v*, where V* = Vz + V,' - 2ViV,pi,.

6.2. The efSect of increased correlation between primary and secondary investments

The option value of the secondary project de- creases as pir increases (for the same values of vi and v,). This can be inferred either from Tables 2-4 or from Fig. 1. An intuitive explanation for this is that for a high value of Pir costs and benefits resulting from the secondary investment vary in a similar pattern. Section 5 discussed factors that affect pir for expansion-flexible FMS. The probability asso- ciated with the revenues from the secondary invest- ment varying in a pattern that is considerably dif- ferent from the primary investment, and resulting in a large profit is low. Positive variations (in the sense of increasing revenues) are of primary interest since the management has an option not to proceed with the secondary project in the event of negative variation.

Table 5 Option values for different T - t values (p, = 0.9)

6.3. The efSect of making decisions before T

If the secondary investment decision can be made anytime before time T, is there an optimal timing for the secondary investment decision? Margrabe [35] argues that it is best to delay the secondary investment decision as much as possible (up to time T ). This is done by comparing the value of an option that can be exercised only at time T (called an European option), with that of an option which can be exercised anytime from the time of purchase until time T (an American option), and demonstrating that the European option al- ways has a value greater than that of the American option for all values of time between t and T. McDonald and Seigel [22] also discuss the value of delaying investment decisions. Management is not obliged to undertake the secondary investment be- cause it has undertaken the primary investment. The secondary investment is an opportunity (ob- tained as a result of undertaking the primary in- vestment), but not an obligation. Delaying the sec- ondary investment decisions increases the likeli- hood of obtaining additional information about the investment decision, and hence making better deci- sions. In the context of expansion-flexible FMS delaying investments may facilitate learning based on primary investments. This may influence the secondary investment project. Also delaying invest- ments may facilitate purchase of more current tech- nology, particularly in scenarios where there is a high rate of technological obsolescence.

6.4. The effect of increasing T

The effect of increasing the time period before which a secondary investment decision should be made is shown below in Table 5, for pir = 0.9. Option values for T - t = 2 time periods are

” 0.10 0.20 0.40 0.60 0.80 1.00 +--t=2 164.56 134.34 134.45 123.59 118.42 122.66 T-t=1 137.59 130.50 119.11 110.26 107.31 110.25

R.L. Kumar/lnt. J. Production Economics 38 (1995) 281-291 289

higher than those for T - t = 1. This result would be expected from the discussion in Section 6.3 that highlights the benefits of delaying investment. Sim- ilar results can be obtained for other values of pir. A situation where there is negligible time gap be- tween primary and secondary FMS investments is an extreme situation where management has no options to stop or otherwise alter the pattern of investment. The benefits of learning from the pri- mary investment do not exist. Larger time gaps between primary and secondary investments allow for learning opportunities based on the primary project. Modification of the secondary project based on the primary project is possible.

Increasing the time period available for decision making between primary and secondary invest- ments facilitates better decisions. This time period can be used to learn from the primary FMS invest- ment and assimilate the new technology into or- ganizational functioning. Feedback from the primary project can be used for future expansion decisions. The passage of time reveals more in- formation that can be used to alter the course of the secondary project. Projects with larger time gaps between primary and secondary investments there- fore have a higher option value.

It is important to note that the cost structure used in this example and resulting conclusions are realistic for the following reasons:

(a) The term FMS is associated with a wide range of equipment [3]. Actual costs of FMS range from hundreds of thousands of dollars to several millions of dollars [S, 333. Also, values of model parameters (Pi*, Vi, v,) are likely to be situation specific. It is therefore extremely difficult to charac- terize a “typical” FMS investment. Details regard- ing different types of FMS can be found in [S, 34, 37-393.

(b) The conclusions presented in Sections 6 and 7 of this paper are not specific to the cost structure (values of Si, s,, pir, Vi, and v,) used in the example. The numerical values presented are mere- ly chosen for illustrative purposes. Results are pre- sented for a range of parameter values.

(c) The model presented in this paper uses ex- pected values of costs and benefits of the secondary project (Si and s,). Calculation of expected values depends on a number of factors that were discussed

in this section. Realism of estimates would be in- fluenced by the realism of components of the cash flow streams used for expected value calculations.

7. Managerial implications

This section focuses on the managerial implica- tions of the model discussed in earlier sections. Specifically, the use of the option valuation model in the evaluation of FMS investment decisions is illustrated. Major managerial issues are high- lighted.

Consider an investment in FMS involving pri- mary and secondary investments. The traditional method of economic evaluation using NPV is as follows:

Value (FMS project) =

NPV(primary project) + NPV(secondary project)

(2)

Let us assume that the NPV of the primary project is $50000. In the example presented in Section 6, expected values of secondary investment (Si) and secondary returns (s,) were $200000 and $300000 respectively. The NPV of the secondary project would therefore be $100000 ($300000 - $200000). The value of the FMS project would be $150000 (50000 + 100 000). Thus, considering the primary as well as secondary projects makes the FMS investment more attractive than considering the primary project alone.

The method proposed in this paper for evalu- ation of investment in expansion-flexible FMS is as follows:

Value (FMS project) = NPV (primary project)

+ Option value (secondary project) (3)

It is interesting to note that the option values cal- culated in Tables 2-4 are all greater than $100000 (the present value of the secondary project). This indicates that there could be several scenarios where use of NPV undervalues an FMS investment project. The options-based approach includes the value of expansion flexibility, while the NPV ap- proach does not. The value of expansion flexibility

290 R.L. Kumarllnt. J. Production Economics 38 (1995) 281-291

is given by the difference between the results of Eqs. (3) and (2), and could be either low or substan- Gal depending on the specific investment situation. The value of expansion flexibility ranges from $7 310 [(107.31* 1000 - lOO* lOOO)] to $64 560 [164.56* 1000 - lOO* lOOO] (Option value (sec- ondary project) - NPV (secondary project)} in Table 5. In other words, the value of flexibility ranges between approximately 5% and 43% of the NPV of the FMS project ($150000).

In practice, it is suggested that managers who are uncertain about specific parameter values experi- ment with a range of reasonable values in order to determine if the value of flexibility is significant in a particular investment scenario. The calculations involved can easily be programmed and are not time consuming. If the value of flexibility is a high percentage of the NPV of the FMS project, then one may want factor this into the investment deci- sion. Also, FMS investment situations may involve other considerations besides those discussed in this paper. These include strategic issues, relationship between different types of flexibility, social and be- havioral factors [3, 51.

This model illustrates that projects with high uncertainties can have high option values which would justify investments even though traditional financial measures (like NPV) would lead to rejec- tion of the project. Projects where there is a high correlation between primary and secondary invest- ments have a lower option value than those in which the degree of correlation is relatively low.

Also, the model reinforces the intuitive belief that it is worth waiting till the last moment before mak- ing a decision regarding taking up or abandoning a second-stage investment project. The motivation for waiting is that delaying the secondary investment decision increases the likelihood of obtaining more information about the second-stage project, and thus making more informed decisions. By the same logic, projects which allow a larger time gap between pri- mary and secondary investments are more desir- able than those in which the primary and second- ary investment decisions are fairly close together in terms of time, provided other factors are the same.

The model presented in this paper need not be restricted in applicability to expansion-flexible manufacturing systems. It is valid in other contexts

that lend themselves to being modeled as primary and secondary investments.

8. Conclusions and future research

A model for exchanging risky assets originally proposed in the context of financial applications has been adapted to the valuation of flexibility in expansion-flexible manufacturing systems. This represents a novel approach for the valuation of flexibility. We illustrate that using the NPV ap- proach can often undervalue the value of an FMS investment project, and present a method of including the value of expansion flexibility in the economic evaluation of the project. The undervalu- ation is because traditional financial methods like NPV do not recognize managerial options avail- able to delay, stop, or otherwise alter the course of the project. Several managerial implications of this approach for investment decisions relating to expansion-flexible manufacturing systems were presented. This paper has focused on two-period investment scenarios. In general, a primary invest- ment could result in an option to make a secondary investment, a secondary investment could result in an option to make a tertiary investment, and so on. This is analogous to a compound financial option. It is possible to use the model presented in this paper as a rolling two-period model to deal with compound options. Alternatively, this scenario could be examined using the theory of compound financial options. More sophisticated model devel- opment may be possible.

Acknowledgements

The author would like to thank the anonymous referees and the editor for their comments and suggestions. This research was supported in part by funds provided by the University of North Carolina at Charlotte.

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