European Journal of Operational Research 202 (2010) 196–203

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

Uncertainty and stepwise investment

Peter M. Kort a,b,*, Pauli Murto c, Grzegorz Pawlina d

a Department of Econometrics and Operations Research and CentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlandsb Department of Economics, University of Antwerp, Prinsstraat 13, 2000 Antwerp 1, Belgiumc Department of Economics, Helsinki School of Economics, P.O. Box 1210, FIN-00101, Finlandd Department of Accounting and Finance, Lancaster University, Lancaster LA1 4YX, UK

a r t i c l e i n f o

Article history:Received 21 August 2008Accepted 8 May 2009Available online 27 May 2009

Keywords:Investment analysisReal optionsCapital budgetingProject flexibilityDynamic programming

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2009.05.027

* Corresponding author. Address: Department of EResearch and CentER, Tilburg University, P.O. BoThe Netherlands. Tel.: +31 13 4662062; fax: +31 13 4

E-mail address: kort@uvt.nl (P.M. Kort).

a b s t r a c t

We analyze the optimal investment strategy of a firm that can complete a project either in one stage at asingle freely chosen time point or in incremental steps at distinct time points. The presence of economiesof scale gives rise to the following trade-off: lumpy investment has a lower total cost, but stepwiseinvestment gives more flexibility by letting the firm choose the timing individually for each stage. Ourmain question is how uncertainty in market development affects this trade-off. The answer is unambig-uous and in contrast with a conventional real-options intuition: higher uncertainty makes the single-stage investment more attractive relative to the more flexible stepwise investment strategy.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The relationship between economic uncertainty and investmentis one of the central issues in modern capital budgeting, as re-flected in the theory of real options (see, e.g., d’Halluin et al.(2007), Clark and Easaw (2007), Fontes (2008), Bøckman et al.(2008), and De Reyck et al. (2008)). The literature on real optionsemphasizes various forms of managerial flexibility in this context(see, e.g., Triantis and Hodder (1990) and Trigeorgis (1996)). Onthe basis of that literature, we are used to thinking that the valueof flexibility is positively related to the level of uncertainty. Forexample, the notion of a positive value of flexibility in an uncertainenvironment is presented in Chapter 1 of Trigeorgis (1996): ‘‘Asnew information arrives and uncertainty about market conditionsand future cash flows is gradually resolved, management may havevaluable flexibility to alter its initial operating strategy in order tocapitalize on favorable opportunities.”

In this paper we consider a particular (but common) form offlexibility and provide a result that calls for a refinement of thisview. We analyze a continuous-time model of investment that fol-lows closely the standard real-options framework. As in McDonaldand Siegel (1986), a firm can choose the optimal time to invest inan irreversible project whose present value depends on the sto-

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conometrics and Operationsx 90153, 5000 LE Tilburg,663280.

chastic market environment. The difference with the standardmodel is that, besides the possibility of undertaking the whole pro-ject at a single, freely chosen point in time, the firm can also com-plete it through incremental steps. We assume that there are scaleeconomies that give rise to the following trade-off: completing theinvestment in a single stage saves on total costs, while proceedingstepwise gives additional flexibility as the firm can respond toresolving uncertainty by choosing the investment timing individu-ally for each step.

Concerning the effect of uncertainty on this trade-off, the basicreal-options intuition seems to suggest that uncertainty favorsflexibility at the expense of scale economies. Dixit and Pindyck(1994) devote Section 2.5 of their book to this issue and indicatethat it is uncertainty that makes flexibility relevant in the firstplace: ‘‘When the growth of demand is uncertain, there is atrade-off between scale economies and the flexibility that is gainedby investing more frequently in small increments to capacity asthey are needed”. Yet our main result is the opposite: the higherthe level of uncertainty, the more attractive the lumpy investmentstrategy relative to the stepwise investment. In this context, there-fore, the payoff of the project being uncertain actually favors scaleeconomies at the expense of flexibility.

To understand this seemingly paradoxical result, it is necessaryto look beyond a superficial real-options intuition and to carefullyconsider what is meant by flexibility in different contexts. The clas-sical result is that the level of uncertainty is positively related tothe value of an option to undertake an irreversible investment.One may express this by saying that the option to invest is aflexible asset, and the value of this flexibility increases with

Fig. 1. The timeline of the lumpy and stepwise investments (the top and the bottomaxis in each pair, respectively) with a commitment to undertake stage 1 immedi-ately (Panel A) and without any constraints on the timing of stage 1 (Panel B).

2 In a similar context, Grenadier and Weiss (1997) show in a model with sequentiatechnological innovations that increased uncertainty favors waiting until the finatechnology is invented. The difference is that in their model uncertainty concerns thearrival time of an improved technology, whereas in our model it concerns the markeenvironment. Consequently, in their model the improved technology is adopted at anexogenously determined moment, while in our model the timing is endogenouslydetermined. It should also be noted that Grenadier and Weiss derive their result bynumerical simulations, while our results are derived analytically.

3 The difference is that in their model two projects of different scale are mutually

P.M. Kort et al. / European Journal of Operational Research 202 (2010) 196–203 197

uncertainty (option to invest has a positive vega1). In this state-ment, flexibility refers to the presence of an option to choose theinvestment timing freely as opposed to a commitment to undertakea particular course of action at a given moment. Similarly, considerthe choice between two investment strategies illustrated in Fig. 1(Panel A): either invest in the whole project now (lumpy investment)or complete a fraction of the project now and keep an option to fin-ish the remaining fraction later (a flexible alternative). Here flexibil-ity stands for the increased degree of freedom in adding capacity,when the firm is allowed to do this in two steps rather than in onelump. We study the relative value of the two investment strategiesin case the project has an uncertain payoff. This uncertainty can becaused, for instance, by (future) demand being stochastic. Again,higher uncertainty increases the relative value of flexibility via itsimpact on the value of the second-stage option. However, this com-parison rests crucially upon the fact that the lumpy investment facesa restriction on its timing (the implicit now-or-never assumption)and, by construction, cannot benefit from a higher payoff volatility.

In contrast, the model presented in this paper allows for anoptimal timing decision for both steps of the flexible project with-out placing any constraints on the timing of the lumpy projecteither, as illustrated in Fig. 1 (Panel B). So, in this paper flexibilityrefers to the possibility of choosing the timing of each step individ-ually as opposed to the more restrictive case where a single timingdecision must apply for the entire project. This differs from a ‘‘stan-dard” notion of flexibility that refers to the possibility to choose thetiming for a project (or a part of a project) freely as opposed to thecommitment to invest immediately.

Why is it then that the value of flexibility reduces with uncer-tainty in our setting? We suggest two intuitions. The first one fol-lows from the theory of option pricing. Both investmentopportunities, the lumpy and the stepwise one, can be viewed asperpetual American options to acquire the same underlying asset(the final full-scale project). When uncertainty increases, the val-ues of both options converge towards the value of the underlyingasset. Consequently, the advantage of being able to choose the tim-ing of each stage separately instead of choosing the timing for bothstages jointly diminishes with uncertainty.

1 Recall that an option vega is the first-order derivative of the option value withrespect to the volatility of the underlying asset and is positive in a typical case.

exclusive: the stepwise investment strategy requires full replacement of the smalplant by the large plant. Our model allows one to proceed to the final ‘‘large” projeceither through one lump or through incremental steps with an arbitrary relationshipbetween investment costs of various alternatives.

The second intuition builds on the well known insight of real-options theory according to which increased uncertainty resultsin more inertia, that is, a decision maker is more reluctant to makecostly switches between states as a response to changes in theunderlying stochastic variable (cf. the hysteresis effect in the en-try–exit model of Dixit (1989)). Our problem can be viewed as aspecial kind of a capacity switching problem. Stepwise investmentstrategy contains two switches: the first switch increases capacityfrom zero level to intermediate level, and a later switch from inter-mediate to the final level. Lumpy investment strategy, on the otherhand, skips the intermediate level, and instead switches directly tothe full capacity mode, albeit after a longer waiting time. As uncer-tainty is increased, the firm is less willing to incur switching costs,which favors the lumpy investment strategy.

While our model is stylized, the result relates to many potentialapplications. As an example, think of a firm that considers entryinto a new market segment and faces the choice between twoalternative strategies: either proceed in multiple stages (for exam-ple, starting with the most profitable geographical market), or waituntil demand has grown enough to justify a single large-scale en-try. A related example is the construction of production capacity:the choice may be between installing one big unit and installinga number of smaller units. Or, consider the adoption of a new tech-nology: a firm can either start with an investment in an intermedi-ate technology, which allows a subsequent implementation of thenext-generation technology at a lower cost, or the firm may saveon total costs by waiting and later ‘‘leap-frogging” directly to thenext-generation technology. Our results indicate that, dependingon the context, increased uncertainty favors large-scale entry, largeproduction plants, and technology leapfrogging.2 Note also that in-stead of referring to one project to be undertaken in one or twosteps, our framework may just as well be interpreted as two distinctprojects either carried out separately or pooled together at a dis-counted total cost. A similar trade-off also appears in the purchasedecision of a consumer, who may buy different goods separatelyeach at its individually optimal time, or purchase them together ata discounted price. With that interpretation, our results indicate thatincreased uncertainty favors bundling of projects and purchases.

Our model is closely related to a number of earlier papers. It hasbeen recognized elsewhere that the degree of uncertainty influ-ences the firm’s choice between alternative investment strategies.Dixit (1993) analyzes the choice between mutually exclusive pro-jects of different sizes, and shows that increased uncertainty favorsa larger project. Related results on the relationship between uncer-tainty and sizes of investment projects appear already in Manne(1961), and later in Capozza and Li (1994), Bar-Ilan and Strange(1999) and Dangl (1999). Décamps et al. (2006), on the other hand,complete the analysis of Dixit (1993) by considering the state-con-tingent investment policy over the entire state space. Their mainfinding is that the interaction between mutually exclusive optionsgives rise to a dichotomous option exercise region (a property thatholds also in our model). While their research question is thus en-tirely different, the model they use is technically related to ours.3

ll

t

lt

198 P.M. Kort et al. / European Journal of Operational Research 202 (2010) 196–203

Another literature stream analyzes investment strategies in a R&Dcontext (see, e.g., Childs et al. (1998), Weitzman et al. (1981), Rob-erts and Weitzman (1981)). That literature, like our paper, high-lights the value of stepwise investment strategies, but its focus ison an entirely different mechanism than ours. In that context themain advantage of stepwise investment is the additional informa-tion that completing a step generates for choosing subsequent ac-tions. In our model there is no learning; the advantage is generatedby the improved flexibility to choose investment timings as a re-sponse to an exogenously changing environment.

The remainder of the paper is organized as follows. Section 2describes the model, while Section 3 derives the optimal invest-ment policy. Section 4 explains how the policy is affected by thekey model parameters, Section 5 discusses some generalizations,and Section 6 concludes. All proofs are relegated to the Appendix.

2. Model

Our framework is based on the standard model of irreversibleinvestment under uncertainty as presented in McDonald and Siegel(1986), and further analyzed in a large number of contributions (anextensive overview of this literature is provided in Dixit and Pin-dyck (1994) as well as in Brennan and Trigeorgis (2000)).

Consider a risk-neutral, all-equity financed firm of given size,which operates in continuous time with an infinite horizon anddiscounts its revenues with a constant rate r. (Instead of risk neu-trality, we could assume that the payout from the project can bereplicated by a portfolio of traded assets.) The firm earns initiallyno revenue, but has an opportunity to invest in a single capitalbudgeting project. This project can be accomplished either in onestage (which is referred to as lumpy investment) or in two separatestages (stepwise investment). The timing of the project and the typeof investment (lumpy vs. stepwise) is to be chosen optimally in or-der to maximize the value of the firm.

Denote by t 2 ½0;1Þ the time index. The uncertain payoff of theproject is characterized by state variable Yt that follows a geomet-ric Brownian motion:

dYt ¼ lYtdt þ rYtdxt; ð1Þ

where Y0 > 0;l < r;r > 0, and the dxs are independently andidentically distributed according to a normal distribution withmean zero and variance dt.

We assume that the initial value Y0 is so low that at time t ¼ 0 itis not yet optimal for the firm to undertake the project (in neitherlumpy nor stepwise fashion). However, since from (1) we learnthat Y is fluctuating over time, with positive probability it will beoptimal for this firm to invest at a later point of time. The fact thatthe firm will not invest at time zero will allow us to make our com-parisons between various investment strategies simply on the ba-sis of their respective value functions calculated at Y0. We thinkthat the assumption of a low initial value for Y is very natural inthe current context: to analyze how uncertainty affects the choicebetween various investment strategies, we want to model the con-ditions under which the investment becomes optimal for the firsttime. If the initial value were higher than that, there should besome reason for why the project has not yet been implemented be-fore the ‘‘initial” time. We recognize, however, that analyzing thewhole investment policy valid for arbitrary initial states wouldbring in additional aspects (see Décamps et al. (2006)).

The firm’s revenues are modeled as follows. Initially, the firmgenerates no revenues. Once k 2 f1;2g stages of the project arecompleted, the firm earns an instantaneous revenue flow:

pt ¼ Yt

Xk

i¼1

Ri; ð2Þ

where Ri is a constant denoting the deterministic part of the reve-nue increment corresponding to stage i. Define R � R1 þ R2. Byaccomplishing the project in a single step, the firm moves at somestopping time tL directly from revenue flow 0 to YtL R (lumpy invest-ment), while by splitting the project, the firm moves first at somestopping time t1 from 0 to Yt1 R1, and at a later stopping time t2 fromYt2 R1 to Yt2 R (stepwise investment). The cost of investment is deter-ministic and depends on whether the project is accomplished in oneor two steps. In case of lumpy investment, the total investment costis simply I. In case of stepwise investment, the associated invest-ment costs for the first and second steps are I1ð< IÞ and I2, respec-tively. The firm’s optimal strategy can thus be characterized asthe following maximization problem (recall that Y0 is so low thatit is not optimal to invest at t ¼ 0):

FðY0Þ ¼maxfFLðY0Þ; FSðY0Þg

¼max suptLP0

E

Z 1

tLe�rtYtRdt � e�rtL

I� �

;

(

supt1P0

E supt2Pt1

E

Z t2

t1e�rtYtR1dt � e�rt1

I1

þZ 1

t2e�rtYtRdt � e�rt2

I2

!!); ð3Þ

where tL; t1, and t2 are stopping times adapted to Yt . The first term inthe brackets, FLðY0Þ, is the expectation of the discounted future reve-nues if the lumpy investment is chosen. Here, the firm chooses thestopping time tL at which the project is undertaken. The second term,FSðY0Þ, corresponds to the stepwise investment. Then, the firm choosestwo stopping times, t1 and t2, corresponding to stages 1 and 2 of theproject, respectively. Whether the firm chooses the lumpy or the step-wise alternative depends on which of the two terms is greater.

We adopt the following assumptions on the investment costsand revenues. First, we assume that completing the project intwo steps is more costly than investing in one lump and define

j � ðI1 þ I2Þ=I P 1: ð4Þ

Consequently, j represents the premium for flexibility that the firmmust pay in order to be able to split the project. Second, withoutloss of generality, we assume that

I1

I2<

R1

R2: ð5Þ

As it becomes clear later, this implies that even if we interpret themodel so that the firm is free to choose the order in which the twosteps are undertaken, the step that will be optimally completed firstis the one labelled with subscript 1. We only assume away the triv-ial case R1

R2¼ I1

I2, which would imply that it is always optimal to

undertake the two steps at the same point of time. In that casethe firm does not benefit from the possibility to split the project,and the lumpy project with no cost premium would always domi-nate. (The latter conclusion would also hold if the firm were notable to first invest in the more profitable stage of project due to,for example, technological constraints.) Our aim is thus to focuson the effect of uncertainty on the trade-off between economicsof scale and flexibility. To keep the analysis as simple as possible,we abstract from incorporating elements like operating costs,although we are well aware that operating costs can influence thistrade-off. This is because a single-stage investment will put muchstress on the organization with increased operating costs as a result.On the other hand, investing stepwise will allow for gradual adap-tation of routines and a gradual build up of internal competence.

To facilitate the communication of our results, we divide theparameters of the model into two classes. First, parameters l;r,and r describe the economic environment in which the firmoperates, and we call them the market-specific parameters. Second,

4 See Bar-Ilan and Strange (1998) for a more complex model of sequentiainvestment that incorporates investment lags.

5 This result is due to the special structure of optimal stopping problems that alsounderlies the main conclusions of Leahy (1993) and Baldursson and Karatzas (1997)according to which an investor, who must take into account subsequent investmentsof the competitors, employs the same investment policy as a monopolist who is nothreatened by such future events.

P.M. Kort et al. / European Journal of Operational Research 202 (2010) 196–203 199

the parameters R1;R2; I1; I2, and I describe the project under consid-eration, and we call them the project-specific parameters. Our pur-pose is to show how changes in market-specific parametersaffect the regions in the space of project-specific parameters inwhich each of the two alternative investment strategies (lumpyvs. stepwise) dominates. This approach will provide an unambigu-ous answer to our main question, that is, how the degree of uncer-tainty affects the optimal choice between the stepwise and lumpyinvestment and, hence, the relative value of flexibility.

3. Optimal investment policy

In this section, we derive the optimal solution to the projectselection problem (3) in three steps. First, we consider the casewhere only the lumpy investment alternative is available. Second,we derive the optimal investment policy for the stepwise invest-ment case. Finally, we consider the general problem where the firmhas to decide about both the timing and the type of investment.

3.1. Lumpy investment

Consider the case in which the project can be undertaken in asingle step only. Then the value of the investment opportunity isthe first term between the brackets in (3), that is, the problem isto choose tL optimally to yield the value FLðY0Þ:

FLðY0Þ ¼ suptLP0

E

Z 1

tLe�rtYtRdt � e�rtL

I� �

:

This case corresponds exactly to the basic model of investment un-der uncertainty as described in McDonald and Siegel (1986). Theoptimal investment policy is a trigger strategy such that it is optimalto invest whenever the current value of Y is above a certain thresh-old level, which we denote by YL. Thus, the optimal investment timeis tL ¼ infft P 0jYt P YL:g. The standard procedure to solve theproblem is to set up the dynamic programming equation for the va-lue function FLðY0Þ, where the application of Itô’s lemma and appro-priate boundary conditions are used to determine the exact form ofFLðY0Þ and the value of YL. We merely state the result here, see, e.g.,Dixit and Pindyck (1994) for details. The investment threshold is

YL ¼b

b� 1IRðr � lÞ; ð6Þ

where

b ¼ 12� l

r2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12� l

r2

� �2

þ 2rr2

s> 1; ð7Þ

and the value of the option to invest is

FLðY0Þ ¼YLR

r � l� I

� �Y0

YL

� �b

: ð8Þ

Expression (8) is valid when initial demand is so low that it is notoptimal to invest right away, that is when Y0 < YL.

3.2. Stepwise investment

Now, consider the case in which the firm splits the project intotwo stages. Then the value of the investment opportunity is thesecond term between the brackets in (3), that is, the problem isto choose t1 and t2 optimally to yield the value FSðY0Þ:

FSðY0Þ

¼ supt1P0

E supt2Pt1

E

Z t2

t1e�rtR1dt � e�rt1

I1 þZ 1

t2e�rtRdt � e�rt2

I2

! !:

ð9Þ

The option to invest in the first stage may be seen as a compoundoption, since accomplishing it generates an option to proceed tothe next stage.4 However, since the instantaneous revenue (2) isadditive in the revenue flows associated with each stage, the prob-lem can be represented as two separate investment problems. Thiscan be seen by re-writing (9) as

FSðY0Þ ¼ supt1P0

E

Z 1

t1e�rtR1dt � e�rt1

I1

� �þ sup

t2Pt1E

Z 1

t2e�rtR2dt � e�rt2

I2

� �: ð10Þ

Expression (10) implies that the problem is decomposed into twostopping problems, which are only linked through the constraintt2 P t1. For the moment, ignore this constraint, and note that eachof the two resulting problems is identical to the one considered inSection 3.1. Therefore, without constraint t2 P t1, the solution mustconsist of two investment thresholds, Y1 and Y2, given by

Y1 ¼b

b� 1I1

R1ðr � lÞ; ð11Þ

Y2 ¼b

b� 1I2

R2ðr � lÞ: ð12Þ

Comparing these expressions, one can see immediately thatY1 < Y2 under our assumption that R1

R2> I1

I2. Therefore, concerning

the corresponding stopping times ti ¼ infft P 0jYt P Yi:g; i 2f1;2g, it must hold that t2 > t1, which means that the constraintt2 P t1 is automatically satisfied. We conclude that the first stageis accomplished strictly earlier than the second stage, and thatthe existence of stage 2 has no effect on the optimal exercise timeof stage 1, meaning that the two stages can be considered sepa-rately.5 We denote the values of the options to invest separatelyfor the two stages as F1ðY0Þ and F2ðY0Þ. Analogously to (8), thesecan be written as

F1ðYÞ ¼Y1R1

r � l� I1

� �Y0

Y1

� �b

; ð13Þ

F2ðYÞ ¼Y2R2

r � l� I2

� �Y0

Y2

� �b

; ð14Þ

and they are applicable for Y0 < Y1 and Y0 < Y2, respectively. The va-lue of the (compound) option to invest sequentially can be written as:

FSðY0Þ ¼ F1ðY0Þ þ F2ðY0Þ

¼ Y1R1

r � l� I1

� �Y0

Y1

� �b

þ Y2R2

r � l� I2

� �Y0

Y2

� �b

; ð15Þ

which is again valid as long as Y0 is low enough, in particular, whenY0 < Y1.

3.3. General problem

So far, we have determined the option values and the optimalinvestment thresholds for the two investment strategies (lumpyand stepwise) separately. Now we consider the general problem(3). Since we have assumed that the initial value Y0 is so low thatit is not optimal to undertake the project (in any mode) at t ¼ 0, thevalue of the firm is simply FðY0Þ ¼maxfFLðY0Þ; FSðY0Þg, where theexpressions for FLðY0Þ and FSðY0Þ are given by (8) and (15), respec-

l

,

t

200 P.M. Kort et al. / European Journal of Operational Research 202 (2010) 196–203

tively. Our aim is to establish conditions that determine which ofthese expressions is greater. Since we are interested in the trade-off between the economies of scale and flexibility, we want to statethe relation of the option values in terms of the parameter j thatrepresents the cost premium that must be paid by the firm forthe flexibility of splitting the investment.

The following proposition states that there is a single cut-off va-lue such that if j is below that level, the option value correspond-ing to the stepwise investment strategy dominates that of lumpyinvestment, while the reverse is true for j above that level. Notethat a similar interpretation of the domination relation of mutuallyexclusive options is implicitly adopted, for example, in Dixit(1993).

Proposition 1. Let Y0 < Y1. There exists a critical level of theinvestment cost premium j > 1 such that when j ¼ j, we haveFSðY0Þ ¼ FLðY0Þ, that is, the lumpy and stepwise investment strategiesare equally good. The critical premium j can be expressed as follows:

j ¼ cPb�11 þ ð1� cÞPb�1

2

h i 1b�1; ð16Þ

where

c � R1

Rð17Þ

is the fraction of the payoff of the project generated after completingstage 1, and

Pi �Ri=Ii

R=ðI1 þ I2Þð18Þ

is the relative profitability of (moneyness of the option associated with)stage i. For j < j, we have FSðY0Þ > FLðY0Þ, whereas for j > j, wehave FLðY0Þ > FSðY0Þ.

Proof. See the Appendix. h

Proposition 1 gives us an unambiguous dominance relation be-tween the lumpy and stepwise investment. Note that bj dependson the market-specific parameters l;r, and r through their effecton parameter b only. Hence, b aggregates the effect of the environ-ment in which the firm operates on the choice between the lumpyand stepwise investment strategies.

The influence of the project-specific factors is captured by threeparameters: the fraction of the total payoff from the project thatcan be attributed to the stage 1 investment, c, and the profitabilityof each stage relative to the profitability of the project as a whole,Pi. This profitability can also be directly interpreted as the relativemoneyness of the option to invest in stage i.

4. Effect of uncertainty

Our main objective is to show how the choice between lumpyand stepwise investment strategies depends on market-specificparameters. An increase (a decrease) in j is equivalent to a reduc-tion (an expansion) of the set of project-specific parameter valuesunder which the lumpy investment dominates (is dominated by)the stepwise investment. This leads to the interpretation that jrepresents the cost advantage of the lumpy investment requiredto compensate for the loss of flexibility in timing each step of theproject separately. Thus, an increase (a decrease) in j is equivalentto a higher (lower) relative value of additional flexibility associatedwith stepwise investment.

From (16), we see that b captures the effects of all market-spe-cific parameters. The next proposition states our main result:

Proposition 2. Consider the critical cost premium j as a function ofb. Then, the following relationship holds:

@j@b

> 0:

This implies that the relative value of stepwise investment is negativelyrelated to the volatility and to the drift rate of process (1), but positivelyrelated to the interest rate:

@j@r

< 0; ð19Þ

@j@l

< 0; ð20Þ

@j@r

> 0: ð21Þ

Proof. See the Appendix. h

Eq. (19) embodies the main result of this paper: increaseduncertainty reduces the premium that the firm is willing to payfor the additional flexibility that the stepwise investment gives.The effects of the drift rate and interest rate, as given by (20) and(21) respectively, are less surprising, but they are neverthelessrevealing. The intuition for (20) is that if the rate at which the pres-ent value of the project grows is reduced (l is reduced), the cost ofdelaying investment until it is optimal to undertake both stages to-gether is increased, which makes the stepwise investment moreattractive. This is quite obvious, but it completes our main argu-ment: it is rather the fact that the growth is gradual (slow) thatmakes stepwise investment valuable in this context, not the factthat growth is uncertain. Of course, the effect of the interest rate,as expressed in (21), can be explained in a similar way: discountingrevenues more heavily makes it more costly to delay investmentuntil both stages are optimally undertaken together. Therefore,the relative value of stepwise investment is higher.

To understand the main result given in (19), it is helpful to usethe following analogy. First, notice that (16) can be rearranged as:

jb�1 ¼ cPb�11 þ ð1� cÞPb�1

2 : ð22Þ

The left-hand side reflects the premium on the investment costwhen selecting the stepwise investment, whereas the right-handside is the benefit from timing the investment in each stage of theproject optimally. The cut-off level of the premium, j, can be there-fore interpreted as the certainty equivalent of random payoutPi; i 2 f1;2g, (occurring with probability c and 1� c), when theutility function is of the form uðxÞ ¼ xb�1. It is straightforward tosee that the certainty equivalent decreases with the concavity ofthe utility function. In the analyzed case, the latter is negatively re-lated to b.

It is useful to think of our main result in yet another way. Forj ¼ 1 and finite r, the value of the lumpy investment opportunityis always smaller than the value of the option to invest in a step-wise fashion (cf. (8) and (15)). When uncertainty tends to infinity,the values of both investment opportunities converge to the valueof the underlying asset, that is, the present value of revenues gen-erated in perpetuity by the full-scale project. (In fact, it is easy toshow that the limit of (8) and (15) for r tending to infinity is equalto YR=ðr � lÞ.) Therefore, it is not surprising that for higher uncer-tainty the willingness to pay a premium for the ability to invest in astepwise fashion is lower. (For a discussion of the limiting proper-ties of j as volatility tends to infinity and zero, see the supplemen-tary web appendix.)

Fig. 2 provides a numerical illustration of the model. It depictsthe relationship between the cut-off level of the flexibility pre-mium, j and the volatility of the revenue flow, r, with the follow-ing parameter values. The riskless interest rate is assumed to be 5%(which is approximately equal to the average yield on long-termtreasury bonds based on the period 1926–1999, as reported in Bod-ie et al. (2002)) and the drift rate of the project is assumed to be

Fig. 2. Flexibility premium cut-off level, j, as a function of payoff volatility, r, fordifferent proportions of the payoff generated by stage 1, c. Other parameter valuesare as follows: a ¼ 0:5; r ¼ 0:05, and l ¼ 0:015.

Fig. 3. Flexibility premium cut-off level, j, as a function of payoff convexity, h, fordifferent numbers of stages, n. Other parameter values are as follows:a ¼ 0:5; ci=ciþ1 ¼ 1:5; r ¼ 0:05;l ¼ 0:015, and r ¼ 0:2.

P.M. Kort et al. / European Journal of Operational Research 202 (2010) 196–203 201

equal to 1.5% (implying a 3.5% return shortfall, which is approxi-mately consistent with a long-term dividend yield as reported byAllen and Michaely (2002)). The proportion of the total investmentcost incurred in each stage is one half ða ¼ 0:5Þ. Consistent withthe results of our model, we can see that the relationship betweenproject flexibility (measured by j) and uncertainty ðrÞ is negative.Assuming that I ¼ $100, and that between three and four fifths ofthe revenue is generated following the completion of stage 1, themaximum amount that the firm is willing to pay for flexibilityranges between $6 and $46 in the case of no uncertainty, and be-tween $3 and $32 for r ¼ 0:2 (the latter corresponds to the stan-dard deviation of equity returns of a representative S&P 500 firm,see Bodie et al. (2002)).

5. Extensions

Admittedly, the model is very simple and it is not obvious towhat extent our main result is driven by the current specificationof the project’s payoff. For example, one might suspect that if, in-stead of a linear dependence like in our basic model, the project va-lue is concave in the stochastic payoff Yt , then the sign of therelationship between uncertainty and the value of flexibility willbe reversed. This could be the case if the effect of a higher upsidepotential associated with a higher payoff volatility was dominatedby its decreasing marginal contribution to the present value of theproject. On the other hand, the notion of project flexibility is oftenassociated with the possibility to revise investment decisions fre-quently (Trigeorgis, 1996), rather than merely adding a second stepto an indivisible investment project. To deal with these issues, thissection extends our result by considering a more general, powerspecification of the revenue function and by allowing for an arbi-trary number of steps for which the timing can be independentlychosen.

To account for a nonlinear impact of the market environment onthe present value of the project, we allow the revenue flow to havea power specification (cf. (2)):

pt ¼ Yt

Xn

i¼1

Ri

!h

; h 2 ð0;bÞ; ð23Þ

with h < b required for obtaining finite valuations. For h < 1, thefirm is facing a concave revenue function, which may reflect, forexample, constraints on input availability or, to some extent, imper-fect hedging opportunities of risk averse shareholders. The case ofh > 1 may arise because of the firm’s ability to respond to uncer-tainty, for example, by changing both price and quantity in responseto demand fluctuations. In particular, one could think of a monop-

olist facing an isoelastic inverse demand function pt ¼ Ytq�1=ht (pt

is price, qt is quantity). Assuming that the firm has constant mar-ginal production cost ci, and no capacity constraint, the revenueflow with optimally chosen output level is given by pt ¼Yhc1�h

i h�1 1� 1h

� �h�1, which is the same as (23) with an appropriatelydefined correspondence between Ri and ci. In that case, investmentoutlays should be interpreted as adoptions of different technologiesthat allow different marginal production costs ci.

Adopting the payoff function (23) with n ¼ 2 instead of (2) is infact equivalent to transforming the stochastic process from Y to Yh

(with the relabeling of the parameters R1 and R2 as R01 � Rh1 and

R02 � Rh � Rh1). Hence, the way to see that the results of Sections 3

and 4 generalize to the power function is to observe that Yh alsofollows a geometric Brownian motion but just with different driftand volatility terms than the original process Y . In fact, one canshow (by applying Itô’s lemma) that if Y follows (1), the processYh is a geometric Brownian motion with drift and volatility termsequal to l0 ¼ hlþ 1

2 hðh� 1Þr2 and r0 ¼ hr, respectively. Hence,

we may write b0 ¼ 12�

l0r02 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12�

l0r02

� �2 þ 2rr02

q¼ b

h. The condition

h < b ensures that l0 < r, that is, that all the relevant valuationsare finite.

In order to incorporate the notion that project flexibility isequivalent to the possibility of revising investment decisions fre-quently (Trigeorgis, 1996), we consider the optimal capital budget-ing strategy when the firm is able to split the project into anarbitrary number of stages. The instantaneous revenue flow isnow given by (23) with n P 2.

Define

Sk �Xk

i¼0

Ri ð24Þ

for k > 0, with R0 � 0. In order to ensure that an investor chooses toaccomplish stage k at a strictly earlier time than stage kþ 1, wemust have

Ik

Ikþ1<

Shk � Sh

k�1

Shkþ1 � Sh

k

� R0kR0kþ1

; 8k < n: ð25Þ

Now, it is possible to generalize the formula for the cut-off value ofthe flexibility premium (see the supplementary web appendix):

j ¼Xn

i¼1

c0iP0bh�1i

! 1bh�1

; ð26Þ

where c0i � R0i=R0 and P0 is defined in the same way as P with RðRiÞreplaced by R0ðR0iÞ. The proof that j increases with b, thus decreaseswith r is now analogous to the proof of Proposition 2.

202 P.M. Kort et al. / European Journal of Operational Research 202 (2010) 196–203

To provide an illustration, Fig. 3 depicts the cut-off level of theflexibility premium as a function of payoff convexity h for differentvalues of n. The cut-off level j decreases with the convexity of therevenue function. This implies that a firm holding a project with amore concave payoff function will, in fact, be willing to incur ahigher additional costs to preserve flexibility in the investment pro-cess. The intuition for that result is that concavity increases the rel-ative value of the initial stages of the project. This effect increasesthe value of the sequential strategy relative to the lumpy invest-ment, simply because the stepwise strategy makes it possible toundertake the (more profitable) initial stages only. Furthermore,consistent with intuition, the value of the project flexibility is high-er for a larger number of stages. For I ¼ $100; ci ¼ 1:5ciþ1 andai ¼ 1=n, the maximum level of flexibility premium ranges be-tween $40 and $117 for a concave ðh ¼ 0:5Þ, $3 and $17 for a linearðh ¼ 1Þ, and between less than $1 and $6 for a convex payoffðh ¼ 1:25Þ.

6. Conclusion

We analyze the optimal investment strategy of a firm that caneither accomplish the project in one lump or proceed in steps.The former strategy has the advantage of scale economies: the to-tal investment cost is lower. The latter benefits from additionalflexibility: the timing of each step can be chosen independentlyof each other. Our focus is on the effect of uncertainty on thetrade-off between the two strategies. The result is that increaseduncertainty favors the lumpy investment relative to the stepwiseinvestment.

One way to express our main result is to say that increaseduncertainty reduces the value of flexibility associated with step-wise investment. This is in contrast with the standard real-optionsintuition, which says that increased uncertainty typically favorsflexibility. Our key message is that such a statement – that is,uncertainty favors managerial flexibility – is not to be taken as ageneral fact. In particular, one must be careful in specifying whatis meant by flexibility. If flexibility refers to the presence of an op-tion to act later as opposed to the commitment to act now, then thestandard result certainly holds: the value of flexibility increaseswith uncertainty. But if, as we show in this paper, flexibility refersto the possibility to split a project into a number of steps and tochoose the timing of each step individually, then we have theopposite result: increased uncertainty reduces the value offlexibility.

Acknowledgement

We thank two anonymous referees, Engelbert Dockner, DavidHobson, Ashay Kadam, participants of the Annual InternationalReal Options Conference (Montréal), EFA (Ljubljana), EFMA (Mi-lan), and seminar participants at Antwerp, Cass (London), Exeter,Tilburg, Warsaw and Warwick for helpful comments and sugges-tions. All remaining errors are ours.

Appendix A

Proof of Proposition 1. We begin by comparing the two optionvalues FLðYÞ and FSðYÞ. It holds (cf. (8) and (15)) that

FSðYÞFLðYÞ

¼Y1R1r�l � I1

� �Y

Y1

� �bþ Y2R2

r�l � I2

� �Y

Y2

� �b

YLRr�l� I� �

YYL

� �b

¼ R�b I1 þ I2

j

� �b�1 Rb1

Ib�11

þ Rb2

Ib�12

!; ðA:1Þ

since I1 þ I2 � jI, and investment thresholds YL;Y1, and Y2, are gi-ven by (6), (11), and (12), respectively. Eq. (16) follows directly from(A.1). Since b is always greater than 1, and all terms in (A.1) are po-sitive, it holds that @

@jFSðYÞFLðYÞ

� �< 0. This implies that FSðYÞ

FLðYÞ¼ 1 if and

only if j ¼ j and that the inequalities stated in the propositionhold.

Now, in order to prove that j > 1, we show that theFSðYÞ > FLðYÞ for j ¼ 1. Recall that a and c are defined (for an arbi-trary j) as

I1 � aIj; ðA:2ÞR1 � cR: ðA:3Þ

Of course, (A.2) and (A.3) imply that I2 ¼ ð1� aÞIj andR2 ¼ ð1� cÞR. Define

DðbÞ � cb

ab�1 þð1� cÞb

ð1� aÞb�1

!; ðA:4Þ

which equals the ratio of FSðYÞ and FLðYÞ for j ¼ 1. It can easily beseen that limb!1DðbÞ ¼ 1. To show that DðbÞ > 1 for b > 1, we calcu-late the following derivative:

@DðbÞ@b

¼ @

@bcb

ab�1 þð1� cÞb

ð1� aÞb�1

!

¼ cca

� �b�1ln

caþ ð1� cÞ 1� c

1� a

� �b�1

ln1� c1� a

: ðA:5Þ

For c # a, (A.5) is equal to zero. Therefore, in order to prove that(A.5) is positive, it is sufficient to show that

@

@cc

ca

� �b�1ln

caþ ð1� cÞ 1� c

1� a

� �b�1

ln1� c1� a

" #ðA:6Þ

is positive. Differentiating (A.6) and rearranging yields

ca

� �b�1b ln

ca� 1� c

1� a

� �b�1

b ln1� c1� a

þ ca

� �b�1� 1� c

1� a

� �b�1

> 0:

The last inequality results from the fact that the first three compo-nents are positive and that c

a > 1 > 1�c1�a. Consequently, for j ¼ 1 and

b > 1, the value of the sequential investment opportunity is higherthan the value of the lumpy project. Since (A.1) decreases with j; jis greater than 1. h

Proof of Proposition 2. j can be expressed as

j ¼ cPb�11 þ ð1� cÞPb�1

2

� � 1b�1: ðA:7Þ

Let us choose two arbitrary values of b, say b0 and b00, such thatb0 > b00, and define

d � b0 � 1b00 � 1

> 1:

It holds that

cPb00�11 þ ð1� cÞPb00�1

2 ¼ cPb0�1

d1 þ ð1� cÞP

b0�1d

2

< cPb0�11 þ ð1� cÞPb0�1

2

� �1d;

where the last inequality results from the fact that y1d is a concave

function. This implies the following inequality:

cPb00�12 þ ð1� cÞPb00�1

2

� �d< cPb0�1

1 þ ð1� cÞPb0�12 :

It follows immediately that

cPb00�11 þ ð1� cÞPb00�1

2

� � 1b00�1

< cPb0�11 þ ð1� cÞPb0�1

2

� � 1b0�1:

P.M. Kort et al. / European Journal of Operational Research 202 (2010) 196–203 203

Defining b0 � b00 þ Db1 and letting Db tend to zero leads to the con-clusion that @j=@b > 0: Results (19)–(21) follow from the fact that@b=@r < 0; @b=@l < 0, and @b=@r > 0, respectively.

Appendix B. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.ejor.2009.05.027.

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