Discussion Papers In Economics
And Business

Valuation of sequential R&D investment undertechnological, market, and rival preemption uncertainty
Michi NISHIHARA
Discussion Paper 14-13

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANValuation of sequential R&D investment undertechnological, market, and rival preemption uncertainty
Michi NISHIHARA
Discussion Paper 14-13

April 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANValuation of sequential R&D investment undertechnological, market, and rival preemption uncertainty∗
Michi NISHIHARA†

Abstract
We develop a real options model for evaluating and optimizing an R&D project. The
model can capture key features of R&D, including research duration, growth opportunity,
debt ﬁnancing, and uncertainty of technological, demand market, and rival preemption.
Nevertheless, it is computationally tractable and thus helps practitioners to evaluate various cases of R&D investment. Further, by analyzing the model with a wide range ofparameter values, we unveil the interactions of key R&D features. The eﬀect of duration
on investment depends on whether there is a possibility of rival preemption. Higher uncertainty of research duration speeds up project inception in the presence of rival preemption.
Higher uncertainty of technological success, combined with a growth opportunity embedded in the R&D project, accelerates investment. Debt ﬁnancing can greatly decrease timelag between the ﬁrst stage project and growth project. These results are consistent withthe empirical evidence.
JEL Classifications Code: G31; G33; 032.
Keywords: R&D; technological uncertainty; investment lag; real options; compound
option.

∗†This is the ﬁrst version on 1 April, 2014.
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,

Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277

1

1Introduction

Research and development (R&D) investment does not only bring the progress of society
via innovation and spillover eﬀects but also is a key determinant of a ﬁrm’s long-term
growth. It is critical for a ﬁrm’s management to accurately evaluate and execute an R&D
project. This paper contributes both theory and practice by developing a tractable modelfor evaluating and optimizing R&D investment as well as unveiling interactions of several
features associated with R&D investment.
The diﬃculties of R&D management lie in that three types of uncertainty are inevitably embedded in R&D projects. The ﬁrst is technological uncertainty. Technological
speciﬁcations, time schedule, and budget are planned before project initiation. However,in many cases, the outcome does not go successfully as planned (e.g. Raz, Shenhar, andDvir (2002)). The risks of technological success, research duration, and investment costs
are called technological uncertainty. The second is market uncertainty. This stands for
uncertain cash ﬂow, which a newly developed technology will generate. The dynamics ofcash ﬂow is not deterministic but aﬀected by product-speciﬁc and macroeconomic shocks
on demand in the product market. The third is a risk of rival preemption. For instance,
if a competitor takes out a patent for a technology ﬁrst, an R&D project concerning thetechnology will be aborted.
In practice, it is challenging to evaluate an R&D project and make a managerial
decision of R&D involving these risks (e.g. Raz, Shenhar, and Dvir (2002)). Although
the Net Present Value (NPV) method remains dominant for project valuation in the realworld, there is an observed growing trend that the real options method1 is adopted as
a complement (e.g. Hartmann and Hassan (2006) and Baker, Dutta, and Saadi (2011)).
Without doubt, the academic literature argues that the real options methodology adds
value to project valuation involving high uncertainty and managerial ﬂexibility, such as
R&D projects. There have been a lot of case studies that apply the real options method to
R&D project valuation (e.g., Perlitz, Peske, and Schrank (1999), Loch and Bode-Greuel(2001), Lee and Paxson (2001), and Cassimon, Backer, Engelenc, Wouwee, and Yordanovf(2011)).
Instead of examining a case study for a speciﬁc company or project, this paper develops
a generic and tractable model of R&D investment so that one can analyze various casesof R&D investment with it. We extend the framework of American compound option as
follows. Consider a ﬁrm that has the timing option to initiate an R&D project by paying
a sunk cost. The project will take time to complete, and after completion, the developed
technology will generate cash ﬂow and a growth opportunity. When the ﬁrm exercises
the growth option by paying a sunk cost, it will increase cash ﬂow from the technology.
1

The real options method is no longer a new concept because it has already been developed for morethan thirty years. For details, we recommend a recent textbook Guthrie (2009) to both academicians andpractitioners.

2

We consider this sort of compound option model, because growth opportunities, such as
expanding production, starting a license business, and applying the new technology toother products, are frequently associated with R&D investment (e.g., Loch and BodeGreuel (2001), Ho, Tjahjapranata, and Yap (2006), and Cassimon, Backer, Engelenc,
Wouwee, and Yordanovf (2011)).
Furthermore, our model takes into consideration the three types of uncertainty, namely,
technological, market, and rival preemption uncertainty. The level of technological successthat inﬂuences cash ﬂow, the lag between project inception and completion (henceforthresearch duration), and total costs are not deterministic but random, and the ﬁrm knows
only their prior distributions. These technological risks are speciﬁc of R&D project valuation, but the existent real options models miss any of the three points. We do not propose
any new setup with regard to market uncertainty. Instead, based on the standard literature, we assume that the dynamics of cash ﬂow from the technology follows a stochastic
process. We assume that the project value is potentially eliminated by rival preemption
before project completion. The rival preemption occurs randomly, and the ﬁrm knows
only its distribution. In other words, the ﬁrm does not possess the full information of its
rival ﬁrms’ R&D progress. We also extend the base model to a case with debt ﬁnancing
because an increasing number of papers, both theoretically and empirically, have pointed
out the signiﬁcance of ﬁnancing sources.2
Before elaborating on the results, we diﬀerentiate our model from the most related anddominant models by Bar-Ilan and Strange (1998) and Weeds (2002). The model by BarIlan and Strange (1998), like our model, can capture the eﬀects of a growth opportunity
and investment lag, but it does not include uncertainty of technological success, total costs,
duration, and rival preemption. Weeds (2002) proposed a model with uncertain research
duration and rival preemption, but the model does not include a growth opportunity anduncertainty of technological success and total costs. In addition, her assumptions of full
information of rival ﬁrms and Poisson distributed duration are not very practical. We also
note that the above models, unlike our model, do not include a case with debt ﬁnancing.
Thus, our model better helps evaluating an R&D project involving the three types of
uncertainty compared to the previous models.
Our analysis of the model yields several empirical implications. First, we unveil interactions between research duration and rival preemption. In the absence of rival preemption,
the ﬁrm tends to accelerate investment in the R&D project with longer duration because
market demand at completion time is expected to be higher. This result is known ininvestment timing models with investment lags (e.g., Bar-Ilan and Strange (1996) and
Bar-Ilan and Strange (1998)). Notably, we show that a slight possibility of rival preemption changes the duration eﬀect. Longer duration increases the possibility of rival2

An incomplete list includes Mauer and Sarkar (2005), Sundaresan and Wang (2007), and Nishihara andShibata (2013) in the real options studies, and Ho, Tjahjapranata, and Yap (2006), Brown, Fazzari, and Petersen(2009), and Hall and Lerner (2009) in the empirical studies of R&D.

3

preemption and decreases the project value. As this negative eﬀect is much stronger thanthe positive eﬀect, the eﬀect of duration on investment becomes negative in the presenceof rival preemption.
Next, we show that higher uncertainty of research duration and technological success drives the ﬁrm to launch the R&D project earlier. This leads to the lesson that
uncertainty-investment sensitivity depends on the type of uncertainty. Indeed, the eﬀectof market uncertainty on investment timing tends to be negative because higher market
uncertainty increases the incentive for the ﬁrm to delay investment and receive additional
information. On the other hand, technological uncertainty will be dissolved, not by waiting but by ﬁnishing the project. The R&D project value is likely to be convex with respect
to research duration and levels of technological success. The former is caused primarily
by the possibility of rival preemption, whereas the latter stems mainly from the growth
option embedded in the R&D project. Because of this convexity, the higher uncertainty
of research duration and technological success increases the project value, and hence, encourages investment in the R&D project. Our result is novel and can potentially accountfor empirical ﬁndings by Driver, Temple, and Urga (2006). They found that industries
with high R&D intensity tend to indicate a positive eﬀect of uncertainty on investment.
As for debt ﬁnancing, we have empirical implications as follows. Access to debt ﬁnancing for the growth project increases the project value and accelerates investment. Thisis straightforward and consistent with previous results in the literature (e.g., Hennessy(2004), Mauer and Sarkar (2005), and Sundaresan and Wang (2007)). More interestingly, the eﬀects of debt ﬁnancing on the ﬁrst stage project are weaker than that on
the growth project, which implies that the ﬁrm can reap the growth opportunity fromthe R&D project earlier by debt ﬁnancing. A large number of empirical studies showedthat large ﬁrms have advantages over small ﬁrms in proﬁts from R&D projects (e.g., Ho,
Tjahjapranata, and Yap (2006)). Our result may be related to the empirical ﬁnding because larger ﬁrms, due to less ﬁnancing costs, are more likely to take the optimal capital
structure.
The remainder of this paper is organized as follows. Section 2 illustrates the model
setup and solutions. In Section 3, we exercise numerical analysis and provide empiricalimplications. In particular, we focus on interactions between research duration and rival
preemption in Section 3.2, interactions between the growth option and uncertainty oftechnological success in Section 3.3, and eﬀects of leverage in Section 3.4. Section 4
brieﬂy summarizes the paper.

4

2

The model

2.1

Setup

We consider a ﬁrm that has an option to initiate an R&D project by paying the sunk cost
I0 . The project will take T1 years until completion. We call T1 research duration. Aftercompletion, the ﬁrm will receive an instantaneous cash ﬂow A1 X(t) at time t.3 Furthermore, we assume that the ﬁrm has a growth opportunity, such as expanding production,
starting a license business, and applying the new technology to other products (e.g., Lochand Bode-Greuel (2001), Ho, Tjahjapranata, and Yap (2006), and Cassimon, Backer, Engelenc, Wouwee, and Yordanovf (2011)). After the ﬁrm exercises the growth option by
paying the sunk cost I2 , it will receive an increased cash ﬂow A1 A2 X(t).
Our model includes three types of uncertainty. First at all, we represent market uncertainty by the cash ﬂow X(t), which dynamically changes by project-speciﬁc and macroeconomic shocks on demand in the product market. Following the standard literature, we
assume that X(t) follows a geometric Brownian motion:dX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,where B(t) denotes the standard Brownian motion deﬁned in a probability space (Ω, F, P)
and µ, σ(> 0) and x(> 0) are constants. For convergence, we assume that r > µ, wherea positive constant r is the discount rate. For the economic rationale behind these assumptions, refer to standard textbooks such as Dixit and Pindyck (1994) and Guthrie(2009).
Second, we represent technological uncertainty by a random vector (A1 , T1 , I1 ). Technical speciﬁcations, time schedule, and total costs, which are planned and estimated ex
ante, may not be accomplished ex post (e.g., Raz, Shenhar, and Dvir (2002)). To deal
with the technological risks, we consider the level of technological success, A1 , and research duration T1 , as random variables. The ﬁrm will observe the realized value of A1 atcompletion time. At the same time, the ﬁrm will observe an extra cost I1 . For tractability, we assume that (A1 , T1 , I1 ) are independent of X(t), which means that technologicaland market risks are irrelevant. There is less technological uncertainty for the growth
project, and then, we assume that A2 (> 1) and I2 for the growth investment are positive
constants.
The last uncertainty lies in rival preemption. The project value will be greatly destroyed when a competitor completes the similar product ﬁrst, especially in competitions
for patents and standardizations (e.g., Weeds (2002)). We assume that the ﬁrm does
not have information of the R&D progress of rival ﬁrms. Instead, the ﬁrm knows that
3

Some papers distinguish the lag between project inception and completion (the gestation lag) and projectcompletion and commercial application (the application lag) (e.g., Pakes and Schankerman (1984)). For simplicity, we assume that the total lag is equal to T1 .

5rival preemption may occur following an exponential distribution (Poisson arrival) withintensity λ, where λ is a positive constant. Technically, until project completion, the cash
ﬂow X(t) will be potentially killed at an instantaneous rate λdt.
The model diﬀers from the compound real options model by Bar-Ilan and Strange(1998) in the sense that we incorporate technological uncertainty (A1 , T1 , I1 ) and rival
preemption λdt.4 Weeds (2002) also examined rival preemption and uncertainty of research duration. The model, however, does not include technological uncertainty (A1 , I1 )and a growth opportunity.5 The model adopts a game-theoretic framework with full information and assumes that research duration follows an exponential distribution. Although
the setup is suﬃcient to provide economic implications, it seems almost impossible toapply the model to a real-world case. Actually, no ﬁrm wishes to inform its R&D progress
to competitors, and no one plans Poisson distributed time schedule. In this paper, we
model using a Poisson arrival not research duration, which can be estimated to a degree
but a rival’s R&D success, which will be an unexpected event.

2.2

Base problem

In this subsection, we provide technical instructions for how to evaluate and optimize the
R&D project with the three types of uncertainty. We formulate the problem of ﬁnding the
project value and the optimal policy as a two stage optimal stopping problem. Similar toBar-Ilan and Strange (1998), we need to solve the problem backward. Suppose that theinvestment time T ∗ , the level of technological success, A1 , research duration T1 , and extra
cost I1 in the ﬁrst stage project are all known. At time s, which is later than completiontime T ∗ + T1 , the problem of ﬁnding the optimal investment time T ∗∗ for the growth
project is expressed as the following optimal stopping problem:
V2 (X(s), A1 ) = sup E
T ∗∗ ≥sX(s)∫ T ∗∗[sdt

cash flow before growth investment∫ ∞+e−r(t−s) (1 − τ )A1 X(t)|{z}e−r(t−s) (1 − τ )A1 A2 X(t) dt − e−r(T{z}T ∗∗ |∗∗ −s)

I2 ],(1)

cash flow after growth investment

where the investment time T ∗∗ is optimized over all stopping times later than s. The notation EX(s) [·] denotes the expectation conditional on X(s), and for later use we introduce
the corporate tax rate τ , which is a positive constant. The value function V2 (X(s), A1 )in the problem (3) stands for the project value at time s between completion of the ﬁrst
stage and initiation of the growth project.
4

As a minor diﬀerence, Bar-Ilan and Strange (1998) does not consider proﬁts before completion of thesecond stage. They consider duration of the second stage investment, but it is not essential because there is no
investment after completion of the second stage.
5
Nishihara and Ohyama (2008) extended Weeds (2002) to a case involving two alternative technologies, but
the model, like Weeds (2002), does not include either technological uncertainty or a growth opportunity.

6

Because the ﬁrm’s expected proﬁts after the investment time T ∗∗ are equal to∫ ∞(1 − τ )A1 A2∗∗
X(T ∗∗ )E[
e−r(t−T ) (1 − τ )A1 A2 X(t)dt] =X(T ∗∗ ),
r−µ
T ∗∗

(2)

we can rewrite (3) asV2 (X(s), A1 ) =(1 − τ )A1
X(s) + U2 (X(s), A1 ) ,
|
{z
}
r−µ
|
{z
} growth option value

(3)

NPV

where U2 (X(s), A1 ) is the growth option value expressed by
U2 (X(s), A1 ))(1 − τ )A1 (A2 − 1)∗∗
= sup E[e
X(T ) − I2 ]r−µT ∗∗ ≥s ()()(1 − τ )A1 (A2 − 1)x∗∗ (A1 )X(s) β− I2(X(s) < x∗∗ (A1 ))r−µx∗∗ (A1 )= (1 − τ )A1 (A2 − 1)X(s) − I2
(X(s) ≥ x∗∗ (A1 )).
r−µ
X(s)

−r(T ∗∗ −s)((4)

The investment trigger x∗∗ (A1 ) and the positive characteristic root β are deﬁned byx∗∗ (A1 ) =β(r − µ)I2.
(β − 1)(1 − τ )A1 (A2 − 1)andµ1β= − 2+
2 σ

√(

µ
1
−
2
σ
2

)2
+

2r
(> 1),
σ2

(5)

(6)

respectively. For A1 = 0 (failure of the ﬁrst stage), we deﬁne x∗∗ (A1 ) = ∞ and U2 (X(s)) =0. The ﬁrm’s optimal policy, T ∗∗ = inf{t ≥ s | X(t) ≥ x∗∗ (A1 )}, is called the thresholdpolicy, which means that the ﬁrm invests in the growth project as soon as the cash ﬂow
X(t) hits the threshold x∗∗ (A1 ). Note that if X(T ∗ + T1 ) ≥ x∗∗ (A1 ) is satisﬁed, the ﬁrm
proceeds to the second stage immediately after completion of the ﬁrst stage.
Next, we turn to a problem of evaluating and optimizing the ﬁrst stage investment.
At this moment, the level of technological success, A1 , research duration T1 , and thecost I1 are random variables, and the prior distributions of (A1 , T1 , I1 ) are known. At
time s(≥ 0), the project value is expressed as the value function of the optimal stopping
problem as follows:{}∗
V1 (X(s)) = sup EX(s) [e−(r+λ)(T −s) e−(r+λ)T1 V2 (X(T ∗ + T1 ), A1 ) − e−rT1 I1 − I0 ],T ∗ ≥s(7)
where λ is the intensity of rival preemption and V2 (X(T ∗ + T1 ), A1 ) is given by (3). In (7)the expectation is taken over all random variables (X(T ∗ + T1 ), A1 , T1 , I1 ). The expected
total costs are I0 + E[e−rT1 I1 ]. As (T1 , I1 ) are independent of X(t), we can remove I1 by
regarding the estimated total costs I0 + E[e−rT1 I1 ] as the initial cost I0 . From now on,
without loss of generality, we assume that I1 = 0.

7Because the expected proﬁts after the investment time T ∗ are equal to∗U1 (X(T ∗ )) = EX(T ) [e−(r+λ)T1 V2 (X(T ∗ + T1 ), A1 )],(8)

we can reduce (7) to∗V1 (X(s)) = sup EX(s) [e−(r+λ)(T −s) (U1 (X(T ∗ )) − I0 )].
T ∗ ≥s(9)

Although we like to solve the optimal stopping problem (9), it is computationally hard.
For, the payoﬀ function in (9), U1 (·), is not analytically derived but numerically computed.
Accordingly, it is impossible to check whether U1 (·) satisﬁes some regularity conditions,
under which the optimal stopping time exists in a class of threshold policies.
We propose the following tractable method for computing the project value and optimal policy. In the ﬁrst place, we restrict our attention within a class of threshold policies,
which are of the form, Tx∗∗ = inf{t ≥ 0 | X(t) ≥ x∗ }. This restriction is not very strong
because U1 (·) is an increasing function. We now maximize the right-hand side of (9) bymoving threshold x∗ rather than by moving stopping time T ∗ . At the initial time s = 0,
the problem of ﬁnding the project value and the optimal policy is equal to6∗

V1 (x) = sup Ex [e−(r+λ)Tx∗ (U1 (X(Tx∗∗ )) − I0 )]
x∗

= sup

x∗ ≥x

( x )β̃x∗(U1 (x∗ ) − I0 ),

(10)

∗

because we have Ex [e−(r+λ)Tx∗ ] = (x/x∗ )β̃ and X(Tx∗∗ ) = x∗ for x∗ ≥ x. The characteristicroot β̃ is deﬁned by (6) replaced r with r + λ.
Problem (10) is more tractable than problem (9) because it is an optimization problem
of one-dimensional function. Nevertheless, in general, computing U1 (·) in (10) requires
multiple integration, deﬁned by (8), which makes problem (10) diﬃcult to compute. Whenrandom variables (A1 , T1 ) have discrete distributions, (8) can be reduced to a single integral with respect to X(t), which makes problem (10) computationally tractable. We
suppose N scenarios, in which (A1 , T1 ) take (Ai1 , T1i ) with probability pi for i = 1, . . . , N .
Then, problem (10) is reduced to(N)( x )β̃ ∑∗ipi Ex [e−(r+λ)T1 V2 (X(T1i ), Ai1 )] − I0 ,V1 (x) = sup∗x∗ ≥x x(11)

i=1

because (A1 , T1 ) are independent of X(t). Problem (11) is computable so that it can help
practitioners to evaluate and optimize various cases of R&D investment. In section 3, we
will conduct numerical analysis by solving problem (11).

6

To avoid unnecessary disorder, we use the same notation V1 (x) in (10) as that of (9). Strictly, the projectvalue in (10) can be smaller than that of (9) because we restrict the ﬁrm’s policies within the threshold policies.

8

2.3

Levered caseIn this subsection, as a supplement, we extend the base model in Section 2.2 by allowing
a ﬁrm to take the optimal capital structure in the growth project. In practice, an initialstage of an R&D project, involving quite high risks, is diﬃcult to be ﬁnanced by debt
issuance (e.g., Brown, Fazzari, and Petersen (2009) and Hall and Lerner (2009)). After the
success in the ﬁrst stage, a ﬁrm is more likely to use debt ﬁnancing for a growth projectwith less risks. For instance, Ho, Tjahjapranata, and Yap (2006) empirically showedthat the ﬁrm size and ﬁnancial leverage interactions inﬂuence growth opportunities from
R&D investments. We explore the eﬀects of the capital structure on R&D investment by
supposing that the ﬁrm can optimally use debt ﬁnancing for the growth project.
We build the levered setup on the dominant models by Fan and Sundaresan (2000),
Goldstein, Ju, and Leland (2001), and Sundaresan and Wang (2007). Following Fan andSundaresan (2000) and Sundaresan and Wang (2007), we assume that the ﬁrm will be able
to avoid costly liquidation. Indeed, equity and debt holders negotiate and reduce coupon
payments when the cash ﬂow X(t) is lower than a critical level. The ﬁrm’s operation will
return to normal when X(t) restores beyond the critical level.
As in Section 2.2, we ﬁrst consider the growth investment problem by supposing thatthe investment time TL∗ , the level of technological success, A1 , and research duration T1
are all known. We denote the levered case by subscript L. The expected proﬁts after thesecond investment time TL∗∗ becomes(1 − τ )A1 A2 ϕ
X(TL∗∗ )r−µ(12)

in the levered case with the optimal capital structure. Note that (12) = ϕ× (2). The
multiplier ϕ, which results from debt ﬁnancing, is a positive constant deﬁned byϕ=1+

τ (1 − ηα)(> 1),(1 − τ (1 − η))h

(13)

where
] 1
β(1 − γ) − γ
(> 1)
h=
β−γ
√(
)
1
µ
µ
1 2 2rγ= − 2−−+ 2 (< 0).
2 σ
σ2 2
σ
[

(14)
(15)

Constants α ∈ (0, 1) and η ∈ [0, 1] denote the proportion of the ﬁrm value which is lostby liquidation and the bargaining power of equity holders in the renegotiation between
equity and debt holders, respectively. For details in derivation of (12)–(15), refer to Fan
and Sundaresan (2000) and Sundaresan and Wang (2007).

9

The growth option value, which corresponds to (4) in the unlevered case, becomesU2L (X(s), A1 ))(1 − τ )A1 (A2 ϕ − 1)∗∗[e
X(TL ) − I2 ]= sup Er−µ
TL∗∗ ≥s ()()(1 − τ )A1 (A2 ϕ − 1)x∗∗
X(s) βL (A1 )

− I2
(X(s) < x∗∗
L (A1 ))

r−µ
x∗∗
(A
)
1
L
=



 (1 − τ )A1 (A2 ϕ − 1)X(s) − I2
(X(s) ≥ x∗∗
L (A1 )),r−µX(s)−r(TL∗∗ −s)

(

(16)

where the investment threshold x∗∗
L (A1 ) is deﬁned byx∗∗
L (A1 ) =β(r − µ)I2.
(β − 1)(1 − τ )A1 (A2 ϕ − 1)(17)Note that (17) = (A2 − 1)/(A2 ϕ − 1)× (5). Because of the optimal capital structure,
the growth option becomes more valuable, and then, the growth investment takes placeearlier. Because the ﬁrst stage project is executed by all-equity ﬁnancing, we can performthe same discussion as that of Section 2.2 for the ﬁrst stage project by replacing U2 (·)
with U2L (·). In a similar fashion to (7)–(11), we can compute the project value and the
optimal policy, and hence, we omit the description.

3

Numerical analysis and implications

3.1

Basic analysis

Our focus in this paper is not a case study of a speciﬁc R&D project. Instead, we show
numerical results for a wide range of parameter values, demonstrating several propertiesof the project value and the optimal policy in a generic R&D project. We set the baseparameter values as
r = 0.08, µ = 0.06, σ = 0.2, τ = 0.15, α = 0.3,

(18)

following Leland (2004) and Sarkar (2008) based on the market data. There are several
methods for estimating the market parameters and discount rate in a real options model(e.g., using the capital asset pricing model). For instance, Chapter 3 of Guthrie (2009)
explains the details of standard calibration methods.
On the other hand, technological parameter values, such as levels of technologicalsuccess, research duration, and investment costs, can be estimated by a project team.
These values greatly diﬀer over the project types and the industries. For example, ina project of developing a new drug, research duration is quite long, and the probabilityof technological success is quite low (Kellogg and Charnes (2000),Loch and Bode-Greuel(2001), Hartmann and Hassan (2006)). Considering the fact that the average durationis around 2 to 4 years in a majority of the literature (e.g., Pakes and Schankerman10

(1984)), in the base case, we assume that T1 takes a value in {2, 3, 4} with probability
1/3. We consider three levels of technological success; A1 = 0 (failure), A1 = 1 (successas expected), and A1 = 2 (success more than expected), and we assume that A1 takes avalue in {0, 1, 2} with probability 1/3.
Following the standard literature of economics, we assume the convexity of the investment costs, i.e., 1/I0 > (A2 − 1)/I2 . Note that if 1/I0 ≤ (A2 − 1)/I2 , the growth option
is likely to be exercised immediately after the success of the ﬁrst stage, which makes theanalysis uninteresting. For expositional purposes, we set I0 = 10, A2 = 2, and I2 = 20 in
the base case. Then, A1 (A2 − 1)/I2 equals 1/20(< 1/I0 ) and 1/10(= 1/I0 ) for A1 = 1
and A1 = 2, respectively. This implies that the ﬁrm tends to postpone the growth project
when the ﬁrst stage results in the average success.
[Insert Table 1 and Figure 1 here.]
Table 1 presents the project values and investment triggers in the base case; the casewith no growth option (we set A2 = 1); the case in the presence of rival preemption (weset λ = 0.2); and the levered case (we set η = 0.5). We show the project value, V1 (x),for x = 0.2.7 Figure 1 shows the value functions V1 (X(s)) (see (11)) in the cases that
correspond to Table 1.
First, we elaborate the result in the base case. The ﬁrm invests in the ﬁrst stage
project as soon as the cash ﬂow X(t) hits the investment threshold x∗ = 1.17, and the
R&D project will take T1 ∈ {2, 3, 4} years until completion. At completion time T ∗ + T1 ,the ﬁrm will know the level of technological success A1 ∈ {0, 1, 2}. If the R&D project
fails (A1 = 0), the ﬁrm will not receive any cash ﬂow and will be ﬁnished with the
project. Consider the case in which the ﬁrst stage turns out to be successful. The ﬁrm
will receive cash ﬂow A1 X(t), and at the same time, it attains the growth opportunity. If
X(T ∗ + T1 ) is larger than x∗∗ (1) = 2.46, the ﬁrm will immediately invest in the growth
project and receive increased cash ﬂow 2A1 X(t). According to our computation, due toX(T ∗ ) = 1.16 < 2.46, the probability of this scenario is very low (it is approximately 0.03).
If X(T ∗ +T1 ) takes a value in [1.23, 2.46), the growth option will be immediately exercised
only in the case of great success (A1 = 2). In the case of average success (A1 = 1), the ﬁrm
postpones the growth project until X(t) hits the investment trigger x∗∗ (1) = 2.46. The
probability that X(T ∗ + T1 ) ∈ [1.23, 2.46) is 0.53. The last scenario is that X(T ∗ + T1 )is less than x∗∗ (2) = 1.23. In this scenario, the ﬁrm delays the second stage investment
until X(t) hits either 1.23 or 2.46, depending on A1 . The probability of this scenario is
0.44.
7

If one considers the expected waiting time until the project initiation, in the cases with no rival preemption

it is more practical to set x suﬃciently close to the investment trigger x∗ . However, for the purpose of comparing
all cases with the same parameter values, we used x = 0.2. The results are qualitatively robust as long as x is
smaller than x∗ (see Figure 1).

11

We now brieﬂy explain the other cases and compare the results with those of the basecase. We can see from Figure 1 that a higher X(s) increases the gap between the basecase and the case with no growth option. This is because the value of the growth option
increases with a higher X(s). Because the project value becomes lower in the absence ofgrowth opportunity, the investment trigger x∗ = 1.31 is higher than that of the base case.
The result is consistent with the stylized fact that growth opportunities increase a projectvalue and encourages investment.
The third row in Table 1 shows the case with preemption. Table 1 and Figure 1 present
the case of λ = 0.2, which means that the expected time up to rival preemption is 5 years,
although we will show the results with varying levels of λ later in this section. Figure 1
shows that the possibility of rival preemption greatly decreases the project value, especiallyfor a low X(s). Despite the decreased value, the ﬁrm tries to speed up investment because
rival preemption can occur before completion of the ﬁrst stage. Indeed, with λ = 0.2,
the investment trigger x∗ decreases to 0.44 from 1.16. There is no possibility of rival
preemption after completion of the ﬁrst stage, and then we have the same investment
triggers as those of the base case. Accordingly, the time lag between project completionof the ﬁrst stage and the second investment is expected to be long.
We see from the last row in Table 1 that access to debt ﬁnancing increases the projectvalue and decreases the investment triggers. These leverage eﬀects are consistent withprevious results by Hennessy (2004), Mauer and Sarkar (2005), and Sundaresan and Wang(2007). In the levered case, we assume that the bargaining powers of equity and debt
holders are equal, i.e., η = 0.5. The multiplier , caused by debt ﬁnancing, is equal to 1.14.
We will closely explore the eﬀects of leverage in Section 3.4.
[Insert Figure 2 here.]
Figure 2 represents the project values V1 (0.2) and investment triggers x∗ , x∗∗ (1), andx∗∗ (2) with varying levels of λ. The other parameter values are the same as the basecase. Note that second investment triggers x∗∗ (1) and x∗∗ (2) do not depend on λ. We
see that V1 (0.2) greatly decreases with λ. As expected from the expression (9), the graphof V1 (0.2) shows convexity. In other words, for a lower λ, the sensitivity is greater. For
instance, V1 (0.2) decreases from 8.35 to 3.79 with a very small λ = 0.05. In contrast to
the monotonic decrease in V1 (0.2), the investment trigger x∗ decreases until λ = 0.12,
and after this point, it increases. The possibility of rival preemption decreases the projectvalue, but at the same time, it increases the ﬁrm’s incentive to invest early and complete
the technology before its rivals. The non-monotonicity in x∗ results from the trade-oﬀ. For
a lower λ, the sensitivity of x∗ with respect to λ is negative because the latter dominates
the former. On the other hand, for a higher λ, the sensitivity is positive because theformer dominates the latter.
[Insert Figure 3 here.]

12Figure 3 plots the project values V1 (0.2) and investment triggers x∗ , x∗∗ (1), and x∗∗ (2)with varying levels of the cash ﬂow volatility σ.8 The other parameter values are set at thebase case. Similar to Chapter 5.2.A of Dixit and Pindyck (1994), we can easily prove that
U2 (X(s), A1 ) and x∗∗ (A1 ) (see (4) and (5)) monotonically increase with σ. We can see
from Figure 3 that V1 (0.2) and x∗ also increase with σ. This result shows the robustnessof the standard theory that higher market uncertainty increases an option value anddelays the exercise of the option (e.g., Dixit and Pindyck (1994)) even if we incorporate
several characteristics of R&D investment. We also examined the comparative statics
for a wide range of parameter values, including positive λ. We found that the standard
result is robust. Relatedly, Bar-Ilan and Strange (1996) and Bar-Ilan and Strange (1998)
showed the similar eﬀects, although their models do not include technological and rival
uncertainty.9
So far, we have explained the results that were more or less known in the prior literature. We also found several ﬁndings that have not been addressed by previous studies.
In the following subsections, we will elaborate key ﬁndings, logic behind the ﬁndings, andempirical implications.

3.2

Interactions between research duration and rival pre-emptionIn this subsection, we reveal the eﬀects of research duration combined with rival preemption. In the presence of rival preemption, longer duration increases the probability that
other ﬁrms will complete the same technology ﬁrst. Then, we can expect some interactions between research duration and rival preemption. Although Bar-Ilan and Strange(1996) and Bar-Ilan and Strange (1998) examined the eﬀects of duration in the absenceof rival preemption, to our knowledge, there are no papers that examine the mixed eﬀects
of duration and rival preemption. We summarize new ﬁndings below.
[Insert Figure 4 here.]
The upper and lower panels of Figure 4 show the project values V1 (0.2) and investment
triggers x∗ , respectively, with varying levels of research duration T1 . In order to clarify theeﬀects of T1 , we change levels of T1 from 1 to 5, instead of taking T1 as random variables.10
In addition to the case with no rival preemption (λ = 0), the panels plot V1 (0.2) and x∗ in
8

For simplicity, we change σ, taking all other parameters, r and µ as constants. This means that changes in σhave only an idiosyncratic risk component. Most of the literature, including Dixit and Pindyck (1994), presents
the comparative statics under this assumption, although some papers, including Wong (2007), examines thecomparative statics assuming the relation between µ and σ.
9Bar-Ilan and Strange (1996) highlight that the standard volatility eﬀect does not always hold for investment
with an abandonment option in addition to investment lag.
10
We have similar results for varying levels of E[T1 ] even if we consider T1 as a random variable.

13the cases with rival preemption (λ = 0.1, 0.2, and 0.3) so that we can see the interactionsof research duration and rival preemption. The other parameter values are set at the basecase. The investment triggers x∗∗ (A1 ) are independent of T1 and λ and remain unchanged
from those of the base case in Table 1, and hence we omit plots of x∗∗ (A1 ).
The upper panel of Figure 4 indicates the straightforward results that V1 (0.2) monotonically decreases with T1 and that the decrease is intensiﬁed with λ. That is, research
duration is more crucial to the ﬁrm as rival preemption is expected to occur earlier. As
expected from the expression (8), we ﬁnd stronger convexity of a graph for a higher λ. Notably, the lower panel shows that the sensitivity of x∗ with respect to T1 depends greatlyon λ. In the absence of preemption x∗ monotonically decreases with T1 . The ﬁrm speeds
up investment considering that X(T ∗ ) is expected to go up to eµT1 X(T ∗ ) at completion
time.11 This result is well known in the previous researches such as Bar-Ilan and Strange(1998) and Bar-Ilan and Strange (1998).
However, as seen from (8), a longer T1 decreases the project value especially when λis high. This decrease in the project value leads to that x∗ monotonically increases with
T1 for λ = 0.1, 0.2, and 0.3 in the lower panel of Figure 4. According to our numericalanalysis, with the slightest threat of rival preemption (λ = 0.01), the sensitivity of x∗ withrespect to T1 remains positive, while in the case with no rival preemption the sensitivity
is negative. Thus, we conclude that in the presence of rival preemption, longer duration
delays the ﬁrm’s project initiation due to the decreased project value. This result is in
sharp contrast with that of the case with no preemption.
Next, we explore the eﬀects of uncertainty of research duration T1 . Actually, research
duration can be extended from that of the initial schedule (e.g., Raz, Shenhar, and Dvir
(2002)), although previous studies do not examine the eﬀects. In order to clarify the eﬀects
of uncertainty of T1 , we consider random variables T1 taking 1, 3, and 5 with probability
wT /2, 1 − wT , and wT /2, respectively, and vary levels of weight wT .12 The variance of T1
monotonically increases with wT , while we maintain that E[T1 ] = 3.
[Insert Figure 5 here.]
The upper and lower panels of Figure 5 show the project values V1 (0.2) and investment
triggers x∗ , respectively, with varying levels of weight wT . We exclude the graphs for λ = 0,
which require very diﬀerent scales, and we show the cases of λ = 0.1, 0.2, and 0.3 in thepanels. The other parameter values are set at the base case. We can recognize from thepanels that a higher wT enhances V1 (0.2) and decreases x∗ . The eﬀects become stronger
11

We do not consider the case of a negative µ because a negative growth rate is not practical for an R&D

project.
12
We have the same results for the base parameter value T1 ∈ {2, 3, 4}, but in order to highlight the results,
we presented the results for T1 ∈ {1, 3, 5} (of course, as the range of T1 is wider, the eﬀects are clearer.) For
robustness, we also examined the comparative statics with respect to variance by changing the range of T1
rather than wT . The results are unchanged.

14

as λ is higher. The reasoning is as follows. As explained with regard to the upper panelof Figure 4, U1 (·) is convex with respect to T1 and the convexity is stronger with a higher
λ. By Jensen’s inequality and the convexity, U1 (·) increases with a higher wT , leading tothat a higher wT increases V1 (0.2) and decreases x∗ . The impact of wT is magniﬁed by ahigher λ.
In consequence, we have a key result that, combined with rival preemption, uncertainty of research duration plays a role in improving the project value and speeding up
investment. Our result contrasts with the standard result that greater uncertainty tends
to enhance the value of waiting and delaying investment (cf. Figure 3). Our result
regards technological uncertainty that will not be dissolved by waiting, whereas the standard volatility eﬀect is based on market uncertainty that will be dissolved by waiting.
This diﬀerence causes the opposite eﬀects of uncertainty. In an R&D project with hightechnological uncertainty, unlike in a case with only market uncertainty, a ﬁrm possesses
incentive to dissolve technological uncertainty by accelerating investment. Our result is
consistent with the empirical evidence in Driver, Temple, and Urga (2006). They observed
positive eﬀects of uncertainty on investment in industries with high R&D intensity.

3.3

Interactions between the growth option and uncertainty

of technological successIn this subsection, we examine the eﬀects of uncertainty of technological success A1 .
Although the real options literature have stressed the eﬀects of market uncertainty on
investment, few papers have examined the eﬀects of technological uncertainty on investment. Below, we reveal the eﬀects of the variance of A1 on project value and investment
timing. We consider random variables A1 taking 0, 1, and 2 with probability wA /2, 1−wA ,and wA /2, respectively, and vary levels of weight wA . Note that the variance of A1 monotonically increases with wA and E[A1 ] = 1 is always satisﬁed.
[Insert Figure 6 here.]
The upper and lower panels of Figure 6 show the project values V1 (0.2) and investment
triggers x∗ , respectively, with varying levels of wA . In order to examine interactions
between technological uncertainty and a growth opportunity, we plots the graphs fordiﬀerent sizes of growth opportunity, i.e., A2 = 1.5, 2, 2.5, and 3. The other parametervalues are set at the base case. Note that in the case with no growth option (A2 = 1),
V1 (0.2) and x∗ are 4.16 and 1.31, respectively, for all wA . The second investment triggersx∗∗ (1) and x∗∗ (2) do not depend on wA ; indeed, we have x∗∗ (2) = 2.46, 1.23, 0.82, and0.62 for A2 = 1.5, 2, 2.5, and 3, respectively. The investment triggers x∗∗ (1) are twice aslarge as x∗∗ (2).
In Figure 6, we recognize that V1 (0.2) monotonically increases with wA , while x∗
monotonically decreases with wA . These eﬀects of wA become greater as A2 increases.

15

The intuition behind the results is as follows. The growth option value U2 (X(s), A1 )is not negative but zero even if the ﬁrst stage fails. On the other hand, U2 (X(s), A1 )monotonically increases with A1 (> 0). Because of this convexity of the growth option
and Jensen’s inequality, a higher wA increases U2 (X(s), A1 ). Considering the increased
value of the growth option, the ﬁrm decreases the investment trigger x∗ for a higher wA .
Accordingly, the eﬀects of uncertainty of A1 become stronger as the size of the growth
option, A2 , is larger.
We conclude that higher uncertainty of technological success improves the project value
and encourages R&D investment by enhancing the growth option value. This argumentaligns with that of compound option cases, in which greater uncertainty may speed up theexercise by increasing the embedded option value, although it contrasts with the standard
eﬀect of uncertainty in the real options literature. Furthermore, we highlight that thetypes of uncertainty determine the sensitivity of uncertainty on investment. Similar to
uncertainty of research duration in Section 3.2, uncertainty of technological success will
never be dissolved by waiting, and hence, the ﬁrm has no incentive to delay investment
and obtain extra information. Our result is also consistent with Driver, Temple, andUrga (2006) who empirically showed that industries with high R&D intensity tend tohave positive eﬀects of uncertainty on investment.
On the other hand, because x∗∗ (1) and x∗∗ (2) are constants, a higher wA increases
the lag between completion of the ﬁrst stage and initiation of the second stage project
for a ﬁxed A1 . For instance, we take a look at the probability that the ﬁrm proceeds tothe second stage immediately after completion of the ﬁrst stage project. As explainedin Section 3.1, in the base case (wA = 2/3), the ﬁrm proceeds to the growth project
immediately after the success of the ﬁrst stage with probability 0.03, and it proceeds
immediately to the growth project only in the case of great success with probability
0.53. For wA = 0.1, due to the increased trigger x∗ = 1.26, the probabilities go up to
0.06 and 0.6, respectively. Although one considers from this that a higher wA lengthens
the waiting time, the interpretation is not correct in terms of the prior probabilities.
Indeed, the prior probabilities that the ﬁrm invests in the second stage project right after
completion of the ﬁrst stage project are 0.03 × 2/3 + 0.53 ∗ 1/3 = 0.2 for wA = 2/3
and 0.06 × 19/20 + 0.6 × 1/20 = 0.09 for wA = 0.1. The ex ante expected waiting time
also decreases with wA . Therefore, we can state that greater uncertainty of technologicalsuccess accelerates R&D investment.

3.4

Eﬀects of leverage

Lastly, we explore the eﬀects of the use of debt ﬁnancing. Note that debt ﬁnancing isavailable only for the second stage project with no risks of technological failure. We
readily see from (16) and (17) that leverage increases the growth option value U2L (X(s))
and decreases the second investment trigger x∗∗
L (A1 ). These correspond to the results

16

in Sundaresan and Wang (2007). We now turn to the initial project value V1 (0.2) andinvestment trigger x∗ .
[Insert Figure 7 here.]
The upper and lower panels of Figure 7 show the project values V1 (0.2) and investment
triggers x∗ , respectively, in both unlevered and levered cases with varying sizes of growth
opportunity A2 . We take the size of growth opportunity A2 as the horizontal axis because
leverage inﬂuences the ﬁrm through the growth project. Naturally, the leverage eﬀects
increase with A2 . For comparison, in the lower panel, we present the second investment
triggers x∗∗ (2) in addition to x∗ , but we omit x∗∗ (1), which is twice as high as x∗∗ (2).
We can see from the upper panel of Figure 7 that V1 (0.2) in the levered case is 1.16 to
1.17 times higher than that of the base case. The lower panel shows that x∗ and x∗∗ (2)in the levered case are lower than those of the base case. Thus, access of debt ﬁnancing
increases the project value and accelerates investment in the R&D project. This leverage
eﬀect is consistent with the prior literature (e.g., Hennessy (2004), Mauer and Sarkar(2005), and Sundaresan and Wang (2007)).
We now look at the lower panel of Figure 7 more closely. We ﬁnd that the impactof leverage on x∗ is weaker than that on x∗∗ (2). This is probably because debt ﬁnancing
inﬂuences the growth investment directly, while it inﬂuences the ﬁrst stage investment at
second hand. As a result, leverage tends to decrease the waiting period between successof the ﬁrst project and initiation of the growth project. Indeed, for A2 = 2, we havex∗ = 1.05 > x∗∗ (2) = 0.96 in the levered case, while we have x∗ = 1.16 < x∗∗ (2) = 1.23
in the base case. Most of empirical studies stated that large ﬁrms have advantages oversmall ﬁrms in growth opportunities from R&D investment (e.g., Ho, Tjahjapranata, andYap (2006)). Our result could be related to the evidence because larger ﬁrms that suﬀer
less ﬁnancing costs are more likely to approximate the optimal capital structure.

4

Conclusion

In this paper, we developed a real options model for evaluating and optimizing an R&D
project. The model can capture the eﬀects of various features of R&D investment suchas research duration, growth opportunity, and technological, demand market, and rival
preemption uncertainty; nevertheless, the model is computationally tractable so that it
can help real-world decision-making process of R&D investment. Further, we presented
numerical results for a wide range of parameter values and unveiled several interactionsof the key features of R&D. Below, we summarize notable results.
In the absence of rival preemption, the ﬁrm speeds up investment in a project withlonger research duration, but a slight possibility of rival preemption reverses this eﬀect of
duration. Indeed, longer duration drives the ﬁrm to delay investment because it increases

17the possibility of rival preemption and decreases the project value. As uncertainty of
research duration and technological success of the R&D project is higher, the projectvalue increases and the optimal investment time is earlier. A higher possibility of rival
preemption intensiﬁes the eﬀects of uncertainty of duration because it strengthens theconvexity of the project value with respect to duration. On the other hand, the eﬀects
of uncertainty of technological success are ampliﬁed by an increase in the growth option.
Access to debt ﬁnancing in the growth project increases the project value and accelerates
investment, but the impact on the ﬁrst stage is relatively weak. Our results can potentiallyaccount for several empirical ﬁndings.

Acknowledgment
This work was supported by the Public Trust for Intellectual Property Research Support
Fund and KAKENHI (23310103).

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20

Table 1: Project values and investment triggers.
V1 (0.2)x∗x∗∗ (1)x∗∗ (2)

Base

8.35

1.16

2.46

1.23

No growth

4.16

1.31

N/A

N/A

Preemption

0.63

0.44

2.46

1.23

Leverage

9.77

1.05

1.93

0.96

180
Base
No growth
Preemption
Leverage160
140

V1(X(s))

120
100
80
60
40
20
0

0.2

0.4

0.6

0.8

1
X(s)

1.2

1.4

1.6

1.8

2

Figure 1: Value functions. The ﬁgure plots the project value functions V1 (X(s)) in the basecase, the case with no growth option (A2 = 1), the case with rival preemption (λ = 0.2), andthe levered case (η = 0.5).

21

9
V1(0.2)8x∗
Project Values and Triggers

7x∗∗(1)x∗∗(2)

6
5
4
3
2
1
0

0

0.05

0.1

0.15

0.2

0.25
λ

0.3

0.35

0.4

0.45

0.5

Figure 2: Eﬀects of rival preemption. This ﬁgure plots the project values V1 (0.2) and investment
triggers x∗ , x∗∗ (1), and x∗∗ (2) with varying levels of λ. The other parameter values are set atthe base case.

12

10
Project Values and Triggers

V1(0.2)x∗8x∗∗(1)x∗∗(2)
6

4

2

0
0.1

0.15

0.2

0.25

0.3
σ

0.35

0.4

0.45

0.5

Figure 3: Eﬀects of market uncertainty. This ﬁgure plots the project values V1 (0.2) and investment triggers x∗ , x∗∗ (1), and x∗∗ (2) with varying levels of market uncertainty σ. The otherparameter values are set at the base case.

22

9
8
7

λ=0

6

λ=0.2

λ=0.1

V1(0.2)

λ=0.3
5
4
3
2
1
0

1

1.5

2

2.5

3
T1

3.5

4

4.5

5

2.5

3
T1

3.5

4

4.5

5

1.3
1.2
1.1

λ=0
λ=0.1

1

λ=0.2
λ=0.3

x∗

0.9
0.8
0.7
0.6
0.5
0.4

1

1.5

2

Figure 4: Eﬀects of research duration. The upper and lower panels plot the project values
V1 (0.2) and investment triggers x∗ with varying levels of research duration T1 , respectively.
The ﬁgure shows the cases of λ = 0, 0.1, 0.2, and 0.3. The other parameter values are are set
at the base case.

23

2.2
2
λ=0.1

1.8

λ=0.2
λ=0.3

1.6

V1(0.2)

1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.1

0.2

0.3

0.4

0.5
wT

0.6

0.7

0.8

0.9

1

0.54
λ=0.1
λ=0.2
λ=0.3

0.52

0.5

x∗

0.48

0.46

0.44

0.42

0.4

0

0.1

0.2

0.3

0.4

0.5
wT

0.6

0.7

0.8

0.9

1

Figure 5: Eﬀects of uncertainty of research duration. The upper and lower panels plot the
project values V1 (0.2) and investment triggers x∗ with varying levels of wT , respectively. The
ﬁgure shows the cases of λ = 0, 0.1, 0.2, and 0.3. The other parameter values are set at thebase case.

2418
A2=1.5A2=2

16

A2=2.5A2=3

V1(0.2)

14

12

10

8

6

4

0

0.1

0.2

0.3

0.4

0.5
wA

0.6

0.7

0.8

0.9

1

0.3

0.4

0.5
wA

0.6

0.7

0.8

0.9

1

1.4

1.3

1.2

x∗

1.1

1

A2=1.50.9

A2=2
A2=2.5

0.8

A2=3
0.7

0

0.1

0.2

Figure 6: Eﬀects of uncertainty of technological success. The upper and lower panels plot the
project values V1 (0.2) and investment triggers x∗ with varying levels of wA , respectively. The
ﬁgure shows the cases of A2 = 1.5, 2, 2.5, and 3. The other parameter values are set at the base
case.

25

18
Base
Leverage
16

V1(0.2)

14

12

10

8

6

4
1.5

2

2.5

3

A2

2.5
x∗ [Base]x∗ [Leverage]x∗ ∗(2) [Base]x∗∗(2) [Leverage]
L

Triggers

2

1.5

1

0.5
1.5

2

2.5

3

A2

Figure 7: Eﬀects of leverage. The upper and lower panels plot the project values V1 (0.2) andinvestment triggers x∗ and x∗∗ (2), respectively, in both base and levered cases with varying
sizes of growth opportunity A2 . The other parameter values are set at the base case.

26