Discussion Papers In Economics
And Business

Valuation of an R&D project with three types of
uncertainty
Michi NISHIHARA
Discussion Paper 17-15

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANValuation of an R&D project with three types of
uncertainty
Michi NISHIHARA
Discussion Paper 17-15

June 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANValuation of an R&D project with three types of
uncertainty ∗Michi NISHIHARA†

Abstract
This paper develops an R&D decision-making model in the real options framework.
The model is generic enough to capture three types of uncertainty in an R&D project,
namely, uncertainty of research duration and costs, market value of technology, and a competitor’s technology development. I derive analytical solutions, which help practitioners
and researchers to evaluate various cases of R&D investment. Further, by analyzing themodel with a wide range of parameter values, I reveal the following eﬀects of the three
types of uncertainty on R&D investment: Higher uncertainty of research duration andcosts, unlike market value uncertainty, speeds up investment, especially combined witha higher risk of competition. The investment timing can be U-shaped in the strength ofcompetition because of the trade-oﬀ between the preemptive investment eﬀect and thedecreased project value eﬀect. These results can account for empirical ﬁndings about theuncertainty-investment relation in industries with high R&D intensity and severe competition.
JEL Classifications Code: G31; G33; 032.
Keywords: Capital budgeting; Decision analysis; Risk; R&D; Real options.

∗This is the ﬁrst version written on 23 June, 2017. This work was supported by the Microsoft Public

Trust for Intellectual Property Research Support Fund and the JSPS KAKENHI (Grant number JP17K01254,
JP17H02547).
†
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 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 to both theory and practice by developing a tractablemodel for evaluating and optimizing R&D investment as well as unveiling the interactionsof several key features associated with R&D investment.
The diﬃculties of R&D decision-making lie in three types of uncertainty, namely,
uncertainty of technology, market value, and competitors (e.g., see Raz, Shenhar, andDvir (2002)). In an R&D project, technological speciﬁcations, time schedule, and budget
are planned before project initiation, but, in many cases, the outcome is not as successful
as planned. This is called technological uncertainty. Market uncertainty means that themarket value of a newly developed technology is not deterministic but aﬀected by bothtechnology-speciﬁc and macroeconomic shocks on the market. An R&D project also faces
a risk of competition. For example, if a competitor develops an alternative technology
ﬁrst, the market value of technology in progress will greatly lower.
In reality, the R&D decision-making process involving the three risks is challenging
for managers. Although the net present value (NPV) method has been widely employed
for R&D decision-making, a more sophisticated method, which is called the real options
method, has gradually spread in the real world (e.g., Hartmann and Hassan (2006) andBaker, Dutta, and Saadi (2011)). In fact, the literature argues advantages of the real
options method over the NPV method in R&D decision-making with high uncertainty.
Many case studies apply the real options method to R&D project valuation (e.g., Lochand Bode-Greuel (2001), Lee and Paxson (2001), Cassimon, Backer, Engelen, Wouwe,
and Yordanov (2011), and Pennings and Sereno (2011)). To my knowledge, however, theexistent real options models miss any of the three types of uncertainty explained above.
Instead of a speciﬁc case study, I develop a generic and tractable real options modelfor R&D decision-making with the three types of uncertainty so that researchers andpractitioners can analyze various cases of R&D investment using the model. This paper
considers the following model: A ﬁrm can initiate an R&D project by paying a sunk costat an arbitrary time. There is a lag between project initiation and completion (henceforth,called research duration). The ﬁrm also continues to pay costs during research duration.
When the ﬁrm completes the project, the ﬁrm receives the market value of the technology.
The model takes into consideration the three types of uncertainty as follows. Technological uncertainty can be modeled by a random variable standing for research duration.
Note that total cost, which increases with longer research duration, is also stochastic.
With regard to market uncertainty, following the standard literature on real options, I
assume that the dynamics of the market value of the technology follows a geometric Brownian motion. Lastly, as in the works of Armada, Kryzanowski, and Pereira (2011) and
Lavrutich, Huisman, and Kort (2016), the model assumes hidden competition, i.e., the2

ﬁrm faces the risk of a competitor’s technology development, which follows the Poisson
arrival process. If a competitor completes an alternative technology ﬁrst, a fraction of thetechnology value is lost.
By analytically deriving the model solutions as well as conducting numerical analysis
with a wide range of parameter values, I reveal several new eﬀects of the three typesof uncertainty on R&D investment decisions. First, I show that higher uncertainty of
research duration improves the project value and accelerates R&D investment. Thisimplies that the uncertainty-investment sensitivity depends on the type of uncertainty. It
is well known that the market uncertainty-investment sensitivity is negative because higher
market uncertainty increases the incentive for a ﬁrm to wait for additional information.
On the other hand, technological uncertainty, which will not be dissolved by waiting,has positive eﬀects on R&D investment because the R&D project value is convex withrespect to research duration. A higher risk of a competitor’s technology development
intensiﬁes the convexity. Because of the convexity, higher uncertainty of research duration
increases the project value and speeds up R&D investment, especially combined withsevere competition.1
These results can potentially explain several empirical ﬁndings. For instance, Grullon,
Lyandres, and Zhdanov (2012) and Kraft, Schwartz, and Weiss (2015) showed that thesensitivity of uncertainty to ﬁrm value increases for a ﬁrm with higher R&D intensity.
Driver, Temple, and Urga (2006) showed that industries with high R&D intensity andsevere preemptive competition tend to have positive uncertainty-investment sensitivities.
Second, I show that the eﬀects of hidden competition on investment are not monotonic
in an R&D project with research duration. It is well known that, in the absence of research
duration, a greater threat of competitors accelerates investment because it decreases thevalue of waiting. With research duration, however, severe competition decreases the
expected project value at project initiation because a competitor can potentially develop
technology before project completion. When the latter eﬀect dominates the former eﬀect,
severe competition delays R&D investment. Indeed, I show that the investment timing
has U-shaped relation with the arrival rate of a competitor’s technology development. I
also show that the investment timing can be non-monotonic with respect to the remaining
value after a competitor’s technology development.
Lastly, I explain key diﬀerences from the related literature to date. A seminal workby Weeds (2002) is the most similar to this paper. She examined a real options modelwith uncertain research duration and rival preemption. However, in her model, a ﬁrm
has complete information about its competitor while research duration is exponentiallydistributed. Unlike Weeds (2002), I assume that not research duration, which can be
estimated internally in the ﬁrm, but a competitor’s technology development, which can
1

Although Huchzermeier and Loch (2001) also showed that higher uncertainty of time schedule can accel-

erate R&D investment, their model does not consider the interactions between technological uncertainty andcompetition.

3

be an unexpected and exogenous event, follows an exponential distribution. Recently,
Cassimon, Backer, Engelen, Wouwe, and Yordanov (2011) and Pennings and Sereno (2011)
conducted case studies of R&D projects in the pharmaceutical industry taking account ofboth technological and market uncertainty, but their models, which are based on European
options, do not entail any implications of the optimal R&D investment timing. Their
papers do not consider a risk of competition either. Thus, this paper, more so than theprevious works, helps R&D decision-making with the three types of uncertainty.
The remainder of this paper is organized as follows. After Section 2 introduces themodel setup, Section 3 derives the model solutions analytically. In Section 4, with numerical examples, I analyze the model in full detail and provide empirical implications.
Section 5 brieﬂy summarizes the paper.

2

Model setup

Consider a ﬁrm that has an option to initiate an R&D project by paying sunk cost I0 , suchas investment costs in new facilities and equipment, which is a positive constant.2 The
project will take T years until completion, where T is a constant in Sections 3.1 and 3.2.
I will also consider random variable T in Section 3.3. Throughout the paper, T is called
research duration.3 For T years, the ﬁrm continuously pays cost I1 , such as personnel
expenses and experimental costs, which is a positive constant. I deﬁne total cost by∫ T
I = I0 +e−rt I1 dt = I0 +0(1 − e−rT )I1
,
r

(1)

where a positive constant r is the discount rate. Total cost (1) increases in T . WhenT follows a random variable, total cost (1) is also stochastic. By introducing random T
in Section 3.3, the model can capture technological uncertainty of research duration andcosts.
At project completion, the ﬁrm receives one-shot proﬁt X(t) as the market value ofthe technology,4 where as in the standard real options literature (e.g., Dixit and Pindyck
2This paper focuses on a ﬁxed-size investment project for developing a new technology. It could be interestingfor future research to incorporate the project type and/or size in the model. For instance, Nishihara and Ohyama(2008), based on Weeds (2002), investigated the choice between two alternative technologies, while Nishihara(2012) studied dynamic management of multiple investment projects with synergies. Several papers including
Huisman and Kort (2015), Shibata and Nishihara (2015a), and Lukas, Spengler, Kupfer, and Kieckhafer (2017)
examined both investment timing and sizing decisions.
3
Some papers distinguish the lag between project inception and completion (the gestation lag) and the lag
between project completion and commercial application (the application lag) (e.g., Pakes and Schankerman(1984)). For simplicity, I assume that the total lag is equal to T .
4X(t) may be interpreted as the expected discounted value of cash ﬂows generated by the technology. I caneasily extend the model solutions in the setup, allowing for a stream of cash ﬂows after the completion time
T rather than the one-shot proﬁt. When cash ﬂows follow a geometric Brownian motion, the solutions in theextended model are essentially the same as the solutions in this paper.

4

(1994)), X(t) follows a geometric Brownian motion:dX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,(2)where B(t) denotes the standard Brownian motion deﬁned in the ﬁltered probability
space (Ω, F, {Ft }, P) and µ, σ(> 0) and x(> 0) are constants. For convergence, I assume
that r > µ.5 By introducing X(t), the model can capture market uncertainty, which
dynamically changes due to speciﬁc and macroeconomic shocks on the demand market.
As in the works by Armada, Kryzanowski, and Pereira (2011) and Lavrutich, Huisman, and Kort (2016), the model considers hidden competition, which means that theﬁrm cannot observe a potential competitor until a technology developed by a competitor
appears in the market. The ﬁrm knows the ex-ante risk of competition that a hidden
competitor completes an alternative technology at probability λdt for inﬁnitesimal timeinterval dt, where arrival rate λ is a positive constant. In other words, a competitor’s
technology development follows an exponential distribution with arrival rate λ.6 This
paper assumes that the exponential distribution is independent of the stochastic process
X(t). If a competitor completes a technology ﬁrst, the technology value X(t) will decrease
to αX(t), where constant α ∈ [0, 1) denotes the remaining fraction after a competitor’s
success. Note that for α > 0, the ﬁrm has an option of investing in the R&D project evenafter a competitor’s success. By introducing λ and α, the model can capture the risk of
a competitor’s technology development.7
In the last part of this section, I explain key diﬀerences from the related models in theprevious literature. Weeds (2002) also focused on uncertainty of research duration anda competitor’s development in the real options R&D model. The model adopts a gametheoretic framework with full information and assumes that research duration follows
an exponential distribution. In reality, however, a ﬁrm does not exactly know the R&D
progress of its competitors (cf. incomplete information in Lambrecht and Perraudin (2003)
and Nishihara and Fukushima (2008)) and may not recognize which ﬁrm is a potential
competitor for the project (cf. hidden competition in Armada, Kryzanowski, and Pereira(2011) and Lavrutich, Huisman, and Kort (2016)). In addition, managers do not plan
exponentially distributed research duration. Because an exponential distribution has theproperty of being memoryless, it is used for modelling an unexpected and exogenous event
5

For the economic rationale behind these assumptions, refer to standard textbooks such as Dixit and Pindyck(1994) and Guthrie (2009).
6
An alternative approach for modeling competition is a game theoretic real option model (e.g., Huisman(2001), Weeds (2002), Pawlina and Kort (2006), Nishihara and Shibata (2010), Shibata (2016)). However,this approach typically imposes stronger assumptions about information of competitors. As was discussed by
Armada, Kryzanowski, and Pereira (2011), a hidden competition model better ﬁts R&D decision-making in theabsence of information about competitors.
7
A competitor’s technology development may reduce research duration T and costs I1 and I2 . This is because
the ﬁrm can potentially develop a technology by utilizing the competitor’s technology. I can easily derive themodel solutions in the extended model, although I omit demonstrating the details.

5such as a natural disaster. Accordingly, unlike Weeds (2002), I assume that not research
duration, which can be estimated internally in the ﬁrm, but a competitor’s technology
development, which can be an unexpected and exogenous event, follows an exponential
distribution.
Cassimon, Backer, Engelen, Wouwe, and Yordanov (2011) and Pennings and Sereno(2011) have conducted case studies of pharmaceutical R&D projects. Their models include
both market and technological uncertainty. However, their models, which are based on
European compound options, cannot provide any implications about the optimal R&D
investment timing. In addition, their models do not consider uncertainty of research
duration and competition.

3

Model solutions

3.1

Project value and investment timing after a competi-

tor’s success
In Sections 3.1 and 3.2, I assume that T is a constant. I consider the problem backward.
Suppose that a competitor has already developed a technology in this subsection. The
expected project value at the project initiation time τ is calculated as follows:E[e−rT αX(τ + T ) | X(τ )] = Ac X(τ ),(3)

Ac = αe−(r−µ)T .

(4)

where

The subscript c denotes the value after a competitor’s success. By (3), the project valuefunction is expressed asVc (x) = sup E[e−rτ (e−rT αX(τ + T ) − I)]τ= sup E[e−rτ (Ac X(τ ) − I)](5)τ

where the investment time τ is optimized over all stopping times.
As is well known (e.g., see Dixit and Pindyck (1994)), in the continuation region, Vc (x)satisﬁes the ordinary diﬀerential equation:µx

dVc (x) σ 2 x2 d2 Vc (x)+= rVc (x),
dx
2
dx2

(6)

where µxd/dx + σ 2 x2 /2d2 /dx2 corresponds to the generator of the geometric Brownian
motion (2). A general solution to (6) is expressed as Vc (x) = B1 xβ + B2 xγ , where B1 and6B2 are constants, and
β=

1
µ
−
+
2 σ2

1
µ
γ= − 2−
2 σ

√(
√(

µ
1
−
σ2 2
µ
1
−
2
σ
2

)2
+

2r
(> 1),
σ2

(7)

+

2r
(< 0).
σ2(8))2

By the trivial boundary condition, i.e., Vc (0) = 0, I have B2 = 0. For the investment
threshold x∗c , I have the boundary conditions:∗
Vc (x∗c ) = B1 x∗βc = Ac xc − I(9)dVc (x∗c )= B1 βx∗β−1= Ac ,cdx(10)which are called the value matching and smooth pasting conditions, respectively (e.g., see
Dixit and Pindyck (1994)). By solving (9) and (10), I can derive B1 and x∗c as follows.
Proposition 1 Suppose that a competitor has already developed a technology. The projectvalue function Vc (x) is given by ( )βx(Ac x∗c − I) (x < x∗c )x∗cVc (x) =
Ac x − I(x ≥ x∗c ),(11)

where the investment trigger x∗c is defined by
x∗c =

βI
(β − 1)Ac(12)

The optimal investment time τc∗ is given byτc∗ = {t ≥ 0 | X(t) ≥ x∗c }.

(13)

In Proposition 1, the upper equation in (11) stands for the value of the option to invest
in the R&D project after a competitor develops a technology. Because the option value
is higher than the investment value, the ﬁrm waits until the technology value X(t) hits
the investment trigger x∗c . Once X(t) hits x∗c , the ﬁrm initiates the R&D project, and the
expected project value becomes Ac x∗c . By ∂Ac /∂α > 0, ∂Ac /∂T < 0, ∂β/∂σ < 0, (11), and(12), I have ∂Vc (x)/∂α > 0, ∂Vc (x)/∂T < 0, ∂Vc (x)/∂σ ≥ 0, ∂x∗c /∂α < 0, ∂x∗c /∂T > 0,and ∂x∗c /∂σ > 0. In other words, a higher remaining value after a competitor’s successand shorter research duration increase the project value and accelerate R&D investment,
whereas higher volatility increases the option value and delays investment.

7

3.2

Project value and investment timing before a competi-

tor’s successIn this subsection, I consider the problem before a competitor’s success. Suppose that acompetitor has not yet developed a technology at the project initiation time τ . Because
the technology value falls to αX(τ + T ) for τ + T > Tc , where Tc denotes the time of acompetitor’s technology development, I can calculate the expected project value at τ asfollows:E[e−rT (1{τ +T <Tc } + 1{τ +T ≥Tc } α)X(τ + T ) | X(τ ), τ < Tc ]=E[e−rT αX(τ + T ) | X(τ )] + E[e−rT (1{τ +T <Tc } (1 − α)X(τ + T ) | X(τ ), τ < Tc ]=αe−(r−µ)T X(τ ) + (1 − α)e−(r−µ)T X(τ )E[1{τ +T <Tc } | τ < Tc ](14)

=AX(τ ),

(15)

where in (14) I used the independence between Tc and X(t), and in (15) I have
A = αe−(r−µ)T + (1 − α)e−(r+λ−µ)T(16)using the distribution function of the exponential distribution.
By (15), the project value function is expressed as
V (x) = sup E[e−rτ 1{τ <Tc } {e−rT (1{τ +T <Tc } + 1{τ +T ≥Tc } α)X(τ + T ) − I} + e−Tc 1{τ ≥Tc } Vc (X(Tc ))]τ= sup E[e−rτ 1{τ <Tc } {AX(τ ) − I} + e−Tc 1{τ ≥Tc } Vc (X(Tc ))]

(17)

τ

where the investment time τ is optimized over all stopping times, and Tc stands for thetime of a competitor’s technology development. Note that in (17), the project valuechanges to Vc (X(Tc )), which is deﬁned by (11), at time Tc , if Tc ≤ τ holds.
As is well known, in the continuation region, V (x) satisﬁes the ordinary diﬀerential
equation:µx

dV (x) σ 2 x2 d2 V (x)++ λ(Vc (x) − V (x)) = rV (x),
dx
2
d2 x

(18)

where λ(Vc (x) − V (x)) in (18) corresponds to the fact that V (x) changes to Vc (x) at
probability λdt in the inﬁnitesimal time interval dt in the continuation region.
Now, suppose that the investment threshold x∗ before a competitor’s success is lower
than the investment threshold x∗c after a competitor’s success. This will be veriﬁed inAppendix A. Because of x∗ < x∗c , a general solution to (18) is expressed as V (x) =Vc (x) + B̃1 xβ̃ + B̃2 xγ̃ , where B̃1 and B̃2 are constants, and β̃ and γ̃ are deﬁned by
√(
)
1
µ
µ
1 2 2(r + λ)β̃ = − 2 +−+(> 1),
(19)
2 σ
σ2 2
σ2
√(
)
1
µ
µ
1 2 2(r + λ)γ̃ = − 2 −−+(< 0).
(20)
2 σ
σ2 2
σ2

8

Note that Vc (x) satisﬁes (6) for x < x∗ (< x∗c ). As in Section 3.1, by the trivial boundary
condition, i.e., V (0) = 0, I have B̃2 = 0. For the investment trigger x∗ , the value matchingand smooth pasting conditions become( ∗ )βx(Ac x∗c − I) + B̃1 x∗β = Ax∗ − I,x∗c( ∗ )β−1dV (x∗ )x
β=(Ac x∗c − I) + B̃1 βx∗β−1 = A,
dx
x∗c
x∗c

V (x∗ ) =

(21)
(22)

respectively. By solving (21) and (22), I can derive B̃1 and x∗ as follows.
Proposition 2 Suppose that a competitor has not developed a technology. The projectvalue function V (x) is given by

( )β̃

 Vc (x) + x(Ax∗ − I − Vc (x∗ )) (x < x∗ )∗x
V (x) =
Ax − I(x ≥ x∗ ),(23)

where Vc (x) is the upper equation in (11), and the investment trigger x∗ ∈ (0, x∗c ) is definedby the solution to
β̃I
+
x =
(β̃ − 1)A∗( ∗ )βx(β̃ − β)(Ac x∗c − I).
x∗c(β̃ − 1)A(24)

The optimal investment time τ ∗ is given byτ ∗ = {t ≥ 0 | X(t) ≥ x∗ }.

(25)

In the upper equation in (23), the project value V (x) is decomposed into the projectvalue after a competitor’s success, i.e., Vc (x), and the value of the option of investing in
the project before a competitor’s success, i.e., (x/x∗ )β̃ (Ax∗ − I − Vc (x∗ )). Proposition
2 implies the R&D investment policy as follows. The ﬁrm waits for R&D investment
until the technology value X(t) hits the investment trigger x∗ . If a competitor develops atechnology ﬁrst, the ﬁrm increases the investment trigger from x∗ to x∗c . When X(t) hitsx∗ before a competitor’s technology development, the ﬁrm initiates the R&D project, andthe expected project value becomes Ax∗ , where A includes a discount due to a risk of acompetitor’s success during research duration T .
By ∂Vc (x)/∂α > 0, ∂Vc (x)/∂T < 0, ∂Vc (x)/∂σ ≥ 0, ∂A/∂α > 0, ∂A/∂λ < 0, ∂A/∂T <0, ∂ β̃/∂λ > 0, ∂ β̃/∂σ < 0, and (23), I have ∂V (x)/∂α > 0, ∂V (x)/∂λ < 0, ∂V (x)/∂T < 0,and ∂V (x)/∂σ ≥ 0. In other words, a lower risk of competition, shorter research duration,and higher volatility increase the R&D project value. On the other hand, the impactsof α, λ, T, and σ on the investment trigger x∗ are not clear from (24). I will reveal theimpacts numerically in Section 4.

9

3.3

Extended model

Although so far I have treated research duration T as a constant, it is easy to extend theresults in Sections 3.1 and 3.2 to a case with random variable T . In this subsection, I
assume that T follows a nonnegative random variable. For tractability, I assume that T
is independent of technology value X(t) and a competitor’s technology development time
Tc . This assumption means that risks of technology, market value, and competition are
not directly related to each other.
By taking expectations of (1), (4), and (16), I deﬁne(1 − E[e−rT ])I1,
I¯ = I0 +r
Āc = αE[e−(r−µ)T ],Ā = αE[e−(r−µ)T ] + (1 − α)E[e−(r+λ−µ)T ].

(26)
(27)
(28)

¯ Āc , and Ā, respectively, and tracing the discussions in
By replacing I, Ac , and A with I,
Sections 3.1 and 3.2, I can easily derive the model solutions. Indeed, the project valuefunction and investment trigger after a competitor’s success are equal to (11) and (12),replacing I and Ac with I¯ and Āc , respectively. The project value function and investment
trigger before a competitor’s success are equal to (23) and (24), replacing I, Ac , and A¯ Āc , and Ā, respectively. Clearly, the comparative statics results are unchanged
with I,
from those explained after Propositions 1 and 2.
The model can capture uncertainty of technology, market value, and competitors in
the R&D decision-making process, while previous models miss any of the three features.
Nevertheless, I can analytically derive the R&D project value and investment timing,
which help in a real-world decision-making process of R&D investment. It is one of thecontributions of this paper to develop such a general and tractable model and derive
the analytical solutions. In the next section, I will make another contribution to theliterature by analyzing the model and revealing the interactions of the three risks innumerical examples.

4

Numerical analysis and implications

4.1

Baseline analysis

This paper does not focus on a case study of a speciﬁc R&D project. Instead, I show
numerical results for a wide range of parameter values, demonstrating several propertiesof the project value and the investment policy in the model. I set the base parametervalues in Table 1. There are several methods for estimating the market parameter values,i.e., r, µ, and σ, 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. I set r, µ, and σ in Table 1 following the standard literature.

10

On the other hand, the technological parameter values, i.e., research duration T andcosts I1 and I2 , as well as the competition risk parameter values, i.e., arrival rate λand remaining fraction α, can be internally estimated by a ﬁrm. These values greatly
diﬀer depending on the project types and the industries. For example, in a project ofdeveloping a new drug, research duration is long (e.g., Kellogg and Charnes (2000), Lochand Bode-Greuel (2001), and Hartmann and Hassan (2006)). However, taking account of
the fact that the average research duration is around 2 to 4 years in most of the literature(e.g., Pakes and Schankerman (1984)), I assume that T takes a value in [2, 4] following auniform distribution. The parameter values λ = 0.2 and α = 0.5 mean that a competitor
is expected to develop a technology 1/λ = 5 years later and that the technology value
decreases by half if a competitor develops prior to the ﬁrm.
Table 2 presents the investment triggers and project values, while Figure 1 shows
the value functions V (X(t)) and Vc (X(t)). In the baseline case, the ﬁrm initiates the
R&D project when the technology value X(t) hits the investment trigger x∗ = 11.593
before a competitor’s technology development. The R&D project takes random duration
T ∈ [2, 4] years until completion. This delay decreases the project value by the multiplierE[e−(r−µ)T ] = 0.942. A competitor may potentially develop a technology during research
duration T . This risk reduces the project value further by the multiplier Ā/E[e−(r−µ)T ] =¯
0.777. As a result, the NPV at the investment time becomes Āx∗ − I¯ = 4.823 = 1.319I.
In other words, considering the option value of waiting, the ﬁrm waits until the expected
proﬁt exceeds 1.319 times higher than the expected investment cost. At completion timeτ ∗ + T , the ﬁrm receives the technology value X(τ ∗ + T ) if a competitor has not yet
developed a technology. The ﬁrm receives the discounted technology value 0.5X(τ ∗ + T )if a competitor develops a technology during research duration.
If a competitor develops a technology before X(t) hits x∗ = 11.593, the ﬁrm changes
the R&D investment policy to the following. The ﬁrm invests in the R&D project once
X(t) hits x∗c = 40.658(> x∗ = 11.593). There is no longer risk of competition. The NPV¯ which means that, considering
at the investment time becomes Āc x∗c − I¯ = 15.49 = 4.236I,
the option value of waiting, the ﬁrm waits until the expected proﬁt exceeds 4.236 times
higher than the expected investment cost. Note that 4.2360 is much higher than thecorresponding value 1.3188 before a competitor’s success. This is because due to λ = 0.2,β̃ = 2.873 is much higher than β = 1.236 (cf. (7) and (19)). It can be interpreted thata risk of competition greatly decreases the value of waiting. At completion time τc∗ + T ,the ﬁrm receives the discounted technology value 0.5X(τc∗ + T ).
In the following subsections, I will examine the comparative statics with respect touncertainty of market value, research duration, and a competitor’s technology development. I reveal how the interactions of the three risks aﬀect the R&D project value andinvestment policy.

11

4.2

The impacts of market value uncertainty

This subsection presents the comparative statics with respect to market value volatility
σ. Figure 2 depicts investment triggers x∗ and x∗c as well as project values V (x) and Vc (x)for varying levels of σ.8 The other parameter values are set in Table 1.
I can see from Figure 2 that higher σ increases x∗ , x∗c , V (x), and Vc (x). The reason isthat β and β̃ decrease from 1.3007 and 3.7870 to 1.1327 and 2, respectively, for σ = 0.1
to 0.4, which means that the value of waiting increases for a higher σ. These results are
straightforward and consistent with the standard theory that higher market uncertainty
increases the option value of waiting and delays the exercise of the option (e.g., Dixit andPindyck (1994)). For a wide range of parameter values, I also investigated the impacts ofh, λ, and α on the volatility eﬀects. However, I could not ﬁnd any signiﬁcant interactions
between uncertainty of market value and uncertainty of research duration or competition.

4.3

The impacts of uncertainty of research duration

This subsection presents the comparative statics with respect to uncertainty of T . I
assume that T follows a uniform distribution between [3 − h, 3 + h] with E[T ] = 3 years.
Figure 3 depicts investment triggers x∗ and x∗c as well as project values V (x) and Vc (x)for varying levels of h. The other parameter values are set in Table 1.
I can see from Figure 3 that higher h decreases x∗ and x∗c and increases V (x) and Vc (x).
The reason is explained as follows. Higher h increases T with respect to the convex order(e.g., see Chapter 3.A in Shaked and Shanthikumar (2007)). Then, higher h decreases
I¯ = I0 + (1 − E[e−rT ])I1 /r and increases Āc = αE[e−(r−µ)T ] and Ā = αE[e−(r−µ)T ] + (1 −α)E[e−(r+λ−µ)T ] because e−rT , e−(r−µ)T , and e−(r+λ−µ)T are convex functions of T . Thus,
interestingly, I can argue that higher uncertainty of research duration plays a positive rolein decreasing the expected investment cost and increasing the expected project values,and hence, it accelerates R&D investment.
Next, I explain how a risk of competition inﬂuences the eﬀects of uncertainty of T .
Figure 4 compares the results for λ = 0.2, 0.4, 0.6, and 0.8. I omit depicting x∗c and Vc (x)because they are independent of λ. I can see from the ﬁgure that higher λ intensiﬁes
the impacts of h on x∗ . The reason is that higher λ intensiﬁes the convexity of Ā =αE[e−(r−µ)T ] + (1 − α)E[e−(r+λ−µ)T ] with respect to T . Similarly, lower α intensiﬁes theimpacts of h, although I omit depicting a ﬁgure. These results suggest that a higher riskof competition magniﬁes the positive impacts of higher uncertainty of research duration
on R&D investment.

8

For simplicity, I 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 a few papers, including Wong (2007), examine thecomparative statics assuming a relation between µ and σ.

12

Although uncertainty of research duration and costs is recognized as a key characteristic of an R&D project (e.g., Raz, Shenhar, and Dvir (2002)), few studies have revealed its
impacts on R&D investment timing and project value.9 In this paper, I demonstrate thathigher uncertainty of research duration increases project value and accelerates investment
and that severe competition intensiﬁes the positive eﬀects. These results contrast with theimpacts of market uncertainty (cf. Section 4.2). Technological uncertainty, unlike market uncertainty, will never be dissolved by waiting, and hence, the ﬁrm has no incentive
to delay investment and obtain extra information. This diﬀerence leads to the opposite
eﬀects of uncertainty.
Although my results about the positive eﬀects of technological uncertainty are novel,
many empirical observations support the results. For example, Driver, Temple, and Urga
(2006) showed positive sensitivities of uncertainty to investment in industries with highR&D intensity. They also documented that positive sensitivities are observed in industries
with strong ﬁrst-mover advantages, i.e., high risks of competition. My results are also
consistent with empirical ﬁndings by Grullon, Lyandres, and Zhdanov (2012) and Kraft,
Schwartz, and Weiss (2015). Indeed, they showed that the sensitivities of uncertainty to
ﬁrm values are higher for ﬁrms with higher R&D intensity.

4.4

The impacts of competitor riskThis subsection presents the comparative statics with respect to the arrival rate λ ofa competitor’s technology development and the remaining value α after a competitor’s
success. Figure 5 depicts investment triggers x∗ and x∗c as well as project values V (x) andVc (x) for varying levels of λ. The other parameter values are set in Table 1. Note thatx∗c and Vc (x) do not depend on λ. As explained after Proposition 2, I can see from theﬁgure that V (x) decreases in λ.
Notably, in the ﬁgure, I ﬁnd that x∗ is U-shaped in λ. This result can be explained by
the tradeoﬀ between the two conﬂicting eﬀects. Because of ∂ Ā/∂λ < 0, higher λ lowers
the expected project value at project initiation. This eﬀect delays R&D investment. On
the other hand, because of ∂ β̃/∂λ > 0, higher λ decreases the value of waiting. Thiseﬀect accelerates R&D investment. For λ < 0.135, the latter eﬀect dominates the former,
and hence, x∗ decreases in λ. For λ > 0.135, the former eﬀect dominates the latter, andhence, x∗ increases in λ. In my computations for a wide range of parameter values, I
always found the U-shaped relation for positive research duration T , although the levelof λ for the two eﬀects oﬀsets changes from 0.135 with changing parameter values. For
instance, shorter T decreases the former eﬀect and hence increases the oﬀsetting threshold
λ.
9

A notable exception is Huchzermeier and Loch (2001). They studied the eﬀects of uncertainty of project

delay in a discrete time model, but they did not reveal any interactions between uncertainty of research durationand competition.

13

Next, I explore the comparative statics with respect to α. Figure 6 depicts investment
triggers x∗ and x∗c as well as project values V (x) and Vc (x) for varying levels of λ. As
explained after Propositions 1 and 2, I can see from the ﬁgure that V (x) and Vc (x)increases in α, while x∗c decreases in α.
The impacts of α on x∗ are interesting in Figure 6. As in the eﬀects of λ, two conﬂicting
eﬀects exist. Because of ∂ Ā/∂α > 0, higher α increases the expected project value atproject initiation. This eﬀect accelerates R&D investment. On the other hand, becauseof ∂Vc (x)/∂α > 0, higher α increases the project value after a competitor’s success andthen increases the value of waiting. This eﬀect delays R&D investment. For the baseline
parameter values, the latter eﬀect dominates the former, and hence, x∗ increases in α.
However, the tradeoﬀ changes with parameter values. For instance, Figure 7 shows theresults for λ = 0.2, 0.4, 0.6, and 0.8. For λ = 0.4 and 0.6, x∗ is U-shaped in α, whereas for
λ = 0.8, x∗ decreases in α. This is because the former eﬀect, which is stronger for higher
λ, can dominate the latter eﬀect.
In summary, I ﬁnd that R&D investment timing can be U-shaped in the strength ofcompetition. This result is strongly contrasted with previous ﬁndings in the literature.
Indeed, the standard literature has argued that a higher risk of competition decreases thevalue of waiting and accelerates investment. Because they do not consider any interactions between competition and research duration, the monotonic results hold. Although∂ Ā/∂λ = ∂ Ā/∂α = 0 holds in the absence of research duration, ∂ Ā/∂λ < 0 and Ā/∂α > 0
hold in the presence of research duration. This paper complements the previous literatureby showing that a risk of competition can adversely aﬀect R&D investment through theinteractions with research duration.

5

Conclusion

In this paper, I developed a generic R&D decision-making model involving three types of
uncertainty in an R&D project, namely, uncertainty of research duration, market valueof technology, and technology development by a competitor. By deriving the analyticalsolutions in the model, this paper helps practitioners and researchers to evaluate andoptimize various cases of R&D investment. In addition, by analyzing the model solutions
for a wide range of parameter values, I obtained novel implications about the eﬀects ofthe three risks on R&D decision making.
I showed that higher uncertainty of research duration increases the project value andspeeds up R&D investment through the convexity of the project value with respect toresearch duration. I also showed that severe competition increases the convexity and thenintensiﬁes the positive eﬀects of uncertainty of research duration on R&D investment.
These results are novel and contrasted with the eﬀects of market uncertainty, and they canaccount for several empirical ﬁndings about positive eﬀects of uncertainty in industries

14with high R&D intensities and severe competition. Furthermore, I showed that R&D
investment timing has a U-shaped relation with the strength of competition. Comparedto the monotonic results in the previous literature, I showed that the non-monotonic
results can occur through the interactions between competition and research duration.
A limitation of the model is the assumption that the three risks are independent.
Without this assumption, the model does not allow analytical solutions. In practice,
however, competitor risk may increase as the market value of technology increases. In
such a case, one may ﬁnd that the eﬀects of market uncertainty, combined with a risk ofcompetition, change from the normal eﬀects. Further, the model does not consider how toﬁnance the R&D project. In recent years, many papers, including Nishihara and Shibata(2013), Shibata and Nishihara (2015b), and Sundaresan, Wang, and Yang (2015), haveinvestigated the eﬀects of ﬁnancial frictions on investment. R&D investment timing andproject value could depend on how eﬀectively a ﬁrm ﬁnances its project. These issues
could be interesting topics for future research.

References
Armada, M., L. Kryzanowski, and P. Pereira, 2011, Optimal investment decisions for two
positioned ﬁrms competing in a duopoly market with hidden competitors, European
Financial Management 17, 305–330.
Baker, H., S. Dutta, and S. Saadi, 2011, Management views on real options in capitalbudgeting, Journal of Applied Finance 21, 18–29.
Cassimon, D., M. Backer, P. Engelen, M. Wouwe, and V. Yordanov, 2011, Incorporating
technical risk in compound real option models to value a pharmaceutical R&D licensing
opportunity, Research Policy 40, 1200–1216.
Dixit, A., and R. Pindyck, 1994, Investment Under Uncertainty (Princeton University
Press: Princeton).
Driver, C., P. Temple, and G. Urga, 2006, Real options – delay vs. pre-emption: Do
industrial characteristics matter?, International Journal of Industrial Organization 26,
532–545.
Grullon, G., E. Lyandres, and A. Zhdanov, 2012, Real options, volatility, and stock returns, Journal of Finance 67, 1499–1537.
Guthrie, G., 2009, Real Options in Theory and Practice (Oxford University Press: Oxford).
Hartmann, M., and A. Hassan, 2006, Application of real options analysis for pharmaceutical R&D project valuation – empirical results from a survey, Research Policy 35,
343–354.

15

Huchzermeier, A., and C. Loch, 2001, Project management under risk: Using the real
options approach to evaluate ﬂexibility in R&D, Management Science 47, 85–101.
Huisman, K., 2001, Technology Investment: A Game Theoretic Real Options Approach(Kluwer Academic Publishers: Boston).
Huisman, K., 2001, and P. Kort, 2015, Strategic capacity investment under uncertainty,
RAND Journal of Economics 46, 376–408.
Kellogg, D., and J. Charnes, 2000, Real-options valuation for a biotechnology company,
Financial Analyst Journal 56, 76–84.
Kraft, H., E. Schwartz, and F. Weiss, 2015, Growth options and ﬁrm valuation, NBER
Working Paper 18836.
Lambrecht, B., and W. Perraudin, 2003, Real options and preemption under incomplete
information, Journal of Economic Dynamics and Control 27, 619–643.
Lavrutich, M., K. Huisman, and P. Kort, 2016, Entry deterrence and hidden competition,
Journal of Economic Dynamics and Control 69, 409–435.
Lee, J., and D. Paxson, 2001, Valuation of R&D real American sequential exchange options, R&D Management 31, 191–201.
Loch, C., and K. Bode-Greuel, 2001, Evaluating growth options as sources of value forpharmaceutical research projects, R&D Management 31, 231–248.
Lukas, E., T. Spengler, S. Kupfer, and K. Kieckhafer, 2017, When and how much toinvest? Investment and capacity choice under product life cycle uncertainty, European
Journal of Operational Research 260, 1105–1114.
Nishihara, M., 2012, Real options with synergies: Static versus dynamic policies, Journalof the Operational Research Society 63, 107–121.
Nishihara, M., and M. Fukushima, 2008, Evaluation of ﬁrm’s loss due to incomplete
information in real investment decision, European Journal of Operational Research 188,
569–585.
Nishihara, M., and A. Ohyama, 2008, R&D competition in alternative technologies: A
real options approach, Journal of the Operations Research Society of Japan 51, 55–80.
Nishihara, M., and T. Shibata, 2010, Interactions between preemptive competition and aﬁnancing constraint, Journal of Economics and Management Strategy 19, 1013–1042.
Nishihara, M., and T. Shibata, 2013, The eﬀects of external ﬁnancing costs on investment
timing and sizing decisions, Journal of Banking and Finance 37, 1160–1175.
Pakes, A., and M. Schankerman, 1984, The rate of obsolescence of patents, researchgestation lags, and the private rate of return to research resources, in Z. Griliches, ed.:R&D, Patents, and Productivity . pp. 73–88 (University of Chicago Press: Chicago).

16

Pawlina, G., and P. Kort, 2006, Real options in an asymmetric duopoly: Who beneﬁts
from your competitive disadvantage?, Journal of Economics and Management Strategy
15, 1–35.
Pennings, E., and L. Sereno, 2011, Evaluating pharmaceutical R&D under technical andeconomic uncertainty, European Journal of Operational Research 212, 374–385.
Raz, T., A. Shenhar, and D. Dvir, 2002, Risk management, project success, and technological uncertainty, R&D Management 32, 101–109.
Shaked, M., and J. Shanthikumar, 2007, Stochastic Orders (Springer: New York).
Shibata, T., 2016, Strategic entry in a triopoly market of ﬁrms with asymmetric cost
structures, European Journal of Operational Research 249, 728–739.
Shibata, T., and M. Nishihara, 2015a, Investment-based ﬁnancing constraints and debt
renegotiation, Journal of Banking and Finance 51, 79–92.
Shibata, T., and M. Nishihara, 2015b, Investment timing, debt structure, and ﬁnancing
constraints, European Journal of Operational Research 241, 513–526.
Sundaresan, S., N. Wang, and J. Yang, 2015, Dynamic investment, capital structure, anddebt overhang, Review of Corporate Finance Studies 4, 1–42.
Weeds, H., 2002, Strategic delay in a real options model of R&D competition, Review of
Economic Studies 69, 729–747.
Wong, K., 2007, The eﬀect of uncertainty on investment timing in a real options model,
Journal of Economic Dynamics and Control 31 31, 2152–2167.

A

Proof of the existence of x∗ ∈ (0, x∗c )Deﬁne the function F (x) byβ̃I
F (x) =+(β̃ − 1)A

(

x
x∗c

)β

(β̃ − β)(Ac x∗c − I)− x.
(β̃ − 1)AFor x = 0, I have
F (0) =For x = x∗c , I have
F (x∗c ) =
=

β̃I
> 0.
(β̃ − 1)A

(29)

β̃I
(β̃ − β)(Ac x∗c − I)+− x∗c(β̃ − 1)A(β̃ − 1)AβI + (β̃ − β)Ac x∗c − (β̃ − 1)Ax∗c(β̃ − 1)AβI + (β̃ − β)Ac x∗c − (β̃ − 1)Ac x∗c(β̃ − 1)AβI − (β − 1)Ac x∗c=(β̃ − 1)A
<

(30)

= 0,

(31)

17

where in (30) I used β̃ > 1 and Ac < A, and in (31) I used (12). By β > 1, F (x) is aconvex function. Then, by (29), (31), and the convexity of F (x), the equation F (x) = 0
has a unique solution x∗ in the interval (0, x∗c ).

18

Table 1: Baseline parameter values.
rµσ

0.08 0.06 0.2

x∗

T

λ

α

I1

I2

x

[2, 4] 0.2

0.5

1

1

8

Table 2: Investment triggers and project values.
x∗
I¯
Ā
Āc
V (x)
Vc (x)
c11.5927 40.6582 3.6566 0.7314

35

0.47092.076435
Vc(X(t))V(X(t))
30

30

25

25

20

20

15

15

10

10

5

5

0

2.6063

10

20

30

40

0

50

X(t)

10

20

30

40

50

X(t)

Figure 1: Value functions. The figure plots V (X(t)) and Vc (X(t)) as functions of X(t). The
parameter values are set in Table 1.

19

18

70
∗

∗

16

x

xc

60

14
50
12
40

10
8
0.1

0.2

0.3

30
0.1

0.4

0.2

σ

0.3

0.4

0.3

0.4

σ

3.2

2.6
V(x)

3

Vc(x)

2.4

2.8
2.2
2.6
2

2.4
2.2
0.1

0.2

0.3

1.8
0.1

0.4

0.2

σ

σ

Figure 2: The impacts of market value uncertainty. This figure plots x∗ , x∗c , V (x), and Vc (x)for varying levels of σ. The other parameter values are set in Table 1.

11.8

41
∗

∗

xc

x

11.6

40.5

11.4
11.2

40

11
10.8

0

1

2

39.5

3

0

1

h

2

3

2

3

h

2.8

2.09
V(x)

2.75

Vc(x)

2.085

2.7
2.08
2.65
2.075

2.6
2.55

0

1

2

2.07

3

0

1

h

h

Figure 3: The impacts of uncertainty of research duration. This figure plots x∗ , x∗c , V (x), andVc (x) for varying levels of h, where research duration T follows a uniform distribution with
[3 − h, 3 + h]. The other parameter values are set in Table 1.

20

x∗ λ=0.2

24

2.8

λ=0.4
λ=0.6

22

2.7

λ=0.8
2.6

20

V(x) λ=0.2

2.5

λ=0.4

18

λ=0.6

2.4

λ=0.8

16
2.3
14

2.2

12
10

2.1

0

1

2

2

3

0

1

2

h

3

h

Figure 4: The impacts of uncertainty of research duration combined with the arrival rate of acompetitor’s technology development. The left and right panels plot x∗ and V (x), respectively,
for varying levels of h and λ, where research duration T follows a uniform distribution with
[3 − h, 3 + h]. The other parameter values are set in Table 1.

22

42
∗

20

x

∗

xc

41.5

18

41

16
40.5

14

40

12
10

0

0.1

0.2

0.3

0.4

39.5

0.5

0

0.1

0.2

λ

0.3

0.4

0.5

λ

5

3.5
V(x)

Vc(x)

3

4

2.5
2

3

1.5
2

0

0.1

0.2

0.3

0.4

1

0.5

λ

0

0.1

0.2

0.3

0.4

0.5

λ

Figure 5: The impacts of the arrival rate of a competitor’s technology development. This figure
plots x∗ , x∗c , V (x), and Vc (x) for varying levels of λ. The other parameter values are set in Table
1.

21

18

250
∗

∗

xc

200

x

16

150
14
100
12
10

50
0.2

0.4

0.6

0

0.8

0.2

0.4

α

0.6

0.8

0.6

0.8

α

5

5
V(x)

4

Vc(x)

4
3

3
2
2
1

1
0.2

0.4

0.6

0

0.8

0.2

0.4

α

α

Figure 6: The impacts of the remaining value after a competitor’s technology development.
This figure plots x∗ , x∗c , V (x), and Vc (x) for varying levels of α. The other parameter values are
set in Table 1.

35

V(x) λ=0.2

4.5

λ=0.4

∗

x λ=0.2
λ=0.4
λ=0.6
λ=0.8

30

4

λ=0.6
λ=0.8

3.5
3

25
2.5
2
20
1.5
1

15

0.5
10

0.2

0.4

0.6

0

0.8

α

0.2

0.4

0.6

0.8

α

Figure 7: The impacts of the remaining value after a competitor’s technology development
combined with the arrival rate of a competitor’s technology development. The left and rightpanels plot x∗ and V (x), respectively, for varying levels of α and λ. The other parameter values
are set in Table 1.

22