The Relationship between Uncertainty and Investment

under Various Stochastic Processes

George Y. Wang and Ciaran Driver

Tanaka Business School, Imperial College University of London, UK

Abstract The conventional belief on a negative relationship between uncertainty and investment has dominated investment theory for a long time. Researchers on real options argue that increased uncertainty will cause a decrease in the current level of investments by raising the value of option of waiting. This paper postulates an important idea that increased uncertainty has two opposing effects on investment: the variance effect and the realization effect, to such an extent that increased uncertainty, in certain situations, may actually encourage investment due to a higher probability of investing. Earlier studies mostly base the argument on the assumption of geometric Brownian motion (GBM), while it has been found that the argument also holds for the situations where the underlying variable follows an alternative stochastic process such as mean reversion, mixed diffusion-jump, and jump amplitude, with the third effect, e.g., mean-reverting effect and jump effect, under consideration. The economic implication is that uncertainty does not always discourage investment even under several sources of uncertainty which has different risk profiles. It is also obvious that the “high-risk” projects are not always dominated by the “low-risk” projects because the “high-risk” projects may have a positive realization effect, i.e., a higher probability of exceeding the optimal triggers, which may encourage investment. Keywords: real options, variance effect, realization effect, geometric Brownian

motion, mean reversion, mixed diffusion-jump, jump amplitude JEL Classification: D81 G31 Correspondence: yungchih.wang@imperial.ac.uk

1

The Relationship between Uncertainty and Investment

under Various Stochastic Processes

1. Introduction The relationship between uncertainty and investment has fascinated financial economists for a long time. Researchers on real options argue that increased uncertainty will cause a decrease in the current level of investment by raising the value of option of waiting. For example, Cukierman (1980) presents a Bayesian framework to address the idea that an investment opportunity can be more valuable by waiting longer for more information arrivals. Pindyck (1988 and 1991) and Dixit (1989 and 1992) also find that a higher level of uncertainty not only increases option value, but also brings about a higher optimal investment trigger to such an extent that uncertainty may in effect discourage investment.

Extending the standard real options theory, Sarkar (2000) and Rhys et al. (2002) explore the relationship between uncertainty and investment by asking the question

whether a project value, V, would reach an optimal investment trigger, V ∗ , given that the project value evolves as a GBM. Both studies apply a similar probability function, and find that the uncertainty-investment relationship is not always negative. They show that in certain situations, increased uncertainty in a GBM setting may encourage investment due to a higher probability of investing or an earlier time of first passage. This paper aims to further investigate the uncertainty-investment relationship under the assumption of a variety of stochastic processes based on the techniques of Monte Carlo simulation. The earlier studies such as Sarkar (2000) and Rhys et al. (2002) apply a probability function to measure the probability of reaching a critical value under a GBM.1 Compared to this approach, Monte Carlo simulation is relatively flexible when the underlying variable follows an alternative process. In addition, the standard real options theory makes two implicit assumptions for tractable solutions: firstly, the investment opportunity of interest is held by the monopolistic firm or the investment opportunity can exist in an infinite time horizon; secondly, the underlying variables are assumed to follow a GBM. In the relaxation of these two assumptions, a general investment framework is proposed to derive an optimal

1 See Harrison (1985).

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investment trigger, based on the techniques of Monte Carlo simulation. Appendix 1 illustrates the detailed procedure of the framework for deriving the optimal triggers. The investment framework is then applied to examine the relationship between uncertainty and investment under a jump amplitude process. The rest of the paper is organized as follows: Section 2 introduces the specifications of a variety of stochastic processes both in continuous time and in discrete time, serving as a foundation for the subsequent sections. Section 3 extends standard real options theory to investigate the relationship between uncertainty and investment under three different stochastic processes, namely, GBM, mean-reversion (MR), and mixed diffusion-jump (MX). Specifically, the research is directed at investigating the overall effects of uncertainty on investment by measuring the probability of reaching an optimal investment trigger. By the proposed general investment framework in Appendix 1, Section 4 illustrates an application of the investment framework to derive the optimal triggers under a jump amplitude (JA) process and to explore the relationship between uncertainty (in terms of stochastic jump) and investment. Section 5 gives concluding remarks.

2. Alternative Stochastic Processes

Since stochastic processes are regarded as major sources of uncertainty in the evaluation of capital investments, in this section we introduce a variety of stochastic processes for applications in subsequent sections. In each subsection, the specifications of each stochastic process both in continuous time and in discrete time are presented. Graphical illustrations are also provided in comparison with the common GBM assumption.

2.1 Geometric Brownian Motion There are several stochastic processes of interest. The first, most widely applied process is geometric Brownian motion which accounts for a continuous form of random walk. The main property of this class of stochastic process is that the rate of return is normally distributed, implying a lognormal distribution of the project value. The continuous-time version of a GBM is given below: dV Vdt Vdzα σ= + (1) where α , σ , and dz denote drift rate, instantaneous volatility, and an increment of

3

a standard Wiener process, respectively. For the simulation purpose, the discrete-time version of GBM is expressed as follows:

lnV v t tσ ε∆ = ∆ + ∆ 2 (2) where t∆ and ε represent a small interval of time and a random drawing from a

standard normal distribution, respectively, and 212

v α σ= − .

2.2 Mean Reversion

Another class of stochastic process is a mean-reverting process which is often used to describe the price behavior of commodity and natural resources. The most prominent property of a mean-reverting process is that its growth rate is not a constant but instead a function of a difference between current value and long-run mean, suggesting that growth rate in effect responds to disequilibrium. Dixit and Pindyck (1994, Ch. 5) examine the value of an investment opportunity whose value follows a mean-reverting process. The specification of this commonly used mean-reverting process is given below:

( )dV V V Vdt Vdzη σ= − + (3)

where η denotes a speed of mean reversion and V is a long-run mean.

As there are many ways to specify a mean-reverting process, Dixit and Pindyck’s specification is somewhat arbitrary but convenient to find a “quasi-analytical” solution for the value of the project. Equation (3) can be discretized into the following equation:

( )lnV V V t tη σ ε∆ = − ∆ + ∆ (4)

For a graphical comparison, we give a simulated sample path of mean reversion

2 Since GBM is log-normally distributed, a more explicit form of Equation (2) is given below:

( )1

tv t tt tV V e σ ε∆ + ∆+

� �=� �

4

according to Equation (4) with the GBM as a comparison in Figure 1.

Note: 0 100, 5%, 20%, 1/ 52, 4, =0.03V V t Tα σ η= = = = ∆ = =

Figure 1 A Graphical Comparison of Mean Reversion and GBM

2.3 Pure Jump Cox and Ross (1975) propose a pure jump process to examine the option problem in that the underlying asset may change discontinuously due to unexpected shocks. In the context of capital investments, the pure jump process is often suggested to characterize technology advances in IT investment. It is important to point out that the specification of a pure jump process in Cox and Ross (1975) differs from other jump processes in the restriction of positive jumps of a fixed size. Let u denote a fixed proportional jump size. A pure jump process in continuous time is expressed by 2dV Vdt Vdqα= − + (5)

where 2dq denotes an increment of a pure jump process with a parameter of jump

intensity λ such that

5

2

with a probability of 0 with a probability of 1-u dt

dqdt

λλ

�= ��

(6)

For the purpose of simulation, Equation (5) can be changed into the discrete-time form as follows: 2lnV t Dα∆ = − ∆ + (7)

where 2D denotes an increment of a pure jump in discrete time and

2

with a probability of 0 with a probability of 1-u t

Dt

λλ∆�

= � ∆� (8)

Figure 2 displays a simulated sample path of a pure jump process in comparison with a GBM.

Note: 0 100, 5%, 20%, 1/ 52, 4, =2, 10%V t T uα σ λ= = = ∆ = = =

Figure 2 A Graphical Comparison of Pure Jump and GBM

2.4 Mixed Diffusion-Jump In general, a mixed diffusion-jump process which combines a GBM and a

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Poisson jump process. There are a variety of forms of a mixed diffusion-jump process, one of which is proposed by Merton (1976) in the financial option pricing problem and then applied by Trigeorgis (1990) in the context of evaluating an investment opportunity with competitive arrivals. A mixed diffusion-jump process in continuous time is expressed as follows:

( ) 3dV k Vdt Vdz Vdqα λ σ= − + + (9)

where 3dq is an increment of a Poisson jump process with a mean arrival rate λ

such that

3

with a probability of 0 with a probability of 1-

dtdq

dt

ϕ λλ

�= ��

(10)

where ( , )N k ϕϕ σ∼ denotes a proportional jump relative to V if a jump occurs.

Note that the Poisson jump term 3dq is assumed to be independent of dz such

that ( )3 0E dq dz = . Equation (10) also reveals that the actual growth rate of such a

mixed diffusion-jump process is not α but instead kα λ− in order to adjust the

influence of a Poisson event. For the simulation purpose, the discrete-time version

of the mixed diffusion-jump process is given as follows:

3lnV v t t Dσ ε∆ = ∆ + ∆ + (11)

where 3D denotes an increment of a Poisson jump in discrete time with a mean

arrival rate λ such that

3

with a probability of

0 with a probability of 1-

tD

t

ϕ λλ

∆�= � ∆�

(12)

It is worth noting that McDonald and Siegel (1986) and Dixit and Pindyck (1994)

also propose a mixed diffusion-jump process with the sign of the jump term changed

into negative to describe the situation in that the project becomes suddenly worthless

when a major competitor of the same product enters the market. For a graphical

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comparison, Figure 3 exhibits simulated sample paths of both mixed diffusion-jump

processes with a GBM as a comparison.

Note: 0 100, 5%, 20%, 1/ 52, 4, 2 ( 1),V t T Mixedα σ λ= = = ∆ = = = as 20%, 10%, 20%k ϕλ σ= = =

Figure 3 A Graphical Comparison of GBM and Mixed Diffusion-Jump

2.5 Jump Amplitude Process

Another type of jump processes are jump amplitude processes which are

suggested by Pennings and Lint (1997) to describe the characteristics of R&D

investments. The jump amplitude process differs from other jump processes in that

the impacts of information arrivals can not be foreseen such that jump direction and

jump size are stochastic by nature. A jump amplitude process can be mathematically

expressed as follows:

4dV Vdt Vdqα= + (13)

where 4dq an increment of a stochastic jump process. The jump term, 4dq , is

characterized by a parameter of jump intensity λ such that

8

4

with a probability of

0 with a probability of 1-

dtdq

dt

ψ λλ

�= ��

(14)

where ψ denotes a proportional jump relative to V.

By definition, Xψ = Γ where ( )1 or 1, 1X P X p= − = = , and )2,(~| XWeiX γΓ .

The discrete-time version of a jump amplitude process is modeled as follows:

4lnV v t D∆ = ∆ + (15)

where 4D denotes an increment of a stochastic jump component in discrete time

with a mean arrival rate λ , and 4D is expressed by

4

with a probability of

0 with a probability of 1-

tD

t

ψ λλ

∆�= � ∆�

(16)

Since a jump amplitude process allows both positive and negative jumps, the

estimation of the probability of up-jumps and down-jumps is important in specifying

the process. Figure 4 presents a simulated jump amplitude process, by assuming 5.0)1 ==P(X , i.e., a 50-50 chance of u-jump and down-jump. As shown in Figure

4, a jump amplitude process evolves in a very different way compared to a GBM.

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Note: 0 100, 5%, 20%, 1/ 52, 4, 2, 50%, 0.1XV t T pα σ λ γ= = = ∆ = = = = =

Figure 4 A Graphical Comparison of GBM and Jump Amplitude

2.6 Random Switch between Two Processes In general, a random switch process is characterized by a particular feature of random switching between two different processes governed by a Poisson switching intensity. In principle, a random switch process is very much similar to a regime-switching process which characterizes the underlying variable with various stochastic processes from different data-generating distributions. There is a minor difference between these two processes in that the former emphasizes different “processes” and the latter highlights different “distributions” or “regimes”. As in the literature on investment, few studies have examined the random switch process in the context of real options while there are already several studies on irreversible investments under the regime-switching process due to discontinuous information arrivals (Naik, 1993) or capability of switching among various distributional regimes (Gray, 1996; Bollen, 1998). Here, we are interested in random switching between a GBM and a mean-reverting process. The random switch process is expressed as follows: dV YVdt Vdzσ= + (17) where Y denotes a growth rate which randomly switches between a GBM and a mean-reverting process, expressed by

( ) with a prob. of switching from GBM to MR

with a prob. of switching from MR to GBMGM

MG

v dtY

V V dt

λη λ

��= � −�� (18)

where GMλ and MGλ denote the parameters of switching intensity from a GBM to

an MR and that from an MR to a GBM, respectively. The set of switching probabilities can be expressed by a switching probability matrix, Π , as follows:

11

GM GM

MG MG

dt dt

dt dt

λ λλ λ

−� �Π = −� �

(19)

10

For the simulation purpose, Equation (17) and (18) can be changed into the discrete-time forms, respectively:

lnV y t tσ ε∆ = ∆ + ∆ (20)

where ( ) with a prob. of switching from GBM to MR

with a prob. of switching from MR to GBMGM

MG

v ty

V V t

λη λ

∆��= � − ∆�� (21)

According to Equation (20) and (21), we simulate a random switch process in comparison with a GBM in Figure 5. For the simulated random switch process, the shaded area represents the time that the process switches from a GBM to a mean-reverting process while the light area stands for the time that the process switches from a mean-reverting process to a GBM. From Figure 5, it is not noticeable to visually distinguish a random switch process from a GBM as they are in fact different.

Note: 1. For the random switch process, the shaded area indicates a mean reversion while the

non-shaded area characterizes a GBM.

2. 0 100, 5%, 20%, 1/ 52, 4, 1, 100, 0.03.GM MGV t T Vα σ λ λ η= = = ∆ = = = = = =

Figure 5 A Graphical Comparison of GBM and Random Switch Process Switching between GBM and Mean Reversion

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3. The Relationship between Uncertainty and Investment An investment opportunity in the standard real options approach is treated as the problem of American-style option to invest, which must be exercised optimally by determining a critical value or optimal investment trigger. Investments are then initiated when the value of a project exceeds optimal investment trigger. As the general optimal investment policy, an optimal investment trigger should not only cover direct investment cost but also maximize the value of option to invest. Since option pricing theory suggests that an increase in uncertainty raises the option value, some researchers therefore argue that uncertainty may in effect discourage investment.3 In this section, we would like to extend the existing line of research by gauging the overall effects of uncertainty on investment given a specific stochastic process. We aim to demonstrate that the conventional belief of a negative uncertainty-investment relationship is not always correct.

3.1 Optimal Investment Triggers

The standard capital investment theory on real options studies the problem of optimal investment timing to pay an investment cost, I, in return for a project, V, characterized by irreversibility and uncertainty. V is considered to be the major source of uncertainty and is normally assumed to follow a GBM as in Equation (1) due to the ease of deriving a tractable solution. The value of an investment opportunity is determined by an optimal investment policy that maximizes the option

value. Let ( )F V denote the value of the investment opportunity and the

superscript * denote optimality. Under the assumption that V follows a GBM process, the optimal investment trigger is given by 4

1

1 1GBM

bV I

b∗ � �

= �−� � (22)

where GBMV ∗ and I denote the optimal GBM trigger and the investment cost,

respectively, and

3 Cukierman (1980), Caballero (1991), Mauer and Ott (1995), and Metcalf and Hassett (1995). 4 McDonald and Siegel (1986), Pindyck (1991), and Dixit and Pindyck (1994).

12

2

1 2 2 2

1 1 22 2

r r rb

δ δσ σ σ− −� � � �= − + − + � �

� � � � (23)

where δ represents the convenience yield of holding a project, which also implies the opportunity cost of deferring a project. For an investment opportunity whose value follows a mixed diffusion-jump process (denoted by MX), McDonald and Siegel (1986) and Dixit and Pindyck (1994) show that when the value of the project may be appropriated by competitive arrivals such that the project becomes suddenly worthless, the solution of optimal trigger

under such a mixed diffusion-jump process, MXV ∗ , has the same form as Equation (22)

with 1b substituted by 2b as follows:

2

2 2 2 2

1 1 2( )2 2

r r rb

δ δ λσ σ σ− − +� � � �= − + − + � �

� � � � (24)

where λ denotes the jump intensity of competitive arrivals. Given the underlying mean-reverting process (denoted by MR), Dixit and Pindyck (1994) provide the solutions of an investment opportunity and optimal investment trigger, respectively, as follows:

( )( ) ; ,F V BV G x gθ θ= (25)

( )MR MRV F V I∗ ∗= + (26)

where 2

2 2 2

1 1 22 2

V V rη ηθσ σ σ

� �= − + − + � �

,

2

2x V

ησ

= ,

2

22

Vg

ηθσ

= + , and

13

( )2 3( 1) ( 1)( 2)

; , 1( 1) 2! ( 1)( 2) 3!

x xG x g x

g g g g g gθ θ θ θ θ θθ + + += + + + +

+ + +� .

Since ( ); ,G x gθ stands for an infinite confluent hypergeometric function, both

Equation (25) and (26) must be solved numerically by an iterative procedure.

3.2 The Uncertainty-Investment Relationship under a GBM In this subsection, we examine the overall effect on investment of two opposing forces due to increased uncertainty. The first force is termed the “variance effect” which states the consequence that an increase in instantaneous volatility raises the level of optimal investment trigger and thus delays investment. The second force is called the “realization effect”, which describes an increase in the likelihood of reaching optimal investment trigger due to a higher level of instantaneous volatility. By combining these two effects and exploring the probability of initiating a project, the conventional wisdom that uncertainty discourages investment can be further verified under an alternative stochastic process. The investigation of the relationship between uncertainty and investment is an extension of Sarkar’s (2000) study, with a more flexible approach applied and more stochastic processes considered. As mentioned earlier, Sarkar applies a probability function of reaching a critical level to compute the probability of investing, given that the project value follows a GBM. While the probability function can not be applied in the situation where the underlying variable follows an alternative process, Monte Carlo simulation is suggested to measure the probability of simulated random paths reaching an investment trigger. The techniques of Monte Carlo simulation are especially advantageous when the underlying variable follows an alternative process rather than a GBM. The procedure is to simulate a large number of sample paths, given a particular stochastic process. The stochastic processes under consideration are GBM, mixed diffusion-jump process, and mean-reverting process, which will be simulated according to Equation (2), (11), and (4), respectively. It is important to note that the actual drift rate of a simulated stochastic process must be reduced by a convenience yield, i.e., an opportunity cost of holding a project. In the risk-neutral world, this adjustment of convenience yield is termed the “equivalent risk-neutral valuation”, which will be discussed later. If at any time in a finite horizon the

project value V is greater than V ∗ , this simulation trial is counted as a case of taking on the project. The probability of investing is obtained by computing the total cases

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of taking on the project out of the simulation trials. The total number of simulation trials should be large enough to ensure the robustness of a result. Thus, a higher probability of investing implies a greater chance of project acceptance, hence a positive impact on investment, and vice versa. Since real options matter especially for a near “at-the-money” project, the base case is assumed to be 0 100V I= = for a situation of concern.5 For each stochastic

process of interest, the optimal triggers are first derived to illustrate the variance effect,

given the base case parameters. For the numerical analysis of GBMV ∗ , the other

parameter values are given as 8%, 1/ 52, 5r t T= ∆ = = . Figure 6 presents the

variance effect on GBMV ∗ , given the project value follows a GBM process. As

displayed in Figure 6, it is obvious that GBMV ∗ increases with σ , i.e. 0GBMVσ

∗∂ >∂

and

decreases with δ , i.e., 0GBMVδ

∗∂ <∂

. The intuition underlying this observation is that

as investment uncertainty increases, management should defer the project even longer until the market condition becomes favorable. However, as the opportunity cost of holding a project increases, it is not sensible to postpone the project any longer, hence lowering the optimal investment triggers.

5 Copeland and Antikarov (2001) contend that real options matter in investment decisions only when the NPV of the project is close to zero, i.e., at-the-money in the terminology of financial options.

15

Note: 0 100, 8%, 1/ 52, 5, 10,000V I r t T Number of Trials= = = ∆ = = =

Figure 6 The Variance Effect under a GBM

Since the realization effect can not be shown graphically, Monte Carlo simulation based on 10,000 trials is then conducted to exhibit the combining effect of two opposing forces, the variance effect and the realization effect, by computing the probability of investing. Visual Basic codes of Monte Carlo simulation for calculating the probability of investing under a GBM process is available from the author on request. The simulation result is displayed in Figure 7. From the diagram, it can be seen that the probability of investing is initially an increasing function of volatility, but after a certain point it becomes a decreasing function of volatility. This result can be alternatively illustrated by the partial derivative of the

probability of investing with respect to volatility, ( ) GBMP Inv

σ∂

∂, where ( ) GBMP Inv

denote the probability of investing under a GBM. Given 6%δ = , it is easy to see

( )P0GBMInv

σ∂

>∂

at 30%σ < and ( )P

0GBMInv

σ∂

<∂

at 30%σ > . Therefore,

the result in Figure 7 indicates that for a lower level of volatility, an increase in

uncertainty actually raises the probability of investing and thus has a positive

influence on investment, while an increase in uncertainty, on the other hand,

discourages investment for a higher level of volatility. The turning point moves to

the higher level of volatility as δ decreases.

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It is also interesting that the probability of investing increases with the

opportunity cost of holding a project, given volatility being unchanged, i.e.,

( )0GBMP Inv

δ∂

>∂

. Thus, an increased convenience yield may have a positive impact

on investment.

Note: 0 100, 8%, 1/ 52, 5, 10,000V I r t T Number of Trials= = = ∆ = = =

Figure 7 The Probability of Investing as a Function of Volatility Given a GBM Process

To sum up, the variance effect has a negative impact on investment due to the

higher optimal investment triggers, while the realization effect can have a positive or

negative impact on investment, depending on the combinations of parameter values.

Consequently, the overall effect of these two offsetting forces on investment is not

always negative. In the preceding numerical analysis, it is demonstrated that in

certain situations where a lower level of volatility exists, uncertainty may in effect

encourage investment. However, for a higher level of volatility, the higher the level

of the opportunity cost of investment, the more likely it is that an increase in

uncertainty will decrease investment.

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3.3 The Uncertainty-Investment Relationship under an MX Process

In this subsection, the relationship between uncertainty and investment under a

mixed diffusion-jump process is examined. The mixed diffusion-jump process of

interest is a mixture of a GBM and a Poisson down jump, proposed both in McDonald

and Siegel (1986) and Dixit and Pindyck (1994). Since the mixed diffusion-jump

process contains an additional source of uncertainty, Poisson down jumps, it is

therefore crucial to analyze the “jump effect” on investment in addition to the

variance effect. The “jump effect” on investment can be defined as the effect of

increased jumps on optimal investment triggers, other parameters being constant. As

a comparison to the preceding analysis, the same parameter values in the base case are

also applied in the numerical analysis. Figure 8 illustrates the result of the jump

effect. According to this diagram, an increase in the rate of jump intensity, holding

the volatility unchanged, lowers the optimal investment triggers, MXV ∗ , i.e., 0MXVλ

∗∂ <∂

.

This result suggests that an increase in jump intensity leads to a positive effect on

investment. The intuition behind the result is that management should undertake

investment sooner when there is an increasing probability of jump due to a higher

likelihood of competitive arrivals. Recall 0GBMVσ

∗∂ >∂

in a GBM process.

According to Figure 8, the variance effect also holds for a mixed diffusion-jump

process, i.e., 0MXVσ

∗∂ >∂

. It is also worth noting that the jump effect on investment is

positive as opposed to the variance effect under a mixed diffusion-jump process.

This is possibly because there are only down jumps allowed in this specific form of

mixed diffusion-jump process.

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Note: 0 100, 8%, 4%, 1/ 52, 5, 10,000V I r t T Number of Trialsα= = = = ∆ = = =

Figure 8 The Jump Effect under a Mixed Diffusion-Jump Process

Since the realization effect of jump intensity can not be directly displayed in a

diagram, Monte Carlo simulation is then conducted to examine the overall effect of

three forces on investment, namely, the variance effect, the jump effect, and the

realization. Figure 9 provides the result of the probability of investing as a function

of jump intensity. According to Figure 9, it is obvious that the probability of

investing appears to be a hump-shaped curve as jump intensity increases, holding the

volatility constant. For a lower level of jump intensity, the probability of investing is

initially an increasing function of jump intensity, but after a certain point the

probability of investing becomes a slowly decreasing function of jump intensity.

Given 20%σ = , the probability of investing is increasing for 30%λ < , i.e.,

( )0MXP Inv

λ∂

>∂

, and decreasing for 30%λ > , i.e., ( )

0MXP Inv

λ∂

<∂

. This result

reveals that the overall effect of three forces on investment under a mixed

diffusion-jump process can be positive or negative, depending on the level of jump

intensity. Consequently, increased jump uncertainty, similar to volatility, may

encourage investment in certain situations. Figure 9 also reveals the result that the

probability of investing under an MX process is significantly higher than that under a

GBM process. The intuition behind the result is that as long as there is a positive

probability of competitive entry, management cannot defer the project infinitely and

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thus is forced to initiate the project sooner in order to preempt potential competitors.

Note: 0 100, 8%, 4%, 1/ 52, 5, 10,000V I r t T Number of Trialsα= = = = ∆ = = =

Figure 9 The Probability of Investing as a Function of Jump Intensity Given an MX Process

To sum up, the jump effect under the specific mixed diffusion-jump process has

an inverse relationship with the optimal investment triggers, thus implying a positive

effect on investment. The jump effect is found to be opposite to the variance effect,

which has a negative effect on investment. The overall effect of combining the

variance effect, the jump effect, and the realization effect, on investment can be either

positive or negative. Consequently, increased jump uncertainty under a mixed

diffusion-jump process can encourage investment in a similar way to increased

volatility uncertainty. It is also demonstrated that as the additional source of

uncertainty, the competitive entry as a down jump, is taken into account, the

probability of investing increases relative to the GBM case, thus indicating a positive

impact on investment.

3.4 The Uncertainty-Investment Relationship under an MR Process

The studies on the relationship between uncertainty and investment under a

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mean-reverting process are Metcalf and Hassett (1995) and Sarkar (2003): the former

argues that mean reversion has two opposing effects, the variance effect and the

realized price effect, on investment, and the overall effect of these two forces are

appropriately equal to such an extent that mean reversion can be justified by the

common assumption of a GBM process; the latter extends their analytical framework

by considering the third effect of mean reversion, the risk-discounting effect of

systematic risk, and contends that mean reversion, in the presence of systematic risk,

does have a significant (either positive or negative) impact on investment, depending

on the combining effect of various factors such as project duration, cost of investing,

and interest rate.

In this subsection, Sarkar’s analysis is further extended to argue that mean

reversion does have a major impact on investment even in the risk-neutral world

where no systematic risk exists. Since a mean-reverting process is seen as a specific

form of a GBM process, it will be demonstrated that mean reversion has the third

effect, the “mean-reverting effect”, in addition to the variance effect and the

realization effect under a common GBM process. The “mean-reverting effect” is

defined as the influence of increased mean-reverting speed on the optimal trigger,

holding the other parameters constant.

To examine the mean-reverting effect on the optimal trigger, the same parameter

values as in the preceding subsection are applied for the analysis. Figure 10 displays

the mean-reverting effect by illustrating the sensitivity of MRV ∗ to the mean-reverting

speed. As revealed from the diagram, MRV ∗ indicates a decreasing function of the

mean-reverting speed, i.e., 0MRVη

∗∂ <∂

, implying that an increase in the mean-reverting

speed leads to a decrease in the optimal trigger. This negative relationship between

the mean-reverting speed and the optimal trigger suggests that the mean-reverting

effect has a positive impact on investment. The mean-reverting effect on lowering

the optimal trigger at a lower level of the mean-reverting speed is more sensitive than

that at a higher level of the mean-reverting speed. This result is mainly because the

optimal trigger at a higher level of the mean-reverting speed is very close to the

investment cost, i.e., the conventional investment rule, representing a smaller option

value and thus less space left to bring the optimal trigger even closer.

21

Note: 0 100, 8%, 1/ 52, 4%, 5, 10,000V I r t T Number of Trialsδ= = = ∆ = = = =

Figure 10 The Mean-Reverting Effect under a Mean-Reverting Process

It is important to note that our “mean-reverting effect” works in a similar way to

the “variance effect” in Metcalf and Hassett (1995), which describes the consequence

that mean reversion decreases the long-run variance and thus lowers the optimal

trigger needed for initiating a project. However, the term “variance effect” can be

misleading in regard to the variance causing instantaneous volatility in the

terminology of financial options. It is therefore crucial to distinguish instantaneous

volatility (or conditional volatility) from long-run volatility (or unconditional

volatility). The literature has demonstrated that even though a GBM process and a

mean-reverting process have the same instantaneous volatility (or conditional

volatility), mean reversion tends to reduce the long-run volatility (or unconditional

volatility). Furthermore, the instantaneous volatility under a mean-reverting process

can still have a negative impact on investment in a similar way to a GBM.

According to Figure 10, it is convenient to observe that an increase in the

instantaneous volatility leads to an increase in MRV ∗ , with the mean-reverting speed

and the other parameters being constant. Therefore, the variance effect discussed in

this paper has the same negative impact on investment by raising the optimal trigger,

i.e., 0Vσ

∗∂ >∂

. This result of the variance effect is consistent across a variety of

22

stochastic processes.

In a similar way to a GBM or an MX process, increased uncertainty

(instantaneous volatility) under an MR process can have a positive influence on

investment by raising the likelihood of reaching the optimal trigger, also termed the

realization effect. Since the realization effect can not be explicitly shown in a

diagram, Monte Carlo simulation is then conducted to directly observe the overall

effect on investment. The probability of investing is illustrated in Figure 11 as a

function of (instantaneous) volatility. As illustrated in Figure 11, the probability of

investing under a mean-reverting process is in general an increasing function of

volatility for the base case, i.e., ( )

0MRP Inv

σ∂

>∂

. This result indicates that the

relationship between uncertainty and investment under a mean-reverting process can

be positive, thus suggesting a positive impact of uncertainty on investment.

Note: 0 100, 8%, 1/ 52, 4%, 5, 10,000V I r t T Number of Trialsδ= = = ∆ = = = =

Figure 11 The Probability of Investing under a Mean-Reverting Process as a Function of Volatility

It is also interesting to observe the overall effect of mean reversion on

investment. Figure 12 displays the probability of investing as a function of

23

mean-reverting speed. As illustrated in Figure 12, the probability of investing

appears to be a convex, decreasing function of the mean-reverting speed for all the

levels of instantaneous volatility, i.e. ( )

0MRP Inv

η∂

<∂

, suggesting that mean

reversion may have a negative impact on investment. It is important to point out that

the result in Figure 12 is consistent with the finding in Sarkar (2003, p.388).6

However, it has been demonstrated here that the result in Sarkar (2003) holds, with or

without the “risk-discounting effect”.

( )η

Note: 0 100, 8%, 1/ 52, 4%, 5, 10,000V I r t T Number of Trialsδ= = = ∆ = = = =

Figure 12 The Probability of Investing under a Mean-Reverting Process as a Function of Mean-Reverting Speed

To sum up, it has been argued that mean reversion has three effects on

investment, namely, the mean-reverting effect, the variance effect, and the realization

effect. The mean-reverting effect has an inverse influence on the optimal trigger,

hence leading to a positive impact on investment. The variance effect, same as that

under a GBM or an MX process, has a positive influence on the optimal trigger, thus

6 Sarkar (2003, p.388) states that mean reversion tends to have a positive (negative) impact on investment for long-lived (short-lived) projects, holding others constant. The short-lived project in his numerical analysis is 5 years, same as the base case our study here.

24

implying a negative impact on investment. By combining the realization effect, it is

found that increased uncertainty under a mean-reverting process may have a positive

impact on investment. This result is very consistent across three stochastic processes

of interest. However, even though mean reversion reduces the optimal investment

trigger, increased mean-reverting speed also decreases the likelihood of reaching the

optimal trigger to such an extent that mean reversion may have a negative impact on

investment as described in the previous analysis, holding the instantaneous volatility

and the other parameters constant,.

4. Uncertainty-Investment Relationship under a Jump Amplitude Process

In this section, the proposed investment framework in Appendix 1 is applied with

Monte Carlo simulation to examine the relationship between uncertainty and

investment under a jump amplitude process (denoted by JA), because there is no

analytical solution for an optimal investment trigger under a JA process ( JAV ∗ ). Since

the main source of uncertainty is stochastic jumps under a jump amplitude process,

the research is directed at investigating the overall effect of two opposing forces, the

“jump effect” and the “realization effect”. The “jump effect” under a jump

amplitude process is to describe how stochastic jumps influence optimal triggers and

the realization effect is to describe the likelihood that a realized project value exceeds

an optimal trigger. It is important to point out that the term “stochastic jump” under

a jump amplitude process is referred to as the setting that both jump direction and

jump size are stochastic.

With the same values of parameter as before, Figure 13 presents the jump effect

on the optimal investment triggers by varying jump intensity (λ ) for three different levels of mean jump size (γ ). As shown in Figure 13, stochastic jumps under a

jump amplitude process have a positive influence on raising optimal investment

triggers, i.e., 0JAVλ

∗∂ >∂

, thus suggesting a negative impact on investment. Also, an

increase in jump size, holding jump intensity and the other parameters constant, leads

to an increase in the optimal trigger, i.e., 0JAVγ

∗∂ >∂

. As both jump size and jump

intensity increase, the jump effect on raising the optimal trigger is even more

substantial because of an increasing option value, hence leading to a convex,

increasing function of both jump intensity and jump size.

25

Note: 0 100, 8%, 4%, 1/ 52, 5, 10,000V I r t T Number of Trialsδ= = = = ∆ = = =

Figure 13 The Jump Effect under a Jump Amplitude Process

Since the realization effect can not be exhibited graphically, Monte Carlo

simulation is conducted to examine the overall effect on investment by calculating the

probability of investing. Figure 14 presents the sensitivity of the probability of

investing to jump intensity for three different levels of jump size. According to the

diagram, the probability of investing appears to be a hump-shaped curve as jump

intensity increases. The probability of investing indicates an increasing function of

jump intensity at a lower level of jump intensity, but after a certain point of jump

intensity, the probability of investing becomes a decreasing function of jump intensity.

Thus, it is found that ( )

0JAP Inv

λ∂

>∂

for λ λ∗< and ( )

0JAP Inv

λ∂

<∂

for λ λ∗> ,

where λ∗ denotes the point of jump intensity which peaks the probability of investing. In the case of 10%γ = , λ∗ is equal to 1. Figure 14 also shows that as

λ∗ increases with γ . Consequently, it can be concluded that increased uncertainty,

in the form of stochastic jumps, may have a major impact on investment, depending

on the set of risk profiles under consideration.

26

Note: 0 100, 8%, 1/ 52, 4%, 5, 10,000V I r t T Number of Trialsδ= = = ∆ = = = =

Figure 14 The Probability of Investing under a Jump Amplitude Process as a Function of Jump Intensity

5. Concluding Remarks

The conventional belief on a negative relationship between uncertainty and

investment has dominated investment theory for a long time. This paper postulates

an important idea that increased uncertainty, in certain situations, may actually

encourage investment due to a higher probability of investing. Earlier studies mostly

base the argument on the GBM assumption, while it is found that the argument also

holds for the situations where the underlying variable follows an alternative stochastic

process. The economic implication is that uncertainty does not always discourage

investment even under several sources of uncertainty which has different risk profiles.

It is also obvious that the “high-risk” projects are not always dominated by the

“low-risk” projects because the “high-risk” projects may have a positive realization

effect, i.e., a higher probability of exceeding the optimal triggers, which may

encourage investment.

It is important to note that the uncertainty-investment relationship within the

framework in the paper should hold only at the firm-level. To extend the result to

the broader aggregate investment level, more industry-specific or economic-specific

factors should be taken into account in the analysis.

27

Reference:

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Derivatives 6 (1998), 38-49.

[2]. Cox, John C. and Ross, Stephen A. “The Pricing of Options for Jump

Processes.” Working Paper 2-75, Rodney L. White Center for Financial

Research, University of Pennsylvania (1975), USA.

[3]. Cukierman, Alex. “The Effect of Uncertainty on Investment under Risk

Neutrality with Endogenous Information.” Journal of Political Economy 88

(1980), 462-475.

[4]. Dixit, Avinash “Entry and Exit Decisions under Uncertainty.” Journal of

Political Economy 97 (1989), 620-638.

[5]. Dixit, Avinash “Investment and Hysteresis.” Journal of Economic

Perspectives 6 (1992), 107-132.

[6]. Dixit, Avinash K. and Pindyck, Robert S. Investment under Uncertainty (1994).

Princeton University Press, New Jersey, USA.

[7]. Gray, Stephen F. “Modeling the Conditional Distribution of Interest Rates as a

Regime-Switching Process.” Journal of Financial Economics 42 (1996),

27-62.

[8]. Harrison, J. Michael. Brownian Motion and Stochastic Flow Systems (1985),

Wiley, New York, USA.

[9]. McDonald, Robert and Siegel, Daniel. “The Value of Waiting to Invest.”

Quarterly Journal of Economics 101 (1986), 707-728.

[10]. Merton, Robert C., “Option Pricing When Underlying Stock Returns are

Discontinuous.” Journal of Financial Economics 3 (1976), 125-144.

[11]. Metcalf, Gilbert E. and Hassett, Kevin A. “Investment under Alternative

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[12]. Naik, Vasanttilak. “Options Valuations and Hedging Strategies with Jumps in

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of the Firm.” American Economic Review 78, 5 (1988), 969-985.

28

[15]. Pindyck, Robert S. “Irreversibility, Uncertainty, and Investment.” Journal of

Economic Literature 29 (1991), 1110-1148.

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Exercise: Some Recent Development.” Engineering Economist 47, 4 (2002),

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[17]. Sarkar, Sudipto. “On the Investment-Uncertainty Relationship in a Real

Options Model.” Journal of Economics Dynamics and Control 24 (2000),

219-225.

[18]. Sarkar, Sudipto. “The Effect of Mean Reversion on Investment under

Uncertainty.” Journal of Economics Dynamics and Control 28 (2003),

377-396.

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on Deferrable Real Investment Opportunities.” Working Paper, Boston

University (1990).

29

Appendix 1 A General Investment Framework for Deriving Optimal Investment Trigger

Since valuing an investment opportunity is akin to the problem of pricing an

American option to invest, Dixit and Pindyck (1994) solve this problem by assuming

that the investment opportunity can exist infinitely or that the firm has a monopoly

power. The consequence of imposing this implicit assumption is that the partial

derivative term of ( )F V with respective to time can be dropped such that the

closed-form solution of optimal investment trigger can be derived. However, the

assumption is not quite realistic as a competitor may enter the market to diminish the

investment opportunity. Vandenbroucke (1999) points out that the closed-form

solution in Dixit and Pindyck (1994) is in fact the limiting case of the quadratic

approximation of American options in Barone-Adesi and Whaley (1987).

A1. A General Investment Framework

Since irreversibility complicates capital investments in that closed-form

expressions for optimal investment triggers seldom exist under an alternative process,

in this section an investment framework for deriving optimal trigger under an

alternative process in a finite time horizon is proposed. As it is known that a firm

can either defer the project in the unfavorable market condition or launch the project

in the favorable market condition, an investment opportunity is equivalent to a call

option. It is assumed that the investment opportunity will disappear at a finite future

time T, if the firm does not take any actions. Therefore, the value of an investment

opportunity at time T, given the information set Tφ , is expressed as follows:

( ) ( )max ,0T T T TF V V I φ= − (A.1)

According to Equation (A.1), the value of investment opportunity at time t can

be given by

( ) ( )max ,0T tP

t TF E e V Iρ− −� �= −� � (A.2)

30

where PE denotes an expectation operator in a risk-adjusted world, P a risk-adjusted

probability measure, and ρ a risk-adjusted discount rate.

In the risk-neutral world, tF can be derived from

( ) ( )max ,0T t rQt TF E e V I− −� �= −� � (A.3)

where r denotes a risk-free rate and Q a risk-neutral probability measure.

It is worth noting that when the market is complete or the investor is risk-neutral,

there exists a unique risk-neutral probability measure Q such that F can be

evaluated by Equation (A.3). If the market is incomplete or the investor is

risk-averse, there does not exist such a unique Q and thus F can be evaluated by

Equation (A.2).

If the risk-free rate is assumed to be constant over the time horizon between t and

the expiration time T, Equation (A.3) can be rewritten as follows:

( ) ( )max ,0T t r Qt TF e E V I− −= −� �� � (A.4)

Equation (A.2), (A.3), or (A.4) states a fundamental equation for valuing an

investment opportunity in a numerical procedure of Monte Carlo simulation, given

any tV .

To determine the optimal investment rule, we need to search for an investment

trigger tV ∗ such that the net present value of taking the project, tV I∗ − , can

compensate the loss of option of waiting, ( )t tF V ∗ . This optimal investment policy

can be described by a value-matching condition as follows:

( )t t tF V V I∗ ∗= − (A.5)

Equation (A.5) can be alternatively rearranged to set the investment trigger equal to

the total investment cost, including the direct investment cost plus the option value, as

31

follows:

( )t t tV F V I∗ ∗= + (A.6)

To rule out the possibility of an arbitrage opportunity or the “kinked” situation7, the

first derivative of the value-matching condition with respect to the state variable at the

maximum must be equal on both sides. This is the famous Samuelson

smooth-pasting condition given below:

( ) 1t

tVF V∗

∗ = (A.7)

By substituting Equation (A.4) into Equation (A.6), we have optimal investment

trigger, tV ∗ , expressed as follows:

( ) ( )max ,0

t t

T t r Qt T V V

V e E V I I∗− −∗

== − +� �� � (A.8)

In a risk-adjusted world, optimal investment trigger is given by

( ) ( )max ,0t t

T t Pt T V V

V e E V I Iρ∗

− −∗=

= − +� �� � (A.9)

Equation (A.8) or (A.9) represents a fundamental equation to derive optimal

investment trigger, tV ∗ . There are two important implications in this framework for

deriving the optimal investment rules. First, these formulas hold regardless of the

assumption of stochastic process. As mentioned earlier, McDonald and Siegel

(1986), Pindyck (1991) and Dixit and Pindyck (1994, Ch. 5) have already provided

the analytical solutions of trigger price for an investment opportunity under a GBM.

For the projects whose underlying process follows an alternative process, the

closed-form solutions for optimal triggers are generally unavailable. The investment

framework in this section is advantageous in the situations in that project values

follow a stochastic process other than a GBM. As to the choice of the valuation

equation, the decision relies on the assumption of risk attitude or market completeness.

7 See Dixit and Pindyck (1994, Ch. 4).

32

If all investors are assumed to be risk-neutral or the market is complete such that the

risk-neutral valuation principle holds, Equation (A.8) is good enough to serve the

valuation purpose. In the other cases, we need to apply Equation (A.9), with risk

premium properly treated, to derive optimal investment trigger.

Second, Equation (A.8) or (A.9) can be readily applied in the situation in that the

investment opportunity will disappear in a known expiration of time in future.

Conventional literature on optimal investment rules mostly makes an implicit

assumption that the investment opportunity can exist in an infinite time horizon for

the convenience in deriving analytical solutions. This assumption is not quite

realistic in practice, especially when the factor of technology obsolesce is involved

with the project or the deferral option has an expiration date.

It is worth noting that growth rate (or drift rate) must be assumed to be less than

discount rate (either risk-adjusted discount rate or risk-free rate), otherwise it will be

never optimal to early exercise an investment opportunity before the expiration time.

By setting growth rate less than discount rate, it is equivalent to assume that there

exists a positive convenience yield which accounts for an opportunity cost (denoted

by δ ) of delaying the construction of a project. In a risk-neutral world, when

convenience yield plays a role in real options valuation, the actual growth rate of an

underlying process must be adjusted by reducing an amount of convenience yield.8

Therefore, as the opportunity cost of delaying a project becomes larger, the actual

growth rate of the underlying process becomes smaller.

Since the framework is in essence based on the valuation of a European-style

option, one may ask whether the early exercise premium matters in real options with

the American nature. According to Barone-Adesi and Whaley (1987), for an

at-the-money option with a moderate opportunity cost ( 4%δ = ) and a short time

horizon ( 0.25 or 0.5T = ), early exercise premium is estimated to be 0.00%.9 For an

8 If the world is assumed to be risk-neutral, the discount rate in the framework becomes risk-free rate. Thus, the actual drift in the stochastic process equals risk-free rate less convenience yield. This treatment is called “equivalent risk-neutral valuation”. See the discussions in Dixit and Pindyck (1994, Ch. 4, p.121-125). 9 The risk-free rate is assumed to be 8%. Refer to Table 2 in Barone-Adesi and Whaley (1987).

33

at-the-money option with a longer time horizon ( 3T = ), early exercise premium is

estimated to be 1.16%.10 Therefore, it is practical to conclude that the effect of early

exercise premiums is minimal and may be negligible in the situations where the

at-the-money project is of interest.

A2. The Implementation – Monte Carlo Simulation and its Algorithm

In the preceding subsection, an investment framework for deriving optimal

investment trigger under an alternative process is proposed, yet the determination of

optimal trigger is not straightforward. It is suggested that the investment framework

can be implemented by the technique of Monte Carlo simulation. In this subsection,

the technique of Monte Carlo simulation and the algorithm of an iterating procedure

for the trigger value are discussed.

In Monte Carlo simulation, a large number of random paths from a finite sample

space, given a specific stochastic process, must be generated. These random paths

enable us to compute terminal payoffs which are then discounted backward at a

discount rate. As mentioned before, if risk-neutral valuation is applied, the discount

rate equals a risk-free rate. In a risk-adjusted world, the discount rate must be

risk-adjusted. If the discount rate is not certain over the investment horizon, an

interest rate process needs to be simulated simultaneously. For a reasonable short

time horizon, we can assume that the discount rate is constant for simplicity. The

value of an investment opportunity can be computed from the mean of discounted

payoffs. The value of optimal investment trigger must be derived from an iterative

procedure which equates V ∗ and ( )F V I∗ + .

To begin the iterative procedure, it is necessary to specify the first two initial

values 1V and 2V , where 1V and 2V are two guessed numbers which are lower

than V ∗ . Next, 1V and 2V are then applied to evaluate the right-hand side of

Equation (A.8). Since it is very unlikely that any of the two numbers would equate

the value-matching condition, we then compute the slope ( χ ) of the line connecting

10 Refer to Table 5 in Barone-Adesi and Whaley (1987).

34

both numbers as follows:

( ) ( )2 2

2 1

F V F V

V Vχ

−=

− (A.10)

Obviously, χ is smaller than 1 because both initial numbers are smaller than V ∗ .

It is then assumed that there is a larger number 3V , i.e., 3 2 1V V V> > such that the

following relationship holds:

( ) ( )3 2 3 2V F V I V Vχ= + + −� �� � (A.11)

By rearranging Equation (A.11) for 3V , the next guessing number is given by

( )2 2

3 1

F V I VV

χχ

+ −� �� �=−

(A.12)

Now 3V can be used to compute a new option value, ( )3F V , and a new slope

with respect to 2V . Since it is very unlikely that both sides of the value-matching

condition are exactly equal during the iteration, an acceptance criterion must be

established in the iterative procedure. By setting a 0.5% as an acceptable tolerance,

an example of an acceptance criterion is given as follows:

1 0.005χ − ≤ (A.13)

Equation (A.13) implies that the iterative procedure must continue until the new slope

meets this criterion.