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The overall effect of volatility on investment
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Economics Letters 99 (2008) 324 – 327 www.elsevier.com/locate/econbase
The overall effect of volatility on investment ☆ Pai-Ta Shih a , Weifeng Hung b,⁎ a
Department of Economics, National Dong Hwa University, No. 1, Sec. 2, Da Hsueh Rd., Shoufeng, Hualien 97401, Taiwan b Department of Finance, Da-Yeh University,112 Shan-Jiau Rd, Da-Tsuen,Chang-hua, Taiwan 515 Received 18 January 2007; received in revised form 14 July 2007; accepted 24 July 2007 Available online 1 August 2007
Abstract We introduce a new channel called random delay effect, through which volatility influences real investment. We show that random delay effect is not negligible in determining the sign of the volatility–investment relationship. © 2007 Elsevier B.V. All rights reserved. Keywords: Real options; Volatility; Investment JEL classification: G12; O31; O34
1. Introduction The relationship between uncertainty and irreversible investment has long drawn a great deal of attention from economists and policy makers (Hartman, 1972; Abel, 1983). Pindyck (1988) showed that for irreversible investments, increasing market volatility reduces investment. Caballero (1991) indicated that the relationship between market volatility and investment is always positive if the firm is operating in a perfectly competitive product market with constant return to scale. He suggested that the negative relationship between volatility and investment is related to the monopolistic market condition. In this study, in addition to the volatility about future profits, we incorporate another source of uncertainty associated with the project period. More specifically, how long it takes to successfully accomplish the project is probabilistic, and the probability is assumed to be conditional on the amount of R&D investment. In practice, R&D-intensive industries, such as software and pharmaceutical companies, seem to fit the scheme.
☆
The authors are grateful for Cheng F. Lee, Eric Maskin (The editor), and an anonymous referee. Any remaining errors are our responsibility. ⁎ Corresponding author. Tel.: +886 4 8511888x3522; fax: +886 4 8511510. E-mail address: [email protected] (W. Hung). 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.07.014
Our main objective is to show that the increase in market volatility will raise the trigger value, and therefore the higher expected marginal project value of the investment. The higher expected marginal project value of the investment motivates the firm to invest more R&D into the project, so as to accomplish the project and enter the market early. We call this a “random delay” effect, which was ignored in previous studies.1 We are going to show that the relationship between volatility and investment is ambiguous if the expected present value of investment cost is taken as the criterion, and the random delay effect is not negligible in determining the sign of the volatility– investment relationship. 2. The model Suppose that a risk-neutral firm is considering investing a R&D project that generates a stochastic net cash flow (xt) at a future time study can be expressed with the geometric Brownian motion (GBM) process below: dx ¼ axdt þ rxdz;
ð1Þ
where the drift rate α is assumed to be less than the risk-free rate r, and the instantaneous market volatility is denoted as r. The 1
We are grateful to the referee for suggesting the term of the effect to be random delay effect.
P.-T. Shih, W. Hung / Economics Letters 99 (2008) 324–327
increment of a standard Wiener process is denoted as dz. Once the project has been successfully accomplished, it will generate the value v (as described below). Since the net cash flow follows GBM, v also follows GBM. Z
l
v¼ 0
x : Eðxt Þert dt ¼ ra
The market value of the project is subject to the trigger value v⁎, such that f ðvÞ ¼ Avb
if v V v⁎
¼
if v z v⁎ :
ð2Þ
bvI I r þ bI a
ðr þ bIðvÞ aÞ2
P ¼ kð I Þekð I Þt ;
Avb ¼
where 1/λ(I) is the expected period to successfully accomplish the project. For analytical purposes, we adopt a linear function for λ(I) = bI, where b is a positive constant.2 That is, the project completion time is related to the amount of investment I. Given a fixed b, we assume that the greater amount of investment will shorten the completion time.3 Based on the above settings, let's first consider the situation where the project has been implemented. The expected present value of the project f(v) can be expressed as: Z f ð vÞ ¼
l
kvðuÞe
ku ru
0
e
bvI : du ¼ r þ bI a
ð4Þ
Second, when the project is idle or postponed, the value of the option to invest follows an ordinary differential equation specified in Dixit and Pindyck.4 The value of this idle project is equal to the value of the option to invest, also known as Avβ, where A is a constant to be determined, and b ¼ 1=2 a=r2 þ
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða=r2 1=2Þ2 þ2r=r2 :
ð5Þ
We divide the total investment cost into two types: R&D investment and non-R&D cost. For R&D intensive industries, R&D investment I⁎NN nonR&D cost F, and therefore we set the non-R&D cost F to zero for simplicity. For a general linear arrival rate k(I) = a + bI without non-R&D cost, a = 0 implies that it is impossible to successfully implement this project without any cost. However, we are able to show that even if non-R&D cost F N 0 and a N 0 with this general linear arrival rate, the random delay effect still exists, i.e., I⁎ is still an increasing function of r2. The detailed proof is available upon request from the authors. 3 In fact, we use b to normalize this linear arrival rate model. For example, if we have a project with average period of 5 “years” and the amount of investment is 0.2 “billion”, we set b to 1/(5year × 0.2billion). In other words, if the unit of b is changed to be 1/(year × million), then b is equal to 1/(5year × 200million. In practice, according to Carmelo et al. (2005), the average R&D project period takes approximately 16 “years” and costs $802 “million” to bring a single new chemical entity to market. Thus, on average, the reasonable b for pharmaceutical industry is about 0.078/(year × billion). 4 Dixit and Pindyck (1994), chapter 5.
ð6Þ
The objective function is to maximize Eq. (6), where I is endogenously determined as a function of v. Applying the first order condition, the optimal I⁎(v) satisfies the following condition:
We consider an alternative uncertainty associated with the time required to accomplish the project. In a similar spirit of Lee and Wilde (1980), we assume that at the initiation of the project, successful implementation depends on a Poisson distribution with the arrival rate λ(I), where I is the R&D investment. Therefore, the probability of successfully implementing the project at time t is P, defined as: ð3Þ
325
bvðr aÞ
¼ 1:
ð7Þ
Based on the boundary conditions (value matching and smooth pasting), we have bIðvÞv IðvÞ; r þ bIðvÞ a
ð8Þ
and bAvb1 ¼
h i bvI d max rþbIa I dv
:
ð9Þ
Using the envelope theorem, Eq. (9) can be expressed as: bAvb1 ¼
bIðvÞ : r þ bIðvÞ a
ð10Þ
By Eqs. (7), (8), and (10), we obtain closed-form functions for the trigger value v⁎ as well as the optimal amount of investment I⁎:
2
v ¼
b b1
I⁎ ¼
ðr aÞ : bð b 1Þ
⁎
ð r aÞ ; b
ð11Þ
ð12Þ
Particularly noteworthy is that β is a decreasing function of σ2,5 so, from Eqs. (11) and (12), the increase in market volatility leads to the rise in v⁎ and I⁎. The reason why I⁎ is the increasing function of σ2 is that higher trigger value v⁎ implies higher expected marginal project value of the R&D investment, and thus the firm has stronger incentives to invest more on R&D in order to shorten the completion time, as well as to realize higher market value of the project.6 This effect is called the “random delay” effect, which is first proposed in the literature. It can also be easily verified from (12) that as it becomes more difficult to successfully implement a project, i.e., when b becomes smaller, the random bðb1Þ Ab It is well known that Ar 2 ¼ ð2br2 r2 þ2aÞ b 0. To verify this inference, we can first differentiate Eq. (4) with respect to I to have a marginal expected project value of the investment, and then differentiate it with respect to v to have a positive relationship between project value (v) and marginal expected project value of the investment. Besides, the policy implication is that any reducing realized volatility due to price control will decrease trigger value and the optimal amount of R&D investment. 5 6
326
P.-T. Shih, W. Hung / Economics Letters 99 (2008) 324–327
Table 1 The overall effect of market volatility on the expected present value of investment Market volatility
Random delay effect Ab 1 N0 2 Ar ðb 1Þ
Variance effect Ab 2 b0 Ar2 ðb 1Þ
Realization effect 4 Ab v 2 ln N0 Ar v0
Overall effect
Aln I 4 Eðert Þ Ar2
0.22 0.32 0.41 0.56 0.61 0.71 0.81 0.91
13.09 6.68 4.13 2.84 2.09 1.61 1.28 1.04
−26.17 −13.35 −8.26 −5.68 − 4.15 − 3.21 − 2.55 − 2.08
20.38 9.26 5.10 3.13 2.06 1.42 1.02 0.75
7.29 2.58 0.97 0.29 0.00 −0.18 −0.26 −0.29
Note: Parameters of the model are α = 0, r = 0.1, b = 1/(year × billion), and v0 = 0.1 × billion.
delay effect becomes larger.7 In the next section, we will show the importance of random delay effect in determining the sign of the volatility–investment relationship. 3. The overall effect of volatility on investment We employ a criterion called the “expected present value of investment” (EPVI, thereafter) to investigate the relationship between volatility and investment–higher EPVI implies higher investment.8 EPVI ¼ E ½I ⁎ ert ¼ I ⁎ E ½ert ;
ð13Þ
where t is the first-passage time for v from the initial value v0 to the trigger value v⁎. Following Dixit and Pindyck (1994, Chapter 9), we have: E ½I ⁎ ert ¼ I ⁎ ðv0 =v⁎ Þb :
ð14Þ
Taking the natural logarithm and differentiation on the EPVI with respect to σ2, we can decompose the result into three components to facilitate the analysis. They are: Ab 2 b 0; Ar2 ð b 1Þ
Ab ½ ln ðv⁎ =v0 Þ N 0; Ar2
and
Ab ½lnðv⁎ =v0 Þ N 0; Ar2
ð15Þ ð16Þ ð17Þ
In a similar spirit of Sarkar (2000, 2003), the first two components, Eqs. (15) and (16), are the “variance” effect and the “realization” effect respectively. Furthermore, the third effect, Eq. (17), is the “random delay” effect.9 First, the variance effect indicates that an increase in market volatility results in an increase in the trigger value, and hence, there is a negative impact on the EPVI. Second, the realization effect reflects the fact that an increase in market volatility increases the likelihood of v reaching the trigger value v⁎, and hence, a positive impact on the EPVI happens. Finally and most importantly, the random delay effect proposed here implies that an increase in volatility may raise the trigger value of the project. Moreover, higher expected marginal project value of the R&D investment induces greater amount of R&D investment, and hence, has a positive impact on the EPVI. Adding together these three effects, the overall effect of market volatility on the EPVI is still mixed. However, we will show by an experiment that random delay effect is important in determining the impact of market volatility on the EPVI. The simulation results based on Eqs. (15), (16, and (17)) are shown in Table 1.10 It is noted that for the case of market volatility from 0.22 to 0.56 (shown in bold), if the random delay effect is not incorporated, the market volatility has a negative impact on the EPVI, which is from − 5.79 to − 2.55. In contrast, if the random delay effect is taken into account, the market volatility has a positive effect on the EPVI. As seen in the last column of Table 1, the overall volatility effect on the EPVI ranges from 7.29 to 0.29. Nevertheless, not taking into account the random delay effect may mislead the conclusion about the relationship between market volatility and investment. 4. Conclusion
AI ⁎ ra A 1 AI ⁎ Here we have Ar 2 ¼ b Ar2 b1 N0 and Ar2 becomes larger as b becomes smaller. 8 To investigate the relationship between volatility and investment, Sarkar (2000) proposed that “the probability of investment” would take place within a specific period. There are three problems in the method proposed by Sarkar. First, while the probability of hitting trigger value is calculated within a specified maturity time of the project, the trigger value computed by Sarkar, however, is based on a perpetual investment opportunity. Second, using only the probability of investment implies ignoring the time value of future investment. Third, it is unrealistic to hold a fixed amount of investment cost. 7
We contribute to the literature in three aspects. First, there exists the random delay effect if the uncertainty associated with the project completion period exists. Especially in practice, 9
Sarkar (2003) used the model of mean reversion process to investigate the relationship, and found variance effect, realization effect, and a third effect called risk discounting effect. 10 For the experiment conducted here, b = 1/(year × billion). Therefore, the unit of v0 is billion.
P.-T. Shih, W. Hung / Economics Letters 99 (2008) 324–327
R&D-intensive industries, such as software and pharmaceutical companies, seem to fit the description. Second, it is numerically found that the relationship between market volatility and investment is ambiguous and the random delay effect is not negligible in determining the sign of the market volatility– investment relationship. Finally, we offer a simple model, which can be easily extended to a general linear arrival rate model with non-R&D cost (as stated in footnote 2), so that the volatility– investment relationship and other related issues for R&Dintensive industries can be empirically investigated in a more compact way. References Abel, A.B., 1983. Optimal investment under uncertainty. American Economic Review 73, 228–233.
327
Caballero, R.J., 1991. On the sign of the investment–uncertainty relationship. American Economic Review 81, 279–288. Carmelo, G., Santerre, Rexford E., John, A.V., 2005. Drug prices and research and development investment behavior in the pharmaceutical industry. Journal of Law and Economics 48, 195–214. Dixit, A.K., Pindyck, R.S., 1994. Investment under uncertainty. Princeton University Press, Princeton. Hartman, R., 1972. The effect of price and cost uncertainty on investment. Journal of Economic Theory 5, 258–266. Lee, T., Wilde, L., 1980. Market structure and innovation: a reformulation. Quarterly Journal of Economics 94, 429–436. Pindyck, R.S., 1988. Irreversible investment, capacity choice, and the value of the firm. American Economic Review 78, 969–985. Sarkar, S., 2000. On the investment–uncertainty relationship in a real options model. Journal of Economic Dynamics & Control 24, 219–225. Sarkar, S., 2003. The effect of mean reversion on investment under uncertainty. Journal of Economic Dynamics & Control 28, 377–396.
Economics Letters 99 (2008) 324 – 327 www.elsevier.com/locate/econbase
The overall effect of volatility on investment ☆ Pai-Ta Shih a , Weifeng Hung b,⁎ a
Department of Economics, National Dong Hwa University, No. 1, Sec. 2, Da Hsueh Rd., Shoufeng, Hualien 97401, Taiwan b Department of Finance, Da-Yeh University,112 Shan-Jiau Rd, Da-Tsuen,Chang-hua, Taiwan 515 Received 18 January 2007; received in revised form 14 July 2007; accepted 24 July 2007 Available online 1 August 2007
Abstract We introduce a new channel called random delay effect, through which volatility influences real investment. We show that random delay effect is not negligible in determining the sign of the volatility–investment relationship. © 2007 Elsevier B.V. All rights reserved. Keywords: Real options; Volatility; Investment JEL classification: G12; O31; O34
1. Introduction The relationship between uncertainty and irreversible investment has long drawn a great deal of attention from economists and policy makers (Hartman, 1972; Abel, 1983). Pindyck (1988) showed that for irreversible investments, increasing market volatility reduces investment. Caballero (1991) indicated that the relationship between market volatility and investment is always positive if the firm is operating in a perfectly competitive product market with constant return to scale. He suggested that the negative relationship between volatility and investment is related to the monopolistic market condition. In this study, in addition to the volatility about future profits, we incorporate another source of uncertainty associated with the project period. More specifically, how long it takes to successfully accomplish the project is probabilistic, and the probability is assumed to be conditional on the amount of R&D investment. In practice, R&D-intensive industries, such as software and pharmaceutical companies, seem to fit the scheme.
☆
The authors are grateful for Cheng F. Lee, Eric Maskin (The editor), and an anonymous referee. Any remaining errors are our responsibility. ⁎ Corresponding author. Tel.: +886 4 8511888x3522; fax: +886 4 8511510. E-mail address: [email protected] (W. Hung). 0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.07.014
Our main objective is to show that the increase in market volatility will raise the trigger value, and therefore the higher expected marginal project value of the investment. The higher expected marginal project value of the investment motivates the firm to invest more R&D into the project, so as to accomplish the project and enter the market early. We call this a “random delay” effect, which was ignored in previous studies.1 We are going to show that the relationship between volatility and investment is ambiguous if the expected present value of investment cost is taken as the criterion, and the random delay effect is not negligible in determining the sign of the volatility– investment relationship. 2. The model Suppose that a risk-neutral firm is considering investing a R&D project that generates a stochastic net cash flow (xt) at a future time study can be expressed with the geometric Brownian motion (GBM) process below: dx ¼ axdt þ rxdz;
ð1Þ
where the drift rate α is assumed to be less than the risk-free rate r, and the instantaneous market volatility is denoted as r. The 1
We are grateful to the referee for suggesting the term of the effect to be random delay effect.
P.-T. Shih, W. Hung / Economics Letters 99 (2008) 324–327
increment of a standard Wiener process is denoted as dz. Once the project has been successfully accomplished, it will generate the value v (as described below). Since the net cash flow follows GBM, v also follows GBM. Z
l
v¼ 0
x : Eðxt Þert dt ¼ ra
The market value of the project is subject to the trigger value v⁎, such that f ðvÞ ¼ Avb
if v V v⁎
¼
if v z v⁎ :
ð2Þ
bvI I r þ bI a
ðr þ bIðvÞ aÞ2
P ¼ kð I Þekð I Þt ;
Avb ¼
where 1/λ(I) is the expected period to successfully accomplish the project. For analytical purposes, we adopt a linear function for λ(I) = bI, where b is a positive constant.2 That is, the project completion time is related to the amount of investment I. Given a fixed b, we assume that the greater amount of investment will shorten the completion time.3 Based on the above settings, let's first consider the situation where the project has been implemented. The expected present value of the project f(v) can be expressed as: Z f ð vÞ ¼
l
kvðuÞe
ku ru
0
e
bvI : du ¼ r þ bI a
ð4Þ
Second, when the project is idle or postponed, the value of the option to invest follows an ordinary differential equation specified in Dixit and Pindyck.4 The value of this idle project is equal to the value of the option to invest, also known as Avβ, where A is a constant to be determined, and b ¼ 1=2 a=r2 þ
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða=r2 1=2Þ2 þ2r=r2 :
ð5Þ
We divide the total investment cost into two types: R&D investment and non-R&D cost. For R&D intensive industries, R&D investment I⁎NN nonR&D cost F, and therefore we set the non-R&D cost F to zero for simplicity. For a general linear arrival rate k(I) = a + bI without non-R&D cost, a = 0 implies that it is impossible to successfully implement this project without any cost. However, we are able to show that even if non-R&D cost F N 0 and a N 0 with this general linear arrival rate, the random delay effect still exists, i.e., I⁎ is still an increasing function of r2. The detailed proof is available upon request from the authors. 3 In fact, we use b to normalize this linear arrival rate model. For example, if we have a project with average period of 5 “years” and the amount of investment is 0.2 “billion”, we set b to 1/(5year × 0.2billion). In other words, if the unit of b is changed to be 1/(year × million), then b is equal to 1/(5year × 200million. In practice, according to Carmelo et al. (2005), the average R&D project period takes approximately 16 “years” and costs $802 “million” to bring a single new chemical entity to market. Thus, on average, the reasonable b for pharmaceutical industry is about 0.078/(year × billion). 4 Dixit and Pindyck (1994), chapter 5.
ð6Þ
The objective function is to maximize Eq. (6), where I is endogenously determined as a function of v. Applying the first order condition, the optimal I⁎(v) satisfies the following condition:
We consider an alternative uncertainty associated with the time required to accomplish the project. In a similar spirit of Lee and Wilde (1980), we assume that at the initiation of the project, successful implementation depends on a Poisson distribution with the arrival rate λ(I), where I is the R&D investment. Therefore, the probability of successfully implementing the project at time t is P, defined as: ð3Þ
325
bvðr aÞ
¼ 1:
ð7Þ
Based on the boundary conditions (value matching and smooth pasting), we have bIðvÞv IðvÞ; r þ bIðvÞ a
ð8Þ
and bAvb1 ¼
h i bvI d max rþbIa I dv
:
ð9Þ
Using the envelope theorem, Eq. (9) can be expressed as: bAvb1 ¼
bIðvÞ : r þ bIðvÞ a
ð10Þ
By Eqs. (7), (8), and (10), we obtain closed-form functions for the trigger value v⁎ as well as the optimal amount of investment I⁎:
2
v ¼
b b1
I⁎ ¼
ðr aÞ : bð b 1Þ
⁎
ð r aÞ ; b
ð11Þ
ð12Þ
Particularly noteworthy is that β is a decreasing function of σ2,5 so, from Eqs. (11) and (12), the increase in market volatility leads to the rise in v⁎ and I⁎. The reason why I⁎ is the increasing function of σ2 is that higher trigger value v⁎ implies higher expected marginal project value of the R&D investment, and thus the firm has stronger incentives to invest more on R&D in order to shorten the completion time, as well as to realize higher market value of the project.6 This effect is called the “random delay” effect, which is first proposed in the literature. It can also be easily verified from (12) that as it becomes more difficult to successfully implement a project, i.e., when b becomes smaller, the random bðb1Þ Ab It is well known that Ar 2 ¼ ð2br2 r2 þ2aÞ b 0. To verify this inference, we can first differentiate Eq. (4) with respect to I to have a marginal expected project value of the investment, and then differentiate it with respect to v to have a positive relationship between project value (v) and marginal expected project value of the investment. Besides, the policy implication is that any reducing realized volatility due to price control will decrease trigger value and the optimal amount of R&D investment. 5 6
326
P.-T. Shih, W. Hung / Economics Letters 99 (2008) 324–327
Table 1 The overall effect of market volatility on the expected present value of investment Market volatility
Random delay effect Ab 1 N0 2 Ar ðb 1Þ
Variance effect Ab 2 b0 Ar2 ðb 1Þ
Realization effect 4 Ab v 2 ln N0 Ar v0
Overall effect
Aln I 4 Eðert Þ Ar2
0.22 0.32 0.41 0.56 0.61 0.71 0.81 0.91
13.09 6.68 4.13 2.84 2.09 1.61 1.28 1.04
−26.17 −13.35 −8.26 −5.68 − 4.15 − 3.21 − 2.55 − 2.08
20.38 9.26 5.10 3.13 2.06 1.42 1.02 0.75
7.29 2.58 0.97 0.29 0.00 −0.18 −0.26 −0.29
Note: Parameters of the model are α = 0, r = 0.1, b = 1/(year × billion), and v0 = 0.1 × billion.
delay effect becomes larger.7 In the next section, we will show the importance of random delay effect in determining the sign of the volatility–investment relationship. 3. The overall effect of volatility on investment We employ a criterion called the “expected present value of investment” (EPVI, thereafter) to investigate the relationship between volatility and investment–higher EPVI implies higher investment.8 EPVI ¼ E ½I ⁎ ert ¼ I ⁎ E ½ert ;
ð13Þ
where t is the first-passage time for v from the initial value v0 to the trigger value v⁎. Following Dixit and Pindyck (1994, Chapter 9), we have: E ½I ⁎ ert ¼ I ⁎ ðv0 =v⁎ Þb :
ð14Þ
Taking the natural logarithm and differentiation on the EPVI with respect to σ2, we can decompose the result into three components to facilitate the analysis. They are: Ab 2 b 0; Ar2 ð b 1Þ
Ab ½ ln ðv⁎ =v0 Þ N 0; Ar2
and
Ab ½lnðv⁎ =v0 Þ N 0; Ar2
ð15Þ ð16Þ ð17Þ
In a similar spirit of Sarkar (2000, 2003), the first two components, Eqs. (15) and (16), are the “variance” effect and the “realization” effect respectively. Furthermore, the third effect, Eq. (17), is the “random delay” effect.9 First, the variance effect indicates that an increase in market volatility results in an increase in the trigger value, and hence, there is a negative impact on the EPVI. Second, the realization effect reflects the fact that an increase in market volatility increases the likelihood of v reaching the trigger value v⁎, and hence, a positive impact on the EPVI happens. Finally and most importantly, the random delay effect proposed here implies that an increase in volatility may raise the trigger value of the project. Moreover, higher expected marginal project value of the R&D investment induces greater amount of R&D investment, and hence, has a positive impact on the EPVI. Adding together these three effects, the overall effect of market volatility on the EPVI is still mixed. However, we will show by an experiment that random delay effect is important in determining the impact of market volatility on the EPVI. The simulation results based on Eqs. (15), (16, and (17)) are shown in Table 1.10 It is noted that for the case of market volatility from 0.22 to 0.56 (shown in bold), if the random delay effect is not incorporated, the market volatility has a negative impact on the EPVI, which is from − 5.79 to − 2.55. In contrast, if the random delay effect is taken into account, the market volatility has a positive effect on the EPVI. As seen in the last column of Table 1, the overall volatility effect on the EPVI ranges from 7.29 to 0.29. Nevertheless, not taking into account the random delay effect may mislead the conclusion about the relationship between market volatility and investment. 4. Conclusion
AI ⁎ ra A 1 AI ⁎ Here we have Ar 2 ¼ b Ar2 b1 N0 and Ar2 becomes larger as b becomes smaller. 8 To investigate the relationship between volatility and investment, Sarkar (2000) proposed that “the probability of investment” would take place within a specific period. There are three problems in the method proposed by Sarkar. First, while the probability of hitting trigger value is calculated within a specified maturity time of the project, the trigger value computed by Sarkar, however, is based on a perpetual investment opportunity. Second, using only the probability of investment implies ignoring the time value of future investment. Third, it is unrealistic to hold a fixed amount of investment cost. 7
We contribute to the literature in three aspects. First, there exists the random delay effect if the uncertainty associated with the project completion period exists. Especially in practice, 9
Sarkar (2003) used the model of mean reversion process to investigate the relationship, and found variance effect, realization effect, and a third effect called risk discounting effect. 10 For the experiment conducted here, b = 1/(year × billion). Therefore, the unit of v0 is billion.
P.-T. Shih, W. Hung / Economics Letters 99 (2008) 324–327
R&D-intensive industries, such as software and pharmaceutical companies, seem to fit the description. Second, it is numerically found that the relationship between market volatility and investment is ambiguous and the random delay effect is not negligible in determining the sign of the market volatility– investment relationship. Finally, we offer a simple model, which can be easily extended to a general linear arrival rate model with non-R&D cost (as stated in footnote 2), so that the volatility– investment relationship and other related issues for R&Dintensive industries can be empirically investigated in a more compact way. References Abel, A.B., 1983. Optimal investment under uncertainty. American Economic Review 73, 228–233.
327
Caballero, R.J., 1991. On the sign of the investment–uncertainty relationship. American Economic Review 81, 279–288. Carmelo, G., Santerre, Rexford E., John, A.V., 2005. Drug prices and research and development investment behavior in the pharmaceutical industry. Journal of Law and Economics 48, 195–214. Dixit, A.K., Pindyck, R.S., 1994. Investment under uncertainty. Princeton University Press, Princeton. Hartman, R., 1972. The effect of price and cost uncertainty on investment. Journal of Economic Theory 5, 258–266. Lee, T., Wilde, L., 1980. Market structure and innovation: a reformulation. Quarterly Journal of Economics 94, 429–436. Pindyck, R.S., 1988. Irreversible investment, capacity choice, and the value of the firm. American Economic Review 78, 969–985. Sarkar, S., 2000. On the investment–uncertainty relationship in a real options model. Journal of Economic Dynamics & Control 24, 219–225. Sarkar, S., 2003. The effect of mean reversion on investment under uncertainty. Journal of Economic Dynamics & Control 28, 377–396.