- Email: [email protected]
Investment timing with information-processing constraints
Finance Research Letters xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Investment timing with information-processing constraints☆ ⁎,c
Congming Mua, Jinqiang Yangb,c, Yuhua Zhang a
Shanghai University of Finance and Economics, Shanghai Institute of International Finance and Economics, Shanghai, China Shanghai Key Laboratory of Financial Information Technology, Shanghai Institute of International Finance and Economics, Shanghai, China c School of Finance, Shanghai University of Finance and Economics, Shanghai, China b
ARTICLE INFO
ABSTRACT
Keywords: Real options Optimal investment Information-processing constraints Channel capacity
This paper investigates the implications of information-processing constraints (rational inattention) for the optimal investment timing and the corresponding value loss based on the real option framework with incomplete information. We find that an agent with rational inattention tends to overinvest and that the loss in option value decreases in the information channel capacity.
JEL classification: D81 D83 G32
1. Introduction In the standard literature about the real options (see, McDonald and Siegel, 1986; Dixit and Pindyck, 1994), it is typically assumed that the economic agents are able to process information with lightning speed. In reality, it always takes them a lot of time and attention to process information about the economic conditions and incorporate this knowledge into their real investment decisions. For instance, Pashler and Johnston (1998) summarize the evidence that the central cognitive-processing capacity of the human brain has its limits. In this paper, we study the effects of the agent’s information-processing constraints (or rational inattention, see Sims, 2003) on the optimal investment decisions and the incurred value loss based on the real option framework. Specifically, the agent with information-processing constraints has an option to start a project, which can produce one commodity. Following Shibata (2008) and Zhang et al. (2015), we propose that the commodity price evolves according to an Ornstein–Uhlenbeck process.1 In order to introduce the agent’s rational inattention, we consider an incomplete information situation in which the agent cannot observe the true commodity price but a noisy signal. Therefore, the agent has to process the information provided by the noisy signal to infer the true dynamics of the commodity price. However, the capacity constraints limit the amount of
☆ Congming Mu acknowledges the support from the China Postdoctoral Science Foundation (#2018M640370). Jinqiang Yang acknowledges the support from the National Natural Science Foundation of China (#71202007, #71522008, #71532009 and #71573033), Innovation Program of Shanghai Municipal Education Commission (#13ZS050), ‘Chen Guang’ Project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation (#12CG44), Innovative Research Team of Shanghai University of Finance and Economics (#2016110241), and Fok Ying-Tong Education Foundation of China (#151086). ⁎ Corresponding author. E-mail addresses: [email protected] (C. Mu), [email protected] (J. Yang), [email protected] (Y. Zhang). 1 Shibata (2008) uses an Ornstein–Uhlenbeck process to model the underlying uncertainty for carrying out the investment opportunity. Zhang et al. (2015) utilize an Ornstein–Uhlenbeck process to model the commodity price to determining the optimal price threshold of mining activation.
https://doi.org/10.1016/j.frl.2019.01.001 Received 2 September 2018; Received in revised form 28 December 2018; Accepted 7 January 2019 1544-6123/ © 2019 Elsevier Inc. All rights reserved.
Please cite this article as: Mu, C., Finance Research Letters, https://doi.org/10.1016/j.frl.2019.01.001
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
information that the agent can process.2 Consequently, the uncertainty in the commodity price perceived by the agent is different from the true one, and the degree of difference between them greatly depends on the agent’s information-processing capacity. In what follows, we show it is the case that when the agent has the infinite capacity, the estimated price process coincides with the true one. Note that in the standard real option model, the volatility of the underlying state variable is the main determinants of the investment timing and option value, ceteris paribus. Thus, in this paper, by incorporating the capacity constraints, we can investigate how the agent’s information-processing constraints influence his optimal investment decisions and the corresponding value loss. The conclusions show that as the information-processing capacity decreases, the agent exercises the investment opportunity earlier but the loss in value increases. There is a growing literature that studies the information-process constraints. For instance, Sims (2003) studies the limited attention of an agent and applies it to the dynamic programming problems in macroeconomics. Maćkowiak and Wiederholt (2009) present a model in which price setting firms decide what to pay attention to, subject to the constrained information flows. Based on the permanent income model, Luo (2008) studies the implications of rational inattention for optimal consumption, saving, and welfare. Luo and Young (2010) examine the implications of limited information-processing capacity for asset prices. Moreover, Luo (2016) explores how interactions of model uncertainty and state uncertainty due to rational inattention affect strategic consumption-portfolio rules and precautionary savings. Luo et al. (2017) study the impacts of rational inattention for the cross-sectional dispersion of consumption and wealth in general equilibrium. Differing from these contributions, we focus on the effects of information capacity constraints on the investment timing and the incurred value loss. 2. The model We assume a risk-neutral agent has an option to invest at any time τ by paying the fixed cost I. Once the option is exercised, the project can continuously produce one commodity with the quantity Q, whose price is denoted by Xt (t ≥ τ). Thus, after undertaking the project, the agent can continuously receive flow payoff QXt (t ≥ τ). Following Shibata (2008) and Zhang et al. (2015), we propose that the commodity price Xt evolves according to an Ornstein–Uhlenbeck process given by
dXt = (
Xt ) dt +
Xd
X t ,
t
(1)
0,
where θ > 0 and σX > 0 are the long-run mean and volatility of the price X, respectively, constant λ > 0 controls the speed of reversion, and tX is a standard Brownian motion defined on a probability space ( , , ) . In addition, the agent can borrow or lend at the risk-free interest rate r > 0 in the financial market. In order to focus on the effects of the agent’s rational inattention, we set Q = 1 for simplicity. Thus, at the initial time 0, the agent chooses the optimal exercising time τ to maximize 0
e
r
max
+
e
r (s
) X ds s
I, 0
.
(2)
In order to incorporate the information-processing constraints, we assume that the agent cannot observe the commodity price process (1) perfectly but observe a noisy signal, X˜t , given by3
dX˜t = Xt dt +
Sd
(3)
S t,
where is another standard Brownian motion independent of and the variance, σS, is a choice variable for the agent in the rational inattention setting (see, Peng, 2005; Luo, 2016). The unobservable commodity price implies that the agent has to infer the commodity price X based on the noisy signal X˜ and the available information set Gt = {X˜s : s t } . However, his information-processing capacity is limited. Following Luo (2016), the limited information-processing capacity is defined as S t
(Xt + t |Gt )
X t ,
(Xt + t |Gt + t )
(4)
t,
where κ is the agent’s channel capacity and represents the maximum amount of information that can be transmitted via the information channel per unit of time; (Xt + t |Gt ) denotes the entropy of the state prior to observing the new signal at t + t and (Xt + t |Gt + t ) the entropy after observing the new signal at t + t . Thus, (4) implies that the information channel capacity imposes an upper bound on the change in entropy. 3. Solution 3.1. Benchmark: Investment with full information Firstly, we focus on the baseline case in which the agent has unlimited information-processing capacity and can observe the commodity price perfectly. For this case, let F(X) denote the value function of investment option, X* the optimal investment threshold. According to the standard arguments in real option analysis (McDonald and Siegel, 1986; Dixit and Pindyck, 1994), F(X) 2
In Information Theory, information is measured by entropy. Formally, for a random variable X with the density function f(X), its entropy is (X ) [ln(f (X ))]. 3 As pointed out by Kasa (2006) and Reis (2011), with the finite information-processing capacity, dXt is not suitable to state uncertainty in that in any finite interval, arbitrarily large amounts of information can be passed through the information channel. 2
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
Fig. 1. Option value G(m) for different channel capacities and comparative statics with respect to r, σX, I, θ and λ, respectively. Table 1 The value loss VL caused by the information-processing constraints. 4
κ
10
VL
0.39
3
0.01
0.10
1.00
0.35
0.26
0.10
1.45 × 10
10
10.0 2
1.52 × 10
3
satisfies the following ordinary differential equation
1 2
2 X FXX
(X ) + (
X ) FX (X )
rF (X ) = 0,
X
X *,
(5)
subject to the following boundary conditions:
F (X *) = FX (X *) =
X* r+ 1 r+
+
r (r + )
I,
,
lim F (X ) = 0.
(6)
X
In the first equality of (6), the first two terms on the right hand side is the value of the discounted cash flows. So the first boundary condition in (6) represents the value-matching condition at the moment the option is exercised. The second boundary condition is the smooth-pasting condition, which guarantees the optimality of the investment threshold X*. The third condition reflects the fact that once the commodity price X becomes negatively infinite, it is impossible for it to become positive again and hence the investment option is no longer valuable.4 3.2. Investment with information-processing constraints In this case, the agent needs to infer the commodity price X based on the noisy signal X˜ and the available information set Gt = {X˜s : s t } . In fact, given the Gaussian prior X0 ∼ N(m0, γ0), this can be solved by Kalman–Bucy filtering method. Let mt = [Xt |Gt ] and t = [(Xt mt )2|Gt ]. According to Liptser and Shiryaev (2001), we have the following results: 4 In essence, the boundary condition limX standard real option literature.
F (X ) = 0 to some extent is similar to that F (0) = 0 with the geometric Brownian motion in the
3
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
dmt = (
d
t
=
dt
t
mt ) dt +
2
t
2 X
+
d
t,
S
(7)
t 2 ) ,
(
(8)
S
where d t = (dX˜t mt dt )/ S is a standard Brownian motion. In process (8), the first two terms denote the incremental uncertainty induced by the dynamics of the commodity price itself, and the last term represents the reduction in uncertainty due to the more accurate information. Recall that in the rational inattention setting, σS is under the agent’s control. Therefore, the agent can manipulate the reduction in uncertainty according to his information-processing capacity, which will be discussed below. Particularly, in the steady state where dγt/dt ≡ 0, the conditional variance γ can be written as
=
2 2 S
S
2 X
+
2 S.
(9)
Clearly, variance γ depends on the agent’s choice σS. On the other hand, as documented by Luo (2016), it is optimal for rational individuals to use all of their channel capacity to reduce the uncertainty upon new observations. As a result, the condition (4) for information-processing constraints becomes
(Xt + t |Gt )
(Xt + t |Gt + t ) =
(10)
t.
In order to apply (10) to the price transition process (1), we discretize process (1) as
Xt +
t
= (1
t)
e
+e
tX t
+
X
1
e 2
2
t
t+ t ,
(11)
where t + t is a standard normal distribution. Similar to Luo (2016), taking the conditional variances on both sides of (11) and substituting them into (10), we can obtain5 2
ln e
t
t
+
e 2 2
(1
t) 2 X
ln[
t+ t ]
=2
t.
(12)
Taking the limitation Δt → 0, we can get
d
t
dt
=
2( + ) t +
2 X,
(13)
which implies that in the steady state defined by dγt/dt ≡ 0, the variance γ satisfies
=
2 X
2( + )
.
(14)
It is worth noting that under the full channel capacity κ, the variance γ satisfies (14). On the other hand, with the inference of commodity price based on the observed signal X˜t in (3), the variance satisfies (9). In order to utilize his full capacity κ, the agent has to optimally choose σS such that the variance given by (9) is identical to that in (14). Clearly, this represents the optimality of σS with respect to the given channel capacity κ. We collect these results in the next proposition. Proposition 1. Given the agent’s finite information-processing capacity κ, in the steady state, the estimated commodity price m evolves according to
dmt = (
mt ) dt + f ( )
Xd
t
(15)
,
where
f( )=
+
,
(16)
and the choice variable σS in process (3) satisfies S
=
2 X
4 ( + )
.
(17)
Proof. In the steady state where d t / dt = 0, combining (9) and (14), we can easily have the equation (17). Substituting (17) into (7) yields the results of (15) and (16). □ Comparing the perceived price process (15) to the true one (1), we find that the agent’s finite capacity κ changes the magnitude of uncertainty in price. With the infinite channel capacity (e.g., = ), the perceived price process defaults to the true one. However, if the agent has finite capacity, the perceived volatility is less than the true one in that f(κ) < 1. Eq. (17) shows the relation between the choice variable σS and the channel capacity κ. Intuitively, the lower the information-processing capacity κ, the higher the uncertainty 5
The entropy of a random variable X with normal distribution N(μ, σ2) depends only on its variance in that 4
(X ) = 0.5 log
2
+ 0.5 log(2 e ) .
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
in the noisy signal X˜ from the perspective of agent with rational inattention. For the agent with limited information-processing capacity, he chooses the optimal investment time τ to maximize (2) but subject to the perceived price process (15) instead of the fully observed price process (1). Similarly, we let G(m) denote the value function of investment option and m* the optimal investment threshold. Thus, G(m) satisfies the following ordinary differential equation:
1 f( ) 2
2G mm (m)
X
+ (
m) Gm (m)
rG (m) = 0 ,
m* ,
m
(18)
subject to the boundary conditions given by
G (m*) =
m* r+
Gm (m*) =
+
1 r+
r (r + )
I,
,
lim G (m ) = 0.
(19)
m
Comparing this optimization problem (18) and (19) to the problem with full information (5) and (6), we can find that the main difference is the coefficient of the second-order derivative in the ordinary differential equation. The reason for this is that in our Gaussian economic setting, the variance of the distribution determines the entropy, the constraints on the change of entropy in turn influence the variance of the estimated commodity price. 4. Value loss due to rational inattention It is natural to question how the information-processing constraints change the option value, increase or decrease? To this end, we firstly consider what is the value function (denoted by F^ (X ) ) if the full-information agent adopts the investment timing taken by the agent with limited channel capacity. Mathematically, this value function F^ (X ) satisfies
1 2
2 ^ X FXX
X ) F^X (X )
(X ) + (
rF^ (X ) = 0,
X
m*,
(20)
subject to the following boundary conditions
F^ (m*) =
m* r+
+
r (r + )
I,
lim F^ (m) = 0,
(21)
m
where m* is the optimal investment threshold taken by the agent with finite information capacity. Clearly, (21) does not include the smooth-pasting condition as condition (6) in that m* is not the optimal investment decision for the agent with full information. Now we can define the loss in option value caused by the information-processing constraints as follows
VL =
+
F (x )
F^ (x ) f (x ) dx ,
(22)
where f(x) is the probability density function of the steady-state distribution of the commodity price, X, with mean θ and variance σ2/ (2λ).6 5. Numerical analysis In this section, we numerically illustrate the effects of the agent’s rational inattention on the optimal investment decisions and the incurred loss in option value. In our analysis, we choose the baseline parameters as r = 0.05, = 0.1, = 0.15, X = 0.3, I = 1. Panel (a) of Fig. 1 plots the option value against the perceived commodity price m for three different channel capacities (e.g., = 10 3, 0.1, ∞). Obviously, the agent’s channel capacity κ increases not only the option value, but also the investment threshold, implying that the agent with finite information channel capacity is more likely to overinvest. These are consistent with the standard wisdom regarding real options in that the agent’s channel capacity κ is positively related to the volatility of the perceived price (see, Eqs. (15) and (16)) and hence has the positive relation to the option value and investment threshold. The reason is that in the rational inattention scenario, when the agent with larger channel capacity formats the observed signal, he optimally adjusts the signal precision upward as discussed above. This action reduces the uncertainty of the signal and subsequently the perceived price volatility. Panel (b)–(f) provide the comparative statics of the optimal investment threshold m*(κ) with respect different values of r, σX, I, θ and λ, respectively. Clearly, consistent with Panel (a), the investment threshold increases in channel capacity κ. In addition, Panel (b)–(d) show that the comparative statics about the interest rate r, volatility σX of commodity price and the investment cost I are consistent with those in standard real option analysis. For instance, the increase in interest rate promotes investment (lowers the investment threshold), but the increase in the uncertainty of commodity price and sunk cost dampens investment, ceteris paribus. In 6
While the dynamic process of the commodity price is an Ornstein–Uhlenbeck process given in (1), we focus on its steady distribution. 5
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
particular, Panel (e) shows that given all else being equal, the investment opportunity whose commodity price has the larger longterm mean attracts the agent to exercise the option earlier. This is economically intuitive in that with the larger long-run expectation in the commodity price, the project is more valuable on average. Similarly, the agent also prefers to overinvest the project with commodity price which can quickly revert to its long-term expectation as shown by Panel (f). The economic intuition is as follows. While the commodity price fluctuates around its long-run mean, which is known for the agent ex ante, the larger mean-reversion speed makes the commodity price less volatile and allows the agent to be more confident in the option value. Finally, the investment threshold decreases more quickly in the low capacity region than in the high capacity region. Table 1 shows the value loss VL induced by the information-processing constraints. Clearly, as the channel capacity κ increases, the loss in option value declines. This is intuitive in that with the larger information-processing capacity, the agent can process more information per unit of time, which effectively reduces the magnitude of incompleteness in information and hence enhances his timing ability and the option value. 6. Conclusion This paper incorporates the information-processing constraints into the real option model with incomplete information and examines the corresponding implications for the investment timing choice and the incurred loss in option value. The findings show that the agent’s information-processing capacity increases the uncertainty in the perceived commodity price. This increase leads to that both the option value and the optimal investment threshold increase in the agent’s information-processing capacity, which means that the agent with more channel capacity limits tends to overinvest. Furthermore, the increase in option value reduces the value loss caused by the rational inattention. Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.frl.2019.01.001. References Dixit, A.K., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press. Kasa, K., 2006. Robustness and information processing. Rev. Econ. Dyn. 9 (1), 1–33. Liptser, R.S., Shiryaev, A.N., 2001. Statistics of Random Processes I and II. Springer Science & Business Media. Luo, Y., 2008. Consumption dynamics under information processing constraints. Rev. Econ. Dyn. 11 (2), 366–385. Luo, Y., 2016. Robustly strategic consumption–portfolio rules with informational frictions. Manage. Sci. 63 (12), 4158–4174. Luo, Y., Nie, J., Wang, G., Young, E.R., 2017. Rational inattention and the dynamics of consumption and wealth in general equilibrium. J. Econ. Theory 172, 55–87. Luo, Y., Young, E.R., 2010. Asset pricing under information-processing constraints. Econ. Lett. 107 (1), 26–29. Maćkowiak, B., Wiederholt, M., 2009. Optimal sticky prices under rational inattention. Am. Econ. Rev. 99 (3), 769–803. McDonald, R., Siegel, D., 1986. The value of waiting to invest. Q. J. Econ. 101 (4), 707–727. Pashler, H., Johnston, J.C., 1998. Attentional limitations in dual-task performance. In: Pashler, H. (Ed.), Attention. Psychology Press, Hove, UK. Peng, L., 2005. Learning with information capacity constraints. J. Financ. Quant. Anal. 40 (2), 307–329. Reis, R., 2011. When Should Policymakers Make Announcements? Manuscript. Shibata, T., 2008. The impacts of uncertainties in a real options model under incomplete information. Eur. J. Oper. Res. 187 (3), 1368–1379. Sims, C.A., 2003. Implications of rational inattention. J. Monet. Econ. 50 (3), 665–690. Zhang, K., Nieto, A., Kleit, A.N., 2015. The real option value of mining operations using mean-reverting commodity prices. Miner. Econ. 28 (1–2), 11–22.
6
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Investment timing with information-processing constraints☆ ⁎,c
Congming Mua, Jinqiang Yangb,c, Yuhua Zhang a
Shanghai University of Finance and Economics, Shanghai Institute of International Finance and Economics, Shanghai, China Shanghai Key Laboratory of Financial Information Technology, Shanghai Institute of International Finance and Economics, Shanghai, China c School of Finance, Shanghai University of Finance and Economics, Shanghai, China b
ARTICLE INFO
ABSTRACT
Keywords: Real options Optimal investment Information-processing constraints Channel capacity
This paper investigates the implications of information-processing constraints (rational inattention) for the optimal investment timing and the corresponding value loss based on the real option framework with incomplete information. We find that an agent with rational inattention tends to overinvest and that the loss in option value decreases in the information channel capacity.
JEL classification: D81 D83 G32
1. Introduction In the standard literature about the real options (see, McDonald and Siegel, 1986; Dixit and Pindyck, 1994), it is typically assumed that the economic agents are able to process information with lightning speed. In reality, it always takes them a lot of time and attention to process information about the economic conditions and incorporate this knowledge into their real investment decisions. For instance, Pashler and Johnston (1998) summarize the evidence that the central cognitive-processing capacity of the human brain has its limits. In this paper, we study the effects of the agent’s information-processing constraints (or rational inattention, see Sims, 2003) on the optimal investment decisions and the incurred value loss based on the real option framework. Specifically, the agent with information-processing constraints has an option to start a project, which can produce one commodity. Following Shibata (2008) and Zhang et al. (2015), we propose that the commodity price evolves according to an Ornstein–Uhlenbeck process.1 In order to introduce the agent’s rational inattention, we consider an incomplete information situation in which the agent cannot observe the true commodity price but a noisy signal. Therefore, the agent has to process the information provided by the noisy signal to infer the true dynamics of the commodity price. However, the capacity constraints limit the amount of
☆ Congming Mu acknowledges the support from the China Postdoctoral Science Foundation (#2018M640370). Jinqiang Yang acknowledges the support from the National Natural Science Foundation of China (#71202007, #71522008, #71532009 and #71573033), Innovation Program of Shanghai Municipal Education Commission (#13ZS050), ‘Chen Guang’ Project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation (#12CG44), Innovative Research Team of Shanghai University of Finance and Economics (#2016110241), and Fok Ying-Tong Education Foundation of China (#151086). ⁎ Corresponding author. E-mail addresses: [email protected] (C. Mu), [email protected] (J. Yang), [email protected] (Y. Zhang). 1 Shibata (2008) uses an Ornstein–Uhlenbeck process to model the underlying uncertainty for carrying out the investment opportunity. Zhang et al. (2015) utilize an Ornstein–Uhlenbeck process to model the commodity price to determining the optimal price threshold of mining activation.
https://doi.org/10.1016/j.frl.2019.01.001 Received 2 September 2018; Received in revised form 28 December 2018; Accepted 7 January 2019 1544-6123/ © 2019 Elsevier Inc. All rights reserved.
Please cite this article as: Mu, C., Finance Research Letters, https://doi.org/10.1016/j.frl.2019.01.001
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
information that the agent can process.2 Consequently, the uncertainty in the commodity price perceived by the agent is different from the true one, and the degree of difference between them greatly depends on the agent’s information-processing capacity. In what follows, we show it is the case that when the agent has the infinite capacity, the estimated price process coincides with the true one. Note that in the standard real option model, the volatility of the underlying state variable is the main determinants of the investment timing and option value, ceteris paribus. Thus, in this paper, by incorporating the capacity constraints, we can investigate how the agent’s information-processing constraints influence his optimal investment decisions and the corresponding value loss. The conclusions show that as the information-processing capacity decreases, the agent exercises the investment opportunity earlier but the loss in value increases. There is a growing literature that studies the information-process constraints. For instance, Sims (2003) studies the limited attention of an agent and applies it to the dynamic programming problems in macroeconomics. Maćkowiak and Wiederholt (2009) present a model in which price setting firms decide what to pay attention to, subject to the constrained information flows. Based on the permanent income model, Luo (2008) studies the implications of rational inattention for optimal consumption, saving, and welfare. Luo and Young (2010) examine the implications of limited information-processing capacity for asset prices. Moreover, Luo (2016) explores how interactions of model uncertainty and state uncertainty due to rational inattention affect strategic consumption-portfolio rules and precautionary savings. Luo et al. (2017) study the impacts of rational inattention for the cross-sectional dispersion of consumption and wealth in general equilibrium. Differing from these contributions, we focus on the effects of information capacity constraints on the investment timing and the incurred value loss. 2. The model We assume a risk-neutral agent has an option to invest at any time τ by paying the fixed cost I. Once the option is exercised, the project can continuously produce one commodity with the quantity Q, whose price is denoted by Xt (t ≥ τ). Thus, after undertaking the project, the agent can continuously receive flow payoff QXt (t ≥ τ). Following Shibata (2008) and Zhang et al. (2015), we propose that the commodity price Xt evolves according to an Ornstein–Uhlenbeck process given by
dXt = (
Xt ) dt +
Xd
X t ,
t
(1)
0,
where θ > 0 and σX > 0 are the long-run mean and volatility of the price X, respectively, constant λ > 0 controls the speed of reversion, and tX is a standard Brownian motion defined on a probability space ( , , ) . In addition, the agent can borrow or lend at the risk-free interest rate r > 0 in the financial market. In order to focus on the effects of the agent’s rational inattention, we set Q = 1 for simplicity. Thus, at the initial time 0, the agent chooses the optimal exercising time τ to maximize 0
e
r
max
+
e
r (s
) X ds s
I, 0
.
(2)
In order to incorporate the information-processing constraints, we assume that the agent cannot observe the commodity price process (1) perfectly but observe a noisy signal, X˜t , given by3
dX˜t = Xt dt +
Sd
(3)
S t,
where is another standard Brownian motion independent of and the variance, σS, is a choice variable for the agent in the rational inattention setting (see, Peng, 2005; Luo, 2016). The unobservable commodity price implies that the agent has to infer the commodity price X based on the noisy signal X˜ and the available information set Gt = {X˜s : s t } . However, his information-processing capacity is limited. Following Luo (2016), the limited information-processing capacity is defined as S t
(Xt + t |Gt )
X t ,
(Xt + t |Gt + t )
(4)
t,
where κ is the agent’s channel capacity and represents the maximum amount of information that can be transmitted via the information channel per unit of time; (Xt + t |Gt ) denotes the entropy of the state prior to observing the new signal at t + t and (Xt + t |Gt + t ) the entropy after observing the new signal at t + t . Thus, (4) implies that the information channel capacity imposes an upper bound on the change in entropy. 3. Solution 3.1. Benchmark: Investment with full information Firstly, we focus on the baseline case in which the agent has unlimited information-processing capacity and can observe the commodity price perfectly. For this case, let F(X) denote the value function of investment option, X* the optimal investment threshold. According to the standard arguments in real option analysis (McDonald and Siegel, 1986; Dixit and Pindyck, 1994), F(X) 2
In Information Theory, information is measured by entropy. Formally, for a random variable X with the density function f(X), its entropy is (X ) [ln(f (X ))]. 3 As pointed out by Kasa (2006) and Reis (2011), with the finite information-processing capacity, dXt is not suitable to state uncertainty in that in any finite interval, arbitrarily large amounts of information can be passed through the information channel. 2
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
Fig. 1. Option value G(m) for different channel capacities and comparative statics with respect to r, σX, I, θ and λ, respectively. Table 1 The value loss VL caused by the information-processing constraints. 4
κ
10
VL
0.39
3
0.01
0.10
1.00
0.35
0.26
0.10
1.45 × 10
10
10.0 2
1.52 × 10
3
satisfies the following ordinary differential equation
1 2
2 X FXX
(X ) + (
X ) FX (X )
rF (X ) = 0,
X
X *,
(5)
subject to the following boundary conditions:
F (X *) = FX (X *) =
X* r+ 1 r+
+
r (r + )
I,
,
lim F (X ) = 0.
(6)
X
In the first equality of (6), the first two terms on the right hand side is the value of the discounted cash flows. So the first boundary condition in (6) represents the value-matching condition at the moment the option is exercised. The second boundary condition is the smooth-pasting condition, which guarantees the optimality of the investment threshold X*. The third condition reflects the fact that once the commodity price X becomes negatively infinite, it is impossible for it to become positive again and hence the investment option is no longer valuable.4 3.2. Investment with information-processing constraints In this case, the agent needs to infer the commodity price X based on the noisy signal X˜ and the available information set Gt = {X˜s : s t } . In fact, given the Gaussian prior X0 ∼ N(m0, γ0), this can be solved by Kalman–Bucy filtering method. Let mt = [Xt |Gt ] and t = [(Xt mt )2|Gt ]. According to Liptser and Shiryaev (2001), we have the following results: 4 In essence, the boundary condition limX standard real option literature.
F (X ) = 0 to some extent is similar to that F (0) = 0 with the geometric Brownian motion in the
3
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
dmt = (
d
t
=
dt
t
mt ) dt +
2
t
2 X
+
d
t,
S
(7)
t 2 ) ,
(
(8)
S
where d t = (dX˜t mt dt )/ S is a standard Brownian motion. In process (8), the first two terms denote the incremental uncertainty induced by the dynamics of the commodity price itself, and the last term represents the reduction in uncertainty due to the more accurate information. Recall that in the rational inattention setting, σS is under the agent’s control. Therefore, the agent can manipulate the reduction in uncertainty according to his information-processing capacity, which will be discussed below. Particularly, in the steady state where dγt/dt ≡ 0, the conditional variance γ can be written as
=
2 2 S
S
2 X
+
2 S.
(9)
Clearly, variance γ depends on the agent’s choice σS. On the other hand, as documented by Luo (2016), it is optimal for rational individuals to use all of their channel capacity to reduce the uncertainty upon new observations. As a result, the condition (4) for information-processing constraints becomes
(Xt + t |Gt )
(Xt + t |Gt + t ) =
(10)
t.
In order to apply (10) to the price transition process (1), we discretize process (1) as
Xt +
t
= (1
t)
e
+e
tX t
+
X
1
e 2
2
t
t+ t ,
(11)
where t + t is a standard normal distribution. Similar to Luo (2016), taking the conditional variances on both sides of (11) and substituting them into (10), we can obtain5 2
ln e
t
t
+
e 2 2
(1
t) 2 X
ln[
t+ t ]
=2
t.
(12)
Taking the limitation Δt → 0, we can get
d
t
dt
=
2( + ) t +
2 X,
(13)
which implies that in the steady state defined by dγt/dt ≡ 0, the variance γ satisfies
=
2 X
2( + )
.
(14)
It is worth noting that under the full channel capacity κ, the variance γ satisfies (14). On the other hand, with the inference of commodity price based on the observed signal X˜t in (3), the variance satisfies (9). In order to utilize his full capacity κ, the agent has to optimally choose σS such that the variance given by (9) is identical to that in (14). Clearly, this represents the optimality of σS with respect to the given channel capacity κ. We collect these results in the next proposition. Proposition 1. Given the agent’s finite information-processing capacity κ, in the steady state, the estimated commodity price m evolves according to
dmt = (
mt ) dt + f ( )
Xd
t
(15)
,
where
f( )=
+
,
(16)
and the choice variable σS in process (3) satisfies S
=
2 X
4 ( + )
.
(17)
Proof. In the steady state where d t / dt = 0, combining (9) and (14), we can easily have the equation (17). Substituting (17) into (7) yields the results of (15) and (16). □ Comparing the perceived price process (15) to the true one (1), we find that the agent’s finite capacity κ changes the magnitude of uncertainty in price. With the infinite channel capacity (e.g., = ), the perceived price process defaults to the true one. However, if the agent has finite capacity, the perceived volatility is less than the true one in that f(κ) < 1. Eq. (17) shows the relation between the choice variable σS and the channel capacity κ. Intuitively, the lower the information-processing capacity κ, the higher the uncertainty 5
The entropy of a random variable X with normal distribution N(μ, σ2) depends only on its variance in that 4
(X ) = 0.5 log
2
+ 0.5 log(2 e ) .
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
in the noisy signal X˜ from the perspective of agent with rational inattention. For the agent with limited information-processing capacity, he chooses the optimal investment time τ to maximize (2) but subject to the perceived price process (15) instead of the fully observed price process (1). Similarly, we let G(m) denote the value function of investment option and m* the optimal investment threshold. Thus, G(m) satisfies the following ordinary differential equation:
1 f( ) 2
2G mm (m)
X
+ (
m) Gm (m)
rG (m) = 0 ,
m* ,
m
(18)
subject to the boundary conditions given by
G (m*) =
m* r+
Gm (m*) =
+
1 r+
r (r + )
I,
,
lim G (m ) = 0.
(19)
m
Comparing this optimization problem (18) and (19) to the problem with full information (5) and (6), we can find that the main difference is the coefficient of the second-order derivative in the ordinary differential equation. The reason for this is that in our Gaussian economic setting, the variance of the distribution determines the entropy, the constraints on the change of entropy in turn influence the variance of the estimated commodity price. 4. Value loss due to rational inattention It is natural to question how the information-processing constraints change the option value, increase or decrease? To this end, we firstly consider what is the value function (denoted by F^ (X ) ) if the full-information agent adopts the investment timing taken by the agent with limited channel capacity. Mathematically, this value function F^ (X ) satisfies
1 2
2 ^ X FXX
X ) F^X (X )
(X ) + (
rF^ (X ) = 0,
X
m*,
(20)
subject to the following boundary conditions
F^ (m*) =
m* r+
+
r (r + )
I,
lim F^ (m) = 0,
(21)
m
where m* is the optimal investment threshold taken by the agent with finite information capacity. Clearly, (21) does not include the smooth-pasting condition as condition (6) in that m* is not the optimal investment decision for the agent with full information. Now we can define the loss in option value caused by the information-processing constraints as follows
VL =
+
F (x )
F^ (x ) f (x ) dx ,
(22)
where f(x) is the probability density function of the steady-state distribution of the commodity price, X, with mean θ and variance σ2/ (2λ).6 5. Numerical analysis In this section, we numerically illustrate the effects of the agent’s rational inattention on the optimal investment decisions and the incurred loss in option value. In our analysis, we choose the baseline parameters as r = 0.05, = 0.1, = 0.15, X = 0.3, I = 1. Panel (a) of Fig. 1 plots the option value against the perceived commodity price m for three different channel capacities (e.g., = 10 3, 0.1, ∞). Obviously, the agent’s channel capacity κ increases not only the option value, but also the investment threshold, implying that the agent with finite information channel capacity is more likely to overinvest. These are consistent with the standard wisdom regarding real options in that the agent’s channel capacity κ is positively related to the volatility of the perceived price (see, Eqs. (15) and (16)) and hence has the positive relation to the option value and investment threshold. The reason is that in the rational inattention scenario, when the agent with larger channel capacity formats the observed signal, he optimally adjusts the signal precision upward as discussed above. This action reduces the uncertainty of the signal and subsequently the perceived price volatility. Panel (b)–(f) provide the comparative statics of the optimal investment threshold m*(κ) with respect different values of r, σX, I, θ and λ, respectively. Clearly, consistent with Panel (a), the investment threshold increases in channel capacity κ. In addition, Panel (b)–(d) show that the comparative statics about the interest rate r, volatility σX of commodity price and the investment cost I are consistent with those in standard real option analysis. For instance, the increase in interest rate promotes investment (lowers the investment threshold), but the increase in the uncertainty of commodity price and sunk cost dampens investment, ceteris paribus. In 6
While the dynamic process of the commodity price is an Ornstein–Uhlenbeck process given in (1), we focus on its steady distribution. 5
Finance Research Letters xxx (xxxx) xxx–xxx
C. Mu et al.
particular, Panel (e) shows that given all else being equal, the investment opportunity whose commodity price has the larger longterm mean attracts the agent to exercise the option earlier. This is economically intuitive in that with the larger long-run expectation in the commodity price, the project is more valuable on average. Similarly, the agent also prefers to overinvest the project with commodity price which can quickly revert to its long-term expectation as shown by Panel (f). The economic intuition is as follows. While the commodity price fluctuates around its long-run mean, which is known for the agent ex ante, the larger mean-reversion speed makes the commodity price less volatile and allows the agent to be more confident in the option value. Finally, the investment threshold decreases more quickly in the low capacity region than in the high capacity region. Table 1 shows the value loss VL induced by the information-processing constraints. Clearly, as the channel capacity κ increases, the loss in option value declines. This is intuitive in that with the larger information-processing capacity, the agent can process more information per unit of time, which effectively reduces the magnitude of incompleteness in information and hence enhances his timing ability and the option value. 6. Conclusion This paper incorporates the information-processing constraints into the real option model with incomplete information and examines the corresponding implications for the investment timing choice and the incurred loss in option value. The findings show that the agent’s information-processing capacity increases the uncertainty in the perceived commodity price. This increase leads to that both the option value and the optimal investment threshold increase in the agent’s information-processing capacity, which means that the agent with more channel capacity limits tends to overinvest. Furthermore, the increase in option value reduces the value loss caused by the rational inattention. Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.frl.2019.01.001. References Dixit, A.K., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press. Kasa, K., 2006. Robustness and information processing. Rev. Econ. Dyn. 9 (1), 1–33. Liptser, R.S., Shiryaev, A.N., 2001. Statistics of Random Processes I and II. Springer Science & Business Media. Luo, Y., 2008. Consumption dynamics under information processing constraints. Rev. Econ. Dyn. 11 (2), 366–385. Luo, Y., 2016. Robustly strategic consumption–portfolio rules with informational frictions. Manage. Sci. 63 (12), 4158–4174. Luo, Y., Nie, J., Wang, G., Young, E.R., 2017. Rational inattention and the dynamics of consumption and wealth in general equilibrium. J. Econ. Theory 172, 55–87. Luo, Y., Young, E.R., 2010. Asset pricing under information-processing constraints. Econ. Lett. 107 (1), 26–29. Maćkowiak, B., Wiederholt, M., 2009. Optimal sticky prices under rational inattention. Am. Econ. Rev. 99 (3), 769–803. McDonald, R., Siegel, D., 1986. The value of waiting to invest. Q. J. Econ. 101 (4), 707–727. Pashler, H., Johnston, J.C., 1998. Attentional limitations in dual-task performance. In: Pashler, H. (Ed.), Attention. Psychology Press, Hove, UK. Peng, L., 2005. Learning with information capacity constraints. J. Financ. Quant. Anal. 40 (2), 307–329. Reis, R., 2011. When Should Policymakers Make Announcements? Manuscript. Shibata, T., 2008. The impacts of uncertainties in a real options model under incomplete information. Eur. J. Oper. Res. 187 (3), 1368–1379. Sims, C.A., 2003. Implications of rational inattention. J. Monet. Econ. 50 (3), 665–690. Zhang, K., Nieto, A., Kleit, A.N., 2015. The real option value of mining operations using mean-reverting commodity prices. Miner. Econ. 28 (1–2), 11–22.
6