Abstract:
This paper analyses optimal investment timing in a two-stage sequential real options model under asymmetric information between principal and agent. In a model similar to Grenadier and Wang (2005), I show that the principal is able to design incentive compatible contracts to induce the agent to truthfully reveal his private information about the costs of a project. In comparison to a single period model, I show that when the private information of the two periods is unrelated, the principal is able to shift rents from the agent to himself between the periods when contracting ex-ante, resulting in a more efficient investment timing than when both investments are contracted separately. When the private information between the periods is related, I display that contracting ex-ante is at least as efficient as contracting the two investments separately, as long as the principal commits to the ex-ante contract. These findings support the findings of Baron and Besanko (1984).
Keywords: investment timing, asymmetric information, sequential investments.
Daniël de Koning S2099519
Supervisor: Dr. G.T.J. Zwart Date: June 8th, 2017
2
Table of contents
1. Introduction ... 3
2. Literature review ... 4
3. The basic real options model ... 9
3.1 The model ... 9
3.2 Dynamic programming solution ... 10
4. Single investment model under asymmetric information ... 14
4.1 First-best benchmark solution ... 14
4.2 Asymmetric information solution ... 15
5. Sequential real option models ... 18
5.1 First-best benchmark solution (model 1) ... 19
5.2 Sequential real options models under asymmetric information ... 21
6. Discussion of the sequential models ... 32
6.1 Comparison of the results with Baron and Besanko (1984) ... 32
6.2 Practical implications ... 33
6.3 Limitations and future research ... 34
7. Conclusion ... 35
References ... 36
3 1. Introduction
When a rational economic agent makes an investment decision, he allocates his capital in such a way that it maximizes his welfare function. Assuming the investment is irreversible and capital can only be allocated to one investment at the same time, the allocation decision consists of comparing an investment with other mutually exclusive investments. The net-present-value (NPV) approach provides a framework to make such a comparison. The NPV framework focusses on a cross-sectional comparison of mutually exclusive investments, i.e., the agent compares a potential investment with other investment opportunities at the same moment in time. However, when making the investment decision, the agent should also consider when to invest. This consideration requires the agent to make an intertemporal comparison of the investment itself, i.e., the agent compares the value of investing in the project immediately with the value of investing in the project at any other moment in the future. The real options approach with regard to investment timing is the framework that helps the agent to execute an investment at the optimal moment.
In some circumstances, the agent does not make the investment decision himself, but delegates this decision to another rational agent. This delegation may result in agency conflicts between principal and agent (owner and manager), as they both try to maximize their own welfare. One of these agency conflicts concerns asymmetric information between principal and agent. Asymmetric information implies the agent has superior information about the prospects of the investment which enables him to misrepresent this information to the principal and divert cash flows to himself at the expense of the principal. The principal is able to mitigate the asymmetric information problem by designing incentive compatible contracts that induce the agent to truthfully reveal the private information. Grenadier and Wang (2005) show that this can result in a delay in investment timing, meaning there is an efficiency loss when there is asymmetric information present.
4
information between the two periods is unrelated, it is most optimal to design a contract ex-ante. When this is not the case, I show that it is at least as efficient to contract ex-ante as to contract the two investment separately.
The results of the two-staged models and the intuition behind the models could be helpful for firms making delegated investment decisions in industries that face delegated sequential investments. One could think of industries that are research and development intensive and where the product or service can only be fully deployed after a research and development stage. An example of such an industry is the energy sector with a focus on the development of new renewable energy technologies. Besides companies, the results in this paper might be interesting for regulatory authorities that are involved in a multi-period regulation of, for instance, contractors or state controlled/owned firms.
The literature review in the next section introduces the publications that form the backbone of this paper in the following order based on topic: (1) basic real options, (2) real options under asymmetric information, (3) sequential real options, (4) asymmetric information in a sequential model. The remainder of this paper is structured according to the same order. Section 3 introduces a basic real options model on investment timing. Section 4 adds asymmetric information to this model in a setting similar to Grenadier and Wang (2005). Section 5 extends the model by introducing a second investment, resulting in the sequential model under asymmetric information. Section 6 compares the results with existing literature1, discusses its implications and sheds light on the limitations of the model and the model’s underlying assumptions. Section 7 concludes.
2. Literature review
Although real options models under asymmetric information are relatively new, real options where already mentioned in Myers (1977). In his publication, Myers recognizes that a firm’s growth opportunities are similar to call options. Besides growth options, more types of real options are identified in literature: options to switch inputs/outputs, options to defer an investment, options to alter scale etc.2
An early publication in the field of investment timing, which is the topic of this paper, is McDonald and Siegel (1986). McDonald and Siegel (1986) study optimal investment
1 Mainly Baron and Besanko (1984). 2
5
timing of an irreversible investment where the investor has the optionality to invest immediately or defer the investment. The value of the investment evolves according to a continuous time stochastic process. McDonald and Siegel recognize that irreversibility implies that upon investment the investor gives up the opportunity to invest at any other moment in the future. When the investor defers the investment, he still has the opportunity to invest in the future. This results in a trade-off: exercise now and give up the opportunity to invest in the future, or postpone the investment. To find the optimal investment timing, the investor should compare the value of investing immediately with the value of investing at any moment in the future. This valuation approach opposes the traditional NPV rule, where the investor assesses the profitability of an investment on the basis of a now-or-never decision. McDonald and Siegel (1986) show that the optimal investment threshold is higher under the real options approach than under the NPV-rule. When there is zero variance in the pay-offs, the investment thresholds of the real options approach and the NPV-rule would be equal.3
The model of McDonald and Siegel (1986), however, does not take into account the delegation of an investment, while it is not necessarily the owner of an investment who manages it. The delegation of the management of an investment can result in potential problems between principal and agent and may affect the value of an investment. Therefore, there is a need for real options models that consider the conflicts between principal and agent. In this paper, I focus on asymmetric information between principal and agent and the implications for the contracts between these two actors. Asymmetric information arises when the agent, managing the investment, is better informed about the prospects of the investment than the principal. Consequently, the agent could use this information advantage in his own interest at the expense of the principal. In the classic asymmetric information model, the privately known prospects model described by Tirole (2006), the principal can reveal untruthful information about the prospects of the investment and divert free cash flows to himself. Obviously, the principal can anticipate on this behaviour by offering a contract which induces the agent to reveal the prospects of the investment truthfully. These contracts are called incentive compatible contracts and are extensively discussed in Tirole (2006).4
More recent real options studies on investment timing take into account the principal-agent setting and the resulting problems of asymmetric information described above. Grenadier and Wang (2005) develop a single investment real options model where a principal
3 Subsection 3.2 of this paper explains why the real options model yields the same result as the
NPV-rule when the variance in the pay-offs is zero.
6
delegates the execution decision of an investment to a manager. The investment decision in Grenadier and Wang (2005) concerns investment timing and is similar to McDonald and Siegel (1986). The payoff from the investment option is composed of a stochastic component that both the principal and agent observe, and a random payoff component only the agent observes. The privately observable payoff component can take two values: high or low. Since only the manager observes the privately observable component, it introduces asymmetric information between principal and agent. This asymmetric information enables the manager to mispresent the true realization of the privately observable payoff component and divert cash flows to himself, at the expense of the principal. Grenadier and Wang (2005) show that it is possible to design an incentive compatible contract in which two wage/investment threshold combinations are specified and offered to the agent.5 The wage/investment threshold received by the agent depends on the reported value for the privately observable payoff component. A result of the incentive compatible contracts is a delay in the investment timing for a manager reporting a low privately observable payoff component, and an optimal investment timing for the manager reporting a high value. This delay ensures that an agent observing a high random payoff component is not willing to lie as the delay in investment reduces his value. Despite that the incentive compatible contracts can induce the agent to truthfully reveal his private information, there is an efficiency loss compared the case in which asymmetric information is absent (first-best benchmark).
Morellec and Schürhoff (2011), in contrast to Grenadier and Wang (2005), develop a single stage real options model under asymmetric information where a firm seeks outside financing. Instead of the principal-agent setting in Grenadier and Wang (2005), the authors focus on an insider-outsider setting where the firm offers the contract to an outside investor. The asymmetric information in Morellec and Schürhoff (2011) is the result of the firm’s private information about its own type, a high or a low growth firm, while the outside investor is not able to observe the growth prospects of the firm. To credibly signal its type, a high growth firm can speed up the investment as it is more costly for a firm with bad growth prospects to speed up the investment. Similarly to Grenadier and Wang (2005), there is some efficiency loss, as the high growth firm’s investment timing is sub-optimal compared to the case where asymmetric information is absent. The difference in the conclusion about investment timing between Morellec and Schürhoff (2011) and Grenadier and Wang (2005),
5 Grenadier and Wang (2005) show that since there are two possible values for the privately observable
7
speeding up versus delaying investments, is attributable to the specification of the models. It is of importance to determine who offers the contract and who receives the rents of the project. In Morellec and Schürhoff (2011) it is the firm, while it is the principal in Grenadier and Wang (2005).
The literature above focusses on single stage real options models, while my aim is to develop and analyse a sequential real options model under asymmetric information. Pindyck and Dixit (1994) provide an example of a sequential real options model and show that such a sequential option is similar to a financial compound option.6 When the payoff of the first option is only composed of a second option, optimality implies that both the first and second investment options should be exercised simultaneously, reducing the sequential decision to a single execution decision. The feature that the sequential investment reduces to a single investment decision is not convenient as the aim of this paper is to analyse asymmetric information in a sequential model. In section (5) of this thesis, I include a payoff component in the first option and show what assumptions should be made in order to create a model that is sequential and does not reduce to a single decision model. An example of a sequential real options model is Siddiqui and Fleten (2010). Their analysis concerns a firm that has the possibility to invest in competing energy technologies: an existing renewable energy technology or an unconventional energy technology. The development of the unconventional technology is a two-staged process composed of a commercialisation phase and full deployment.
The publications discussed above deal with sequential investments, although not under asymmetric information. Baron and Besanko (1984) model the continuing relationship between a regulatory authority and a regulated firm under asymmetric information, although not in a real options context. The firm submits privately observable information about its marginal costs of each period to the regulator, which the regulator uses to set its policy in terms of an output price for the firm’s product. As the firm privately observes its marginal costs, the firm has the opportunity to misrepresent information and will do so when this is more beneficial to the firm. This misrepresentation is not necessarily the same as fraud, but could result from the firm’s discretion in choosing a measurement mechanism or dataset to base its marginal cost estimates on. Baron and Besanko (1984) analyse the contracts the regulator can offer and commit to ex-ante for more periods, while ensuring that the firm truthfully reveals its marginal costs. These contracts are analysed under different assumptions
8
about the dependency (correlation) between the marginal costs of each period. The two boundary cases they discuss are the case in which the marginal costs for the first and second period are determined independently (no correlation), and the case in which the two costs are perfectly correlated. When the marginal costs of the first and second period are unrelated, the firm has better information about the first period, but has no better information about the second period than the regulator, since the private information on the marginal costs of the first period provides no information to the firm about the marginal costs of the second period. This means that there is no asymmetric information ex-ante (prior to the first period) between the firm and the regulator about the second period. The regulator’s optimal policy is to follow the single period Baron-Myerson policy in the first period.7 The Baron-Myerson policy is a policy in which the regulator maximizes a social welfare function under asymmetric information between itself and the regulated firm. The social welfare function is composed of consumer surplus and a value component for the regulated firm (producer surplus). Since there is asymmetric information between firm and regulator, the regulator has to give up some rents to create an incentive compatible contract to induce a truthful revelation of the marginal costs in the first period. For the second period, the regulator can set the price in the second period at the first-best benchmark price (price equals marginal costs) like there is no asymmetric information between the firm and the regulator. This is a striking result as it means that, although the firm receives private information about the second period marginal costs in the future, the firm is not able to materialize on its informational advantage. The regulator is able to set such a policy as, ex-ante, firm and regulator have the same information on the prospects of the second investment, and the regulator is able to transfer rents without any costs between the first and second period to the firm. When the marginal costs in the first and second period are equal, the two-period model reduces to a static Baron-Myerson model where the regulator needs to take into account that, ex-ante, the firm has better information about the marginal costs of the second period. Although the model of Baron and Besanko (1984) is not a real options model and does not concern investment timing, it proves to be useful when designing a sequential real options model as I do in this paper.
9 3. The basic real options model
In this section, I discuss and solve a basic real options model. In contrast to the widely used NPV rule, the real options approach takes into account the timing of the investment. This is in line with the notion of McDonald and Siegel (1986) that even when the investment is irreversible, the decision to defer the investment is reversible, giving the investor optionality in investment timing. Therefore, the value of the option to invest should not be calculated on a now-or-never basis, but should be compared with the value of investing at any other moment in the future. The model in this section is similar to the basic model developed in Dixit and Pindyck (1994).8 Although the real option in this section bears similarities with financial American call options, there are fundamental differences which will be discussed in the following subsections.
3.1 The model
In the basic real options model, an investor has a perpetual investment opportunity with a payoff upon exercise equal to 𝑅 − 𝐼, where 𝐼 is the investment cost, and 𝑅 is a lump sum payment which evolves according to a geometric Brownian motion:
𝑑𝑅 = 𝛼𝑅𝑑𝑡 + 𝜎𝑅𝑑𝑧. (1)
In equation (1), 𝑑𝑧 is the increment of a Wiener process, 𝑑𝑧 = 𝜖√𝑑𝑡 with 𝜖~𝑁(0,1), 𝛼 is the drift rate, and 𝜎 is a constant volatility factor, time is continuous. Furthermore, the investor is risk-neutral, meaning the investor does not require an increase in expected return for an increase in risk.9
The investor’s goal is to maximize the value of the investment option with respect to the investment threshold. This means that the investor wants to exercise at an investment threshold at which it is more valuable to exercise the option than to wait any longer. This means that the investor compares the values investing at any moment in time with each other, which is similar to the approach suggested by McDonald and Siegel (1986). The optimal investment threshold is denoted by 𝑅𝑇, and is the solution to the following maximization
problem:
𝐹(𝑅𝑇) = 𝑚𝑎𝑥 𝐸[𝑅𝑇− 𝜃 − 𝐼)𝑒−𝑟𝑇. (2)
8
See Dixit and Pindyck (1994) chapter 5.
10
The left-hand side of equation (2) is the option’s value as a function of the underlying project value, 𝑅. On the right-hand side (RHS), 𝐸 denotes the expectation, 𝑇 is the time at which option to investment is exercised, and 𝑟 is the discount rate. As the investor is risk neutral, discount rate 𝑟 equals the risk-free interest rate. Furthermore, it is important that 𝑟 > 𝛼. Otherwise, it would always be more valuable to postpone the investment than to invest immediately, as the expected growth in payoff 𝑅 would be higher than the discount rate. As a result, the option would never be exercised.
Up until now, this setting is similar to an American call option on a stock that does not pay dividends. Although the payoff structure of a real option is similar to a financial option, there are fundamental differences, which are either the result of the underlying assumptions, or the result of the nature of the underlying of the option: a stock versus a real investment project. Firstly, in contrast to a plain vanilla American call option, the real option is assumed to be a perpetual option, i.e. an option that does not mature. Therefore, the value of the real option is not affected by the passage of time itself and is, consequently, not a function of time to maturity. Secondly, real options are exercised, which opposes the logic that American call options, in absence of dividend payments, should never be exercised early but be traded instead. The two assumptions that ensure the exercise of real options is the non-tradability of real options, and 𝑟 > 𝛼. Both the perpetual lifespan and non-tradability of the real option are assumptions that to a certain extend stem from the underlying of the option: a real investment project. Although both assumptions are limiting an may fail to hold in practice, they could be reasonable in certain real investment cases, and they simplify the process of finding a solution for the optimal investment threshold. The next subsection provides a solution to the basic real options model.
3.2 Dynamic programming solution
This section discusses the dynamic programming approach in continuous time10 to find a solution for the optimal investment threshold. Although I do not discuss the contingent claim analysis in this paper, contingent claim analysis is another approach to find an optimal solution for the investment threshold.11
10 See Dixit and Pindyck (1994) chapters 4.1 and 5.2. 11
11
Dynamic programming, in the context the investment timing of a real option, is basically a systematic approach to compare the value of investing immediately with the value of investing at a moment in the future. In continuous time, the time interval over which this comparison is made is infinitesimally small. As the option is perpetual (infinite time horizon), the value function of the option is not a function of time itself anymore, meaning that the option’s value is only a function of the underlying. Both assumptions, infinite time horizon and continuous time, are very convenient to find a solution.
Let 𝐹(𝑅) denote the value of the investment option with underlying value 𝑅. This investment option’s value, in absence of early cash flows, can be expressed as:
𝐹(𝑅) = 𝐸[𝑒−𝑟𝑑𝑡𝐹(𝑅 + 𝑑𝑅)], (3)
which is known as the Bellman equation. This Bellman equation states that the value of the option equals the discounted expected future value of the option. From Itô’s lemma in combination with 𝑅 evolving according to geometric Brownian motion (1), we know that the option’s value function, 𝐹(𝑅), must solve:
1 2𝜎
2𝑉2𝐹′′(𝑅) + 𝛼𝑉𝐹′(𝑅) − 𝑟𝐹(𝑅) = 0, (4)
known as the Hamilton-Jacobi-Bellman (HJB) equation12, where 𝐹′(𝑅) and 𝐹′′(𝑅) denote the
first- and second-order derivative of 𝐹(𝑅) with respect to 𝑅, respectively. The general solution to HJB equation (4) takes the form:
𝐹(𝑅) = 𝐴𝑅𝛽1+ 𝐵𝑅𝛽2, (5)
where 𝛽1 and 𝛽2 are the positive and negative solution to quadratic
𝑟 = 1
2𝜎
2𝛽(𝛽 − 1) + 𝛼𝛽, (6)
respectively.
A viable value function, 𝐹(𝑅), should satisfy three conditions. The first condition is:
𝐹(0) = 0. (7)
This condition follows from geometric Brownian motion (1), according to which 𝑅 evolves. When the value of lump sum payment 𝑅 is zero, it remains zero, meaning that the option’s value should be zero. Component 𝑅𝛽2 of option value function (5) violates this condition as
𝛽2 is negative. Therefore, constant 𝐵 should be equal to zero, resulting in a viable solution to
the option value:
𝐹(𝑅) = 𝐴𝑅𝛽, (8)
12 See appendix A.1 for the derivation of the HJB equation. Furthermore, note that this equation is
12
where 𝛽 corresponds to 𝛽1 (I use 𝛽 throughout the rest of this paper to refer to the positive
value for 𝛽). The positive solution to 𝛽 can be obtained by applying the quadratic formula to equation (6), and is:
𝛽 =1 2− 𝛼 𝜎2+ √( 𝛼 𝜎2− 1 2) 2+2𝑟 𝜎2 > 1. (9) 13
The second condition that must be satisfied is the value-matching condition. This condition states that at optimal investment threshold, 𝑅̂, the value of the option equals the value of investing immediately:
𝐹(𝑅̂) = 𝐴𝑅̂𝛽 = 𝑅̂ − 𝐼. (10)
The third condition is the smooth-pasting condition, ensuring that the value functions of the option and immediate investment fit smoothly at 𝑅̂, i.e. the first-order derivatives of both value functions are equal at 𝑅̂:
𝐹′(𝑅̂) = 𝛽𝐴𝑅̂𝛽−1 = 1. (11)
Solving the system of equations composed of the value-matching condition and the smooth-pasting condition for the optimal investment threshold 𝑅̂ yields:
𝑅̂ = 𝛽
𝛽−1𝐼. (12)
Besides solving the value-matching and smooth-pasting conditions, there is another approach to find 𝑅̂. This approach does not require a smooth-pasting condition and is based on rewriting and maximizing the value-matching condition. Firstly, rewriting value-matching condition (10) and solving for constant 𝐴 yields:
𝐴 = (𝑅̂ − 𝐼)/𝑅̂𝛽. (13)
Substituting 𝐴 into value-function (8) results in the value function: 𝐹(𝑅) = (𝑅̂ − 𝐼) (𝑅
𝑅̂) 𝛽
. (14)
The first term on the RHS of equation (14) is the payoff from the option at the investment threshold. The second term is the discount factor. This discount factor, 𝐷(0, 𝑅∗) is the
solution to equation (4) with boundary conditions:
𝐷(𝑅̂, 𝑅̂) = 𝐴𝑅̂𝛽 = 1, (15)
𝐷(0, 𝑅̂) = 𝐴0 = 0. (16)
The solution to this system of equations is: 𝐷(𝑅, 𝑅̂) = (𝑅
𝑅̂) 𝛽
. (17)
13 Note 𝛽 > 1 is only valid when 𝑟 > 𝛼, which, is assumed to hold throughout the models presented in
13
Note that discount factor 𝐷 is the expected discount factor, 𝐷 = 𝐸(𝑒𝑟𝑇), where 𝐸 denotes the expectation and 𝑇 is the timing of the cash flows for which we want to know the discount factor. Maximizing value function (14) with respect to 𝑅̂ yields14
: 𝑅̂ = 𝛽
𝛽−1𝐼, (18)
which is identical to result (12) for 𝑅̂ obtained from solving the value-matching and smooth-pasting condition. This value maximization approach is a useful way to find the optimal investment threshold when, due to distortions in the model, it is not possible to satisfy the smooth-pasting condition.
The total value function of the investment is: 𝐹(𝑅) = {(𝑅̂ − 𝐼) ( 𝑅 𝑅̂) 𝛽 , 𝑅 < 𝑅̂ 𝑅 − 𝐼, 𝑅 ≥ 𝑅̂ . (19)
In comparison, when applying the traditional NPV rule, the investment would be made when 𝑅 = 𝐼, which is clearly a lower threshold than 𝑅̂ derived from the real options approach. The difference between the NPV rule and the real options approach is represented by the wedge
𝛽
𝛽−1. Additionally, the real options approach also provides information on how volatility
affects the optimal investment threshold and the option value. From equation (9), we learn that 𝛽 is negatively related to volatility 𝜎. From equation (18), we learn that when 𝛽 decreases, the optimal investment threshold 𝑅̂ increases. Therefore, equations (9) and (18) together imply that an increase in volatility 𝜎 results in an increase of the optimal investment threshold 𝑅̂. The relationship between the investment threshold under the real options approach and under the NPV rule can be explained as follows. When volatility approaches zero, beta approaches infinity. In that case, the optimal investment threshold under the real options approach approaches the investment threshold of the NPV-rule. Recall that this conclusion is similar to the conclusion by McDonald and Siegel (1986) that the NPV rule is only valid in absence of variance in the payoffs. Analogously to financial options, volatility is positively related to the value of the option, 𝐹(𝑅). In summary, volatility has two effects on the real option. Firstly, the higher the volatility, the higher the investment threshold. Secondly, the higher volatility, the higher the option value. As mentioned by Dixit and Pindyck (1994), volatility increases the option value, but reduces the amount of actual investment.
14
4. Single investment model under asymmetric information
The model discussed in the previous section is limited as it does not take into account the separation of ownership and control over an investment, and the problems arising of such a separation. Therefore, this section elaborates on the basic real options model by introducing a principal-agent setting, in which the principal, owning the option to invest, hires an agent to manage and execute the investment. This separation of ownership and control of the option is likely to introduce asymmetric information between the principal and the agent about the prospects of the investment. This asymmetric information could entice the agent to reveal untruthful information and divert cash flows to himself, at the expense of the principal.
In the principal-agent model in this section, the addition of privately observable cost component 𝜃 introduces an asymmetric information problem between principal and owner, as only the agent observes 𝜃. Cost component 𝜃 can take two values, high or low, where 𝜃ℎ > 𝜃𝑙. The probability of receiving a low cost component, 𝜃𝑙, is 𝑞, and the probability of
receiving a high cost component, 𝜃ℎ, is 1 − 𝑞. Although the principal does not observe 𝜃, he knows which values of 𝜃 are possible and the accompanying probabilities. The resulting model is similar to Grenadier and Wang (2005), except that Grenadier and Wang introduce a payoff component instead of a cost component, and they include moral hazard in their model, while moral hazard is not discussed in the models presented in my paper.
The next subsection provides a first-best benchmark solution to the model where 𝜃 is introduced. This first-best benchmark is the solution to the model including 𝜃, but in absence of asymmetric information about 𝜃. Subsection 4.2 discusses and solves the model in presence of asymmetric information and shows how contracting can mitigate the asymmetric information problem, although at some efficiency loss. Recall from the introduction that I solve the models from a principal’s perspective, meaning that the principal’s value function is maximized.
4.1 First-best benchmark solution
15
himself. Both interpretations of the first-best benchmark result in the same outcome. The principal’s payoff from the option to invest is:
𝑅 − 𝜃𝑘− 𝐼 − 𝑤𝑘, for 𝑘 = 𝑙, ℎ. (20)
Here, 𝑤𝑘 is the wage that the principal offers to the agent upon exercise. However, as the
principal knows 𝜃𝑘, he can set 𝑤𝑘 equal to zero as there is no possibility for the agent to lie to the principal and divert cash flows. The resulting option value function, which can be derived as discussed in section 3, is:
𝐹(𝑅, 𝜃𝑘) = (𝑅̂𝑘− 𝜃𝑘− 𝐼) (𝑅
𝑅̂𝑘)
𝛽
, for 𝑘 = 𝑙, ℎ. (21)
Maximizing this function results in the first-best investment thresholds: 𝑅̂𝑘 =
𝛽
𝛽−1(𝐼 + 𝜃𝑘), for 𝑘 = 𝑙, ℎ. (22)
This result shows that in absence of asymmetric information, the cost component increases the investment threshold without any other distortions to investment timing. The investment threshold 𝑅̂ℎ is higher than 𝑅̂𝑙 as 𝜃ℎ > 𝜃𝑙. This means that the investor invests later when he observes 𝜃ℎ than when he observes 𝜃𝑙. The next subsection discusses the same model under
asymmetric information and shows how proper contract design can induce the agent to truthfully reveal 𝜃.
4.2 Asymmetric information solution
Under asymmetric information, the agent privately observes cost component 𝜃 ex-ante, while the principal is not able to observe 𝜃. Recall from the introduction of section 4 that there are two possible values for cost component 𝜃, high, 𝜃ℎ, or low, 𝜃𝑙, with Pr(𝜃 = 𝜃𝑙) = 𝑞, and Pr(𝜃 = 𝜃ℎ) = 1 − 𝑞. As mentioned, the resulting problem of the information asymmetry is the possibility that the agent observing a low cost component reports high costs and diverts cash flows equal to ∆𝜃 = 𝜃ℎ− 𝜃𝑙 to himself. The principal can mitigate the asymmetric
16
𝑤(𝑅̂), (23)
where 𝑅̂ is the value of the lump sum payment when the agent exercises the option to invest. The principal specifies two wage/investment threshold combinations from which the agent can choose: exercise at 𝑅̂𝑙 and receive 𝑤𝑙, or exercise at 𝑅̂ℎ and receive 𝑤ℎ. Obviously, the
agent observing 𝜃𝑙 should be induced to exercise at 𝑅̂𝑙, and when he observes 𝜃ℎ he should exercise at 𝑅̂ℎ.
Similar to section 3, finding the optimal investment thresholds requires maximizing a value function. As the principal offers the contract to the agent, this setting focusses on maximizing the principal’s value function. The value function of the principal is:
𝑞(𝑅̂𝑙− 𝜃𝑙− 𝐼 − 𝑤𝑙) (𝑅 𝑅̂𝑙) 𝛽 + (1 − 𝑞)(𝑅̂ℎ− 𝜃ℎ− 𝐼 − 𝑤ℎ) (𝑅 𝑅̂ℎ) 𝛽 . (24)
This value function is the expected present value of the project to the principal, and shows that the project is composed of probability 𝑞 and 1 − 𝑞 of owning an option with a low or high cost component 𝜃, respectively. The principal wants to maximize this value function with respect to 𝑅̂𝑙, 𝑅̂ℎ, 𝑤𝑙, 𝑤ℎ. This maximization problem is subject to a number of constraints, including constraints to ensure that the agent truthfully reveals 𝜃 and executes the investment at a threshold that is optimal for the principal.
The first constraint is that the principal should at least break even in expectation, else, the principal would not invest. This constraint is called the principal’s participation constraint (PC). However, as the maximization problem concerns the value function of the principal, the principal’s PC is unlikely to bind. The other constraints concern the agent. A participation constraint for the agent ensures the wage of the agent cannot be negative:
𝑤𝑘 ≥ 0 for 𝑘 = 𝑙, ℎ. (PC4.1)
If this constraint is not satisfied for an agent (observing 𝜃ℎ or 𝜃𝑙), the agent would not be
willing to manage the investment since he would lose money in expectation.
The other constraints in the model are specific combinations of the wages and investment thresholds that should induce the agent to truthfully reveal the privately observed cost component 𝜃. These constraints are called incentive compatibility constraints (ICCs). The agent observing 𝜃𝑙 has the opportunity to report 𝜃ℎ, exercise at 𝑅̂ℎ, and divert ∆𝜃 = 𝜃ℎ− 𝜃𝑙> 0 to himself. To prevent this scenario, the wage/investment threshold combination offered to an agent reporting 𝜃𝑙 should satisfy the following ICC:
17
Similarly, an agent observing 𝜃ℎ could mimic an agent observing 𝜃𝑙 by giving up ∆𝜃 and
receive 𝑤𝑙. This means that he effectively pays ∆𝜃 to convince the principal that he faces a low cost project. The wage/investment threshold combination for the agent observing 𝜃ℎ should satisfy: 𝑤ℎ( 𝑅 𝑅̂ℎ) 𝛽 ≥ (𝑤𝑙− ∆𝜃) ( 𝑅 𝑅̂𝑙) 𝛽 . (ICC4.2)
Optimization of the principal’s value function (24) implies that both wages should be as low as possible. Constraint (ICC4.1) shows that any increase in 𝑤ℎ should be accompanied by an increase in 𝑤𝑙. In combination with (PC4.1), this implies it is optimal to set 𝑤ℎ = 0,
and make sure (ICC4.1) binds. This is possible as long as 𝑤𝑙≤ ∆𝜃, which follows from (ICC4.2). However, as 𝑅̂𝑙 is expected to be lower than 𝑅̂ℎ, it is unlikely that (ICC4.2) binds. The minimal value for 𝑤𝑙, is the value for which incentive compatibility constraint (ICC4.1)
binds. The resulting principal’s value function is:
𝑚𝑎𝑥 𝑤𝑙,𝑅̂𝑙,𝑅̂ℎ𝑞(𝑅̂𝑙− 𝜃𝑙− 𝐼 − 𝑤𝑙) ( 𝑅 𝑅̂𝑙) 𝛽 + (1 − 𝑞)(𝑅̂ℎ− 𝜃ℎ − 𝐼) (𝑅 𝑅̂ℎ) 𝛽 , (25) s.t. 𝑤𝑙(𝑅0 𝑅̂𝑙) 𝛽 = ∆𝜃 (𝑅 𝑅̂ℎ) 𝛽 . (26)
Plugging (26) into value function (25) yields:
𝑚𝑎𝑥 𝑅̂𝑙,𝑅̂ℎ𝑞(𝑅̂𝑙− 𝜃𝑙− 𝐼) ( 𝑅0 𝑅̂𝑙) 𝛽 + (1 − 𝑞) (𝑅̂ℎ− 𝜃ℎ− 𝐼 − 𝑞 1−𝑞∆𝜃) ( 𝑅 𝑅̂ℎ) 𝛽 . (27)
The solutions to investment thresholds 𝑅̂𝑙 and 𝑅̂ℎ are:
𝑅̂𝑙 = 𝛽 𝛽−1(𝐼 + 𝜃𝑙), (28) 𝑅̂ℎ = 𝛽 𝛽−1(𝐼 + 𝜃ℎ+ 𝑞 1−𝑞∆𝜃). (29)
As 𝑅̂𝑙< 𝑅̂ℎ we learn from constraint (26) that 𝑤𝑙 < ∆𝜃 and therefore satisfies (ICC4.2). From (28), we learn that the optimal investment threshold for an agent who reports 𝜃𝑙,
𝑅̂𝑙, equals the first-best benchmark investment threshold (22). In contrast, when the agent reports 𝜃ℎ, the investment threshold (29) is higher under asymmetric information than the first-best benchmark (22). This means that, compared to the first-best benchmark, the investment is delayed for an agent revealing 𝜃ℎ. By delaying the investment when an agent
reveals 𝜃ℎ, it is less likely that an agent observing 𝜃𝑙 mimics an agent observing 𝜃ℎ, as he must wait longer before he has the opportunity to divert the cash flows. As a result, the delay in investment when an agent reveals 𝜃ℎ reduces the asymmetric information problem, though at some loss in efficiency compared to the first-best benchmark. This efficiency loss in investment timing presents itself by the 𝑞
18
𝑅̂ℎ in equation (29) compared to the first-best benchmark equation (22). Furthermore, the principal needs to set a positive wage for the agent observing 𝜃𝑙, while this was not the case in
absence of asymmetric information. Table 1 provides an overview of the results of the first-best benchmark model and the single investment asymmetric information model.
Table 1
Comparison of the single stage investment models
First-best benchmark Asymmetric information
Wage/ threshold Result Wage/ threshold Result 𝑅̂𝑙 𝛽 𝛽 − 1(𝐼 + 𝜃𝑙) 𝑅̂𝑙 𝛽 𝛽 − 1(𝐼 + 𝜃𝑙) 𝑅̂ℎ 𝛽 𝛽 − 1(𝐼 + 𝜃ℎ) 𝑅̂ℎ 𝛽 𝛽 − 1(𝐼 + 𝜃ℎ+ 𝑞 1 − 𝑞∆𝜃) 𝑤𝑙( R 𝑅̂𝑙 ) 𝛽 Zero 𝑤𝑙(R 𝑅̂𝑙 ) 𝛽 ∆𝜃 (𝑅 𝑅̂ℎ ) 𝛽 𝑤ℎ Zero 𝑤ℎ Zero
5. Sequential real option models
19
investment project, or he can offer a contract prior to each investment separately resulting in two contracts. For the relationship between 𝜃1 and 𝜃2, I analyse the two most extreme cases: (1) 𝜃1 and 𝜃2 are completely independent, (2) 𝜃2 = 𝜃1. A result of these assumptions is that
the contracts offered to the agent differ in each case regarding the investment timing and the wages offered. This kind of analysis is similar to Baron and Besanko (1984), although their analysis is not conducted in a real options context and they focus on a relationship between regulator and firm while I focus on a principal-agent (owner-manager) relationship.
5.1 First-best benchmark solution (model 1)
The first-best benchmark solution is the solution to the sequential real options model in absence of asymmetric information. In this section, I assume the principal knows the values of 𝜃1 and 𝜃2 ex-ante. The solution of the sequential real options investment model can be
obtained by backward induction, starting at the last sequence of investment project. The payoff from the second option to the principal is:
𝑅̂2,𝑖− 𝜃2,𝑖− 𝐼2− 𝑤2,𝑖, for 𝑖 = 𝑙, ℎ, (30) when exercised at 𝑅̂2,𝑖. The payoff from the first option to the principal is:
𝑅̂1,𝑘− 𝜃1,𝑘− 𝐼2 − 𝑤1,𝑘+ 𝐹2(𝑅̂1,𝑘), for 𝑘 = 𝑙, ℎ, (31)
where 𝐹2(𝑅̂1,𝑘) is the value of the second investment option at 𝑅̂1,𝑘. Similarly to previous
sections, 𝑅1 and 𝑅2 are lump sum payments evolving according to a geometric Brownian
motion. Following the steps discussed in subsection 3.2 and section 4, the expected value (ex-ante) of the second investment option to the principal in case of a low cost component, 𝜃2,𝑙, is:
𝐹2(𝑅) = (𝑅̂2,𝑙− 𝜃2,𝑙− 𝐼2− 𝑤2,𝑙) ( 𝑅
𝑅̂2,𝑙)
𝛽
. (32)
In case of a high cost component for the second investment option, 𝜃2,ℎ, the principal’s value function is: 𝐹2(𝑅) = (𝑅̂2,ℎ− 𝜃2,ℎ− 𝐼2− 𝑤2,ℎ) ( 𝑅 𝑅̂2,ℎ) 𝛽 , (33)
Similarly, the value functions for the first option for 𝜃1,𝑙 or 𝜃1,ℎ are respectively:
20
Note that there are no probabilities involved as I assume the principal knows the value of both 𝜃s ex-ante. As there is no asymmetric information between principal and agent in this section, wages can be set equal to zero. The optimal solutions are obtained by maximizing value functions (32) up to and including (35) with wages set equal to zero, with respect to 𝑅̂1,𝑙, 𝑅̂1,ℎ, 𝑅̂2,𝑙 and 𝑅̂2,ℎ. This results in the following optimal investment thresholds:
𝑅̂1,𝑘 = 𝛽 𝛽−1(𝐼1+ 𝜃1,𝑘), for 𝑘 = 𝑙, ℎ, (36) 𝑅̂2,𝑙 = 𝛽 𝛽−1(𝐼2+ 𝜃2,𝑙), (37) 𝑅̂2,ℎ = 𝛽 𝛽−1(𝐼2+ 𝜃2,ℎ). (38)
As there is no asymmetric information present, there are no distortions in the investment thresholds compared to the case in which each investment is made separately in absence of asymmetric information. To ensure the investments are sequential, the investment thresholds should satisfy 𝑅̂1,𝑙 < 𝑅̂2,𝑖 and 𝑅̂1,ℎ < 𝑅̂2,𝑖, for 𝑖 = 𝑙, ℎ,. These conditions are represented by:
(𝐼1+ 𝜃1,𝑙) < (𝐼2 + 𝜃2,𝑖), (39)
(𝐼1+ 𝜃1,ℎ) < (𝐼2+ 𝜃2,𝑖), (40)
for 𝑖 = 𝑙, ℎ. If the optimal investment thresholds do not satisfy condition (39) or (40), the model reduces to a single investment model as the principal would exercise the second option instantaneously after the first option. Such a reduced model would be similar to Dixit and Pindyck (1994)15, where the exercise of the first option does not result in a lump sum payment, and where the model collapses to a single investment decision as well. Therefore, the sequential models in this subsection and in subsection 5.2 are only sequential when both the first and the second option payoff a lump sum, and when the optimal threshold of the second option is larger than the first option’s optimal investment threshold.
Furthermore, if the second option would be exercised instantaneously after the first option, both options would pay off the same lump sum 𝑅̂. However, the second option has higher investment costs, meaning that the return on the second option would be lower than on the first option, which implies there are decreasing returns to scale. This means that although the payment would double and become 2𝑅̂, the investment costs would more than double. The real options approach learns us that it is not optimal to exercise the second investment instantaneously after the first option. It is optimal to wait with exercising the second investment option until the value of the lump sum payment 𝑅 reaches 𝑅̂2,𝑘 for 𝑘 = 𝑙, ℎ.
15
21
Therefore, applying the real options approach to two subsequent investments with their own payoff, of which one has higher investment costs, results in a sequential investment model.
5.2 Sequential real options models under asymmetric information
The sequential investment model under asymmetric information is similar to the model without asymmetric information, except that the principal cannot observe 𝜃. In contrast to the principal, the agent is able to observe 𝜃 prior to each investment. This subsection analyses contracts in the sequential model under asymmetric information for different assumptions about the nature of the asymmetric information. Firstly, in section 5.2.1, the principal and the agent enter into a contract for each investment option separately under the assumption that the agent learns the privately observable cost component before entering into the contract for each investment. The cost components 𝜃1 and 𝜃2 in subsection 5.2.1 are
assumed to be completely independent. In sections 5.2.2 and 5.2.3, principal and agent enter into a contract for both investment options ex-ante (meaning prior to the first investment), where the agent is assumed to have ex-ante knowledge about 𝜃1. I show that the agent’s participation constraint can sometimes be relaxed by offering a contract for both investment options ex-ante. Secondly, the model is analysed for different assumptions about the relationship between privately observable cost components of each investment: 𝜃1 and 𝜃2. The first assumption that 𝜃1 and 𝜃2 are completely independent is discussed in subsection
5.2.2. The other assumption is 𝜃1 = 𝜃2 and is analysed in subsection 5.2.3. These two assumptions form the boundaries of the possible assumptions of about the relationship between 𝜃1 and 𝜃2. Throughout this subsection (5.2), the probabilities of receiving 𝜃𝑙 or 𝜃ℎ are Pr(𝜃 = 𝜃𝑙) = 𝑞, and Pr(𝜃 = 𝜃ℎ) = 1 − 𝑞, for each investment. Furthermore, although the principal is not able to observe 𝜃1 and 𝜃2, he is aware of the two possible values and the accompanying probabilities.
5.2.1 Separate contracting, 𝜃1 and 𝜃2 are completely independent (model 2)
22
The principal’s value function for both investment options combined is: 𝐹(𝑅) = 𝑞(𝑅̂1,𝑙− 𝜃1,𝑙− 𝐼1− 𝑤1,𝑙) ( 𝑅 𝑅̂1,𝑙) 𝛽 + (1 − 𝑞)(𝑅̂1,ℎ− 𝜃1,ℎ− 𝐼1− 𝑤1,ℎ) ( 𝑅 𝑅̂1,ℎ) 𝛽 + 𝑞(𝑅̂2,𝑙− 𝜃2,𝑙− 𝐼2− 𝑤2,𝑙) ( 𝑅 𝑅̂2,𝑙) 𝛽 + (1 − 𝑞)(𝑅̂2,ℎ− 𝜃2,ℎ− 𝐼2− 𝑤2,ℎ) ( 𝑅 𝑅̂2,ℎ) 𝛽 . (41)16 This function should be maximized subject to the following constraints, which should ensure the agent truthfully reveals 𝜃 for both periods and is willing to enter the contract:
𝑤2,𝑙( 𝑅 𝑅̂2,𝑙) 𝛽 ≥ (∆𝜃2+ 𝑤2,ℎ) ( 𝑅 𝑅̂2,ℎ) 𝛽 , (ICC5.1) 𝑤2,ℎ( 𝑅 𝑅̂2,ℎ) 𝛽 ≥ (𝑤2,𝑙− ∆𝜃2) ( 𝑅 𝑅̂2,𝑙) 𝛽 , (ICC5.2) 𝑤1,𝑙( 𝑅 𝑅̂1,𝑙) 𝛽 ≥ (∆𝜃1+ 𝑤1,ℎ) ( 𝑅 𝑅̂1,ℎ) 𝛽 , (ICC5.3) 𝑤1,ℎ( 𝑅 𝑅̂1,ℎ) 𝛽 ≥ (𝑤1,𝑙− ∆𝜃1) ( 𝑅 𝑅̂1,𝑙) 𝛽 , (ICC5.4) 𝑤𝑡,𝑘 ≥ 0, for 𝑡 = 1,2 and 𝑘 = 𝑙, ℎ, (PC5.1)
where ∆𝜃2 = 𝜃2,ℎ− 𝜃2,𝑙, and ∆𝜃1 = 𝜃1,ℎ− 𝜃1,𝑙. (ICC5.1) and (ICC5.2) should induce the agent to report 𝜃2 truthfully, while (ICC5.3) and (ICC5.4) serve the same purpose for the first
investment option. As principal and agent enter into a contract prior to each investment, assuming the agent learns 𝜃 prior to each investment, the agent has better information when entering into both contracts. A rational agent only accepts to manage an investment when the wage is at least zero or higher, this is represented by (PC5.1). (PC5.1) show that for both the first and second investment, wages cannot be negative as the agent would reject a contract with a negative wage.
As it is optimal for the principal to set the wages as low as possible, he can set 𝑤1,ℎ
and 𝑤2,ℎ equal to zero without violating any of the ICCs. The optimal wages 𝑤2,𝑙 and 𝑤1,𝑙 are
the wages for which (ICC5.1) and (ICC5.3) bind given 𝑤1,ℎ = 𝑤2,ℎ = 0:
𝑤2,𝑙( 𝑅 𝑅̂2,𝑙) 𝛽 = ∆𝜃2( 𝑅 𝑅̂2,ℎ) 𝛽 , (42) 𝑤1,𝑙( 𝑅 𝑅̂1,𝑙) 𝛽 = ∆𝜃1( 𝑅 𝑅̂1,ℎ) 𝛽 . (43)
Plugging (42) and (43) into value function (41) and maximizing (41) with respect to investment thresholds 𝑅̂1,𝑙, 𝑅̂1,ℎ, 𝑅̂2,𝑙 and 𝑅̂2,ℎ yields:
𝑅̂1,𝑙 = 𝛽
𝛽−1(𝐼1+ 𝜃1,𝑙), (44)
23 𝑅̂1,ℎ = 𝛽 𝛽−1(𝐼1+ 𝜃1,ℎ+ 𝑞 (1−𝑞)∆𝜃1), (45) 𝑅̂2,𝑙 = 𝛽 𝛽−1(𝐼2+ 𝜃2,𝑙), (46) 𝑅̂2,ℎ = 𝛽 𝛽−1(𝐼2+ 𝜃2,ℎ+ 𝑞 (1−𝑞)∆𝜃2). (47)
The resulting thresholds are the same thresholds described in section 4.2 as this case is similar. Furthermore, as 𝑅̂1,𝑙< 𝑅̂1,ℎ and 𝑅̂2,𝑙 < 𝑅̂2,ℎ, these results also satisfy (ICC5.2) and (ICC5.4), making the solution viable.
Although the thresholds are similar to the thresholds discussed in section 4.2, there are implications for the sequential model under asymmetric information compared to the first-best benchmark, as the investment thresholds 𝑅̂1,ℎ and 𝑅̂2,ℎ are higher than the first-best benchmark investment thresholds from subsection 5.1. Again 𝑅̂2 should be higher than 𝑅̂1 to ensure the model is sequential. In case an agent observes 𝑅̂1,ℎ, the second investment threshold under asymmetric information should be strictly higher than without asymmetric information. Under the first-best benchmark, when the agent reports 𝑅̂1,ℎ, the investments are sequential when:
𝑅̂1,ℎ = 𝛽
𝛽−1(𝐼1+ 𝜃1,ℎ) < 𝑅̂2. (48)
Under asymmetric information the same condition is: 𝑅̂1,ℎ = 𝛽
𝛽−1(𝐼1 + 𝜃1,ℎ+ 𝑞
(1−𝑞)∆𝜃1) < 𝑅̂2, (49)
which is strictly stronger than under the first-best benchmark. Recall that when (49) fails to hold, the sequential model under asymmetric information reduces to a single decision model where the second option is exercised instantaneously after the first option. For an agent observing 𝜃𝑙 ex-ante, the condition is:
𝑅̂1,𝑙 = 𝛽
𝛽−1(𝐼1+ 𝜃1,𝑙) < 𝑅̂2, (50)
24
𝜃1,ℎ and for an agent reporting 𝜃2,ℎ. This delay is represented by the term 𝑞
(1−𝑞)∆𝜃1 for the
first investment and 𝑞
(1−𝑞)∆𝜃2 for the second investment in equations (45) and (47)
respectively. Furthermore, (42) and (43) show that the wages for an agent reporting 𝜃1,𝑙 and/or 𝜃2,𝑙 are positive, while their wages would be zero in absence of asymmetric information.
5.2.2 Contracting ex-ante, 𝜃1 and 𝜃2 are completely independent (model 3)
In this subsection, the principal offers, ex-ante, a single contract to the agent in which both investment options are included. The agent knows 𝜃1 prior to the first investment, and 𝜃2 prior to the second investment. However, since 𝜃1 and 𝜃2 are completely independent, the
agent has, ex-ante, no better information about 𝜃2 than the principal. This means that upon entering the contract, there is asymmetric information present between principal and agent for the first investment, but there is no asymmetric information for the second investment. Although the principal’s value function is the same as equation (41) from the previous subsection, the intuition and derivation are different and shown in appendix B.2. The value function of the principal is:
𝐹1(𝑅) = 𝑞(𝑅̂1,𝑙− 𝜃1,𝑙− 𝐼1− 𝑤1,𝑙) ( 𝑅 𝑅̂1,𝑙) 𝛽 + (1 − 𝑞)(𝑅̂1,ℎ− 𝜃1,ℎ− 𝐼1− 𝑤1,ℎ) ( 𝑅 𝑅̂1,ℎ) 𝛽 + 𝑞(𝑅̂2,𝑙− 𝜃2,𝑙− 𝐼2− 𝑤2,𝑙) ( 𝑅 𝑅̂2,𝑙) 𝛽 + (1 − 𝑞)(𝑅̂2,ℎ− 𝜃2,ℎ− 𝐼2− 𝑤2,ℎ) ( 𝑅 𝑅̂2,ℎ) 𝛽 . (51)
This value function should be maximized subject to the following ICCs to induce a truthful revelation of the cost components:
𝑤2,𝑙( 𝑅 𝑅̂2,𝑙) 𝛽 ≥ (∆𝜃2+ 𝑤2,ℎ) ( 𝑅 𝑅̂2,ℎ) 𝛽 , (ICC5.1) 𝑤2,ℎ( 𝑅 𝑅̂2,ℎ) 𝛽 ≥ (𝑤2,𝑙− ∆𝜃2) ( 𝑅 𝑅̂2,𝑙) 𝛽 , (ICC5.2) 𝑤1,𝑙( 𝑅 𝑅̂1,𝑙) 𝛽 ≥ (∆𝜃1+ 𝑤1,ℎ) ( 𝑅 𝑅̂1,ℎ) 𝛽 , (ICC5.3) 𝑤1,ℎ( 𝑅 𝑅̂1,ℎ) 𝛽 ≥ (𝑤1,𝑙− ∆𝜃1) ( 𝑅 𝑅̂1,𝑙) 𝛽 , (ICC5.4)
25
The difference between this model and the model from subsection 5.2.1 is that upon entering the contract ex-ante, as a result of the two independent 𝜃s, the principal and agent have symmetric information about 𝜃2. They both know that the expected value for 𝜃2 is:
𝐸𝜃2 = 𝑞𝜃2,𝑙+ (1 − 𝑞)𝜃2,ℎ. (52)
This means that the expected present value of the wage to the agent for the second investment project is: 𝐸𝑤2(𝑅 𝑅̂2) 𝛽 = 𝑞𝑤2,𝑙( 𝑅 𝑅̂2,𝑙) 𝛽 + (1 − 𝑞)𝑤2,ℎ( 𝑅 𝑅̂2,ℎ) 𝛽 . (53)
Ex-ante, the agent should break-even in expectation. This results in two PCs, one for an agent observing 𝜃1,𝑙 ex-ante (PC,L), and one for an agent observing 𝜃1,ℎ ex-ante (PC,H):
𝑤1,𝑙( 𝑅 𝑅̂1,𝑙) 𝛽 + 𝑞𝑤2,𝑙( 𝑅 𝑅̂2,𝑙) 𝛽 + (1 − 𝑞)𝑤2,ℎ( 𝑅 𝑅̂2,ℎ) 𝛽 , (PC,L) 𝑤1,ℎ( 𝑅 𝑅̂1,ℎ) 𝛽 + 𝑞𝑤2,𝑙( 𝑅 𝑅̂2,𝑙) 𝛽 + (1 − 𝑞)𝑤2,ℎ( 𝑅 𝑅̂2,ℎ) 𝛽 . (PC,H) In both (PC,L) and (PC,H), the first part is the present value of the agent’s wage for the first investment option. The second two terms together are the expected present value of the second wage. (PC,L) and (PC,H) are only valid when the incentive compatibility constraints are satisfied as the participation constraints do not account for any value the agent could derive from diverting cash flows.
In contrast to separate contracting, this specification allows negative wages and allows the principal to transfer rents to the agent between the first to the second period. Although the principal should ensure (ICC5.1) and (ICC5.2) are satisfied to create a separating equilibrium for the second investment, he can reduce the agents wages in the first period, as long as (ICC5.3) and (ICC5.4) are satisfied, and as long as (PC,H) and (PC,L) are satisfied. Note that this setting assumes there is no limited liability constraint for the agent, meaning that wages are allowed to be negative.
The solution to this model can be obtained in a similar way as the other models are solved. Firstly, the principal must ensure that the contract is incentive compatible for the second option. Furthermore, optimality implies that (ICC5.1) should bind. The value of the wage for an agent reporting 𝜃2,𝑙 is:
𝑤2,𝑙( 𝑅 𝑅̂2,𝑙) 𝛽 = (∆𝜃2+ 𝑤2,ℎ) ( 𝑅 𝑅̂2,ℎ) 𝛽 . (54)
26
Since participation constraint (PC,L) is stronger than (PC,H) 𝑤1,𝑙 should be larger than 𝑤1,ℎ as implied by (ICC5.3). Furthermore it is optimal to set participation constraint (PC,H) equal to zero, meaning that equation (55) should also be equal to zero. From equation (55) we learn that the wage for an agent reporting 𝜃1,ℎ (first investment option) is:
𝑤1,ℎ( 𝑅 𝑅̂1,ℎ) 𝛽 = −𝑤2,ℎ( 𝑅 𝑅̂2,ℎ) 𝛽 − 𝑞∆𝜃2( 𝑅 𝑅̂2,ℎ) 𝛽 . (56)
It is clear that the present value of 𝑤1,ℎ is specified in such a way that it offsets the wages for
the second investment. This means that the principal is able to set, without any cost, two incentive compatible wages in the second period. With equation (56), in combination with optimality implying that (ICC5.3) should bind, we can find the wage for an agent reporting 𝜃1,𝑙 (first investment option):
𝑤1,𝑙( 𝑅 𝑅̂1,𝑙) 𝛽 = −𝑤2,ℎ( 𝑅 𝑅̂2,ℎ) 𝛽 − 𝑞∆𝜃2( 𝑅 𝑅̂2,ℎ) 𝛽 + ∆𝜃1( 𝑅 𝑅̂1,ℎ) 𝛽 . (57)
The final step is to substitute these wages into the principal value function 𝐹1(𝑅), and maximize with respect to the four investment thresholds.17 This results in the following investment thresholds: 𝑅̂1,𝑙 = 𝛽 𝛽−1(𝐼1+ 𝜃1,𝑙). (58) 𝑅̂1,ℎ = 𝛽 𝛽−1(𝐼1+ 𝜃1,ℎ+ 𝑞 1−𝑞∆𝜃1). (59) 𝑅̂2,𝑙 = 𝛽 𝛽−1(𝐼1+ 𝜃1,𝑙). (60) 𝑅̂2,ℎ = 𝛽 𝛽−1(𝐼1+ 𝜃1,ℎ). (61)
Strikingly, the investment thresholds for the first period, (58) and (59), are equal to the thresholds obtained from a single investment option under asymmetric information while the investment thresholds for the second option, (60) and (61), equal the first-best investment thresholds in absence of asymmetric information. The interpretation of these results is that the agent is not able to benefit from the asymmetric information about 𝜃2 as, at the moment of entering into the contract, the agent has not acquired this information yet. The principal can specify a contract that diverts rents to the agent for the second investment at the expense of the agent’s wage in the first period. Hence, although the agent acquires better information about 𝜃2 in the future, he does not benefit from this opportunity when the principal offers a contract ex-ante.
27
The necessary condition to ensure that the first and second investment are sequential are the same as in the previous section. For an agent observing 𝜃1,𝑙 ex-ante, this condition is:
𝑅̂1,𝑙 = 𝛽
𝛽−1(𝐼1+ 𝜃1,𝑙) < 𝑅̂2. (62)
For an agent observing 𝜃1,ℎ ex-ante, the condition is: 𝑅̂1,ℎ = 𝛽
𝛽−1(𝐼1 + 𝜃1,ℎ+ 𝑞
(1−𝑞)∆𝜃1) < 𝑅̂2. (63)
Again, asymmetric information only affects the condition for an agent observing 𝜃1,ℎ ex-ante.
5.2.3 Contracting ex-ante, 𝜃2 = 𝜃1 (model 4)
In this section, the privately observable cost component of the second option equals the component of the first option, 𝜃2 = 𝜃1, and both the principal and the agent are aware of this relationship. A result of the assumption that 𝜃2 = 𝜃1 is that the agent ex-ante has better information about both investment options. Since the principal knows this relationship as well, the principal is able to set the second period wage equal to zero, as long as the agent truthfully reveals 𝜃1. Note that when the agent truthfully reveals 𝜃1 he automatically reveals
𝜃2. Therefore, the model reduces to a model with two incentive compatibility constraints, as opposed to four in the previous subsection. Furthermore, the resulting investment thresholds equal the ones derived in a separate contracting setting described in section 5.2.1.18
Firstly, as long as the principal commits to the contracts that are entered into ex-ante, only two ICCs need to be satisfied:
𝑤1,𝑙( 𝑅 𝑅̂1,𝑙) 𝛽 ≥ (𝑤1,ℎ+ ∆𝜃1) ( 𝑅 𝑅̂1,ℎ) 𝛽 + ∆𝜃2( 𝑅 𝑅̂2,ℎ) 𝛽 , (ICC5.5) 𝑤1,ℎ( 𝑅 𝑅̂1,ℎ) 𝛽 ≥ (𝑤1,𝑙− ∆𝜃1) ( 𝑅 𝑅̂1,𝑙) 𝛽 − ∆𝜃2( 𝑅 𝑅̂2,𝑙) 𝛽 . (ICC5.6)
(ICC5.5) ensures a truthful revelation of 𝜃1 by an agent observing 𝜃1,𝑙 ex-ante, while (ICC5.6) should induce a truthful revelation of 𝜃1 by an agent observing 𝜃1,ℎ ex-ante. Since 𝜃2 = 𝜃1, the principal knows 𝜃2 when the agent reports 𝜃1. Therefore, the second period wage, 𝑤2, is not included in (ICC5.5) and (ICC5.6). However, it is important that the principal compensates the agent for the possibility of diverting cash flows for both investments. This follows from the notion that truthful revelation of 𝜃1 prevents the agent to divert cash flows
18 Baron and Besanko (1984) find a similar result in a model where a firm is regulated by a regulating
28
for both the first and second investment options. Therefore, the principal should compensate the agent accordingly by accounting for ∆𝜃2 in both (ICC5.5) and (ICC5.6). Furthermore, the principal should commit to the contract to be able to enter into an efficient contract ex-ante. When the principal does not commit to the contract, he could force the agent to follow the first-best benchmark for the second investment, which is not favourable to the agent. However, when the agent takes this possibility into account, the agent would require a higher wage for the first investment to offset the reduction in rents during the second investment. As a result, it is not efficient for the principal not to commit to the ex-ante contract.
As the contracts specify only a wage for the first stage, the participation constraints for an agent observing 𝜃1,𝑙 and 𝜃1,ℎ ex-ante are:
𝑤1,𝑘≥ 0, for 𝑘 = 𝑙, ℎ. (PC5.2)
Optimality implies that wages should be as low as possible, meaning that the principal should set 𝑤1,ℎ equal to zero. Although the derivation is different, the principal’s value function is similar to the principal’s value functions (41) and (51) from two previous sections, except the absence of a wage for the second period.19 The principal’s ex-ante value function is:
𝐹1(𝑅) = 𝑞(𝑅̂1,𝑙− 𝜃1,𝑙− 𝐼1− 𝑤1,𝑙) ( 𝑅 𝑅̂1,𝑙) 𝛽 + (1 − 𝑞)(𝑅̂1,ℎ− 𝜃1,ℎ− 𝐼1− 𝑤1,ℎ) ( 𝑅 𝑅̂1,ℎ) 𝛽 + 𝑞(𝑅̂2,𝑙− 𝜃2,𝑙− 𝐼2) ( 𝑅 𝑅̂2,𝑙) 𝛽 + (1 − 𝑞)(𝑅̂2,ℎ− 𝜃2,ℎ− 𝐼2) ( 𝑅 𝑅̂2,ℎ) 𝛽 . (64)
Maximizing the principal’s value function with respect to the investment thresholds, given 𝑤1,ℎ = 0, results in the following solutions to the optimal investment thresholds:
𝑅̂1,𝑙 = 𝛽 𝛽−1(𝐼1+ 𝜃1,𝑙), (65) 𝑅̂1,ℎ = 𝛽 𝛽−1(𝐼1+ 𝜃1,ℎ+ 𝑞 (1−𝑞)∆𝜃1), (66) 𝑅̂2,𝑙 = 𝛽 𝛽−1(𝐼2+ 𝜃2,𝑙), (67) 𝑅̂2,ℎ = 𝛽 𝛽−1(𝐼2+ 𝜃2,ℎ+ 𝑞 (1−𝑞)∆𝜃2). (68)
The agent’s wage when reporting 𝜃1,𝑙 is:
𝑤1,𝑙( 𝑅 𝑅̂1,𝑙) 𝛽 = ∆𝜃1( 𝑅 𝑅̂1,ℎ) 𝛽 + ∆𝜃2( 𝑅 𝑅̂2,ℎ) 𝛽 . (69)
These optimal investment thresholds are similar to the separate contracting thresholds from subsection 5.2.1. Regarding the agent’s level of private information, this setting is very similar to the separate contracting case in subsection 5.2.1. In subsection 5.2.1, the agent
29
enters into a contract twice, and has better information for each investment than the principal. In this subsection, 5.2.3, the agent enters into a contract once, but has better information about both investments when entering the contract. Effectively, the amount of information the agent has prior to entering the contracts is the same for the separate contracting case and the ex-ante contracting case with 𝜃2 = 𝜃1 described in this section. Therefore, the rents that the principal need to divert to the agent are the same in both cases.
The conditions to ensure the first and second investment are equal to section 5.2.1 and 5.2.2 and are: 𝑅̂1,ℎ = 𝛽 𝛽−1(𝐼1 + 𝜃1,ℎ+ 𝑞 (1−𝑞)∆𝜃1) < 𝑅̂2, (70)
for an agent observing 𝜃1,ℎ ex-ante and: 𝑅̂1,𝑙 = 𝛽
𝛽−1(𝐼1+ 𝜃1,𝑙) < 𝑅̂2, (71)
for an agent observing 𝜃1,𝑙 ex-ante and.
5.2.4 Overview of the results
In the three previous subsections, I developed and solved three sequential real options models under asymmetric information. In this subsection, I compare these models with each other and with the first-best benchmark model from subsection 5.1 in terms of efficiency from a principal’s point of view. Tables 2a and 2b provide an overview of the thresholds and wages of each model where the model numbers correspond to models indicated in the titles of subsections 5.2.1 up to and including 5.2.3.
30
Table 2a
Comparison of the sequential investment models
Model (1) First-best benchmark Model (2) Asymmetric information, separate
contracting, 𝜽𝟏 and 𝜽𝟐 completely independent
Wage/ threshold Result Wage/ threshold Result 𝑅̂1,𝑙 𝛽 𝛽 − 1(𝐼1+ 𝜃1,𝑙) 𝑅̂1,𝑙 𝛽 𝛽 − 1(𝐼1+ 𝜃1,𝑙) 𝑅̂1,ℎ 𝛽 𝛽 − 1(𝐼1+ 𝜃1,ℎ) 𝑅̂1,ℎ 𝛽 𝛽 − 1(𝐼1+ 𝜃1,ℎ+ 𝑞 1 − 𝑞∆𝜃1) 𝑤1,𝑙 Zero 𝑤1,𝑙( 𝑅 𝑅̂1,𝑙 ) 𝛽 ∆𝜃1( 𝑅 𝑅̂1,ℎ ) 𝛽 𝑤1,ℎ Zero 𝑤1,ℎ Zero 𝑅̂2,𝑙 𝛽 𝛽 − 1(𝐼2+ 𝜃2,𝑙) 𝑅̂2,𝑙 𝛽 𝛽 − 1(𝐼2+ 𝜃2,𝑙) 𝑅̂2,ℎ 𝛽 𝛽 − 1(𝐼2+ 𝜃2,ℎ) 𝑅̂2,ℎ 𝛽 𝛽 − 1(𝐼2+ 𝜃2,ℎ+ 𝑞 1 − 𝑞∆𝜃2) 𝑤2,𝑙 Zero 𝑤2,𝑙( 𝑅 𝑅̂2,𝑙 ) 𝛽 ∆𝜃2( 𝑅 𝑅̂2,ℎ ) 𝛽 𝑤2,ℎ Zero 𝑤2,ℎ Zero
Secondly, we learn from model (2) and model (3) (see table 2b) that the timing of entering into a contract, ex-ante or prior to each investment, has implications for the wages and the optimal investment thresholds when 𝜃1 and 𝜃2 are completely independent. The major
