Delegated disinvestment under Uncertainty

Delegated disinvestment under Uncertainty

Key Words: Investment under Volatility, Investment Timing, Real Options

Theory, Delegation, Agency Theory

JEL Codes: D92, P45

Abstract:

The first step in this paper is the general explanation of real options theory and agency theory and building a model from earlier work by both McDonald and Siegel (1986) and Grenadier and Wang (2005). Following that, the model is extended by the option of leaving the investment and receiving a scrap value and also agency problems are introduced in the setting. In this original contribution of the paper a second-best solution that minimizes agency costs is found. This solution is compared to its first-best counterpart and finally the effects of volatility and the magnitude of the hidden information on the second-best solution are graphically represented.

Author:

Jos Zwier

Student Number: 1861786

Supervisor:

Dr. G.T.J. Zwart

Version:

Final version 1

Date:

13

of June, 2016

1. Introduction ... 1

2. Literature review ... 3

3. Base model ... 7

3.1 Real options model ... 7

3.2 Adverse selection base model ... 9

3.2.1 Grenadier and Wang (2005)... 9

3.2.2 Timeline ... 10

3.2.3 Deriving discount factors ... 10

3.2.4 Deriving incentive compatibility constraints ... 11

3.2.5 Solving the model... 12

3.3 Graphical solution ... 13

4. Execution or exit ... 17

4.1 Setting of the model ... 17

4.2 Execution or exit solution without agency conflict ... 17

4.3 Execution or exit with hidden information ... 19

4.4 Wages ... 22

4.5 Graphical representation ... 23

5. Concluding remarks ... 27

5.1 Conclusions ... 27

5.2 Implications for practitioners and academics ... 28

5.3 Shortcomings of this research ... 28

5.4 Place in literature ... 28

5.5 Suggestions for future research ... 29

References ... 30 Appendices ... 31 Appendix A ... 31 Appendix B ... 33 Appendix B.1 solution of λ ... 33 Appendix B.2 solution of V* ... 33 Appendix C ... 35

Appendix C.1 Calculation of thresholds ... 35

Appendix C.2 Calculation of the wage ... 35

Appendix D ... 36

Appendix D.1 Calculation of thresholds... 36

Appendix D.2 Calculation of the wage ... 37

1. Introduction

Volatility in the payoff or the costs of a project increase the threshold at which value a rational, risk-averse investor will invest in a project. Also, practically every decision to invest or to disinvest in a project is accompanied by volatility. Yet, questions pertaining to agency costs (and to most other economic theories) are generally answered in the setting of an NPV-model; a setting in which a project is deemed worthwhile if it returns the risk-free rate or even simply zero. This is clearly an incomplete setting and therefore this paper attempts to construct a more realistic model based on real options literature in which investment in a project and disinvestment in scouting the project are both possibilities and the payoff of the project is uncertain, following that the impact of hidden information will be examined in this model.

To understand the basics of real options theory, one has to see the ability to invest in a project as analogous to an American call option on a stock; you have the option, but not the obligation to invest in this project for a certain time (until competition catches up, a patent expires, etc.). Once this analogy has been made, all pricing models for derivative instruments are available to evaluate uncertain investments in projects, joint ventures, etcetera.

Especially in the presence of uncertainty about the costs or payoff of a project (i.e. practically every decision made by investors or finance scholars) the real options approach has been proven to be superior when it comes to predicting firm behaviour (see for example McDonald & Siegel (1986) or Shibata and Nishihara (2013)) and when it comes to finding the threshold value of the underlying at which investment is desirable (see for example Dixit and Pindyck (1994)), however, current agency cost models work with the less complete NPV-approach to evaluate investments. This paper seeks to expand existing literature on real options theory with a more complete model in which scouting a project is costly (e.g.

opportunity cost of material, direct cost of wages, etc.), introduce agency costs to the model and find intuitive solutions for the effect of volatility and the value of the hidden information on investment thresholds and on the magnitude of the agency costs (read: on the incremental wage to be paid to the manager/agent).

paper by McDonald and Siegel (1986) will be extended to incorporate the opportunity to exit a prospective project and receiving a scrap value (as opposed to basically putting a project on hold forever.) This extension will also be put in a delegated investment setting and the

2. Literature review

This paper will attempt to extend the real options literature with the option to exit a prospective project and incorporating adverse selection. The reader is assumed to be familiar with both general agency theory and the aforementioned real options valuation theory; short explanations of both theories will be provided in the respective subsections, if the reader desires a more extensive elaboration on agency theory the reader is referred to graduate level Corporate Finance course books, (such as The Theory of Corporate Finance by J. Tirole (2006)) and for a more extensive elaboration on real options valuation one is referred to overviews provided by either Dixit and Pindyck (1994) or Trigeorgis (1996).

As mentioned before, real options theory stems from the analogy between an ability to invest and an American call option. General option pricing was pioneered by Black, Scholes and Merton, who manage to construct a portfolio including an option that is risk-free over an infinitesimally small period of time and derive the famous “Black-Scholes formula” (Mertons name is left out because he died before they received the Nobel prize for this work) to

properly value options.

Adverse selection and hidden knowledge are studied in most graduate level Corporate Finance textbooks. J. Tirole (2006) introduces a cash flow that is not completely observable to the principal and then minimizes the costs associated with hidden knowledge by offering the agent a choice of two contracts with wages contingent on either a high or a low cash flow that is certain to make it most profitable for the agent to tell the truth. This is the general approach to solving problems pertaining to hidden knowledge or adverse selection and is also the intuition behind the solutions that will be provided in this paper.

is that compared to traditional methods this approach is superior in the presence of volatility and similar answers if all cash flows are certain.

Another approach taken to researching real options theory is testing hypotheses based on the theory empirically. Folta and Miller (2001) research hypotheses based on hypotheses regarding the effects of uncertainty in the value of developing technologies in the uncertain world of biotechnology takeovers (of partner R&D firms.) They find that lower volatility increases the likelihood of an acquisition of a majority stake (p-value under 0.1), but finally conclude that they cannot find a significant effect confirming their hypotheses (possibly due to a low sample size of 22); Shyam Kumar (2005) examines the value created from acquiring and divesting joint ventures and concludes that decisions that are made with a focus on proper portfolio management with respect to volatility add more value than decisions made with an eye on growth or acquiring market share, in line with real options valuation theory.

Ford and Lander (2011) attempt to explain behaviour by markets and investors using real options theory. Many professional investors do not explicitly use real options theory, but instead intuitively manage uncertainty within given bounds. However, Ford and Lander (2011) compare managerial perceptions of uncertainty with real options theory and found that their perceptions and intuitively taken decisions were often in line with the theory.

As mentioned earlier, the question this paper tries to answer is often answered in more traditional models based on the NPV-method. An example of this can be found in graduate level corporate finance study books such as Tirole (2006). Kruse and Strack (2015) look at a stopping problem in the setting of a monotonic optimization problem. Similar to the model used in this paper the agent observes a private signal, in contrast to this paper though, this signal influences both his and the principal his payoff, rather than being an unobserved cash flow that can be taken by the agent. The consequence of using monotonicity is that the optimal rule only affects the marginal cut-off type, which is the most biased agent. Similarly, in this paper I will find that it is optimal to only distort one type of agent, however in this paper it will be the agent that is initially not incentivized to lie.

Grenadier and Wang (2005) focus on investment timing in their paper, using a real options framework. They start off with a model based on McDonald and Siegel (1986) and add a privately observed benefit 𝜃 to the payoff which might be earned and might not be earned. Ex ante it is possible for the manager of the firm to exert effort to increase the chance of earning 𝜃₁ > 𝜃₂. Grenadier and Wang manage to construct contracts able to dissuade managers of higher quality projects from lying ex post about their payoffs and pocketing the difference themselves. They conclude that, compared to the first-best solution that does not include agency costs, there is a greater inertia in investment as the manager gains more from waiting than the investor does.

This paper starts off by making a model inspired by Grenadier and Wang to look at agency costs in a real options setting and continues to extend the model to allow for stopping the scouting of a project to receive the scrap value of the resources committed to it. The paper of 2005 is his first paper using a model similar to the model in this paper, but Grenadier and other researchers have continued building on this model and using it to answer other

questions as well. In a more recent publication, Grenadier and Malenko (2011) concern a real options game where the decision to exercise gives a signal to outsiders that affects the utility of the investor; examples of these decisions can be investments in large projects, IPO’s, etc. They find that the timing of the decision will be significantly distorted, either by lowering or increasing the thresholds. The investment is sped up if the decision-maker’s utility is

positively related with the outsiders’ belief about the value of the asset and is delayed if the decision-maker’s utility decreases with the outsiders’ belief about the value of the asset.

Furthermore, in a forthcoming paper, Grenadier, Malenko and Malenko (forthcoming) consider a situation where an uninformed principal makes timing decisions based solely on communication with agents that are more informed about the optimal timing but are also biased. They build a model that incorporates the timing bias of informed agents as a

parameter 𝑏. Grenadier et al. (forthcoming) look at the optimal behaviour of the principal for different possible values for 𝑏 when he has a choice between keeping authority over the decision to invest, or delegating this authority to the agent. Their results imply that the sign of 𝑏 is the key determinant of which choice is optimal, where delegation is preferred for

negative values of 𝑏 and holding on to the authority is preferred for positive values.

minimum investment threshold and by simply increasing the expected payoffs. However, he does show that the auditing technology does not necessarily increase social welfare as the agents welfare can drop by a larger amount than is gained by the principal.

Apart from uncertain payoff structures, the theory also allows for uncertainty in investment cost. Shibata and Nishihara (2013) research how a firm which can invest in a small-scale or a large-scale expansion with either internal capital or external funding at a cost C, which reacts to changes in volatility of the underlying; making use of a real options model Shibata and Nishihara (2013) show that differences between the behaviours of immature and mature funds explain “empirical findings that the investment volume has a U-shaped relation with internal funds” and also explain “empirical predictions that the investment threshold has a U-shaped relation with internal funds”. This conclusion is based on analyses on firms with respectively low, intermediate or high amounts of internal funds.

3. Base model

This section will first describe the real options model that was originally proposed by McDonald and Siegel (1986). The next section will describe the extensions by Grenadier and Wang (2005) who tackle agency problems with a wage schedule for the manager. The

sections after that will provide several extensions of the model, which will be the original content of this paper. The final section will conclude and discuss the implications of the paper for the research and business communities.

3.1 Real options model

The solution of the real options model will follow that of Dixit and Pindyck (1996). Investments done by actual firms usually have several defining characteristics. At least a large part of the investment is usually in the form of a sunk cost, and therefore irreversible; and firms or investors usually have the opportunity to wait and see if more information can come out before they make their final decision. The payoff of the investment we will name V and the investment cost I; both of these factors will be discounted, leaving the principal wishing to maximize his payoff P(V) which can be described as follows:

𝑃(𝑉) = Ɛ(𝑉 − 𝐼)𝑒−𝑟𝑇 ₍₁₎

where Ɛ denotes the expectation, T stands for the exercise date, V is the value of the project, I is the investment cost and r is the discount factor. In this model, the value of the project (V) is the parameter that will vary over time. V will follow the following geometric Brownian motion:

𝑑𝑉 = 𝛼𝑉𝑑𝑡 + 𝜎𝑉𝑑𝑧 (2)

where α denotes the drift rate of the value of the project and σ the variance rate, dt denotes a step in time and finally dz denotes a stochastic wiener process. One can also see from this form that if the drift rate 𝛼 would be larger than the discount rate it will always be more valuable to wait.

Holding a portfolio consisting of long an option to invest (worth P(V)) and short a portion P’(V) of the project itself (so not the option). The return on this portfolio in timeframe dt will be:

where 𝛿 stands for the dividends that need to be paid over the equity in the actual project and an apostrophe implies a derivative. This can be rewritten in the following way:

𝑅𝑃 = 1₂𝜎2𝑉2𝑃′′(𝑉) −𝑃′(𝑉)𝛿 (4)

Since there is no risk in this portfolio in the short time interval dt, it must equal the risk-free rate.

𝑅_𝑃 = 𝑟[𝑃(𝑉) − 𝑃′_{(𝑉)𝑉] (5)}

Setting the above two returns equal to each other simplifies to the following differential equation that will always hold for the option value 𝑃(𝑉):

1 2𝜎

2_𝑉2_𝑃′′_{(𝑉) + 𝛼𝑉𝑃}′_{(𝑉) − 𝑟𝑃(𝑉) = 0 (6)}

where an apostrophe implies a derivative. A more thorough derivation of this equation will be included in this paper in appendix A. Furthermore this maximization problem is subject to the following three boundary conditions:

𝑃(0) = 0 (7)

𝑃(𝑉∗_{) = 𝑉}∗_{− 𝐼 (8)}

𝑃′_(𝑉∗_{) = 1 (9)}

where 𝑉∗ _{is the value of V at which investment takes place, or as it will be called throughout} the rest of this paper, the investment threshold. Equation (7) implies that zero is an absolute barrier for the value of the project. If the project value is ever truly zero that implies there is also no more value in waiting, therefore the project is terminated; equation (8) is the value-matching principle, which implies that the value of the project should be equal to exactly its maximum value minus the investment costs. Finally, equation (9) is the smooth-pasting condition, which is the condition that ensures that the value of the threshold that we find is indeed the optimal value to invest at.

In order to satisfy boundary condition (7), the solution must take the following form:

𝑃(𝑉) = 𝐴𝑉𝜆 ₍₁₀₎

where λ is a variable computed to show V*

𝑉∗ ₌ 𝜆

𝜆−1 𝐼 (11)

where λ is given as:

𝜆 = 1₂− _𝜎𝛼₂ + √(_𝜎𝛼₂− 1₂)2₊ 2𝑟

𝜎2 (12)

which will always be larger than one for any positive α and/or discount rate r. This means, using the terminology of Dixit and Pindyck (1994), there is a wedge driven between the investment cost and the investment threshold. Where more traditional methods such as the DCF and NPV approaches will return with solutions of the value of the investment threshold that are equal to the cost of the initial investment, this approach will return a higher solution, because:

 if 𝜆 > 1, then _𝜆−1𝜆 > 1, and thus

 𝑉∗ ₌ 𝜆

𝜆−1 𝐼 > 𝐼

Given any existing variance λ will always be larger than one, and the wedge _𝜆−1𝜆 is strictly increasing in σ. The intuitive way to understand this is that one also gives up an option to invest by investing, effectively increasing the investment cost by the value of this option, and the value of any option is higher when variance is higher.

3.2 Adverse selection base model

The basic intuition behind any agency conflict is the discrepancy between who is paying for a project and receiving the proceeds and who is managing a project. In the case of adverse selection, the problem is with the information presented to both parties. The person managing a project knows more about its costs and its payoffs than the person providing the funds for the investments; this will lead to a model with not one, but two different option values. One option value for the investor (i.e. the principal) and one option value for the manager (i.e. the agent).

3.2.1 Grenadier and Wang (2005)

In their paper Grenadier and Wang (2005) start off with a first-best solution that does not include any agency problems that is nearly identical to the solution derived in the first-best solution in the previous section, since they base their work on the same standard

𝜃 can (initially) take the value 𝜃₁ or 𝜃₂ where 𝜃₁ > 𝜃₂ (they later extend the model to allow for a continuous distribution of 𝜃) furthermore it is possible for the agent to exert effort ex ante to increase the probability of drawing a higher value of 𝜃 and finally the costs of investing in this project are known. Grenadier and Wang develop a contract with two different pairs of a wage and an investment threshold, one pair for the agent to choose when he observes 𝜃1 and one to choose when he observes 𝜃2. In this section our original real options framework will be augmented with an unobservable component 𝜃 and the investment timing decision is delegated to an agent in exchange for a wage w. The only real difference between this model and that of Grenadier and Wang (2005) is that there is no possibility for the agent to exert effort in advance to increase the possibility of drawing a higher 𝜃 (and in continuum, there are no effort costs). To clarify the situation a timeline (of our model, not Grenadier and Wang’s) is provided in the next subsection.

3.2.2 Timeline

Figure 1: Timeline

- At t0 the contracts are offered to the agent, specifying 𝑉1∗, 𝑉2∗, 𝑤1 and 𝑤2, the agent will consider his options and choose the combination of investment threshold and wage that is optimal to him or her;

- at t1 project value passes 𝑉₁∗ and agents that observed 𝜃1 should be enticed to execute their option, consequently they shall be paid out 𝑤₁ (final cash flows to principal: 𝑉₁∗_{+ 𝜃}

1 incoming and 𝐼 + 𝑤1 outgoing);

- at t2 project value passes 𝑉2∗ and agents that observed 𝜃2 should be enticed to execute their option, consequently they shall be paid out 𝑤₂ (final cash flows to principal: 𝑉₂∗_{+ 𝜃}

2 incoming and 𝐼 + 𝑤2 outgoing). 3.2.3 Deriving discount factors

first time. This is the solution to equation (6) subject to the extra boundary conditions (13) and (14).

𝐷(𝑉∗_{, 𝑉}∗_{) = 1 (13)}

𝐷( 0, 𝑉∗_{) = 0 (14)}

The solution to this problem is: 𝐷(𝑉₀, 𝑉∗_{) = (}𝑉0

𝑉∗)𝜆 (15)

which can be found by following the same steps outlined in appendix A. This term, D, is equal to the expected discount rate of each cash flow and will replace the e-rt-terms in our analyses.

𝐷(𝑉₀, 𝑉∗_{) = Ɛ[𝑒}−𝑟𝑇_{] (16)}

3.2.4 Deriving incentive compatibility constraints

Similar to the project without asymmetric information, we have a project with the following payoff function:

𝑃(𝑉) = 𝑉 + 𝜃 − 𝐼 (17)

where 𝜃 is unobservable to the principal and can take on either the value 𝜃1 or the value 𝜃2, 𝐼 is a known investment cost and finally 𝑉 follows the geometric Brownian motion:

𝑑𝑉 = 𝛼𝑉𝑑𝑡 + 𝜎𝑉𝑑𝑧 (18)

where α is the drift parameter, dt is an infinitesimally small increment in time, σ is the variance parameter and finally dz is the increment in a standard Wiener process.

Though the actual value of 𝜃 is unobservable, the probability of drawing 𝜃1 and the probability of drawing 𝜃₂ are known to be respectively 𝑞 and (1 − 𝑞). The principal offers the agent a choice between two pairs of an investment threshold V* and a wage w for when the agent observes either 𝜃₁ or 𝜃₂, leaving the total value for the principal to be built up by a probability 𝑞 that there will be incoming cash flows 𝑉₁∗_{+ 𝜃}

1 and outgoing cash flows 𝑤1 + 𝐼 which will all have to be discounted at rate (𝑉0

𝑉₁∗)

𝜆 _{and a similar probability of (1 − 𝑞) that} there will be incoming cash flows 𝑉₂∗_{+ 𝜃}

2 and outgoing cash flows 𝑤2+ 𝐼 which will all have to be discounted at rate (𝑉0

which he will want to maximize over the variables 𝑉₁∗_{, 𝑉}

2∗, 𝑤1 and 𝑤2. The choice of these variables will however be subject to several constraints. It must induce the agent to exercise at the threshold 𝑉₁∗ _{and provide the principal with a value of 𝑉}

1∗+ 𝜃1− 𝐼 if 𝜃1 is observed, it must induce the agent to exercise at the threshold 𝑉₂∗ _{and provide the principal with a value of} 𝑉₂∗_{+ 𝜃}

2− 𝐼 if 𝜃2 is observed and finally both 𝑤1 and 𝑤2 are subject to non-negativity constraints. The agent can be enticed by means of his payoff functions:

𝑃_𝐴1(𝑉) = (𝑉0 𝑉₁∗) 𝜆 𝑤₁ (19) 𝑃𝐴2(𝑉) = (_𝑉𝑉0 2∗) 𝜆 𝑤2 (20)

where 𝑃𝐴1(𝑉) is the payoff to the agent who executes his or her option at investment threshold 𝑉₁∗_{and 𝑃}

𝐴2(𝑉) is the payoff to the agent who executes his or her option at investment threshold 𝑉₂∗_{. In order to ensure that the agent will exercise at 𝑉}

1∗ when he observes 𝜃1, but at 𝑉2∗ rather than 𝑉1∗ when he observes 𝜃2 we need to account for two more incentive constraints: (𝑉0 𝑉₁∗)𝜆𝑤1 ≥ ( 𝑉0 𝑉₂∗) 𝜆 (𝑤₂+ ∆𝜃) (21) (𝑉0 𝑉₁∗) 𝜆_(𝑤 1− ∆𝜃) ≤ (_𝑉𝑉0 2∗) 𝜆 𝑤₂ (22)

where Δ𝜃 = 𝜃₁ − 𝜃₂.Equation (17) is to ensure that it is more profitable for the agent who observes 𝜃1 tell the truth as his wage, discounted properly, is larger than his alternative wage and the gains from lying, also discounted properly, and equation (18) is to ensure that it is more profitable for the agent who observes 𝜃₂ to tell the truth, by making it too expensive to him or her to lie about the quality of the project. However, since (as we will see later) 𝑤₁ will asymptotically converge to ∆𝜃 but never reach it if we increase the amount of hidden

information; because of that, the left side of equation (18) will always be negative and the constraint is non-binding.

3.2.5 Solving the model

acceptable; for the remainder of the analysis 𝑤₂ will be set equal to 0. Thus the principal needs to maximize equation (14) subject to:

(𝑉0 𝑉₁∗) 𝜆 𝑤₁ = (𝑉0 𝑉₂∗) 𝜆 ∆𝜃 (23)

In order to determine 𝑤₁ we now simply rewrite equation (23): 𝑤₁ = (𝑉1∗

𝑉₂∗)

𝜆

∆𝜃 (24)

And we already concluded earlier that 𝑤₂ = 0. Furthermore, by substituting (𝑉0 𝑉1∗) 𝜆_𝑤 1 by (_𝑉𝑉0 2∗) 𝜆

∆𝜃 and 𝑤2 by zero in equation (18) we get 𝑃_𝑃(𝑉) = 𝑞(𝑉0 𝑉₁∗)𝜆(𝑉1∗+ 𝜃1− 𝐼) − 𝑞 ( 𝑉0 𝑉₂∗) 𝜆 ∆𝜃 + (1 − 𝑞) (𝑉0 𝑉₂∗) 𝜆 (𝑉2∗+ 𝜃2− 𝐼) (25) We can rewrite this equation as:

𝑃_𝑃(𝑉) = 𝑞(𝑉0 𝑉₁∗) 𝜆_(𝑉 1∗+ 𝜃1 − 𝐼) + (1 − 𝑞) (_𝑉𝑉0 2∗) 𝜆 (𝑉₂∗_{+ 𝜃} 2− 𝐼 −_1−𝑞𝑞 ∆𝜃) (26) By comparing equation (26) to the first best solution presented in equation (18) one can see that the first half of this equation is unchanged, however the second half of the equation is distorted by −_1−𝑞𝑞 ∆𝜃 and therefore the second investment threshold is not on the same level as the first-best solution. This is in line with general agency problem theory since the project of the agent who was originally not incentivized to lie is the project which has its investment threshold distorted. Concluding, we have the following investment thresholds:

𝑉₁∗ ₌ 𝜆 𝜆−1(𝐼 − 𝜃1) (27) 𝑉₂∗ ₌ 𝜆 𝜆−1(𝐼 + 𝑞 1−𝑞∆𝜃 − 𝜃2) (28) 3.3 Graphical solution

Equations (24), (27) and (28) are the analytical solutions to the optimization problem that this section attempted to solve. This subsection is meant to show the solution in a

graphical way. First we will solve the problem with the following realistic numbers: I is 100, α is 0.02, σ is 0.2, r is 0.04, q is 0.6, 𝜃₁ is 10 and finally 𝜃₁ is 5. Solving these equations will give us the following values for both thresholds and the wage:

 𝑉1∗ = 308,28

 𝑤1 = 4,31

Figures 2 through 5 will then each illustrate the effect of changing one of these variables; below there are two sets of two graphs, one showing the effect on both 𝑉₁∗ _{and 𝑉}

2∗ and one showing the effect on the necessary wage for the manager of the project that pays off a high 𝜃. Appendix C contains the numerical solution for the scenario with our base numbers as an example of a solution. In the graphs depicting both investment thresholds, the dark grey line represents 𝑉1∗ and the light grey line represents 𝑉2∗.

Figure 2. Effect of 𝜎 on the minimal incremental wage

Figure 3. Effect of 𝜎 on both investment thresholds

The minimum value for volatility in this analysis is 0.000001, this is because λ can only be found through the formula with σ in several denominators and thus 0 is not a possible

value. However, because the term is squared every time, this value can be seen as the value in the limit of 0.

That being said, the effect volatility has on the wage seems to be negligible (it can only be properly estimated when zooming in, look at the y-axis of the top graph,) which we would expect as the wage is mostly dependent on the amount of hidden information and not on the uncertainty of the initial investment. The effect of volatility on the investment

thresholds is also what we would expect, by raising volatility both thresholds show continuously steeper slopes.

Figure 4. Effect of 𝜃1 on the minimal incremental wage

When 𝜃₁ is equal to 𝜃₂, there is no hidden information, and therefore the wage is equal to 0. After 5 is reached it is almost linear, but (slightly) decreasingly increasing. The reason for this is perfectly captured by the second picture; the increasing 𝜃₁ significantly changes the investment thresholds, which in turn change the discount factors of the pay-offs for the manager. As the investment threshold is lowered, the payoff will occur earlier and the necessary wage is lowered.

The investment threshold for the high-payoff projects is lowered as a larger

proportion of the costs will certainly be made back, effectively reducing the investment. The investment threshold for the low-payoff projects is raised as the amount of hidden

4. Execution or exit

This section will first introduce a new setting without delegation of the investment timing decision and then we will add an adverse selection situation again in section 4.3.

4.1 Setting of the model

So far we have only looked at the upper threshold to find at which option value it is optimal for the principal to execute the option; however, in business it is not unrealistic to assume that there are costs associated with keeping an option to invest open (think of opportunity costs of idle production factors, wages of employees who are scouting an idea, time spent on constantly re-evaluating the situation, etc.), so it is relevant to extend the model to one where there are not one, but two thresholds, an upper threshold that triggers execution of the option and a lower threshold that triggers killing the option (i.e. using the idle

production factors for another project or simply stopping consideration for the project altogether).

4.2 Execution or exit solution without agency conflict

First we will find a first-best solution again where there are no agency problems; either the agent perfectly mimics the behaviour the principal would exhibit or there is simply no delegation of the investment. The analysis to find this solution will be done from start to finish, but many elements similar to those in the previous sections some mathematical steps will be skipped and references will be made to the appropriate section. There is a 𝜃 in this model, but for now it is observable.

put, either action will mean giving up on both options. It is obvious that the value of both options depends upon the expected cash flow 𝑉 received when one (dis)invests.

The resulting value that will be maximized in the first-best case is the following: 𝑃(𝑉) = Ɛ[𝑒−𝑟𝑇∗

](𝑉∗_{+ 𝜃 − 𝐼) + Ɛ[𝑒}−𝑟𝑇∗∗

]𝐸 (29)

where Ɛ denotes the expectation and 𝑒−𝑟𝑇 _{is the appropriate discount rate from t}

0 to T, also factoring in the probability of touching the relevant threshold first.

The purpose of this section is to find respectively the upper threshold 𝑉∗ _{at which one} would execute the call option and the lower threshold 𝑉∗∗ _{at which one would execute the put} option. To do this we follow all the steps that we had to previously take to find λ, outlined in appendix B. Only this time we do not discard the negative root because 0 is no longer an absolute barrier, when the disinvestment threshold is reached one will execute the put option, so there is no need to worry about ever reaching 0. Consequently, the solution to 𝑃(𝑉) must have the following form:

𝑃(𝑉) = 𝐴𝑉𝜆1 + 𝐵𝑉𝜆2 ₍₃₀₎

where 𝐴 and 𝐵 are both constants and 𝜆₁ and 𝜆₂ given by:

𝜆1=1₂−_𝜎𝛼2+ √( 𝛼 𝜎2− 1 2)2+ 2𝑟 𝜎2> 1 (31) 𝜆₂=1 2− 𝛼 𝜎2− √( 𝛼 𝜎2− 1 2)2+ 2𝑟 𝜎2< 0 (32)

At the upper threshold of this area, 𝑉∗, the value of the project has to be equal to 𝑉∗_{+ 𝜃 − 𝐼, giving us the first equations to the system that will eventually have to be solved} to find both thresholds. The value-matching condition at the upper threshold is:

𝐴𝑉∗𝜆1_{+ 𝐵𝑉}∗𝜆2 _{= 𝑉}∗_{+ 𝜃 − 𝐼 (33)}

and the corresponding smooth-pasting condition is:

𝐴𝜆1𝑉∗𝜆1−1+ 𝐵𝜆2𝑉∗𝜆2−1= 1 (34)

which can be rewritten as follows:

𝐴𝜆1𝑉∗𝜆1+ 𝐵𝜆2𝑉∗𝜆2 = 𝑉∗ (35)

clean-up costs of the area around a shut-down plant;) in this paper a positive scrap value will be considered. The value-matching condition at the lower threshold is:

𝐴𝑉∗∗𝜆1_{+ 𝐵𝑉}∗∗𝜆2 _{= 𝐸 (36)}

and smooth-pasting condition:

𝐴𝜆1𝑉∗∗𝜆1−1+ 𝐵𝜆2𝑉∗∗𝜆2−1 = 0 (37)

which can be rewritten as follows:

𝐴𝜆1𝑉∗∗𝜆1+ 𝐵𝜆2𝑉∗∗𝜆2 = 0 (38)

We multiply equation (36) by 𝜆₂, subtract equation (38) and rewrite it to define 𝐴 in terms of only 𝑉∗ _{and 𝑉}∗∗ _{in equation (39). We do the same thing with 𝜆}

1 in order to define 𝐵 in equation (40) in similar terms:

𝐴 = 𝜆2𝐸

(𝜆2−𝜆1)𝑉∗∗𝜆1 (39)

𝐵 = 𝜆1𝐸

(𝜆1−𝜆2)𝑉∗∗𝜆2 (40)

We fill these terms in in equations (33) and (35) to obtain following system of equations with just the two unknowns left that we are looking for:

𝜆2𝐸𝑉∗𝜆1 (𝜆2−𝜆1)𝑉∗∗𝜆1 + 𝜆1𝐸𝑉∗𝜆2 (𝜆1−𝜆2)𝑉∗∗𝜆2 = 𝑉 ∗_{+ 𝜃 − 𝐼 (41)} 𝜆2𝐸𝜆1𝑉∗𝜆1 (𝜆2−𝜆1)𝑉∗∗𝜆1 + 𝜆1𝐸𝜆2𝑉∗𝜆2 (𝜆1−𝜆2)𝑉∗∗𝜆2 = 𝑉 ∗ ₍₄₂₎

While it is not possible to solve this system of equations ((41) and (42)) analytically; however it is true that 1 true solution exists and it can be approximated by means of

reiterative mathematical software. The next subsection will first reintroduce an agency

conflict, hidden information, and the subsection following that will show the solutions for the investment thresholds and wage similar to section 3.3.

4.3 Execution or exit with hidden information

is unobservable. 𝜃 will again be able to take on the value 𝜃₁ or 𝜃₂, where 𝜃₁ > 𝜃₂. Before the rest of the analysis is carried out, first all five values of 𝑉 will now be defined.

 𝑉0 denotes the starting value of 𝑉;

 𝑉1∗ denotes the investment threshold for a project with a high 𝜃;

 𝑉₁∗∗ _{denotes the disinvestment threshold for a project with a high 𝜃;}

 𝑉₂∗ _{denotes the investment threshold for a project with a low 𝜃;}

 𝑉₂∗∗ _{denotes the disinvestment threshold for a project with a low 𝜃.} The value for the principal that we will try to maximize is the following: 𝑃𝑃(𝑉) = 𝑞(Ɛ[𝑒−𝑟𝑇1 ∗ ](𝑉₁∗_{+ 𝜃} 1− 𝐼 − 𝑤1∗) + Ɛ[𝑒−𝑟𝑇1 ∗∗ ](𝐸 − 𝑤₁∗∗_{)) + (1 −} 𝑞)(Ɛ[𝑒−𝑟𝑇2∗](𝑉 2∗+ 𝜃2− 𝐼 − 𝑤2∗) + Ɛ[𝑒−𝑟𝑇2 ∗∗ ](𝐸 − 𝑤₂∗∗_{)) (43)}

Solving this system will lead to two systems of two equations each, one system for agents observing 𝜃1 and one system for agents observing 𝜃2, that will be very similar to those in the previous subsection. The subscript of all the thresholds and wages refers to the 𝜃 observed by the agent. Since all terms with subscript 1 and with subscript 2 are perfectly separated optimizing it will lead to separately optimizing the thresholds and respective wages for agents observing the two different 𝜃’s. The expected discount rates, [𝑒−𝑟𝑡_{], are special} parameters that also account for the possibility of never reaching a point (once disinvested in scouting the project one will never be able to invest in the project and vice versa); how these are calculated is explained in section 4.4 and an example calculation is provided in appendix D.2.

Figure 6. Two sample paths for V, the left one leads to 𝑉1∗ and the right one leads to 𝑉1∗∗

which can be seen because 𝑉₁∗ _{is the value at which the agent invests in the project or} “exercises the call option” (left side of the figure) and 𝑉₁∗∗ _{is the value at which the agent} disinvests in scouting the project or “exercises the put option” (right side of the figure).

Depending on which 𝜃 he or she reports to have observed, the agent will be offered a contract with one of two sets of thresholds and wages. Each contract will specify a wage for investing and a wage for disinvesting.

For the agent who observes 𝜃1 it has to be more profitable to admit to this than to lie about observing 𝜃₂ and keeping the difference:

Ɛ[𝑒−𝑟𝑇₁∗

]𝑤₁∗_{+ Ɛ[𝑒}−𝑟𝑇₁∗∗

]𝑤₁∗∗ _{≥ Ɛ[𝑒}−𝑟𝑇₂∗

](𝑤₂∗_{+ 𝛥𝜃) + Ɛ[𝑒}−𝑟𝑇₂∗∗

]𝑤₂∗∗ ₍₄₄₎ and for the agent who observes 𝜃2 the expected value of telling the truth also has to be highest: Ɛ[𝑒−𝑟𝑇2∗]𝑤 2∗+ Ɛ[𝑒−𝑟𝑇2 ∗∗ ]𝑤₂∗∗ _{≥ Ɛ[𝑒}−𝑟𝑇1∗](𝑤 1∗− 𝛥𝜃) + Ɛ[𝑒−𝑟𝑇1 ∗∗ ]𝑤₁∗∗ ₍₄₅₎ It seems intuitive that 𝑤₁∗∗ _{and 𝑤}

2∗∗ are equal to zero and with those filled in we are left with only 1 binding constraint as we know that 𝑤₁∗ _{will converge to 𝛥𝜃 (see equation (46)} and thus equation (45) is not binding. In the end we are left with only one nonzero wage:

𝑤₁∗ ₌ Ɛ[𝑒−𝑟𝑇2∗]

Ɛ[𝑒−𝑟𝑇1∗]𝛥𝜃 (46)

Filling in this value for 𝑤₁∗ _{and zero for all other wages in equation (43) we are left} with the following equation that will be maximized:

𝑃𝑃(𝑉) = 𝑞(Ɛ[𝑒−𝑟𝑇1 ∗ ](𝑉₁∗_{+ 𝜃} 1− 𝐼) + Ɛ[𝑒−𝑟𝑇1 ∗∗ ]𝐸) + (1 − 𝑞)(Ɛ[𝑒−𝑟𝑇2∗](𝑉 2∗+ 𝜃2− 𝐼 − 𝛥𝜃) + Ɛ[𝑒−𝑟𝑇₂∗∗_{]𝐸) (47)}

We can clearly see from here already that similarly to before, the thresholds of the agent that was initially not incentivized to lie are the thresholds that are altered. This is also in line with traditional agency theory. For the sake of completeness, I will provide below the systems of equations that the principal will solve after hearing from the agent how high 𝜃 is.

After hearing from the agent that he or she observed 𝜃1the principal will solve the following system of equations to find 𝑉₁∗ _{and 𝑉}

𝜆2𝐸𝜆1𝑉1∗𝜆1

(𝜆2−𝜆1)𝑉1∗∗𝜆1 +

𝜆1𝐸𝜆2𝑉1∗𝜆2

(𝜆1−𝜆2)𝑉1∗∗𝜆2 = 𝑉1

∗ ₍₄₉₎

And after hearing from the agent that he or she observed 𝜃1the principal will solve the following system of equations to find 𝑉₂∗ _{and 𝑉}

2∗∗: 𝜆2𝐸𝑉2∗𝜆1 (𝜆2−𝜆1)𝑉2∗∗𝜆1 + 𝜆1𝐸𝑉2∗𝜆2 (𝜆1−𝜆2)𝑉2∗∗𝜆2 = 𝑉₂∗_{+ 𝜃} 2 − 𝐼 − 𝛥𝜃 (50) 𝜆2𝐸𝜆1𝑉2∗𝜆1 (𝜆2−𝜆1)𝑉2∗∗𝜆1 + 𝜆1𝐸𝜆2𝑉2∗𝜆2 (𝜆1−𝜆2)𝑉2∗∗𝜆2 = 𝑉₂∗ ₍₅₁₎ 4.4 Wages

In the previous subsection we focussed on finding the (dis)investment thresholds that the contracts offered to the agent should specify, however, we still need to find the wages to offer the agent. We have already found that we should only reward the agent managing a project returning a high 𝜃 and also only when he or she invests, not when the disinvestment threshold is reached first. As we can see in equation (46), to find 𝑤₁∗_{, we need to find the} expected discount rates for 𝑇₁∗ _{and 𝑇}

2∗. Similar to how we found the discount rates in section 3, we need to solve equation (6) subject to two boundary conditions. This solution will have the familiar form:

𝐷 = 𝐴𝑉𝜆1+ 𝐵𝑉𝜆2 ₍₅₂₎

and will be subject to the boundary conditions:

𝐷1(𝐴, 𝐵, 𝑉1∗) = 1 (53)

𝐷₁( 𝐴, 𝐵, 𝑉1∗∗) = 0 (54)

The thresholds 𝑉₁∗ _{and 𝑉}

1∗∗ were already found in the previous subsection and can therefore be treated as constants. We are left with a system of two linear equations with two unknowns which we can solve to find 𝐴 and 𝐵 (which is simplified as much as possible):

Filling these values for 𝐴 and 𝐵 and the appropriate starting value for 𝑉, 𝑉₀, in in equation (52), we can find the appropriate discount rate 𝐷1, which will replace Ɛ[𝑒−𝑟𝑇1

∗ ]: 𝐷1 = 1 𝑉₁∗𝜆2₋₍𝑉1∗∗𝜆2 𝑉1∗∗𝜆1)𝑉1 ∗𝜆1 𝑉0𝜆2 − 𝑉1 ∗∗𝜆2 𝑉1∗𝜆2𝑉1∗∗𝜆1−𝑉1∗𝜆1𝑉1∗∗𝜆2𝑉0 𝜆1 ₍₅₇₎

Similarly, we can find 𝐷2, which will replace Ɛ[𝑒−𝑟𝑇2

∗ ]: 𝐷₂ = 1 𝑉2∗𝜆2−(𝑉2 ∗∗𝜆2 𝑉2∗∗𝜆1)𝑉2 ∗𝜆1 𝑉₀𝜆2₋ 𝑉2∗∗𝜆2 𝑉₂∗𝜆2_𝑉 2∗∗𝜆1−𝑉2∗𝜆1𝑉2∗∗𝜆2 𝑉₀𝜆1 ₍₅₈₎

We can now solve equation (46) to find the incremental wage or bonus to pay out to the agent.

𝑤₁∗ ₌ 𝐷2

𝐷1𝛥𝜃 (59)

Furthermore, unrelated to the wages; with these two discount rates, we can already calculate the value of the principal’s call option on the project. In order to find the value of the put option one has to find Ɛ[𝑒−𝑟𝑇₁∗∗_{] and Ɛ[𝑒}−𝑟𝑇₂∗∗_{], to do this one should repeat the steps} mentioned above, but set equation (53) equal to 0 and equation (54) equal to 1. This is not done in this paper, as we are not looking for the option value, but is mentioned for the sake of completeness.

4.5 Graphical representation

The systems of equations (48)-(49) and (50)-(51) and equation (59) are the analytical solutions to the optimization problemthat this section attempted to solve. This subsection is meant to show the solution in a graphical way. First we will solve the problem with the following realistic numbers: I is 100, 𝑉0 is 150, α is 0.02, σ is 0.2, r is 0.04, q is 0.5, 𝜃1 is 10 and finally 𝜃₁ is 5. Solving these equations will give us the following values for both

Figure 7. Effect of σ on the minimal incremental wage

Figure 8. Effect of 𝜎 on all four (dis)investment thresholds

The above graphs depicts the effect of volatility on the minimal incremental wage and all four (dis)investment thresholds; the effect volatility has on the wage seems again to be negligible (it can again only be properly estimated when zooming in, look at the y-axis of the top graph,) which we would expect as the wage is mostly dependent on the amount of hidden information and not on the uncertainty of the initial investment.

Furthermore, as we would expect we see that the effect of volatility on the investment thresholds is increasing rather than constant, the reason for this is that

rational investor would hold off investing in the project until the value is high enough with respect to the given volatility. Furthermore increasing the volatility lowers the disinvestment thresholds, as the increased volatility makes it more probable that 𝑉 will increase again making the project viable again. To explain these effects intuitively, (dis)investing means giving up the other option as well, volatility

increases the value of the options, and thus it is more costly to (dis)invest. These two effects combined mean that the presence of volatility induces a great inertia in the investment or in the project, as the region of inaction (in between the investment- and the disinvestment thresholds) grows increasingly larger. Though not visible properly on the graph, there are two different disinvestment thresholds, one for agents

observing 𝜃1 and one for agents observing 𝜃2; these two thresholds show up as one line on the graph because they are very close to each other (see appendix D.2 for an example.)

Figure 9. Effect of 𝜃1 on the minimal incremental wage

Figure 10. Effect of 𝜃1 on all four (dis)investment thresholds

Just as was the case before the possibility of exiting the project, the minimal necessary wage is zero if 𝜃1 = 𝜃2 and past that point it almost depends linearly on the magnitude of the hidden information. Its effect only slightly diminishes as the magnitude increases which moves the (dis)investment thresholds and reduces the 𝐷_𝐷2

1-term.

The above graphs depicts the effect of the magnitude of the hidden information on the minimal incremental wage and all four (dis)investment thresholds, the left side y-axis is for the investment thresholds which are represented by the darker lines (and are also the top pair of thresholds) and the right side y-axis is for the disinvestment thresholds which are

represented by the lighter grey lines. The effect of hidden information for the investment thresholds is exactly the same as in the base model. The disinvestment threshold for the agent who observes 𝜃₁ is lowered as the project will be more valuable when its total cash flow increases (𝜃₁ increases, therefore 𝑉 + 𝜃₁ increases) and thus the option to invest that one gives up by disinvesting also increases in value. The disinvestment threshold for the agent who observes 𝜃2 rises as similarly to the case with the investment thresholds, only the thresholds for the agent observing 𝜃₂ are distorted.

5. Concluding remarks

5.1 Conclusions

This paper set off to find the thresholds at which delegated risky projects should be undertaken (or disinvested in) in a way to maximize value to the principal. We started by tweaking the model of Grenadier and Wang (2005) to deal with hidden information only and then extended this model to allow for exiting a prospective project and receiving a scrap value. We have found thresholds that were over three times as high as the cost of investment, in line with current real options theory (see Dixit and Pindyck (1994)). With the use of contingent claims analysis, we were able to find both the optimal thresholds and incremental wages. We found that agency costs in the form of hidden information in the optimal setting distort the investment thresholds for agents observing lower 𝜃’s and also that we only want to give an incremental reward to agents managing a project that both returns a high 𝜃 and that leads to investment (so only if the upper threshold is touched before the lower threshold is.)

The (dis)investment thresholds react increasingly to volatility in both the base model and the execution or exit model. This means that the area of inaction (the area in between the investment threshold and the disinvestment threshold) becomes significantly larger as

volatility increases. This is what we would expect, as the increased volatility increases the value of both the call option on the project and the put option on scouting the project and (dis)investing means giving up both these options, and thus the payoff needs to be

sufficiently high to make up for this loss. In addition, increasing the magnitude of the hidden information pushes down both the investment threshold and the disinvestment threshold of the agent who observes 𝜃1 while pushing up the both the investment threshold and the disinvestment threshold of the agent who observes 𝜃2. The first effect is because a higher 𝜃1 will increase the cash flow gained from investing (which is 𝑉 + 𝜃₁) and the second effect is because the distortion completely falls on the thresholds of the agent who observes 𝜃2, and this distortion is increasing in the magnitude of the hidden information.

The incremental reward does not change significantly by the addition of a possibility to exit the prospective project. As mentioned above, we still only want to pay one kind of agent and even then, only when the underlying value of his or her project touches the

investment threshold before the disinvestment threshold. The wage depends most strongly on the magnitude of the hidden information, 𝛥𝜃, and is almost linear in this parameter; the effect is only slightly diminished by the 𝑉1

∗

project, and by the 𝐷_𝐷2

1-term after the addition. Furthermore, both before and after the

possibility to exit the project the wage does react slightly to the volatility of the cash flows, but this effect is almost negligible.

5.2 Implications for practitioners and academics

For academics this paper has extended the literature on real options by building a model that does not only look at the upside potential, but also at the downside risk and finds the threshold at which to disinvest. Furthermore, it provides a model in which one can evaluate the consequences of hidden information in a more complete model that correctly prices in volatility.

This paper has confirmed that the investment threshold for risky projects should be higher than models ignoring volatility suggest. For practitioners dealing with regularly traded assets (read: assets with known volatilities or that can be spanned by the market) this means that real options theory should be leading in making (dis)investment decisions. Especially in low-volatility investments decisions the effect of the little uncertainty that is still there is often underestimated and this model will be able to objectively price the volatility.

Furthermore this paper has given institutional investors a handle on how to properly handle remuneration of agents.

5.3 Shortcomings of this research

This paper shares at least one major shortcoming with all research in the area of real options theory. The theory is based on the fact one can see an opportunity to invest in a project as an American call option and use all tools option pricing theory has given us in order to evaluate the (timing of) investments; however, this does mean the project either needs to be traded or it needs to be possible to span the assets of the projects on the market, otherwise the exogenous variables in the models (e.g. volatility) cannot be found and need to be “guessed” by the evaluator. Furthermore, though this paper does discuss theory and make predictions it does not test any of these predictions empirically, and though the outcomes seem more realistic, one therefore simply cannot conclude anything until proper empirical follow-up studies have been done.

5.4 Place in literature

continued building on this model, some of the more recent papers of S. Grenadier are also referred to throughout this paper. Furthermore, Dixit (1989) builds a model similar to the model in this paper, only where a firm does not permanently give up the option to abandon a project by investing in it and vice versa, but can return to the original state of “mothballing” by paying a reinvestment cost. Dixit and Pindyck (1994) combine their collective work on models built using real options analysis in a book which is accepted as the current standard and is also the basis for the work in this paper.

This paper is purely theoretical, in that models are built in it and propositions are proposed, but not tested empirically; which is done in more recent literature in the field. Testing empirically is not done in this paper because it would go beyond the objective of finding the optimal thresholds and the wage.

5.5 Suggestions for future research

References

Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. The Journal of Political Economy 81, 637-654

Copeland, T., Weiner, J., 1990. Proactive management of uncertainty. The McKinsey Quarterly, 4, 133-152

Dixit, A. K., 1989. Entry and exit decisions under uncertainty. Journal of political economy, 97, 3, 620-638

Dixit, A. K., Pindyck, R. S., 1994. Investment Under Uncertainty. Princeton University Press Folta, T. B., Miller, K. D., 2002. Real options in equity partnerships. Strategic Management Journal 23, 77-88

Ford, D. N., Lander, D. M., 2011, Real option perceptions among project managers. Risk Management 13, 3, 122-146

Grenadier, S. R., Malenko A., Malenko N., Forthcoming. Timing decisions in organizations: Communication and authority in a dynamic environment. American Economic Review Grenadier, S. R., Malenko, S., 2011. Real options signalling games with applications to corporate finance. The Review of Financial Studies, 24, 12, 3993-4036

Kruse, T., Strack, P., 2015. Optimal stopping with private information. Journal of Economic Theory, 159B, 702-727

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Appendices

Appendix A

I will provide the derivation of equation (6) here. I will use contingent claims analysis which is related to no-arbitrage valuation and as such relies on the same assumptions. The equation we are trying to derive is:

1 2𝜎

2_𝑉2_𝑃′′_{(𝑉) + 𝛼𝑉𝑃}′_{(𝑉) − 𝑟𝑃(𝑉) = 0 (A.1)}

The risk-neutral drift rate of the project α is equal to the true drift rate of the value of the investment (μ) minus implied dividends δ.

𝛼 = 𝜇 − 𝛿 (A.2)

Furthermore since we are valuating this project in a risk-neutral world the expected true drift rate of the project μ is estimated to be equal to the risk-free rate r.

We want to maximize the project value P(V), which is the expected value of the resulting payoff minus the investment cost, both cash flows discounted from the investment date/exercise date.

𝑃(𝑉) = 𝑚𝑎𝑥 Ɛ[(𝑉𝑇− 𝐼)𝑒−𝑟𝑇] (A.3)

where V follows the geometric Brownian motion

𝑑𝑉 = 𝛼𝑉𝑑𝑡 + 𝜎𝑉𝑑𝑧 (A.4)

where α is the risk-neutral drift parameter, dt is an infinitesimally small increment in time, σ is the variance parameter and finally dz is the increment in a standard Wiener process.

Now imagine a portfolio consisting of a long option to invest (worth P(V)) and short a portion n = P’(V) of the project. The value of this portfolio is P(V) – nV, where we will hold n fixed over short intervals of time dt. Therefore the total return from holding the portfolio over the timeframe dt will be:

𝑅_𝑃 = 𝑑𝑃(𝑉) − 𝑃′_{(𝑉)𝑑𝑉 − 𝑃}′_{(𝑉)𝛿 (A.5)}

In order to find dP(V) we use Ito’s lemma: 𝑑𝑃(𝑉) = 𝑃′_{(𝑉)𝑑𝑉 +} 1

We know that (dV)2 = σ2V2dt, making the total return on the portfolio over the timeframe dt:

𝑅_𝑃 = 1₂𝜎2_𝑉2_𝑃′′_{(𝑉) −𝑃}′_{(𝑉)𝛿 (A.7)}

Since this is return is risk-free, it must equal the value of the portfolio multiplied by the risk-free rate, which is just the normal discount rate as we are using the same assumptions as with risk-neutral valuation.

1 2𝜎

2_𝑉2_𝑃′′_{(𝑉) − 𝑃}′_{(𝑉)𝛿 = 𝑟[𝑃(𝑉) − 𝑃}′_{(𝑉)𝑉] (A.8)}

Rewriting this we will get the equality from equation (A.2) that we set out to prove: 1

2𝜎

2_𝑉2_𝑃′′_{(𝑉) + (𝜇 − 𝛿)𝑉𝑃}′_{(𝑉) − 𝑟𝑃(𝑉) = 0 (A.9)}

which reduces to equation (A.10) when one fills in the risk-neutral α from equation (A.2).

1 2𝜎

Appendix B

I will provide the derivation of V* in this appendix, along with the derivation of λ. V* is the value of V at which it is optimal to invest, given the differential equation (equation (3)) and the three boundary conditions (equations 4 through 6).

Appendix B.1 solution of λ

We know that the solution has to take the following form

𝑃(𝑉) = 𝐴𝑉𝜆 _(B.1)

which also leads to the following equalities:

𝑃′_{(𝑉) = 𝜆𝐴𝑉}𝜆−1 _(B.2)

𝑃′′_{(𝑉) = 𝜆(𝜆 − 1)𝐴𝑉}𝜆−2 _(B.3)

Plugging these equalities into equation (3) yields: 1

2𝜎2𝑉2𝜆(𝜆 − 1)𝐴𝑉

𝜆−2_{+ 𝛼𝑉𝜆𝐴𝑉}𝜆−1_{− 𝑟𝑃 = 0 (B.4)} Eliminating the V-terms and dividing through P(V) gives us:

1 2𝜎

2_{𝜆(𝜆 − 1)+ 𝛼𝜆 − 𝑟 = 0 (B.5)}

which can be rewritten as

1

2𝜎2𝜆2+ (𝛼 − 1

2𝜎2) 𝜆 − 𝑟 = 0 (B.6)

We can now find solutions for 𝜆 using the abc-formula

𝜆1,2= −(𝛼−1₂𝜎2_{)±√((𝛼−}1 2𝜎2)2−4( 1 2𝜎2)(𝑟)) 2(1₂𝜎2₎ (B.7) 𝜆₁=1 2− 𝛼 𝜎2+ √( 𝛼 𝜎2− 1 2)2+ 2𝑟 𝜎2> 1 (B.8) 𝜆₂=1 2− 𝛼 𝜎2− √( 𝛼 𝜎2− 1 2)2+ 2𝑟 𝜎2< 0 (B.9)

Since solution λ2 returns a negative value for any set of values for α, σ and r and that does not adhere to boundary equation (4) we can conclude that λ1 is the only true solution and will simply refer to it as λ from now on.

Appendix B.2 solution of V*

We know from equation (B.1) that the solution will take the form 𝑃(𝑉) = 𝐴𝑉λ_{, we} also know that 𝑃(𝑉∗_{) = 𝑉}∗_{− 𝐼; therefore we also know that:}

Furthermore, we know that 𝑃′_(𝑉∗_{) = 1, which implies that:}

λ𝐴𝑉∗λ−1 = 1 (B.11)

Multiplying equation (B.11) with V* will return:

λ𝐴𝑉∗λ = 𝑉∗ (B.12)

We now divide equation (B.12) through λ. 𝐴𝑉∗λ

=𝑉_λ∗ (B.13)

Setting equation (B.10) equal to equation (B.13) yields the following result: 𝑉∗_{− 𝐼 =}𝑉∗ λ (B.14) (1 −1 λ) 𝑉 ∗_{= 𝐼 (B.15)} 𝑉∗₌ λ λ−1𝐼 (B.16)

Appendix C

Both investment thresholds and the wage will be rounded off to 2 decimals. Appendix C.1 Calculation of thresholds

The following is an example of the calculation of the thresholds 𝑉₁∗ _{and 𝑉}

2∗ and the wage 𝑤₁.

We have the following values for our relevant exogenous variables:

 𝛼 = 0.02  𝜎 = 0.2  𝜎2_{= 0.04}  𝑟 = 0.04  𝐼 = 100  𝜃1 = 10  𝜃₂ = 5  ∆𝜃 = 𝜃₁− 𝜃₂ = 5  𝑞 = 0.6

This gives us the following value for 𝜆:

𝜆 =1 2− 𝛼 𝜎2+ √( 𝛼 𝜎2− 1 2)2+ 2𝑟 𝜎2= √2 (C.1)

and the following corresponding values for 𝑉₁∗ _{and 𝑉} 2∗: 𝑉₁∗ ₌ 𝜆 𝜆−1(𝐼 − 𝜃1) = 308.28 (C.2) 𝑉₂∗ ₌ 𝜆 𝜆−1(𝐼 + 𝑞 1−𝑞∆𝜃 − 𝜃2) = 341.42 (C.3)

Appendix C.2 Calculation of the wage

The values mentioned above for the two investment thresholds and ∆𝜃 give us the following value of the minimum incremental wage:

𝑤1 = (𝑉1

∗

𝑉2∗)

𝜆

Appendix D

Both the (dis)investment thresholds and the minimal incremental wage will be rounded off to whole numbers to illustrate the limited accuracy of reiterative software Appendix D.1 Calculation of thresholds

The following is an example of the calculation of all four thresholds. We have the following values for the endogenous variables:

 𝛼 = 0.02  𝜎 = 0.2  𝜎2_{= 0.04}  𝑟 = 0.04  𝐼 = 100  𝑉0= 150  𝑞 = 0.5  𝜃1 = 10  𝜃₂ = 5  ∆𝜃 = 𝜃₁− 𝜃₂ = 5

which gives us the following two values for 𝜆₁ and 𝜆₂:

𝜆₁=1 2− 0.02 0.04+ √( 0.02 0.04− 1 2)2+ 2∗0.04 0.04 = √2 (D.1) 𝜆2=1₂−0.02_0.04− √(_0.040.02−1₂)2+2∗0.04_0.04 = −√2 (D.2) Consequently, after hearing from the agent that he or she observed 𝜃1the principal will solve the following system of two equations and two unknowns (which is simplified as much as possible): 5 (𝑉1∗ 𝑉1∗∗) √2 + 5 (𝑉1∗ 𝑉1∗∗) −√2 = 𝑉₁∗_{− 90 (D.3)} 5√2(𝑉1∗ 𝑉1∗∗) √2 −10_√2(𝑉1∗ 𝑉1∗∗) −√2 = 𝑉₁∗ _(D.4)

resulting in the following two values for 𝑉1∗ and 𝑉1∗∗:

 𝑉₁∗ _{= 308}

 𝑉₁∗∗ _{= 21}

5 (𝑉2∗ 𝑉2∗∗) √2 + 5 (𝑉2∗ 𝑉2∗∗) −√2 = 𝑉₂∗_{− 100 (D.5)} 5√2(𝑉2∗ 𝑉₂∗∗) √2 −10 √2( 𝑉₂∗ 𝑉₂∗∗) −√2 = 𝑉₂∗ _(D.6)

resulting in the following two values for 𝑉₂∗ _{and 𝑉} 2∗∗:

 𝑉₂∗ _{= 342}

 𝑉₂∗∗ _{= 22}

Appendix D.2 Calculation of the wage

The values mentioned above for the (dis)investment thresholds, 𝑉0 and both lambdas give us the following values of the expected discount factors:

𝐷₁ = 1 308−√2₋₂₁−2₃₀₈√2150−√2− 𝑉1∗∗−√2 308−√2₂₁√2₋₃₀₈√2₂₁−√2150√2= 0.36 (D.7) 𝐷₂ = 1 342−√2₋₂₂−2₃₄₂√2150−√2− 22−√2 342−√2₂₂√2₋₃₄₂√2₂₂−√2150√2 = 0.31 (D.8) These values for the expected discount factors and for ∆𝜃 will finally lead us to the minimal incremental wage to pay out to the agent:

Appendix E

This appendix will define all the variables mentioned in this paper in the order they appear.

P(V) payoff of the project

Ɛ denotes expected values

V value of the project

I investment

r discount rate

T time to exercise date

α risk-neutral drift rate

δ (implied) dividends

μ true drift rate

σ variance

dt infinitesimally small step in time

dz stochastic wiener process

λ see equation (8) and appendix B

q probability of drawing a high 𝜃 (𝜃₁)

1 (subscript) parameter w.r.t. agent observing 𝜃₁ 2 (subscript) parameter w.r.t. agent observing 𝜃₂

A parameter of constant (arbitrary) value

𝜃 unobservable part of a cash flow

w wage

D discount factor