Overcoming Adverse Selection within the Real Options Approach
ABSTRACT
This paper studies the timing of irreversible and uncertain investment decisions using the real options approach. This is proven to be a superior method compared to the traditional NPV method when valuing uncertain investments. Moreover, asymmetric information is included in this approach via adverse selection. In the benchmark model it is found that adverse selection distorts the first-best solution and this causes a delay in the investment timing. Several extensions to this benchmark model are provided, namely monitoring and not imposing a limited liability constraint. Additionally, a different principal-agent setting is examined where an investor is needed to fund the investment. It is found that depending on the contract parameters, the owner of the project can decide to execute a different strategy to overcome the hidden information problem compared to delaying the investment trigger. Moreover, when there is an external investor early investment, as opposed to delay, is needed to create a separating equilibrium in order to overcome adverse selection.
JEL-classification: D82; G31; G32
Keywords: Real Options, Adverse Selection, Investment Timing
June, 2015
Special Research Project dr. G.T.J Zwart University of Groningen
Msc. Finance Name: Irene Staal
Student Number: s1977288 Word count1: 14103
1. Introduction & Literature Review
Investment decisions and the timing of these investments are crucial within the field of Finance. Usually when firms face an investment decision, the Net Present Value (NPV) of the investment is calculated. Using this method, the investment is undertaken if the discounted expected cash flows are equal or larger than the initial investment. Based on this relatively simple calculation, firms decide if they make the investment or not. However, this classical NPV calculation fails to take into account investments that face uncertainty or cannot be reversed once they are undertaken. As McDonald and Siegel (1986) state: ‘The NPV-method is only valid if the variance of the present value of the future benefits and costs is equal to zero.’ If this is not the case, future payoffs from the projects are uncertain which needs to be taken into account. Contrary to the NPV-method, the Real Options Approach includes the volatility of investments since this approach is analogous to valuing an American call option (Dixit and Pindyck, 1994).
Moreover, several papers have examined the differences for valuing project using either the traditional NPV approach or the real options approach. For instance, Yeo and Qiu (2003) make an extensive comparison of the traditional investment method, the DCF approach (which uses the Net Present Value and Internal Rate of Return) to value investment decisions and the real options approach. From this comparison it is concluded that the traditional methods of valuing investments are ineffective since businesses nowadays are dynamic and changing quickly. The DCF method does not take this dynamic environment into account but the real options approach does, and therefore this is a better approach for valuing investments within a dynamic and rapidly changing environment. Furthermore, Miller and Park (2002) state that when levels of uncertainty are high, firms should first perform a DCF analysis to calculate the inputs for the real options approach. In their view, the real options approach is a complementary tool for firms when making decisions. The DCF analysis is more suitable for firms that have a more straightforward business structure, while the real options approach allows for more complex and uncertain business decisions.
Because the real options approach has several benefits compared to the tradition DCF approach, various models have been developed where the real options approach is applied in different valuation settings, especially in industries that face high uncertainty. For instance, Ford et al (2002) describe the advantages of using the real options approach for project management in construction planning. They conclude that the implementation of the real options approach can increase returns through the improvement of project planning and management, as a result of using the real options approach compared to traditional valuation techniques. Moreover, Insley (2002) applies the real options approach in determining the optimal tree harvesting decision when there are uncertain future lumber prices. Furthermore, Williams (1991) uses the real options approach to determine the date and degree when to develop or abandon undeveloped property and Titman (1985) applies the real options framework to value uncertain urban land prices. In this model, uncertainty about the type of building that might be constructed in the future is the main factor in valuing the unoccupied land. The option to select the type of building for the unoccupied land is therefore extremely valuable.
However, despite all these advantages the real options approach also has one main disadvantage: it fails to take into account agency problems. In the real options approach it is assumed that the owner of the option is in control of the investment decision. However this is often not the case, which means that the firm faces an agency problem. The agency relationship was first described by Jensen and Meckling (1976) who defined it as “a contract where one person (the principal) appoints another person (the agent) to perform some service on their behalf.’’ This contract involves delegating some decision-making ability from the principal to the agent and this causes agency costs to arise. These agency costs emerge since it is generally impossible that the agent will make the optimal investment decision from the principals’ point of view at zero costs. These costs can either be rents that the principal pays to the agent or monitoring costs incurred by the principal. Since asymmetric information is an essential element in corporate finance, it is very interesting to examine what happens to the real options approach when it is adjusted for these agency problems.
Extensive literature is available on agency problems and information asymmetries. For example Tirole (2006) describes how information asymmetries may affect the agency relationship. This can be due to adverse selection, which is defined as insiders having private information about the firm, or in the case of the real options, having private information about the project. Or there can be moral hazard, which is defined by Tirole (2006) as ‘outsiders not being able to observe the insiders’ carefulness in selecting the projects, the riskiness of investments, or the effort they exert to make the project profitable.’ Apart from information asymmetries, there are more concepts that need to be taken into account when considering the agency relationship, since the incentives of the owner and the manager may not be aligned. For example, investors are protected by limited liability, which implies they cannot be accounted for losses the firm makes beyond their initial investment and this restricts the owner in the types of incentive contracts he can write (Innis, 1990).
managers’ preference for large firms as opposed to smaller, profitable firms. He states that due to empire building, the manager has an incentive to understate payoffs. The paper of Bernardo et al (2001) studies the incentive compensation that managers can receive. They find that managers have an incentive to overstate project quality to secure more capital for themselves. Furthermore, their paper states that managers need to be given an appropriate incentive to provide a sufficient level of effort. Another dimension of the agency problem is that managers have a greater impatience compared to the owner of the project. This concept is known as managerial myopia and is illustrated by the model of Narayanan (1985). Here it is concluded that if managers have private information, they might have an incentive to make decisions based on short-terms gains, which sacrifices long-term shareholder value. All these concepts concerning agency problems need to be taken into account when developing the real option model within the theory of corporate finance. These concepts will later be used in defining the constraints in the model that is used to find the optimal contract under asymmetric information. Already several models have been developed to apply the concept of information asymmetries within the real options approach. For instance, Morellec and Schürhoff (2011) have developed a model that focuses solely on adverse selection. This model considers a firm that needs to raise funds from outside investors who are uninformed about the prospects of the firm. Moreover, firms with bad prospects have an incentive to mimic firms with good prospects, meaning that the bad firms can sell overpriced securities to the investor. But due to this mimicking, the good firm can secure less capital and he therefore wants to separate himself from the firm with bad prospects. Their model shows that this separation can be achieved through the timing of the investment: by speeding up their investment mimicking becomes more costly for the firm with bad prospects since the intrinsic value for the bad firm is lowered. Moreover, it is shown that the cost of changing the investment timing is lower compared to the costs the firm with good prospect faces when mimicking occurs.
Bustamante (2012) finds a similar result is found, in this paper the decision to go public is the real option and firms need to decide on the timing of their initial public offering (IPO). However, the firms have different prospects with regard to their investment and might want to signal this to outsiders. The model assumes that firms can only use the timing of their IPO to signal their investment prospects. Moreover, a distinction is made between hot and cold markets where hot markets can be characterized as markets where there is on average high IPO activity2. Bustamente (2012) shows that:
‘When adverse selection is more relevant (cold markets), firms with better investment prospects accelerate their IPO relative to the perfect information benchmark.’ However, in hot markets (where adverse selection is less relevant) the signal to go public is uninformative since there is simultaneous issuing and mispricing of shares in the market.
However, not all models conclude that speeding up the investment decision can help firms
2 A more detailed description about the difference in hot and cold markets is described in Helwege and Liang
overcome hidden information. For instance, Bouvard (2012) develops a model where an entrepreneur needs to raise funds and has private knowledge about his project. In order to transfer information to the investor, investment timing and monetary payments are used. The model proves that entrepreneurs with high prior beliefs about the quality of their project are more willing to postpone their investment beyond the first-best solution. In exchange for delay they will receive a performance-based compensation if the project is indeed profitable. Since they have high beliefs about their project, they are more likely to accept this contract compared to firms with low prior beliefs about the project quality. Therefore, in this model the investment decision is postponed for the firm with good prospect. Furthermore, Grenadier and Melenko (2011) find mixed results regarding adverse selection in the real options approach. Based on their model, it is concluded that the direction of the distortion of the first-best investment timing depends on different aspects: the option can either be delayed or accelerated. Their paper proposes a flexible model and this model is used to examine four different applications, namely: managerial myopia, venture capital grandstanding, cash flow diversion and product entry decisions. In the first two applications the investment decision is speeded up and in the last two applications the investment decision is delayed. However, despite these mixed result they state that the main message of the paper is that adverse selection distorts the timing decision of firms compared to the first-best scenario when no adverse selection is present.
The last model that will be discussed is the model developed by Grenadier and Wang (2005). Later on this model will be explained in detail, where after several extensions will be provided. Although the model concentrates on both adverse selection and moral hazard, this paper will only focus on the adverse selection solution since the extensions in this paper will also examine adverse selection. Their model predicts that when there is adverse selection, the investment decision is delayed in order to overcome the hidden information problem. This paper will examine various extensions that might be alternative solutions compared to distorting the investment trigger. These extensions will focus on monitoring, the limited liability constraint, and the effect on the model when the principal-agent relationship is different. In this different principal-agency setting an external investor is needed to finance to project as opposed to an owner who has sufficient funds. The alternative solutions provided in the extensions will then be compared to the first-best solution in the real options approach and the benchmark case where the investment trigger is delayed.
Their survey among Canadian firms shows that 16.8 percent of the managers use the real options approach for capital budgeting decisions and the use of the real options approach is placed last among other techniques. The main reason for this adaption is that managers lack the knowledge and expertise, which is also in line with the results found by Lander and Pinches (1998) However, it was found that managers are interested in adapting the real options approach if they had the required expertise. These papers suggest that a real options model under asymmetric information is most likely not found and applied in practice. The main reason for this is the general absence of knowledge about the standard real options approach and since adverse selection makes the model more comprehensive it is most likely not applied in real-life situations since it will be too difficult to understand and implement for most managers. However, several papers do show that firms use the timing of their IPO to signal information about their prospects when there is asymmetric information among investors. This is shown, among other things, by the model proposed by Altı (2005) and Allen and Faulhaber (1989). Based on this it can be concluded that firms indeed use investment timing as a signaling strategy when they face agency problems. However, it is unclear if the underlying real options approach is used in determining the optimal investment timing strategy.
The main objective of this paper is to demonstrate how owners of a project can apply the concept of adverse selection in the real options approach when valuing investments. By doing so, it gives owners insight in what techniques can be used to overcome adverse selection while maintaining an option value as high as possible. This paper contributes to existing literature in several ways. Firstly, it provides a clear and more detailed description of the real options approach when there is adverse selection. Additionally, it extensively describes the dynamics behind the model. This more detailed description gives the reader a clear understanding of the model before discussing the extensions. This may allow managers to apply the real options approach in practice, even when they face adverse selection.. Secondly, this paper proposes several extensions to an existing model that are also compared extensively to the first-best solution and the benchmark case. This gives a clear understanding of the model under adverse selection and what techniques can be used to overcome this adverse selection problem at minimum costs. Lastly, this paper also proposes a different principal-agency setting and therefore shows that early investment, as opposed to delay, might also be used in overcoming adverse selection.
2. The Real Options approach
This section will describe the basic model for valuing projects under the real options approach. This model was first developed by McDonald and Siegel (1985), where a firm is considered who faces an uncertain investment opportunity with an option to shut down. Later, Dixit and Pindyck (1994) described the real option approach more in detail in their book ‘Investment under Uncertainty’. This book will be used as a guide to explain the mechanism behind the real options approach.
According to Dixit and Pindyck (1994), within the option to invest lie two important characteristics of the investment. Firstly, the investment is irreversible and secondly the firm has the opportunity to delay the investment decision. Furthermore, the investment consists of two components: the investment costs and the value of the project, which will be called I and V respectively. The payoff from investing in the project for a given amount of V will yield the payoff P(V) = V – I. Since the future value of the project V is uncertain and the firm has the option to delay the investment decision, the investment opportunity is the equivalent of an American call option. This gives the holder the right, but not the obligation, to exercise his option to invest in the project. Therefore, the main problem the firm is facing is the choice between investing in the project today or some day in the future. Moreover, the firm will invest in the project when the option value is maximized which means that the firm faces a maximization problem. Since the future values of the project are unknown and thus uncertain, it is assumed that V evolves as a geometric Brownian motion as presented below.
dV = α V dt + σ V dz (1)
In this equation α represents the growth parameter, σ the variance parameter and dz is the increment of a stochastic Wiener process. The payoff from the investment opportunity will be denoted as P(V) and the firm wants to maximize this payoff for a given value of V. Therefore, the expected present value of the option to invest needs to be maximized and this is presented by the equation below.
P(V) = max Ɛ[(VT – I)e- r T ] (2)
where Ɛ denotes the expectation, T is the time at which the investment is made and r is the appropriate discount rate which should be larger than the growth rate3. The maximization problem in equation 2 is
subject to equation 1 for the value of V. Before the investment takes place, the value of P(V) satisfies the differential equation as presented in equation 3. The derivation for this can be found in Appendix A.
3 If this is not the case, the value of the project is expected to grow faster than its discounted value indicating that
½ σ2 V2 P’’(V) + α V P’(V) – r P = 0 (3) The optimal solution to the maximization problem in equation 2 can be found by solving the differential equation as presented in equation 3 subject to the appropriate boundary conditions. These boundary conditions are presented below.
P(0) = 0 (4)
P(V*) = V* – I (5)
P’(V*) = 1 (6)
The first condition implies that zero is an absorbing barrier: If the value of V goes to zero it will stay at zero and the option will have no value. The second boundary condition in equation (5) is known as the value-matching principle. This states that the payoff of the option, denoted by P(V*), is equal to V* – I once the option is exercised. The last boundary condition in equation (6) is the smooth-pasting principle and this ensures that the investment decision occurs along the optimal path. If this condition is not met, one can realize a higher payoff by exercising the option at a different point. Solving the differential equation for the appropriate boundary conditions yields the following critical value V* at which it is optimal for the firm to invest in the project, or equivalently, exercise the option to invest. The derivations for the values of V* and x can be found in appendix B.
V* = !!! ! I (7)
Where x is equal to:
x = ½ – !!!+ [ ! !! − ! !]!+ !" !! >1 (8)
3. Model under asymmetric information
The model as described in section 2 deals with the investment decision under the real options approach and it concludes that this superior compared to the NPV method when there is uncertainty. However, as stated in the introduction, a disadvantage of the real options approach is that it fails to take into account information asymmetries that may arise when facing an investment decision. This section will focus on the model developed by Grenadier and Wang (2005), where the impact of adverse selection and moral hazard on the real options model is examined.
The underlying assumption in this model is that the owner (principal) delegates the investment decision to a manager (agent). Due to this delegation, the owner of the project is no longer in charge of the timing of the investment and therefore an agency problem arises. Furthermore, it is assumed that the project will generate two payoffs: one that can be observed by both the owner and the manager and one that can only be observed by the manager. This means that the manager has a chance to understate payoffs to the owner and this misrepresenting of cash flows can be characterized as adverse selection. But apart from adverse selection, the manager can decide to put in effort in order to obtain a higher private payoff. By exerting effort before the investment takes place, there is a higher probability of obtaining a higher payoff, but this comes at a cost for the manager. The owner cannot observe the level of effort the manager exerts and therefore the owner faces a hidden action problem, which can be characterized as moral hazard. Due to these hidden information and hidden action problems, the owner needs to write a contract to make sure that the manager will exert effort and will hand over the true value of the unobservable payoff. In this contract, a wage will be promised to the manager in order to induce effort and to make sure he provides the owner with the corresponding private payoff of the project. In order to give a clear insight in this model and the different timing aspects, a time-line is drawn and presented below.
0 1 2 3 time
3.1. First-‐best solution
Before the agency setting is discussed, the first-best solution to this model will be provided since this solution will later on be used as a comparison. In the model, V is the observable part of the project value, δ is the part that is only observed by the manager and the parameter I represents the investment costs. Thus, under symmetric information the total payoff of the project equals P(V;δ) = V + δ – I. Where V is the uncertain, observable, value that evolves as a geometric Brownian motion and δ is a random variable that can either be δH, the higher private value, or δL, the lower private value.
Moreover, the manager can decide to undertake effort and this will affect the likelihood of obtaining a high quality project or a low quality project. By exerting effort at cost c, there is a higher probability for the manager to obtain the higher value δH and the probability of getting the higher value project
when exerting effort rises from qL to qH. In order to exert effort and reveal the true value of δ, the
owner needs to provide the manager with an incentive. This incentive is a wage that will be contingent on the value of V and the private value the manager hands to the owner. When there is no asymmetric information, the value of the option is determined in the same way as described in section 2 and in appendix B. This means that in order to derive the value of the option, the differential equation needs to be solved subject to the appropriate three boundary conditions. When there are no information asymmetries, it is assumed that the owner observes the true value of δ and V*(δ) is the value that triggers investment by the manager, which is presented in equation 9.
V*(δ) = !!!! (I – δ) (9)
where the value of x is defined similarly as in section 2 and is dependent on the growth rate, the discount rate and the variance. Using the same arguments, the owner’s option value at time zero can be derived with the exercise trigger V*(δ). The derivation for this can be found in appendix C.
P(V0; δ) = !!∗(!)!
!
(V*(δ) + δ – I ) for V0 < V*(δ) (10)
Furthermore, it should be noted that the term !!
!∗(!)
!
is the expected discount factor of the option. This discount factor was first represented by e-rT in the maximization problem as described in equation 2. So the option value as presented in equation 10 is the present value of the payoff from the option when it is exercised at the V*(δ) trigger. Moreover it is assumed that V0 < V*(δ), which means that the
initial value of the project (V0) is smaller than the value upon exercise and that there is an option of
waiting to invest. Since δ can take on the value of either δH or δL, the investment trigger V*(δ) can be
defined as V*(δH) = V*H or V*(δL) = V*L where V*H occurs earlier than V*L. Without asymmetric
P*(V0)owner = qH !!∗!!
!
(V*H + δH – I ) + (1 – qH) !!∗!!
!
(V*L + δL – I ) – c (11)
where P*(V0) denotes the optimal value of the option at time zero. Since it is assumed that the
manager will exert effort, he needs to pay the costs of effort upfront as described in the timeline. The owner of the project will need to compensate the manager for these effort costs, since the manager would otherwise have a negative option value. By paying the manager c, the option value for the manager will be equal to zero in this setting. Since there is neither hidden information nor hidden action, no other financial incentives are needed for the manager. Therefore, the manager has no option value since he receives no additional rents apart from the compensation for c.
3.2. Asymmetric Information Benchmark case
3.2.1. Setup of the modelWhen there is asymmetric information the owner needs to write a contract to overcome the hidden information and hidden action problem. Therefore, a wage needs to be promised to the manager and this wage will be made contingent on the investment trigger observed by the owner. The owner will give a wage of wH when the option is exercised at the VH trigger and a wage of wL when the option is
exercised at the VL trigger. In the presence of asymmetric information, the option value for the owner
and the manager can be defined as followed, assuming that the manager will report his information truthfully and that he will exert effort.
P*(V0)owner = qH !!! ! ! (VH + δH – I – wH) + (1 – qH) !!! ! ! (VL + δL – I – wL) (12) P*(V0)manager = qH !!! ! ! wH + (1 – qH) !!! ! ! wL – c (13)
The owner aims to maximize his option value given the contract terms wH, wL, VH and VL which he
can influence. This means that the owner is able deviate from the optimal investment triggers as determined in the case where no asymmetric information is present. Therefore, the owner faces the following maximization problem.
max qH !!! ! ! (VH + δH – I – wH) + (1 – qH) !!! ! ! (VL + δL – I – wL) (14)
and this constraint makes sure that the manager will exert effort. This constraint is defined below in equation 15. q! !!! ! ! w! + (1 – qH) !!! ! ! w! – c ≥ q! !!! ! ! w! + (1 – qL) !!! ! ! w! (15)
The constraint states that the payoff from exerting effort should be equal or larger than the payoff when no effort is exerted. Rewriting yields the following reduced ex-ante incentive constraint.
!! !! ! w! ! !!! ! ! w! !!!! (16)
where Δq is equal to qH – qL, which is larger than zero.
Secondly, the manager of the project should have an option value that is equal or larger than zero in expectation in order to accept the contract ex-ante. This results in the following ex-ante participation constraint. qH !!! ! ! wH + (1 – qH) !! !! ! wL – c ≥ 0 (17)
Additionally, there is a need for an ex-post incentive constraint to make sure that the manager will exercise the option in line with the owner’s expectations. This indicates that the manager will exercise the option at the appropriate trigger and does not have an incentive to hide the true value of the project from the owner. The ex-post incentive constraints can be defined by two equations, these are defined below. !! !! ! w! ≥ !! !! ! (w!+ Δδ) (18) !! !! ! (w!− Δδ) ≤ !!! ! ! w! (19)
where Δδ can be defined as δH – δL, which is larger than zero. Both the constraints ensure that the
manager will not divert value, where equation 18 is for the high quality project manager and equation 19 for the low quality project manager. Based on these constraints, it is more beneficial for the manager to reveal his true type compared to lying about this.
Moreover, apart from the incentive and participation constraints, ex-post enforceability of the contract needs to be ensured. The owner can achieve this by imposing a non-monetary penalty when the manager does not deliver VH + δH when he exercises at the VH trigger. In this case, the owner
corresponding higher private value. Therefore the owner can impose a penalty, such as the manager losing his job, reputational damage or going to prison. Since these are quite severe penalties, the manager would never want to receive such a penalty and will therefore always choose to deliver the corresponding private component when he exercises at the VH trigger. As stated before, this makes
sure the contract is enforced ex-ante. Lastly, due to the limited liability of the manager, the wages wH
and wL need to be positive. If the lower wage is negative the manager could walk away from the
project if he observes that he holds the low quality project. This implies the following limited liability constraint.
wi ≥ 0 i = H, L (20)
However, it can be argued that the wage for the manager of the low quality project should always be equal to zero. That is, if the manager of the low quality project receives a positive wage, this implies that the manager of the high quality project needs to receive a higher wage in order to meet his ex-post incentive constraint as defined in equation 18. Therefore, in order to minimize the wage for the manager of the high quality project subject to the ex-post incentive constraint, the manager of the low quality project will not receive a wage, implying that wL should be equal to zero. By taking all these
constraints into account, the optimal contract for the manager can be derived by maximizing equation 12 subject to the constraints.
However, although five constraints are provided, the solution can be presented by only taking into account two constraints. According to the propositions provided by Grenadier and Wang (2005)4, some of the constraints as defined in the previous section are proven not to bind. These are the ex-ante participation constraint, the limited liability constraint for the manager of the high quality project and the ex-post incentive constraint for the manager of the lower quality project. Therefore, the maximization problem is reduced to the following:
max qH !!! ! ! (VH + δH – I – wH) + (1 – qH) !!! ! ! (VL + δL – I) (21) Subject to: !! !! ! wH ≥ !!! ! ! Δδ (22) !! !! ! wH ≥ !!! (23)
4 The proof for these non-binding constraints is beyond the scope of this paper and can be found in the appendix
where equation 22 is the ex-post incentive constraint for the manager of the high quality project and equation 23 is the ex-ante incentive constraint, which makes sure that the manager will exert effort. At least one of the constraints must bind in order to derive the optimal contract.
3.2.2. Model solution
Three different solutions exist for the optimal contract, where the solution depends on which of the two constraints as defined in equation 22 and 23 bind. As discussed before, this paper will only take into account the hidden information region5 and this is the region where only the ex-post incentive
constraint binds. Therefore, the focus is on adverse selection and consequently it is assumed that the manager will always exert effort, which implies that the effort costs are very low in this solution region. Moreover, the solution regions can be defined by where the cost-benefit ratio of inducing effort, as defined by !
!!, falls relative to the present value of obtaining the wage. In order to determine
the option value for the owner, the values of the different contract terms the owner can influence need to be determined and these are VH, VL, wH and wL.
First, the exercise triggers VH and VL can be determined. This can be done by using the
ex-post incentive constraint for the manager of the high quality project and substituting this in the option value for the owner. From the ex-post incentive constraint is can be noted that:
!! !! ! wH = !!! ! ! Δδ (24)
where the left-hand side of the constraint also appears in the option value for the owner. Therefore, this can be substituted into the option value and this gives the following result.
qH !!! ! ! (VH + δH – I) – qH !!! ! ! Δδ + (1 – qH) !!! ! ! (VL + δL – I) (25)
Rewriting this equation gives:
qH !!! ! ! (VH + δH – I) + (1 – qH) !!! ! ! (VL + δL– !!!!! !Δδ – I) (26)
From this equation it can be noted that the first part of the equation is equal to the first-best solution and therefore the investment trigger for the high quality project can occur at the first-best level V*H.
However, it can also be noted that the payoff for the low quality project is reduced by the wage for the manager of the high quality project. So for the low quality project we have:
5 A detailed description of the other two solutions, the hidden action region and joint region, can be found in the
(1 – qH) !!! ! ! (VL + δZ – I) (27) where δZ = δL – !!!!! !Δδ < δL.
So in the presence of adverse selection, the option value for the low quality project is lowered by the wage for the manager of the high quality project. Therefore, the investment trigger will not occur at the first-best level since the option value is distorted. Moreover, it can be argued that the investment trigger for the manager of the low quality project should occur later. If it occurred earlier, the investment trigger for the low quality project would be closer to the investment trigger for the high quality project. In that case, the manager of the high quality project could at a relative low cost pretend to hold the low quality project since he only needs to distort his investment trigger by a small amount. Therefore, the investment trigger for the low quality project under adverse selection can be defined as:
VL = V*Z > V*L (28)
where V*Z = V*(δZ).
Now, the wages for the managers can be determined. As stated before, the wage for the low quality project is always equal to zero. Furthermore, the wage for the high quality project can be derived by taking into account the ex-post incentive constraint as defined in equation 22. Rewriting this equation for wH and substituting VL for V*Z and VH for V*H gives the following wage for the
manager of the high quality project.
wH = !
∗!
!∗!
!
Δδ (29)
Since the manager only requires compensation to stimulate exercise at the first-best trigger V*H, this
wage is relatively small. But because the wage is small, V*z needs to be significantly later to make
sure the manager of the high quality project will not be tempted to deviate from the optimal exercise trigger. In conclusion, all the contract terms the owner can influence in the hidden information region are defined as followed:
Given these contract terms, the option value for the owner and the manager can be defined. These option values are determined by considering the maximization problem in equation 21, substituting for the contract terms as defined above and rewriting in order to compare it to the first-best solution. This gives the option values at time zero for the owner and the manager respectively.
P*(V0)owner = qH !!∗!! ! (V* H + δH – I ) + (1 – qH) !!∗!! ! (V* Z + δZ – I ) (30) P*(V0)manager = qH !!∗!! ! Δδ (31)
And as stated before, δZ can be defined as:
δZ = δL – !! !!!
! Δδ < δL (32)
Moreover, since the costs of exerting effort are low this hidden action region can be defined as:
! !! < !! !∗! ! Δδ (33)
Which states that the cost-benefit ratio of exerting effort is smaller than the present value of the wage received by the manager. Moreover, the option value derived under adverse selection can be compared to the first-best case as described in equation 11. This is presented below, where equation 34 represents the first-best solution and equation 35 represents the option value under adverse selection.
P*(V0)owner = qH !!∗!! ! (V*H + δH – I ) + (1 – qH) !!∗!! ! (V*L + δL – I ) – c (34) P*(V0)owner = qH !!∗! ! ! (V*H + δH – I ) + (1 – qH) !!∗! ! ! (V*Z + δZ – I ) (35)
4. Extensions to the adverse selection benchmark case
The model described in the previous section focuses on the hidden information problem and the remainder of this paper will continue to focus only on adverse selection. It will provide extensions to the existing model of Grenadier and Wang (2005) and it will look at this model from a different perspective. By comparing these different extensions to the benchmark case of Grenadier and Wang (2005), several solutions are provided to how the owner of the project can overcome adverse selection using other methods. In total, three different extensions will be provided. Firstly, the impact of monitoring will be examined. Thereafter, no limited liability constraint will be imposed which implies that the wages can be negative. Lastly, a different agency setting is considered: one where the manager is the owner of the project who seeks external funding since he does not have sufficient assets on hand. All the extensions will be compared to the first-best solution and there benchmark case under adverse selection as provided in the model of Grenadier and Wang (2005). Based on these comparisons, a conclusion will be provided.
As stated in the previous sections, when no information asymmetries are present, the first-best solution for the owner of the project is the following:
P*(V0)owner = qH !!∗!!
!
(V*H + δH – I ) + (1 – qH) !!∗!!
!
(V*L + δL – I ) – c (36)
Which is the optimal option value at time zero for the owner, the option value for the manager is equal to zero. In the benchmark case where adverse selection plays a key role, this option value is changed to the following option values for the owner and the manager, respectively.
P(V0)owner = qH !!∗!! ! (V* H + δH – I ) + (1 – qH) !!∗!! ! (V* Z + δZ – I ) (37) P(V0)manager = qH !!∗! ! ! Δδ (38)
4.1. Monitoring
The first extension will assess the impact of monitoring. Monitoring to overcome an agency problem is discussed and examined extensively in literature, but the concept has never been applied to the real options model before. For example Townsend (1979) provides a model with asymmetric information where agents can only communicate information to other agents at a cost. Moreover, the realization of an investment is only known when a cost is made, and these costs can be characterized as monitoring costs. Furthermore, Lülfesmann (2002) examines a model with monitoring of managerial effort where it is found that monitoring leads to a reduction in productive efficiency and lower rents.
The model in this extension will focus on similar assumptions as described in Townsend (1979), where it is assumed that the owner can hire an external monitor. This external monitor could for instance be a board of directors, but the owner can also choose to monitor the manager by himself. Nevertheless, no matter how the owner decides to monitor the agent, monitoring comes at a cost. However, monitoring also ensures that the owner is able to observe the true value of the private component at the same time as the manager can. The monitoring costs are represented by τ and the owner needs to pay them before the investment takes place. Since monitoring guarantees that the owner can observe the true value of the private component at the same time as the manager, there is no ex-post incentive constraint. Namely, the owner observes the true value of the private component and this implies that the manager does not have the opportunity to divert this private value. Moreover, since the extension will only focus on adverse selection, the ex-ante incentive constraint is also not relevant in this analysis. The absence of both the ex-ante and ex-post incentive constraints indicate that only the ex-ante participation constraint and the limited liability constraint are relevant for determining the optimal contract value. Since the manager still needs to break even in expectation and the wages cannot be negative. The ex-ante participation constraint is defined as followed:
qH !!! ! ! wH + (1 – qH) !! !! ! wL – c ≥ 0 (39)
This maximization problem can be solved by setting wL equal to zero and solving for wH. This gives
the following wage for the manager of the high quality project.
wH = !!!
!
! !
!! (42)
Given that the costs of effort are low, this indicates that the wage will also be low. Because the effort costs are small, the manager requires very little compensation in order to break even and meet his participation constraint. Moreover, since value diversion cannot occur due to monitoring, no additional compensation is needed to stimulate the manager to exercise the option at the right trigger. In order to determine the exercise triggers, the wage as defined in equation 42 can be substituted in the option value and this gives the following result.
qH !!! ! ! (VH + δH – I) – qH !!! ! ! ! ! !! ! ! !! + (1 – qH) !! !! ! (VL + δL – I) (43) Rewriting gives qH !!! ! ! (VH + δH – I) + (1 – qH) !!! ! ! (VL + δL – I ) – c (44)
which is equal to the first-best solution. Therefore, both exercise triggers can occur at the first-best level. So in conclusion, when the owner decides to monitor the contract parameters are set as followed: VH = V*H VL = V*L WH = !!! ! ! ! !! WL = 0
And therefore, substituting for these values and rewriting, the following option values can be derived for the owner and the manager.
When comparing this option value to the first-best solution, it should be noted that the option value is equal to the first-best solution minus the costs of monitoring, denoted by τ. The only rents the owner pays the manager are to compensate for the costs of effort and therefore the option value for the manager is equal to zero. This is also the case in the first-best solution.
Additionally, the option value under monitoring can also be compared to the benchmark case when there is adverse selection. Since the monitoring costs τ are the only difference compared to the first-best solution, the owner will always choose to monitor the manager if the monitoring costs are sufficiently low. This can be shown by comparing the option value for the owner under monitoring to the benchmark case under adverse selection, which gives the following result.
(1 – qH) !!∗!! ! (V* L + δL – I ) – c – τ ≥ (1 – qH) !!∗!! ! (V* Z + δZ – I ) (47)
This equation states that the owner will decide to monitor the manager if the option value under monitoring is larger compared to the benchmark case. Monitoring lowers the option value through the cost of monitoring and wage the owner needs to pay the manager of the high quality project. However, this wage is equal to the costs of effort and these are assumed to be very low since the manager will always exert effort. Since these effort costs are low, this implies that the monitoring costs are the main determinant in deciding if the owner wants to monitor or not. In the benchmark case with adverse selection, the option value is lowered through a distortion of the optimal exercise trigger for the low quality project. If the monitoring costs are sufficiently low, the option value under monitoring is most likely higher compared to the benchmark case with adverse selection. However, when the owner faces very high monitoring costs it might be optimal for him to delay the exercise trigger as described in the benchmark case. Therefore, it can be concluded that the owner of the project will choose to monitor over distorting the exercise trigger when these monitoring costs are sufficiently low. If the monitoring costs are high, this will lower the option value more compared to the distortion of the exercise trigger and in that case monitoring is not preferred.
4.2. Not imposing a limited liability constraint
Another proposed extension is one where no limited liability constraint is imposed. Such an extension has never been applied in order to solve the adverse selection problem since it is very challenging to apply this in a real-life setting. Nonetheless, it is very interesting to examine what will happen to the option value if no limited liability constraint is imposed for the manager of the low quality project. Normally, the limited liability constraint ensures that the manager of the low quality project does not walk away when he observes his project value. However, this extension makes the assumption that the constraint is not imposed and that the manager will not walk away even though he receives a negative wage. Relaxing the limited liability constraint can be seen as imposing a penalty for the manager of the high quality project when he pretends to hold the low quality project.
As stated before, in a real-life setting it might be very hard to impose a negative wage and to make sure that the manager indeed pays this negative wage ex-post when the quality of the project is low. However, there are some cases that can be considered as a method of not imposing limited liability. For instance, the owner can ask the manager to provide part of the investment costs and if the outcome of the project is high he will get his contribution back. However, if the project fails or when the outcome is low the manager will lose his investment and this can be seen as the equivalent of receiving a negative wage. Another real-life application is one where the manager is obligated to buy stocks in the firm before the investment is made. If the outcome of the project is high, these stocks will rise in value and the manager will therefore gain from this. Consequently, if the quality of the project is low these stocks will decline in value and the manager will lose money. In both examples, the manager is asked by the owner to invest upfront and can therefore not walk away from his obligation ex-post. However, in this extension it is assumed that the owner can persuade the manager to pay this negative wage ex-post when it is observed that the outcome of the project is low. This is not a setting that will be observed empirically, but it is interesting to examine what will happen to the option value if this were possible.
In this extension, when no limited liability is imposed, the owner still wants to maximize his option value. As in the benchmark case, he faces some constraints to this maximization. Namely, even though no limited liability is imposed, the manager still needs to break even in expectation. Since he has a chance of receiving a negative wage when the outcome of the project is low, this needs to be compensated by a higher wage for the high quality project. This implies the following participation constraint. qH !!! ! ! wH + (1 – qH) !! !! ! wL – c ≥ 0 (48)
!! !! ! wH ≥ !!! ! ! (Δδ + wL) (49)
This states that the wage for the higher quality project must be larger than the private value plus the wage for the low quality project. Since the limited liability constraint is not imposed and the wage for the low quality project can be negative, this implies that when the option is exercised at the lower trigger the manager needs to pay an amount equal to wL to the manager. Therefore, the private
component that he would otherwise have kept for himself is lowered since a part of this private component is needed to pay the owner the negative wage. Due to the constraints, the maximization problem the owner faces is the following.
max qH !!! ! ! (VH + δH – I – wH) + (1 – qH) !!! ! ! (VL + δL – I – wL) (50) Subject to qH !!! ! ! wH +(1 – qH) !! !! ! wL – c ≥ 0 (51) !! !! ! wH ≥ !!! ! ! (Δδ + wL) (52)
First, the exercise triggers can be determined by taking the wages out of the brackets in the maximization problem and rewriting. This gives the following:
qH !!! ! ! (VH + δH – I) + (1 – qH) !!! ! ! (VL + δL – I) – qH !!! ! ! wH – (1 – qH) !!! ! ! wL (53)
Moreover, from the participation constraint it can be noted that.
– c = – qH !!! ! ! wH – (1 – qH) !! !! ! wL (54)
Therefore, we can write the option value for the owner as followed.
qH !!! ! ! (VH + δH – I) + (1 – qH) !!! ! ! (VL + δL – I) – c (55)
which is equal to the first-best solution. Therefore, both the exercise triggers VH and VL can occur at
can be defined. These wages can be determined by rewriting the ex-post incentive constraint for wH,
substituting it in the ex-ante participation constraint and rewriting for wL6.
wH = !!! ! ! (Δδ + wL) (56) wL = !!! ! ! c – qH Δδ (57)
Based on the equations it can be noted that the wage for the high quality project depends on the wage for the manager of the low quality project. Besides, the wage for the manager of the low quality project will be negative when !!
!!
!
c is smaller than qH Δδ. Since it is assumed the manager will
always exert effort, the costs of effort are very small and therefore it is likely that the wage for the manager of the low quality project is indeed negative. Moreover, the wage for the manager of the high quality project will always be positive. Specifically, if c is equal to zero, the wage for the manager of the low quality project is equal to – qH Δδ. However, since qH is smaller than one, this means that qH
Δδ is smaller than Δδ implying that the manager of the high quality project always receives a positive wage. Aditionally, both the exercise triggers can occur along the optimal path and therefore at the first-best level. In conclusion, all the parameters the owner can influence in this extension are defined as followed. VH = V*H VL = V*L WH = !∗!∗! ! ! (Δδ + wL) WL = !!! ! ! c – qH Δδ
And therefore, by substituting for the different contract terms and rewriting, the option value for the owner and the manager can be determined. This gives the following result, which is derived in appendix D.
6 It should be noted that there exists a different solution where the ex-post incentive constraint is not binding.
P*(V0)owner = qH !!! ! ! (VH + δH – I) + (1 – qH) !!! ! ! (VL + δL – I) – c (58) P*(V0)manager = 0 (59)
From these equations it should be noted that the option values for the owner and manager are equal to the first-best solution. Therefore, by not imposing a limited liability constraint the owner can overcome adverse selection in this model.
4.3. Manager seeks external funding
The model of Grenadier and Wang (2005) assumes that the owner seeks a manager to execute the project on his behalf. In this setting, the owner is the principal who delegates the investment decision to the manager. However, Tirole (2006) considers a different relationship where a manager owns a project but has insufficient assets to finance this project. Therefore he seeks an investor who can provide him with funding for this project. This extension will focus on this different setting and this is similar to the model as proposed by Morellec and Schürhoff (2011). As described in the introduction, this model makes the assumption that corporate insiders have superior information about the growth prospects of the firm. Although the manager is still the agent and the investor the principal in this setting, the relationship is different. Namely, the manager is in charge of the investment decision but since the investor provides the manager with the full investment costs he can be seen as the owner of the project. Moreover, it is assumed that the project only yields the stochastic, observable part V and that no private component is available. So the only payoff from the project is the observable part V, which evolves as a Geometric Brownian motion described by the following formula.
dV = α V dt + σ V dz (60)
The amount that the project yields will be distributed between the manager and the investor. This will be allocated as such that the investor is at least compensated for his investment costs and therefore breaks even. This will be denoted by VI and the remaining part will be allocated to the manager.
Although there is no private benefit, adverse selection still plays a key role since the manager has private information about the quality of his project. Specifically, the manager is aware of the probabilities of success for the project. The good manager knows that his project has a high probability of success and the bad manager knows that his project has a low probability of success. As stated before, when the project succeeds it will yield V, but if the project fails there will be no payoffs. This is illustrated below for both the good and the bad manager respectively, where pH > pL.
pH V pL V
Good manager Bad manager
1 – pH 0 1 – pL 0
pH VI ≥ I (61)
PL VI ≥ I (62)
Where equation 61 is the participation constraint for the good manager and equation 62 for the bad manager. Moreover, VI can be defined as V – VG for the good manager and V – VB for the bad manager. Furthermore, the probability that the investor faces a good manager can be characterized by q and the probability that he faces a bad manager can be characterized by 1 – q. However, when there are no information asymmetries, the investor can differentiate between the good and the bad manager and he can choose in which project he wants to invest.
The investor is willing to invest in the project as long as his participation constraint is met, since that allows him to break even. For both types of managers it is assumed that the participation constraint is met and therefore both managers will receive funding for their project in the absence of asymmetric information. Therefore, the optimal investment trigger for the good and the bad manager can be determined using the same arguments as described in appendix B. These values are different for the good and bad manager since the managers face different payoffs due to the different probabilities of success. The derivation of the investment triggers can be found in the appendix E.
V*G = ! !
!( ! !!) I (63)
V*B = ! !
!(! !!) I (64)
Since pH is larger than pL it follows from the equations that V*G is lower than V*B and therefore the
good manager will invest earlier than the bad manager. Since the bad manager has a lower probability of success it is optimal for him to wait longer before investing in the uncertain, irreversible project. In conclusion, when there is no adverse selection the option values for the good manager and the bad manager are defined as followed using the same arguments as in appendix C.
P*(V0)good manager = pH !∗!! ! ! VG (65) P*(V0)bad manager = pL !∗!! ! ! VB (66)
where the investment for the good and the bad manager will occur at the investment triggers V*G and
V*B respectively. Moreover, VG is the value the good manager will receive and VB is the value the bad
manager will receive. Where VG is defined as (V*
G – VI) and VB is defined as (V*B – VI). Moreover,
compensated for his investment costs and therefore breaks even. These option values can be considered the first-best option values in this extension and will later on be used for comparison when adverse selection is present.
4.3.1. Asymmetric information – pooling equilibrium
When there is asymmetric information, the investor cannot distinguish between the good and the bad manager. This means that upon investing he has no knowledge about the probabilities of success for the projects and therefore he faces a hidden information problem. Consequently, the bad manager can decide to mimic the good manager since this will increase his own payoff. By pretending to be the good manager, the investor believes the bad manager has a higher probability of success for the project and the investor therefore sets a different participation constraint. Accordingly, the bad manager needs to pledge a lower payoff to the investor when he mimics and this in turn increases his own payoff. This can also be shown using the participation constraints as defined in equation 61 and 62. Substituting VI for V – VG and V – VB and rewriting for VG and VB gives the following payoffs for
the good and the bad manager. VG = V – !!
! (67)
VB = V – !
!! (68)
From these equations it can be noted that VG > VB, which implies that the bad manager has an
incentive to mimic the good manager. When the bad manager chooses to mimic the good manager, a pooling equilibrium will arise. Namely, the investor does not know if he faces a good or bad manager, but he is aware of the proportion between good and bad managers. As introduced before, q can be noted as the probability of facing a good manager and 1 – q is the probability of facing a bad manager. This yields the following expected probability of success for the investor, ppooled, or pP.
pp = q pH + (1 – q) pL (69)
Where pL < pp < pH
Since the investor has no additional information, he will use this pooling probability as a benchmark for setting his participation constraint, which will therefore change to the following.
And rewriting gives the following payoff for both managers, which is represented by VM.
VM = V – !!
! (71)
Since the investor uses a pooling probability both managers need to pledge the same amount to the investor, which means they both receive the same amount of VM. Moreover, it can be noted that VB <
VM < VG. Hence, because of the pooling the good manager receives a lower payoff and the bad manager receives a higher payoff compared to the symmetric information case. Moreover, the optimal investment trigger for both managers will change due to this new pooling probability. This optimal investment trigger can be defined as V*P and is derived using the same arguments as in appendix B
and appendix E. V*p = ! !
!( ! !!) I (72)
Since pL < pp < pH it can be concluded that V*B > V*P > V*G which means that investment in the
pooling equilibrium occurs later for the good manager, but earlier for the bad manager. Furthermore, the amount the good manager receives in this pooling equilibrium is lower compared to his first-best case and therefore his option value is lower. The option values for both the good and bad manager in the pooling equilibrium are defined below.
P*(V0)good manager = pH !∗!! ! ! VM (73) P*(V0)bad manager = pL !∗!! ! ! VM (74)
4.3.2. Asymmetric information – separated
As stated in the previous section, the good manager loses from the pooling equilibrium and would therefore like to create a separating equilibrium. The investor still has no information about the probabilities of success, but the good manager can send an informative signal to convince the investor that he is the good manager. In existing literature, using the investment timing as an informative signal is examined extensively. From this literature it is concluded that firms with private prospects can speed up their investment decision to create a separating equilibrium and therefore they can reveal their prospects to the market. (Bustamente 2012). Moreover, the model of Morellec and Schürhoff (2011) finds a similar result and they state that adverse selection speeds up the investment: Firms with higher growth potentials invest earlier than firms with lower growth potentials.
These models suggest that the good manager in this extension can use the timing of his investment to signal his type to the investor. Specifically, by investing earlier, the option value will be lowered and this is more costly for the bad manager since his optimal investment trigger already occurs later compared to the good manager. Therefore, speeding up the investment, compared to delaying it, would make mimicking more costly for the bad manager since he now has to significantly change his investment strategy compared to his first-best solution. This indicates that the good manager can create a separating equilibrium and maximize his option value by exercising his option earlier at the trigger V. However, this maximization faces some constraints. Firstly, the bad manager faces a trade-off: He can at a higher cost pretend to be the good manager or he can decide to truthfully reveal his type to the investor and exercise at his first-best trigger. This implies a no mimicking-constraint and this is defined as followed.
pL !!! ! VG ≤ pL !∗!! ! ! VB (75)
The left-hand side of the equation represents the payoff for the bad manager if he decides to mimic and the right-hand side represents the payoff from revealing his true type. To make sure the bad manager is not willing to mimic, the payoff he receives from mimicking should be lower compared to the payoff he receives when he truthfully reveals his type. This will give him the lower value VB but at the first-best trigger that maximizes his option value. Furthermore, this maximization must also be subject to the participation constraint of the investor. Since the investment trigger will be changed to V, the payoff from the investment is changed which in turn changes the participation constraint to the following.
pH (V – VG) ≥ I (76)
max pH !!! ! VG (77) Subject to: pL !!! ! VG ≤ p L !∗!! ! ! VB (78) pH (V – VG) ≥ I (79)
The good manager wants to solve this maximization for VG and V in order to determine his option
value in this separating equilibrium. By rewriting the participation constraint, the value for VG can be
determined and this can then be substituted in the no mimicking constraint. This gives the following result, for which the derivation can be found in appendix F.
VG = V – !! ! (80) ! ! ! (V – !! !) = ! !∗! ! (V* B – !! ! ) (81)
Since it is not possible to solve equation 81 for V analytically, a numerical solution will be presented to determine if V is smaller or larger that V*G. In other words, this numerical solution will show if the
investment will occur earlier or later.
From equation 81 it can be noted that the only unknown part of the equation V is. Therefore, assumptions can be made about the values of the other parameters in this equation. The following values are assumed and V*G and V*B can be calculated using equations 63 and 64.
pH = 0.8 pL = 0.5 I = 10 x = 2
Using these values and filling them in for equation 63 and 64 gives the following values for V*G and
V*B.
V*G = !.!(!!!)! 10 = 25 (82)
V*B = !.!(!!!)! 10 = 40 (83)
parameters. This is shown in equation 84. ! !" ! (40 – !" !.!) = 0.0125 (84)
Substituting this back in equation 81 and filling out the remaining known values for the parameters gives the following expression.
! !
!
(V – !.!!") ≤ 0.0125 (85)
This can now be solved for V. The good manager wants to set V as high as possible, but not too high since that will violate the no-mimicking constraint. By using the solver function in Excel, the exact amount of V can be calculated. By setting V = 15.5051 the following result is obtained:
!
!".!"!# !
(15.5051 – !"
!.!) = 0.0125 = 0.0125 (86)
And therefore V should be set equal or smaller than 15.5051 in order to making mimicking too costly for the bad manager. Moreover, it is shown that V < V*G < V*B and consequently the good manager
can create a separating equilibrium by investing earlier. When V is set smaller than 15.5051 it will be too costly for the bad manager to mimic the good manager and therefore he will therefore decide to invest his option according to his first-best solution.
So in conclusion, the option values for the good manager and bad manager in this separating equilibrium are the following:
P*(V0)good manager = pH !!! ! VG (87) P*(V0)bad manager = pL !∗!! ! ! VB (88)
where the payoff for the bad manager is equal to his first-best solution. Moreover, it should be noted that the good manager will only decide to create this separating equilibrium when his option value is higher compared to the option value in the pooling equilibrium. This is shown by the following equation. pH !!! ! VG ≥ pH !∗!! ! ! VM (89)
