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Contracting on real option payoffs
Journal of Economics and Business 58 (2006) 20–35
Contracting on real option payoffs Nicholas Wonder ∗ Department of Finance and Marketing, College of Business and Economics, Western Washington University, Bellingham, WA 98225-9073, USA Received 29 June 2004; received in revised form 25 May 2005; accepted 1 June 2005
Abstract We consider the design of contracts that pay managers on the basis of a project’s payoff. We show that a contract that induces appropriate timing of project investment by a privately informed risk neutral manager will not offer proper incentives to forgo perk consumption at the time of investment. Under the firm’s optimal contract within the class of payments that do not depend on time of investment, the manager waits too long to invest. The optimal contract can be implemented with a call option on the project payoff. © 2005 Elsevier Inc. All rights reserved. JEL classification: G31 Keywords: Real options; Agency; Timing
1. Introduction The real option literature offers prescriptions on value maximization in the presence of various types of flexibility, but there is limited instruction available regarding the problem of motivating self interested managers to exercise real options appropriately. We consider a firm with a risk neutral manager who privately observes a continuously changing signal, which may or may not be noisy, of a potential project’s profitability. On the basis of the signal he observes and the form of his compensation, the manager may at any time invest some of the firm’s money to develop the project. When he invests, he has the opportunity to consume some perquisites, which we can broadly construe as managing the project in a way that appeals to the manager but is not value-maximizing. Perk consumption reduces the project cash flows. Finally, ∗
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the cash flows are realized and the manager is paid on the basis of those cash flows. Assuming the manager lacks sufficient capital to buy the firm, salary functions that induce first best timing of exercise will result in excess perk consumption. The firm’s value-maximizing (net of perks) contract results in some loss associated with both perk consumption and delayed investment. We consider contracts with payoffs that do not explicitly depend on the time of exercise. Within this class, optimal contracts are not found in closed form, but under some assumptions the numerical requirements are minimal. In addition to inducing delayed investment, the optimal contracts in some parameter ranges give the manager a higher valued salary package than the amount he could get by refusing the contract and accepting an outside opportunity. When the manager’s reservation value is very low, it is in the firm’s interest to pay him more than he requires, since incentives can be greatly improved. Since we assume the manager cannot consume perks prior to investment in the project, one might expect him to overinvest. The fact that the manager receives a cash payment that can vary with project payoff, however, means that the manager potentially has a reason to wait. Actually, there are two reasons why, given an optimal contract, the manager waits longer on average to invest than he would absent an agency problem. First, taking as given the result that there will be some perk consumption upon investment in the project, the firm reduces the expected discounted deadweight loss from perks by inducing the manager to wait longer. Second, the sort of salary function that can limit the manager’s desire to consume perks when he finally does invest has the unavoidable effect of delaying the manager’s decision to invest. Given this tradeoff between inducing desired timing and eliminating perks, the firm’s optimal contract does involve a positive expected deadweight loss from perks. The intuition for the tradeoff can be obtained by assuming the manager’s signal of project value is perfect, which is one case we will consider, and assuming that the firm offers him a contract that looks like a call option on the payoff from the project. We will see that such a contract can do as well as any other. Then to completely eliminate the deadweight loss of perk consumption, the firm would like to set the slope of the call option payoff equal to one, so that the manager bears all the cost associated with the perks. On the other hand, the firm does not want to give the manager a contract that is worth more than the manager requires. So it can set the strike price on this contract high to reduce its value. But the optimal exercise level for an American call option rises as the strike price rises, so setting a high strike price means the manager will wait too long to invest. The best contract for the firm sets the strike a little high so the manager waits a little too long, and has a slope of less than one so some perk consumption/mismanagement results. Do the project cash flow-based options derived in this paper match real world management compensation? We have not discovered precisely equivalent contracts, but there are indications that project compensation is growing in general. In any event, our model can be valid in a prescriptive sense even if its suggestions are not currently widely followed. The paper is organized in the following way. Section 2 discusses related literature. Section 3 introduces the assumptions of the model. In Section 4 the problem in the absence of agency problems, the “first best” case, is solved. Sections 5 and 6 determine the manager’s optimal response to any contract and describe requirements that the optimal contract for the firm must satisfy, given that response. Section 7 represents numerical comparative statics, Section 8 presents implementation of the optimal contract, and Section 9 concludes.
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2. Literature review Other agency models of investment timing have also appeared recently. Kim and Yoon (2000), Antle, Bogetoft, and Stark (2001), and Arya, Glover, and Routledge (2002) (the latter with two managers) consider two period settings with various assumptions on the information structure. The most closely related paper to our work is Grenadier and Wang (2005). In their continuous time model, a manager exerts effort at time zero, influencing the likelihood that a privately observed component of project quality will be high or low (there are only two levels of quality). Manager and firm owner both observe innovations to project value. The optimal contract results in delayed investment by managers with unfavorable private information. The model also shares our result that the manager may receive a wage in excess of his reservation level. While both that paper and this one have asymmetric information and moral hazard, the specification is substantially different. First, our perk consumption occurs only at the time of investment, which is at least plausible in that it may be easier to conceal misuse of funds in the context of a large investment expenditure. Second, we have no information asymmetry at the signing of the contract,1 but the manager observes innovations to project value due to day to day monitoring. Time zero asymmetry in Grenadier and Wang’s paper means that, even given a cost of effort such that effort is not a consideration in the optimal contract, the firm still must pay informational rents and therefore fails to achieve first best. Outcomes in our model, in contrast, approach first best as the costs of perk consumption become increasingly convex, while both perk consumption and investment timing drive the contract terms for any parameters short of the limit. Another distinction is that we allow for the manager’s signal to be imperfect, while, at the moment of investment, the manager in Grenadier and Wang (2005) knows exactly the project payoff that the firm owners will observe. Maeland (1999) also presents a continuous time model with time-zero information asymmetry, but his model includes neither effort nor perk consumption. Only limited empirical evidence directly relates to our work. Guay (1999) finds that firms with growth options have managers whose salaries are more strongly increasing in the volatility of firm asset value. That is, their contracts are more convex. Our model involves a firm with a growth option and yields convex optimal contracts. To that extent, Guay’s findings are consistent with our predictions. 3. The model A firm has available a perpetual option to undertake a project at a cost K. A manager observes a signal S of the project’s value, with dS = µSdt + σSdW
(1)
where W is a standard Brownian motion. S may or may not be a noisy signal of project cash flows as we discuss later in this section. The signal changes as demand conditions, probable productions costs, and competitive pressure, for example, change. The manager is given the right to exercise the real option at any time. When he does, the firm invests K in the project. At exercise, the manager can choose φ, the value he receives from perk
1 An early version of our paper considered continuous menus of contracts (the counterpart to Grenadier and Wang’s pair of contracts) when initial project value, in addition to the innovations, was observed only by the manager.
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consumption.2 Given the choice of φ, project gross cash flows are assumed to be C ≡ [S − γ(φ)]ε
(2)
where • γ is an increasing, convex differentiable function representing the cost to the firm of perk consumption. We set γ(0) = 0. If in addition γ (0) = 1, then the first-best level of perks is 0,3 but social loss is small initially. Convexity of loss due to perks is a reasonable assumption. The manager can make use of some of the project’s assets just as well as the firm can, and these are the ones he would divert first. Eventually he will encounter diminishing returns, however, and an effort to hide his activities would reduce the efficiency of asset transfer even more. We will assume the maximum level of perks is such that γ(φ) = S—the manager cannot usurp more than the project itself. • ε is a random variable with non-negative support and, without loss of generality, mean 1. This implies that if the manager did not divert any assets, the signal observed at the instant before exercise would equal the expected cash flow from the project. We can allow the case where ε is always 1, so that the manager faces no residual uncertainty at exercise. If ε is not degenerate at 1, we require it to have a differentiable density function that is positive at all values above 0 and has no mass points except possibly at zero. The manager immediately receives payment M(C). It is assumed that it is not possible to contract on the time that exercise occurs. Also, it will be convenient if the manager’s expected payoff is differentiable with respect to the manager’s signal. Sufficient conditions on M to ensure this are not very onerous. We assume that the manager has no capital with which to buy the project from the firm. Also, we will require that the manager’s salary, at least in equilibrium, is never negative and that the firm makes no salary payments prior to exercise. The latter is not restrictive given our other assumptions. Manager and firm are risk neutral and discount future payoffs at the rate ρ. The manager attempts to maximize the discounted expected value of his cash payment and perks, M + ϕ, while the firm maximizes the discounted expected value of C − M − K (recall that C is the project cash flow net of the effect of the manager’s perk consumption). We will require that the discount rate exceed the drift in project value, so that, absent agency problems, there exists some level of the signal S at which investment should occur. At the beginning the manager’s discounted expected payoff (salary plus assets diverted) must be at least w, the value of his outside opportunity. One last assumption is that the parameters are such that the first best investment policy is not immediate exercise, although most of the analysis holds without that requirement. Finally, note that while our firm is modeled as consisting of an option and nothing else, the presence of other assets is inconsequential, provided we can still contract on the cash flow from this particular project.
2 We assumed that perk consumption is not possible prior to investment. In some settings, it might be more reasonable to assume that perks are gradually consumed over time. Then perk consumption could reduce the drift of the project value. That specification would complicate the analysis, but it may be worthy of further study, as it is not clear to what extent the conclusions here would change. 3 This follows from the fact that the marginal net gain from one additional unit of perk consumption is 1 − ␥ . Due to the convexity of γ, this marginal gain will be negative for φ > 0.
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4. The firm’s problem and first best analysis The firm’s problem is to maximize its discounted expected profits net of perks, subject to the manager receiving at least his reservation utility and choosing his stopping rule and asset diversion optimally: ∞ max e−τρ E[C − M(C) − K]f (τ)dτ, M(C),S ∗ ,φ 0 ∞ e−τρ E[M(C) + φ(τ)]f (τ)dτ ≥ w subject to (3) 0 ∞ ˜ e−τρ E[M(C) + φ(τ)]f (τ)dτ and S ∗ , φ ∈ arg max S˜ ∗ ,φ˜
0
where τ is the investment time, f(τ) the probability density function for exercising at time τ, and S* is the signal level at which the manager invests.4 It will be useful to express the manager’s expected payoff conditional on exercise as Q and his initial discounted expected payoff q. Then Q(τ) = φ(τ) + E[M(C)] Also, E(C) = (S* − γ(φ)) so the maximization can be written as ∞ max∗ e−τρ [S ∗ − K + φ − γ(φ) − Q]f (τ)dτ, M(C),S ,φ 0 ∞ e−τρ Qf (τ)dτ = q ≥ w subject to 0 ∞ e−τρ Qf (τ)dτ and S ∗ , φ ∈ arg max S˜ ∗ ,φ˜
(4)
(5)
0
The term (φ − γ(φ)) in the firm’s objective function represents subtraction of the deadweight loss due to perks. This deadweight loss is the difference between perks received by the manager and loss in project expected payoff attributable to those perks. As a basis for future comparison, we state the first best outcome, which is standard for a perpetual call option (for example, Dixit & Pindyck, 1994, p. 142): Proposition 1. Under the assumption γ (0) ≥ 1 (so that zero diversion of assets is optimal)5 the first best exercise level satisfies ∗ Sfb =
Kβ β−1
and the firm’s net starting value is β 1−β K S0 − w, β β−1
(6)
(7)
4 In principle, the manager’s threshold signal level, choice of perks, and resulting expected salary could depend on time. But the perpetual nature of the investment opportunity and the assumed time-independence of the salary function combine to create a time-independent setting. 5 In the case of γ (0) < 1, the expression for the first-best S* would replace K with K − φ + γ(φ ) < K. fb fb
N. Wonder / Journal of Economics and Business 58 (2006) 20–35
where 1 µ β= − 2 + 2 σ
µ 1 − σ2 2
2 +
2ρ . σ2
25
(8)
∗ is proportional to the required investment K and is decreasing So the critical signal level Sfb in β. β is decreasing in drift µ and volatility σ 2 and increasing in discount rate ρ. The firm waits longer to invest, if by waiting it is more likely to benefit from the drift and diffusion or if it is 2 more patient. Also, if µ − σ2 ≥ 0, investment will eventually occur with probability 1, since the signal will hit any barrier above its initial value under this condition.
5. Second best: the manager’s problem To derive the second best (time independent) contract, we first need to find the manager’s optimal action as a function of the contract the firm offers him. Then we will use those results to write the firm value as a function of certain properties of the contract, and finally maximize that value. The following result employs the manager’s first order conditions to find his optimal choices. Lemma 1. Assuming there is an interior solution for the manger’s problem, his choice of exercise level S* satisfies 1 −β qβ 1−β 1−β ∗ S = S0 (9) Ω and he chooses perks φ such that γ (φ) = where Ω ≡
1 , Ω
∂ ∗ ∂S ∗ [E(M(C)|S , φ)],
(10) evaluated at the manager’s optimal S* .
We are able to write the manager’s optimal investment trigger S* and asset diversion choice φ as functions of the manager’s salary sensitivity Ω and discounted payoff inclusive of perks q, along with the exogenous variables S* and β. For that reason, we will now view the firm’s problem as one of choosing sensitivity and discounted payoff, rather than the full salary function M. From Eq. (9), we see that exercise level is increasing in the ratio (Ω/q). That is, the manager’s timing decision rises as the marginal sensitivity of his expected salary at the optimum increases relative to his total discounted expected value. This relation follows from consideration of what a manager stands to gain or lose by waiting to invest. If the manager waits another period, he loses the time value of his salary and perks. The amount of this loss (the interest he could have earned on his salary) is related to the level of his payoff, and, in turn, to his discounted payoff. The potential gain to waiting depends on the exogenous drift and diffusion terms, but also on the sensitivity of the manager’s payoff to changes in project payoff, Ω. Given a certain level q, a higher sensitivity implies a longer time to investment. Turning to the choice of perks, the condition (10) means the manager diverts assets up until the marginal loss associated with doing so equals the gain. The marginal loss depends on the marginal loss to the whole firm, represented by the slope of the perk cost function, and it also depends on the portion Ω of the firm’s marginal loss that the manager bears. If the manager has an Ω of 1, he will make the first best perk consumption decision.
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The first order condition tells us the optimal choice of perks so long as there is an interior optimum. There do exist parameter values for which a locally optimal perk choice is not the global maximum. This issue is a second reason the firm might need to pay the manager more than his outside option.6 6. Second best: the firm’s optimal contract Given the results on the manager’s optimal behavior, we can remove the incentive compatibility constraint from the earlier expression of the firm’s problem and instead incorporate the manager’s choices of investment time and perks into the quantity the firm is maximizing: ∗ −β S , subject to q ≥ w (11) max(S ∗ − K − Q + φ − γ(φ)) × q,Ω S0 We substitute in for S* , φ, γ(φ), and Q, each of which can be written as functions of q, Ω, and the exogenous variables. For our numerical results, we will need a specific functional form for the loss due to perks. Let the reduction in project value from asset diversion be quadratic: γ = yφ + zφ2
(12)
If y = 1, as we will assume for the numerical calculations, γ (0) = 1, so that small amounts of perks produce very small deadweight losses. From the first order condition (10), the optimal perk choice φ satisfies φ=
(1 − Ωy) 2Ωz
(13)
Differentiating the objective function in (11) with respect to Ω and simplifying produces the following first order condition. 2 −β 1−2β 1 β (y − 2y) Ω 1−β −(qβ) 1−β S01−β + Ω2 −K (1 − β) 4z 2β −β +Ω + =0 (14) 4z(1 − β) 4z(1 − β) −2y) − K ≤ 0, (true for y = 1) this equation has The optimal Ω solves this equation. Provided (y 4z one and only one solution for Ω ∈ (0,1]. At this point, we can summarize the dilemma for the firm. Solving the perk problem necessitates a steeply sloped contract, that is, setting Ω equal to one. But if the manager is not to be given the entire value of the project, his optimal timing policy will then involve waiting too long. The second best contract trades off these two goals. The outcome of the tradeoff is the following proposition, our main claim: the second best contract produces underinvestment and a positive 2
In particular, the manager can pick perks such that γ(φ) = S and exercise the option immediately at S0 . That is, he can divert all the invested assets of the project so that its remaining value is zero. His value from this strategy is γ −1 (S0 ), so any incentive compatible salary offered by the firm must offer at least that value. If, however, the manager’s signal is noiseless, the corner solution is not a significant problem if the contract can threaten punishment (negative M) for low values of C. Even aside from that case, monitoring could remove the possibility of complete asset diversion. 6
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expected deadweight loss from perks. This is not dependent on the quadratic form of the perk cost function. ∗ and φ > φ . Proposition 2. Assume γ (0) ≤ 1.7 Then under the second best contract, S ∗ > Sfb fb
When the reservation wage w is low the firm will offer a salary that gives the manager discounted value q strictly greater than w. If q is quite small, the firm can increase the manager’s pay by a small amount in absolute terms but a large amount in percentage terms. This will change the manager’s optimal decisions significantly, and have an impact on firm value that exceeds the extra amount the manager received. So in this instance q will be set by its first order condition from the firm’s maximization. The next section presents the sort of parameter values that result in q being determined by either the manager’s outside opportunity or the first order condition. 7. Numerical comparative statics In this section we consider how characteristics of the optimal contract depend on different parameters of the model. We start with a base case where the expected project payoff equals required investment: S0 = K = 100. We set the perk cost coefficients y and z equal to 1, with that choice of y implying that a small amount of perk consumption generates only a small deadweight loss. Finally, we set β = 6.391, and the manager’s reservation wage is w = 2. From that point, we vary z, w, k, and β. β = 6.391 can result from various combinations of µ, σ, and ρ, including µ = −.1, σ = .2, ρ = .05, as well as (−.092, .2, .1) and (.01, .089, .2).8 When reporting the probability of eventual investment and the median time to investment, we assume that µ = −.1 and σ = .2. The negative drift might represent a significant probability that the investment project will become obsolete as time passes.9 We do consider different drifts in Table 4. Also, increasing the drift would result in higher values for β if volatility and discount rate were not changed, so that one can roughly characterize the impact of changing drift using our table with changing β. Under the base case parameter values, the first best project value is 6.2521 for a net firm value of 4.2521, ∗ = 118.55, the first best probability that investment ever occurs is .3603, and, conditional on Sfb investment happening eventually, there is a 50% chance that it will occur within .74 years. We do not claim that our parameter choices represent an average project, but the comparative statics do not appear highly sensitive to the base case. The results stated in the tables assume that there is no possibility of immediate diversion of all the assets, the corner solution mentioned earlier. That means they are most likely to apply when the manager faces no residual uncertainty about firm cash flows. Table 1 shows that a higher convexity of perk costs (higher z) implies a lower agency problem, as the firm is able to offer less powerful incentives (a lower Ω) and has a value closer to the first best. In particular, raising z from .01 to 20.48 means Ω falls from .7887 to .323 while firm value rises from 2.448 to 4.234. For z = .02 and below, discounted value to the manager exceeds his outside opportunity (q > 2).
7 We make this assumption (that the social loss due to perks is initially small) simply because alternative assumptions might result in zero perk consumption under the second best case as well. 8 Drift, volatility, and discount rate are captured by the summary parameter β in this model because the discount factor associated with the first passage time to any signal value also depends on those three variables only through β. β is the positive root of a quadratic equation in S that describes that discount factor (see the proof of Proposition 1 in Appendix A). 9 A constant probability through time that the value of the project jumped to zero because of obsolescence would likely yield similar results to a negative drift term, though we have not attempted to solve that case.
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Table 1 Characteristics of second best contract as a function of z, the quadratic coefficient on cost of asset diversion z
Ω
q
V
S*
Pr(Inv)
Med(τ)
γ −φ
.01 .02 .04 .08 .16 .32 .64 1.28 2.56 5.12 10.24 20.48
.789 .703 .618 .544 .480 .428 .388 .360 .343 .332 .326 .323
2.20 2.08 2 2 2 2 2 2 2 2 2 2
2.448 2.635 2.874 3.141 3.413 3.664 3.871 4.024 4.124 4.183 4.216 4.234
137.66 136.17 133.93 130.81 127.80 125.10 122.87 121.20 120.07 119.37 118.98 118.77
.147 .157 .173 .200 .230 .261 .291 .315 .334 .346 .352 .356
1.78 1.70 1.58 1.41 1.24 1.09 .97 .88 .82 .78 .76 .75
1.795 2.222 2.398 2.201 1.839 1.401 .971 .615 .360 .198 .104 .054
Each row doubles z. Table shows results assuming that diversion of all assets is not possible. The first best firm value is 4.252. S* is the manager’s optimal signal value at exercise, Ω is the equilibrium sensitivity of payment at S* , q is the manager’s expected discounted value, V is net firm value after expected payment to manager, γ − φ is deadweight loss at exercise (not discounted) from perk consumption, and Pr(Inv) is the probability that investment ever occurs. Med(τ) is the time such that, conditional on investment occurring at some point, investment will have happened with 50% probability. All rows assume w (dollar value of outside payment) = 2, S0 (initial signal value) = 100, K (strike price) = 100, y (linear coefficient in asset diversion cost function) = 1, β (function of discount rate and signal drift and diffusion) = 6.3912. To compute Pr(Inv) and Median(τ), it is assumed µ = −.1 and σ = .2.
Regarding the time until investment, the probability of eventual investment is as low as .147 for z = .01, with a corresponding median time of 1.78 years to investment. These values approach the first best levels as convexity rises, corresponding to an improved trigger level S* . The relation between z and S* 10 means that firms with high agency costs (low z) will be more profitable than others, if and when investment finally occurs. The table also shows how the deadweight loss realized at exercise, γ − φ, declines as perk cost convexity rises. Given the significance of z in determining the characteristics of the contract (more about this appears in the next section) as well as the actions of the manager, connecting it to observable quantities is desirable. Most directly, z measures how destructive it is to misuse the assets of the firm. When assets are intangible and highly specific, z (and/or y, the linear cost of perks) should be high. When there is a high variance in costs from suppliers, then it is more costly for a CEO to offer a supplier contract to his cousin and z would be high. Linking firm characteristics such as intangible assets to low agency costs is perhaps in conflict with standard views and intuition. More intuitive implications may follow if we regard the quadratic cost component as related to managerial cost of hiding perk consumption. Then well governed firms have high z. Here we can draw on existing literature about characteristics connected to good monitoring and governance. A high z firm might have a small board with independent directors and a chairman who was not its CEO, or it would have large external shareholders. It would be followed by security analysts and it would access the capital market frequently. A high ratio of short-term to total debt is one indicator of the latter characteristic.11 Or, more precisely, the relation between S − γ − M (firm value net of perks and manager salary) and z. Doukas, Kim, and Pantzalis (2000) study the role of security analysts in monitoring, while Comment and Jarrell (1995) suggest the short-term debt ratio as a measure of reliance on external markets. 10
11
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Table 2 Characteristics of second best contract as a function of w, the manager’s reservation wage w
Ω
q
V
Vfb
S*
Pr(Inv)
Med(τ)
γ −φ
.5 1 1.5 2 2.5 3 3.5 4 4.5 5
.265 .265 .304 .369 .437 .507 .580 .654 .730 .806
1.19 1.19 1.5 2 2.5 3 3.5 4 4.5 5
4.304 4.304 4.243 3.976 3.599 3.167 2.705 2.227 1.740 1.247
5.572 5.252 4.752 4.252 3.752 3.252 2.752 2.252 1.752 1.252
126.1 126.1 123.9 121.7 120.5 119.8 119.3 119.0 118.8 118.7
.249 .249 .277 .307 .326 .339 .346 .352 .355 .357
1.15 1.15 1.03 .91 .84 .80 .78 .76 .75 .75
1.92 1.92 1.31 .73 .42 .24 .13 .07 .03 .01
S* is the manager’s optimal signal value at exercise, Ω the equilibrium sensitivity of payment at S* , q the manager’s expected discounted value, V the net firm value after expected payment to manager, Vfb the firm value in the first best case, γ − φ the deadweight loss at exercise (not discounted) from perk consumption, and Pr(Inv) is the probability that investment ever occurs. Med(τ) is the time such that, conditional on investment occurring eventually, investment will have happened with 50% probability. Table assumes S0 (initial signal value) = 100, K (strike price) = 100, y (linear coefficient in asset diversion cost function) = 1, z (quadratic coefficient) = 1, β (function of discount rate and signal drift and diffusion) = 6.3912. To compute Pr(Inv) and Median(τ), it is assumed µ = −.1 and σ = .2. Results assume that diversion of all assets is not possible.
Looking at different outside opportunity (w) levels in Table 2, we see that the firm will offer the manager a package worth 1.190 even when his outside options are less than that. So the maximum possible firm value is 4.304. This table illustrates the fact that, as the manager’s share of firm value becomes large, it becomes easier to align his incentives with the firm, and agency costs become small. In particular, as w approaches the first best project value of 6.252, the manager’s investment threshold approaches first best, deadweight losses from perks approach zero, and firm value approaches its first best level. Table 3 shows how the contract characteristics vary as the exercise price K goes from 85 to 115. For the low levels of K, firm value is of course higher. But agency loss from first best firm value is also higher in absolute and percentage terms. An explanation is that since the manager’s outside opportunity is held fixed, his share of first best firm value is smaller for the lower values Table 3 Characteristics of second best contract as a function of K, the exercise price of the real option K
Ω
q
V
Vfb
S*
∗ Sfb
γ −φ
85 90 95 100 105 110 115
.2780 .2734 .3084 .3690 .4497 .5552 .6900
2.954 2.146 2 2 2 2 2
10.165 7.514 5.565 3.976 2.706 1.709 .936
13.016 9.034 6.244 4.252 2.806 1.740 .943
107.44 113.65 117.75 121.73 126.28 131.32 136.72
100.77 106.69 112.62 118.55 124.48 130.40 136.33
1.69 1.77 1.26 .731 .374 .160 .051
is the equilibrium sensitivity of payment at S* , q is the manager’s expected discounted value and V is the net firm value ∗ are the values obtained in the first best case, and γ − φ is deadweight loss after expected payment to manager. Vfb and Sfb at exercise (not discounted) from perk consumption. All rows assume w (dollar value of outside payment) = 2, S0 (initial signal value) = 100, y (linear coefficient in asset diversion cost function) = 1, z (quadratic coefficient) = 1, β (function of discount rate and signal drift and diffusion) = 6.3912.
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Table 4 Characteristics of second best contract as a function of µ, the drift of the project payoff µ
β
Ω
q
V
Vfb
% Loss
S*
∗ Sfb
Med(τ)
−.14 −.12 −.10 −.08 −.06 −.04 −.02 0 .02 .04
8.30 7.34 6.39 5.46 4.55 3.68 2.87 2.16 1.58 1.16
.469 .419 .369 .319 .268 .218 .189 .161 .128 .080
2 2 2 2 2 2 2.4 3.1 4.0 4.8
2.59 3.20 3.98 5.01 6.44 8.52 11.8 17.7 29.0 55.4
2.72 3.38 4.25 5.43 7.11 9.62 13.6 20.5 33.4 61.0
.046 .054 .065 .078 .094 .114 .133 .140 .130 .091
115.3 118.0 121.7 127.2 135.6 150.0 171.8 215.0 330.5 1046.7
113.70 115.8 118.5 122.4 128.2 137.3 153.5 186.3 272.1 731.7
.48 .65 .91 1.35 2.14 3.79 7.21 17.2 78.5 83.3
Pr(Inv) .321 .314 .307 .301 .295 .296 .338 .465 1 1
Ω is the equilibrium sensitivity of payment at S* , q is the manager’s expected discounted value and V is the net firm value ∗ are the values obtained in the first best case. % Loss is percentage by after expected payment to manager. Vfb and Sfb which V is less than Vfb . All rows assume σ (volatility of project payoff) = .2, ρ (discount rate) = .05, w (dollar value of outside payment) = 2, S0 (initial signal value) = 100, K (required investment) = 100, y (linear coefficient in asset diversion cost function) = 1, and z (quadratic coefficient) = 1.
of K, leading to higher agency costs. However, unreported results show that, even if w is held at a constant fraction of first-best firm value, lower exercise prices lead to firm value deviating by a greater percentage of first-best value. In Table 4 the drift is varied. Other parameters are held fixed, including volatility of .2 and discount rate .05. The effect of increasing drift is somewhat similar to the effect of reducing exercise price, as the manager’s share of firm value falls as firm value rises. However, for high levels of drift, the percentage loss from first best firm value starts to drop. This is something of a surprise. Value lost does consistently rise with drift in absolute terms, however. The median time to investment and probability of eventual investment steadily rise with higher drift.12 In Table 5 we consider different values of β, the summary parameter for drift, diffusion, and discount rate. Recall that β is increasing in the discount rate and decreasing in the drift and volatility. Therefore, this exercise is similar to that portrayed in Table 4, except that we now also vary the manager’s reservation wage so as to keep it at a constant proportion of first best firm value. This may be representative of the variation we would see in the cross section. As β increases (that is, as the discount rate rises or the drift or volatility falls), agency costs (Vfb − V) stay approximately constant, therefore rising as a percentage of the falling first best value. So we have the contrast that projects that are long shots due to high required investment (relative to initial signal S0 ) experience low agency costs, while projects that are long shots due to high β have high agency costs. 8. Implementation of the optimal contract Once the characteristics of the optimal contract have been found, the firm must find a function that implements it. This function will have to fit the expected amount and sensitivity of the salary
Another possible experiment is raising the drift but holding β fixed, by lowering volatility or raising the discount rate. Given either type of adjustment, a higher drift leads to greater probability of eventual investment and longer median wait to investment. 12
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Table 5 Characteristics of second best contract as a function of β, holding w constant as a percentage of first best project value β
w
q
V
Vfb
Ω
S*
∗ Sfb
2.39 2.89 3.39 3.89 4.39 4.89 5.39 5.89 6.39 6.89 7.39 7.89
6.30 4.96 4.09 3.48 3.03 2.69 2.41 2.19 2 1.84 1.71 1.59
6.30 4.96 4.09 3.48 3.03 2.69 2.41 2.19 2 1.84 1.71 1.59
13.105 10.248 8.402 7.110 6.155 5.421 4.839 4.367 3.976 3.647 3.670 3.125
13.388 10.541 8.697 7.405 6.448 5.710 5.124 4.648 4.252 3.919 3.634 3.388
.3323 .3375 .3425 .3474 .3521 .3566 .3609 .3650 .3690 .3729 .3766 .3802
176.6 157.3 145.9 138.5 133.2 129.3 126.2 123.7 121.7 120.1 118.6 117.4
171.9 152.9 141.8 134.6 129.5 125.7 122.8 120.4 118.6 117.0 115.7 114.5
Ω is the equilibrium sensitivity of payment at S* , q is the manager’s expected discounted value and V is the net firm value ∗ are the values obtained in the first best case, and γ − φ is deadweight loss at after expected payment to manager. Vfb and Sfb exercise (not discounted) from perk consumption. All rows assume y (linear coefficient in asset diversion cost function) = 1, z (quadratic coefficient) = 1, K (required investment) = 100, and the initial signal S0 = 100. The table presents only results when diversion of all assets is not possible.
payment at the manager’s optimal S* and φ. That is, we use the results from above to find S* − γ(φ) and Q. Then we need to produce a contract such that E[M(ε(S ∗ − γ(φ)))] + φ = Q
and
∂ (E[M(ε(S ∗ − γ(φ)))]) = Ω. ∂S ∗
Now the distribution of the random variable ε comes into play, as the expected value of a function of ((S − γ)ε) depends on more than just E(ε). Also, we will require that the manager’s payoff at exercise be non-negative, at least in the case of residual uncertainty at the moment of investment. Any function that produces the desired managerial payoff value and sensitivity yields the same value for the firm. Here we consider contracts that take the form of call options on C. The manager’s salary M(C) is zero for C < C1 and is set equal to (C − C1 ) × C2 for C ≥ C1 .13 This solution is not unique, since there exist many functions that have a particular level and derivative at a certain point. But options are the least complex contract that will work, and option-based compensation is commonly observed in practice.14 Again consider the base case parameters {w, β, S, K, y, z} = {2, 6.1239, 100, 100, 1, 1}. The solution includes sensitivity Ω = .3690. We consider one example with residual uncertainty at the moment of investment and one without. (1) Truncated normal distribution: f(ε) = 0, ε < 0, ε ∼ N(.99999, .2763), ε > 0, Pr(ε = 0) = .0001. This distribution has mean 1. Using numerical methods, we find that {C1 ,C2 } = {134.81,
13
Earlier we assumed that the manager’s expected salary is differentiable with respect to his signal. The call option described here violates that condition if the manager faces no residual uncertainty, but the point of non-differentiability will be below the manager’s exercise level such that it is not a concern. 14 Of course, typical real-world employee options do not have payoffs that depend on cash flows of individual projects, nor is their exercise triggered by project investment.
32
N. Wonder / Journal of Economics and Business 58 (2006) 20–35
Fig. 1. Expected salary as a function of (S − γ) Plots display manager’s expected salary as a function of (S − γ), using S0 = K = 100, z = y = 1, β = 6.3912, w = 2, and either truncated normal distribution of ε or no residual uncertainty.
.8595}. If the manager invests at S* = 121.732 and is close to its mean, he will receive a salary of zero. (2) ε = 1 with certainty: Then the slope of the contract, C2 , is simply equal to Ω; {C1 ,C2 } = {103.42, .3690}. Notice that the example with residual uncertainty has a higher slope on the contract. In fact, other parameter choices suggest a positive relation between variance of ε and slope of contract, but that is likely sensitive to the risk neutral model we employ. For these two cases, Fig. 1 shows the manager’s expected salary as a function of S − γ, his signal net of the cost of perks. At S − γ = 120.146 (which occurs given the optimal contract),
Fig. 2. Strike price and slope of salary contract plots display strike price (divided by 100) and slope of manager’s contract in the case of no residual uncertainty and the parameter values S0 = K = 100, y = 1, β = 6.3912, w = 2.
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the two plots are tangent with slope Ω = .3690. Fig. 2 shows the slope and strike price of the manager’s contract under no residual uncertainty as we vary z, the quadratic component of perk cost. In Section 7 we mentioned that an increase in z leads to a reduction in the optimal Ω, which here translates into a lower slope of a call option. Roughly, a less severe agency problem implies a lower pay-for-performance sensitivity. Fig. 2 shows that this reduction in slope of the call option is accompanied by a lower strike price, so that the manager is willing to accept the contract. 9. Conclusion We have characterized optimal contracts in the presence of asset diversion and private signals of real option profitability. The solutions are based on the first order condition of the manager’s maximization, and while not in closed form they are easy to obtain numerically. The resulting compensation function is a call option itself. In our setting managers keep the real option alive too long, and are sometimes paid more than their reservation wages. A final thought on model implications pertains to the movement of share prices for firms with real options. Even if timing of investment is first best, uninformed outsiders view each instant without investment as bad news, since the trigger level of profitability evidently has not been reached. Therefore, if share price is set by outsiders, it will drift at less than the discount rate. When investment occurs, share price jumps. This share price jump will be larger when there is an agency conflict and delayed investment as in our model. There are many generalizations of the assumptions that could make the model more useful for application. In other versions of the paper, we have considered risk aversion, non-constant cost of investment, and differences in discount rates between manager and firm. Other possible extensions include contracts that depend on exercise time and managers who are overoptimistic or have an option to quit. It also might be more realistic to assume that a manager has multiple tasks with multiple payoffs, rather than just one real option signal to watch. These possibilities are left for future research. Acknowledgements I wish to thank two anonymous referees for several useful comments. Likewise, thanks are owed to Editor Sherrill Shaffer and Executive Editor Kenneth Kopecky. In addition, I appreciate the comments and suggestions of my advisor Jonathan Berk, Jeffrey Coles, Mintao Fan, Diego Garcia, Nils Hakansson, Hayne Leland, Spencer Martin, Mark Seasholes, Mark Taranto, Branko Uroˇsevi´c, Nancy Wallace, and others at Arizona State, KMV, U.C. Berkeley, Western Washington, and Wyoming for many helpful suggestions. All errors are mine. Appendix A Proof of Proposition 1. Under the first best, the incentive compatibility constraint is unnecessary. The problem for the firm is ∞ max ∗ e−τρ [S ∗ − K − Q]f (τ)dτ, M(C),S 0 ∞ (15) subject to e−τρ Qf (τ)dτ = q ≥ w 0
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f(τ,S* ) is just the first passage density for geometric Brownian motion. Karlin and Taylor (1975) show that the expected discount factor is15 ∗ −β ∞ S e−τρ f (τ)dτ = (16) S0 0 Under our assumption µ < ρ, β > 1. So the problem can be expressed as ∗ −β S ∗ (S − K − Q) × , max S ∗ ,Q S0 ∗ −β S subject to Q ≥w S0
(17)
The inequality will be binding as nothing is gained by giving the manager extra, and the first order condition of this problem with respect to S* is Eq. (6). Proof of Lemma 1. We can again employ the result (16) for the expected discount factor to write the manager’s problem as ∗ −β S [E(M(C)|S ∗ , φ) + φ] (18) max S ∗ ,φ S0 The first order condition for S* is ∂ β S0 −β(S ∗ )−β−1 Q + (S ∗ )−β ∗ [E(M(C)|S ∗ , φ)] = 0 ∂S
(19)
where we have inserted Q for the manager’s optimal [E(M(C)|S* ,φ) + φ]. Using the definition of ∗ −β Ω, along with q = Q SS0 , we can write (9).The first order condition for φ implies 0= or (10).
S∗ S0
−β
∗ −β ∂ S [E(M(C)|S ∗ , φ)] + 1 = [Ωγ (φ) + 1] ∂φ S0
Proof of Proposition 2. First, write the firm’s net value as Suppose φ = φfb . Under our assumption that ∂φ dV ∂V ∂S ∗ = ∂V fixed, dΩ ∂φ ∂Ω + ∂S ∗ ∂Ω .
γ (0) ≤ 1,
S∗ S0
−β
(20)
(S ∗ − K + φ − γ(φ)) − q.
this implies that Ω = 1. Now holding q
∂V ∂φ = 0. Notice that given a particular φ, the first best exercise level is such that −β−1 β ∗ (S ) S0 [−β(S ∗ − K + φ − γ(φ)) + S ∗ ] = 0.
But at φ = φfb , ∂V ∂S ∗
= If Ω = 1, and q is less than the first best value of the project, which has been a maintained assumption, S* exceeds this value (we saw from Eq. (9) that S* is strictly increasing in Ω q ), so that the value inside the brackets is negative and is not optimal.
15
∂V ∂S ∗
< 0. Since
Note that Karlin and Taylor’s µ equals what we have defined as µ −
∂S ∗ ∂Ω
σ2 2 .
> 0,
dV dΩ
< 0 and the contract
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∗ . This implies Ω < 1 and φ > φ . But the S* that maximizes firm value Now suppose S ∗ ≤ Sfb fb ∗ the optimal S* given φ . Therefore, by raising S* , given q and a choice of φ is greater than Sfb fb we can increase value by both improving timing and reducing φ.
A.1. Computational tools Matlab was used to find the strike price and slope of the call option in the presence of residual uncertainty. Matlab-provided quadrature functions were used to compute the expected payoff and its sensitivity to S* , and Matlab optimization procedures were used to minimize the differences between the computed values and the targeted values. References Antle, R., Bogetoft, P., & Stark, A. W. (2001). Incentive problems and investment timing options. Manchester Business School working paper. Arya, A., Glover, J., & Routledge, B. R. (2002). Project assignment rights and incentives for eliciting ideas. Management Science, 48(July (7)), 886–899. Comment, R., & Jarrell, G. (1995). Corporate focus and stock returns. Journal of Financial Economics, 37(January (1)), 67–87. Dixit, A., & Pindyck, R. (1994). Dynamic investment under uncertainty. Princeton University Press. Doukas, J., Kim, C., & Pantzalis, C. (2000). Security analysis, agency costs, and company characteristics. Financial Analysts Journal, 56(November–December (6)), 54–63. Grenadier, S. R., & Wang, N. E. (2005). Incentives and investment timing: real options in a principal-agent setting. Journal of Financial Economics, 75(March (3)), 493–533. Guay, W. (1999). The sensitivity of CEO wealth to equity risk: an analysis of the magnitude and determinants. Journal of Financial Economics, 53(July (1)), 43–71. Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes. New York: Academic Press. Kim, J., & Yoon, S. H. (2000). Investment timing choice, moral hazard and incentive compatibility, working paper. Hong Kong Polytechnic University. Maeland, J. (1999). Valuation of irreversible investments and agency problems, working paper.
Contracting on real option payoffs Nicholas Wonder ∗ Department of Finance and Marketing, College of Business and Economics, Western Washington University, Bellingham, WA 98225-9073, USA Received 29 June 2004; received in revised form 25 May 2005; accepted 1 June 2005
Abstract We consider the design of contracts that pay managers on the basis of a project’s payoff. We show that a contract that induces appropriate timing of project investment by a privately informed risk neutral manager will not offer proper incentives to forgo perk consumption at the time of investment. Under the firm’s optimal contract within the class of payments that do not depend on time of investment, the manager waits too long to invest. The optimal contract can be implemented with a call option on the project payoff. © 2005 Elsevier Inc. All rights reserved. JEL classification: G31 Keywords: Real options; Agency; Timing
1. Introduction The real option literature offers prescriptions on value maximization in the presence of various types of flexibility, but there is limited instruction available regarding the problem of motivating self interested managers to exercise real options appropriately. We consider a firm with a risk neutral manager who privately observes a continuously changing signal, which may or may not be noisy, of a potential project’s profitability. On the basis of the signal he observes and the form of his compensation, the manager may at any time invest some of the firm’s money to develop the project. When he invests, he has the opportunity to consume some perquisites, which we can broadly construe as managing the project in a way that appeals to the manager but is not value-maximizing. Perk consumption reduces the project cash flows. Finally, ∗
Tel.: +1 360 650 2444; fax: +1 360 650 4844. E-mail address: [email protected].
0148-6195/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jeconbus.2005.06.004
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the cash flows are realized and the manager is paid on the basis of those cash flows. Assuming the manager lacks sufficient capital to buy the firm, salary functions that induce first best timing of exercise will result in excess perk consumption. The firm’s value-maximizing (net of perks) contract results in some loss associated with both perk consumption and delayed investment. We consider contracts with payoffs that do not explicitly depend on the time of exercise. Within this class, optimal contracts are not found in closed form, but under some assumptions the numerical requirements are minimal. In addition to inducing delayed investment, the optimal contracts in some parameter ranges give the manager a higher valued salary package than the amount he could get by refusing the contract and accepting an outside opportunity. When the manager’s reservation value is very low, it is in the firm’s interest to pay him more than he requires, since incentives can be greatly improved. Since we assume the manager cannot consume perks prior to investment in the project, one might expect him to overinvest. The fact that the manager receives a cash payment that can vary with project payoff, however, means that the manager potentially has a reason to wait. Actually, there are two reasons why, given an optimal contract, the manager waits longer on average to invest than he would absent an agency problem. First, taking as given the result that there will be some perk consumption upon investment in the project, the firm reduces the expected discounted deadweight loss from perks by inducing the manager to wait longer. Second, the sort of salary function that can limit the manager’s desire to consume perks when he finally does invest has the unavoidable effect of delaying the manager’s decision to invest. Given this tradeoff between inducing desired timing and eliminating perks, the firm’s optimal contract does involve a positive expected deadweight loss from perks. The intuition for the tradeoff can be obtained by assuming the manager’s signal of project value is perfect, which is one case we will consider, and assuming that the firm offers him a contract that looks like a call option on the payoff from the project. We will see that such a contract can do as well as any other. Then to completely eliminate the deadweight loss of perk consumption, the firm would like to set the slope of the call option payoff equal to one, so that the manager bears all the cost associated with the perks. On the other hand, the firm does not want to give the manager a contract that is worth more than the manager requires. So it can set the strike price on this contract high to reduce its value. But the optimal exercise level for an American call option rises as the strike price rises, so setting a high strike price means the manager will wait too long to invest. The best contract for the firm sets the strike a little high so the manager waits a little too long, and has a slope of less than one so some perk consumption/mismanagement results. Do the project cash flow-based options derived in this paper match real world management compensation? We have not discovered precisely equivalent contracts, but there are indications that project compensation is growing in general. In any event, our model can be valid in a prescriptive sense even if its suggestions are not currently widely followed. The paper is organized in the following way. Section 2 discusses related literature. Section 3 introduces the assumptions of the model. In Section 4 the problem in the absence of agency problems, the “first best” case, is solved. Sections 5 and 6 determine the manager’s optimal response to any contract and describe requirements that the optimal contract for the firm must satisfy, given that response. Section 7 represents numerical comparative statics, Section 8 presents implementation of the optimal contract, and Section 9 concludes.
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2. Literature review Other agency models of investment timing have also appeared recently. Kim and Yoon (2000), Antle, Bogetoft, and Stark (2001), and Arya, Glover, and Routledge (2002) (the latter with two managers) consider two period settings with various assumptions on the information structure. The most closely related paper to our work is Grenadier and Wang (2005). In their continuous time model, a manager exerts effort at time zero, influencing the likelihood that a privately observed component of project quality will be high or low (there are only two levels of quality). Manager and firm owner both observe innovations to project value. The optimal contract results in delayed investment by managers with unfavorable private information. The model also shares our result that the manager may receive a wage in excess of his reservation level. While both that paper and this one have asymmetric information and moral hazard, the specification is substantially different. First, our perk consumption occurs only at the time of investment, which is at least plausible in that it may be easier to conceal misuse of funds in the context of a large investment expenditure. Second, we have no information asymmetry at the signing of the contract,1 but the manager observes innovations to project value due to day to day monitoring. Time zero asymmetry in Grenadier and Wang’s paper means that, even given a cost of effort such that effort is not a consideration in the optimal contract, the firm still must pay informational rents and therefore fails to achieve first best. Outcomes in our model, in contrast, approach first best as the costs of perk consumption become increasingly convex, while both perk consumption and investment timing drive the contract terms for any parameters short of the limit. Another distinction is that we allow for the manager’s signal to be imperfect, while, at the moment of investment, the manager in Grenadier and Wang (2005) knows exactly the project payoff that the firm owners will observe. Maeland (1999) also presents a continuous time model with time-zero information asymmetry, but his model includes neither effort nor perk consumption. Only limited empirical evidence directly relates to our work. Guay (1999) finds that firms with growth options have managers whose salaries are more strongly increasing in the volatility of firm asset value. That is, their contracts are more convex. Our model involves a firm with a growth option and yields convex optimal contracts. To that extent, Guay’s findings are consistent with our predictions. 3. The model A firm has available a perpetual option to undertake a project at a cost K. A manager observes a signal S of the project’s value, with dS = µSdt + σSdW
(1)
where W is a standard Brownian motion. S may or may not be a noisy signal of project cash flows as we discuss later in this section. The signal changes as demand conditions, probable productions costs, and competitive pressure, for example, change. The manager is given the right to exercise the real option at any time. When he does, the firm invests K in the project. At exercise, the manager can choose φ, the value he receives from perk
1 An early version of our paper considered continuous menus of contracts (the counterpart to Grenadier and Wang’s pair of contracts) when initial project value, in addition to the innovations, was observed only by the manager.
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consumption.2 Given the choice of φ, project gross cash flows are assumed to be C ≡ [S − γ(φ)]ε
(2)
where • γ is an increasing, convex differentiable function representing the cost to the firm of perk consumption. We set γ(0) = 0. If in addition γ (0) = 1, then the first-best level of perks is 0,3 but social loss is small initially. Convexity of loss due to perks is a reasonable assumption. The manager can make use of some of the project’s assets just as well as the firm can, and these are the ones he would divert first. Eventually he will encounter diminishing returns, however, and an effort to hide his activities would reduce the efficiency of asset transfer even more. We will assume the maximum level of perks is such that γ(φ) = S—the manager cannot usurp more than the project itself. • ε is a random variable with non-negative support and, without loss of generality, mean 1. This implies that if the manager did not divert any assets, the signal observed at the instant before exercise would equal the expected cash flow from the project. We can allow the case where ε is always 1, so that the manager faces no residual uncertainty at exercise. If ε is not degenerate at 1, we require it to have a differentiable density function that is positive at all values above 0 and has no mass points except possibly at zero. The manager immediately receives payment M(C). It is assumed that it is not possible to contract on the time that exercise occurs. Also, it will be convenient if the manager’s expected payoff is differentiable with respect to the manager’s signal. Sufficient conditions on M to ensure this are not very onerous. We assume that the manager has no capital with which to buy the project from the firm. Also, we will require that the manager’s salary, at least in equilibrium, is never negative and that the firm makes no salary payments prior to exercise. The latter is not restrictive given our other assumptions. Manager and firm are risk neutral and discount future payoffs at the rate ρ. The manager attempts to maximize the discounted expected value of his cash payment and perks, M + ϕ, while the firm maximizes the discounted expected value of C − M − K (recall that C is the project cash flow net of the effect of the manager’s perk consumption). We will require that the discount rate exceed the drift in project value, so that, absent agency problems, there exists some level of the signal S at which investment should occur. At the beginning the manager’s discounted expected payoff (salary plus assets diverted) must be at least w, the value of his outside opportunity. One last assumption is that the parameters are such that the first best investment policy is not immediate exercise, although most of the analysis holds without that requirement. Finally, note that while our firm is modeled as consisting of an option and nothing else, the presence of other assets is inconsequential, provided we can still contract on the cash flow from this particular project.
2 We assumed that perk consumption is not possible prior to investment. In some settings, it might be more reasonable to assume that perks are gradually consumed over time. Then perk consumption could reduce the drift of the project value. That specification would complicate the analysis, but it may be worthy of further study, as it is not clear to what extent the conclusions here would change. 3 This follows from the fact that the marginal net gain from one additional unit of perk consumption is 1 − ␥ . Due to the convexity of γ, this marginal gain will be negative for φ > 0.
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4. The firm’s problem and first best analysis The firm’s problem is to maximize its discounted expected profits net of perks, subject to the manager receiving at least his reservation utility and choosing his stopping rule and asset diversion optimally: ∞ max e−τρ E[C − M(C) − K]f (τ)dτ, M(C),S ∗ ,φ 0 ∞ e−τρ E[M(C) + φ(τ)]f (τ)dτ ≥ w subject to (3) 0 ∞ ˜ e−τρ E[M(C) + φ(τ)]f (τ)dτ and S ∗ , φ ∈ arg max S˜ ∗ ,φ˜
0
where τ is the investment time, f(τ) the probability density function for exercising at time τ, and S* is the signal level at which the manager invests.4 It will be useful to express the manager’s expected payoff conditional on exercise as Q and his initial discounted expected payoff q. Then Q(τ) = φ(τ) + E[M(C)] Also, E(C) = (S* − γ(φ)) so the maximization can be written as ∞ max∗ e−τρ [S ∗ − K + φ − γ(φ) − Q]f (τ)dτ, M(C),S ,φ 0 ∞ e−τρ Qf (τ)dτ = q ≥ w subject to 0 ∞ e−τρ Qf (τ)dτ and S ∗ , φ ∈ arg max S˜ ∗ ,φ˜
(4)
(5)
0
The term (φ − γ(φ)) in the firm’s objective function represents subtraction of the deadweight loss due to perks. This deadweight loss is the difference between perks received by the manager and loss in project expected payoff attributable to those perks. As a basis for future comparison, we state the first best outcome, which is standard for a perpetual call option (for example, Dixit & Pindyck, 1994, p. 142): Proposition 1. Under the assumption γ (0) ≥ 1 (so that zero diversion of assets is optimal)5 the first best exercise level satisfies ∗ Sfb =
Kβ β−1
and the firm’s net starting value is β 1−β K S0 − w, β β−1
(6)
(7)
4 In principle, the manager’s threshold signal level, choice of perks, and resulting expected salary could depend on time. But the perpetual nature of the investment opportunity and the assumed time-independence of the salary function combine to create a time-independent setting. 5 In the case of γ (0) < 1, the expression for the first-best S* would replace K with K − φ + γ(φ ) < K. fb fb
N. Wonder / Journal of Economics and Business 58 (2006) 20–35
where 1 µ β= − 2 + 2 σ
µ 1 − σ2 2
2 +
2ρ . σ2
25
(8)
∗ is proportional to the required investment K and is decreasing So the critical signal level Sfb in β. β is decreasing in drift µ and volatility σ 2 and increasing in discount rate ρ. The firm waits longer to invest, if by waiting it is more likely to benefit from the drift and diffusion or if it is 2 more patient. Also, if µ − σ2 ≥ 0, investment will eventually occur with probability 1, since the signal will hit any barrier above its initial value under this condition.
5. Second best: the manager’s problem To derive the second best (time independent) contract, we first need to find the manager’s optimal action as a function of the contract the firm offers him. Then we will use those results to write the firm value as a function of certain properties of the contract, and finally maximize that value. The following result employs the manager’s first order conditions to find his optimal choices. Lemma 1. Assuming there is an interior solution for the manger’s problem, his choice of exercise level S* satisfies 1 −β qβ 1−β 1−β ∗ S = S0 (9) Ω and he chooses perks φ such that γ (φ) = where Ω ≡
1 , Ω
∂ ∗ ∂S ∗ [E(M(C)|S , φ)],
(10) evaluated at the manager’s optimal S* .
We are able to write the manager’s optimal investment trigger S* and asset diversion choice φ as functions of the manager’s salary sensitivity Ω and discounted payoff inclusive of perks q, along with the exogenous variables S* and β. For that reason, we will now view the firm’s problem as one of choosing sensitivity and discounted payoff, rather than the full salary function M. From Eq. (9), we see that exercise level is increasing in the ratio (Ω/q). That is, the manager’s timing decision rises as the marginal sensitivity of his expected salary at the optimum increases relative to his total discounted expected value. This relation follows from consideration of what a manager stands to gain or lose by waiting to invest. If the manager waits another period, he loses the time value of his salary and perks. The amount of this loss (the interest he could have earned on his salary) is related to the level of his payoff, and, in turn, to his discounted payoff. The potential gain to waiting depends on the exogenous drift and diffusion terms, but also on the sensitivity of the manager’s payoff to changes in project payoff, Ω. Given a certain level q, a higher sensitivity implies a longer time to investment. Turning to the choice of perks, the condition (10) means the manager diverts assets up until the marginal loss associated with doing so equals the gain. The marginal loss depends on the marginal loss to the whole firm, represented by the slope of the perk cost function, and it also depends on the portion Ω of the firm’s marginal loss that the manager bears. If the manager has an Ω of 1, he will make the first best perk consumption decision.
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The first order condition tells us the optimal choice of perks so long as there is an interior optimum. There do exist parameter values for which a locally optimal perk choice is not the global maximum. This issue is a second reason the firm might need to pay the manager more than his outside option.6 6. Second best: the firm’s optimal contract Given the results on the manager’s optimal behavior, we can remove the incentive compatibility constraint from the earlier expression of the firm’s problem and instead incorporate the manager’s choices of investment time and perks into the quantity the firm is maximizing: ∗ −β S , subject to q ≥ w (11) max(S ∗ − K − Q + φ − γ(φ)) × q,Ω S0 We substitute in for S* , φ, γ(φ), and Q, each of which can be written as functions of q, Ω, and the exogenous variables. For our numerical results, we will need a specific functional form for the loss due to perks. Let the reduction in project value from asset diversion be quadratic: γ = yφ + zφ2
(12)
If y = 1, as we will assume for the numerical calculations, γ (0) = 1, so that small amounts of perks produce very small deadweight losses. From the first order condition (10), the optimal perk choice φ satisfies φ=
(1 − Ωy) 2Ωz
(13)
Differentiating the objective function in (11) with respect to Ω and simplifying produces the following first order condition. 2 −β 1−2β 1 β (y − 2y) Ω 1−β −(qβ) 1−β S01−β + Ω2 −K (1 − β) 4z 2β −β +Ω + =0 (14) 4z(1 − β) 4z(1 − β) −2y) − K ≤ 0, (true for y = 1) this equation has The optimal Ω solves this equation. Provided (y 4z one and only one solution for Ω ∈ (0,1]. At this point, we can summarize the dilemma for the firm. Solving the perk problem necessitates a steeply sloped contract, that is, setting Ω equal to one. But if the manager is not to be given the entire value of the project, his optimal timing policy will then involve waiting too long. The second best contract trades off these two goals. The outcome of the tradeoff is the following proposition, our main claim: the second best contract produces underinvestment and a positive 2
In particular, the manager can pick perks such that γ(φ) = S and exercise the option immediately at S0 . That is, he can divert all the invested assets of the project so that its remaining value is zero. His value from this strategy is γ −1 (S0 ), so any incentive compatible salary offered by the firm must offer at least that value. If, however, the manager’s signal is noiseless, the corner solution is not a significant problem if the contract can threaten punishment (negative M) for low values of C. Even aside from that case, monitoring could remove the possibility of complete asset diversion. 6
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expected deadweight loss from perks. This is not dependent on the quadratic form of the perk cost function. ∗ and φ > φ . Proposition 2. Assume γ (0) ≤ 1.7 Then under the second best contract, S ∗ > Sfb fb
When the reservation wage w is low the firm will offer a salary that gives the manager discounted value q strictly greater than w. If q is quite small, the firm can increase the manager’s pay by a small amount in absolute terms but a large amount in percentage terms. This will change the manager’s optimal decisions significantly, and have an impact on firm value that exceeds the extra amount the manager received. So in this instance q will be set by its first order condition from the firm’s maximization. The next section presents the sort of parameter values that result in q being determined by either the manager’s outside opportunity or the first order condition. 7. Numerical comparative statics In this section we consider how characteristics of the optimal contract depend on different parameters of the model. We start with a base case where the expected project payoff equals required investment: S0 = K = 100. We set the perk cost coefficients y and z equal to 1, with that choice of y implying that a small amount of perk consumption generates only a small deadweight loss. Finally, we set β = 6.391, and the manager’s reservation wage is w = 2. From that point, we vary z, w, k, and β. β = 6.391 can result from various combinations of µ, σ, and ρ, including µ = −.1, σ = .2, ρ = .05, as well as (−.092, .2, .1) and (.01, .089, .2).8 When reporting the probability of eventual investment and the median time to investment, we assume that µ = −.1 and σ = .2. The negative drift might represent a significant probability that the investment project will become obsolete as time passes.9 We do consider different drifts in Table 4. Also, increasing the drift would result in higher values for β if volatility and discount rate were not changed, so that one can roughly characterize the impact of changing drift using our table with changing β. Under the base case parameter values, the first best project value is 6.2521 for a net firm value of 4.2521, ∗ = 118.55, the first best probability that investment ever occurs is .3603, and, conditional on Sfb investment happening eventually, there is a 50% chance that it will occur within .74 years. We do not claim that our parameter choices represent an average project, but the comparative statics do not appear highly sensitive to the base case. The results stated in the tables assume that there is no possibility of immediate diversion of all the assets, the corner solution mentioned earlier. That means they are most likely to apply when the manager faces no residual uncertainty about firm cash flows. Table 1 shows that a higher convexity of perk costs (higher z) implies a lower agency problem, as the firm is able to offer less powerful incentives (a lower Ω) and has a value closer to the first best. In particular, raising z from .01 to 20.48 means Ω falls from .7887 to .323 while firm value rises from 2.448 to 4.234. For z = .02 and below, discounted value to the manager exceeds his outside opportunity (q > 2).
7 We make this assumption (that the social loss due to perks is initially small) simply because alternative assumptions might result in zero perk consumption under the second best case as well. 8 Drift, volatility, and discount rate are captured by the summary parameter β in this model because the discount factor associated with the first passage time to any signal value also depends on those three variables only through β. β is the positive root of a quadratic equation in S that describes that discount factor (see the proof of Proposition 1 in Appendix A). 9 A constant probability through time that the value of the project jumped to zero because of obsolescence would likely yield similar results to a negative drift term, though we have not attempted to solve that case.
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Table 1 Characteristics of second best contract as a function of z, the quadratic coefficient on cost of asset diversion z
Ω
q
V
S*
Pr(Inv)
Med(τ)
γ −φ
.01 .02 .04 .08 .16 .32 .64 1.28 2.56 5.12 10.24 20.48
.789 .703 .618 .544 .480 .428 .388 .360 .343 .332 .326 .323
2.20 2.08 2 2 2 2 2 2 2 2 2 2
2.448 2.635 2.874 3.141 3.413 3.664 3.871 4.024 4.124 4.183 4.216 4.234
137.66 136.17 133.93 130.81 127.80 125.10 122.87 121.20 120.07 119.37 118.98 118.77
.147 .157 .173 .200 .230 .261 .291 .315 .334 .346 .352 .356
1.78 1.70 1.58 1.41 1.24 1.09 .97 .88 .82 .78 .76 .75
1.795 2.222 2.398 2.201 1.839 1.401 .971 .615 .360 .198 .104 .054
Each row doubles z. Table shows results assuming that diversion of all assets is not possible. The first best firm value is 4.252. S* is the manager’s optimal signal value at exercise, Ω is the equilibrium sensitivity of payment at S* , q is the manager’s expected discounted value, V is net firm value after expected payment to manager, γ − φ is deadweight loss at exercise (not discounted) from perk consumption, and Pr(Inv) is the probability that investment ever occurs. Med(τ) is the time such that, conditional on investment occurring at some point, investment will have happened with 50% probability. All rows assume w (dollar value of outside payment) = 2, S0 (initial signal value) = 100, K (strike price) = 100, y (linear coefficient in asset diversion cost function) = 1, β (function of discount rate and signal drift and diffusion) = 6.3912. To compute Pr(Inv) and Median(τ), it is assumed µ = −.1 and σ = .2.
Regarding the time until investment, the probability of eventual investment is as low as .147 for z = .01, with a corresponding median time of 1.78 years to investment. These values approach the first best levels as convexity rises, corresponding to an improved trigger level S* . The relation between z and S* 10 means that firms with high agency costs (low z) will be more profitable than others, if and when investment finally occurs. The table also shows how the deadweight loss realized at exercise, γ − φ, declines as perk cost convexity rises. Given the significance of z in determining the characteristics of the contract (more about this appears in the next section) as well as the actions of the manager, connecting it to observable quantities is desirable. Most directly, z measures how destructive it is to misuse the assets of the firm. When assets are intangible and highly specific, z (and/or y, the linear cost of perks) should be high. When there is a high variance in costs from suppliers, then it is more costly for a CEO to offer a supplier contract to his cousin and z would be high. Linking firm characteristics such as intangible assets to low agency costs is perhaps in conflict with standard views and intuition. More intuitive implications may follow if we regard the quadratic cost component as related to managerial cost of hiding perk consumption. Then well governed firms have high z. Here we can draw on existing literature about characteristics connected to good monitoring and governance. A high z firm might have a small board with independent directors and a chairman who was not its CEO, or it would have large external shareholders. It would be followed by security analysts and it would access the capital market frequently. A high ratio of short-term to total debt is one indicator of the latter characteristic.11 Or, more precisely, the relation between S − γ − M (firm value net of perks and manager salary) and z. Doukas, Kim, and Pantzalis (2000) study the role of security analysts in monitoring, while Comment and Jarrell (1995) suggest the short-term debt ratio as a measure of reliance on external markets. 10
11
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Table 2 Characteristics of second best contract as a function of w, the manager’s reservation wage w
Ω
q
V
Vfb
S*
Pr(Inv)
Med(τ)
γ −φ
.5 1 1.5 2 2.5 3 3.5 4 4.5 5
.265 .265 .304 .369 .437 .507 .580 .654 .730 .806
1.19 1.19 1.5 2 2.5 3 3.5 4 4.5 5
4.304 4.304 4.243 3.976 3.599 3.167 2.705 2.227 1.740 1.247
5.572 5.252 4.752 4.252 3.752 3.252 2.752 2.252 1.752 1.252
126.1 126.1 123.9 121.7 120.5 119.8 119.3 119.0 118.8 118.7
.249 .249 .277 .307 .326 .339 .346 .352 .355 .357
1.15 1.15 1.03 .91 .84 .80 .78 .76 .75 .75
1.92 1.92 1.31 .73 .42 .24 .13 .07 .03 .01
S* is the manager’s optimal signal value at exercise, Ω the equilibrium sensitivity of payment at S* , q the manager’s expected discounted value, V the net firm value after expected payment to manager, Vfb the firm value in the first best case, γ − φ the deadweight loss at exercise (not discounted) from perk consumption, and Pr(Inv) is the probability that investment ever occurs. Med(τ) is the time such that, conditional on investment occurring eventually, investment will have happened with 50% probability. Table assumes S0 (initial signal value) = 100, K (strike price) = 100, y (linear coefficient in asset diversion cost function) = 1, z (quadratic coefficient) = 1, β (function of discount rate and signal drift and diffusion) = 6.3912. To compute Pr(Inv) and Median(τ), it is assumed µ = −.1 and σ = .2. Results assume that diversion of all assets is not possible.
Looking at different outside opportunity (w) levels in Table 2, we see that the firm will offer the manager a package worth 1.190 even when his outside options are less than that. So the maximum possible firm value is 4.304. This table illustrates the fact that, as the manager’s share of firm value becomes large, it becomes easier to align his incentives with the firm, and agency costs become small. In particular, as w approaches the first best project value of 6.252, the manager’s investment threshold approaches first best, deadweight losses from perks approach zero, and firm value approaches its first best level. Table 3 shows how the contract characteristics vary as the exercise price K goes from 85 to 115. For the low levels of K, firm value is of course higher. But agency loss from first best firm value is also higher in absolute and percentage terms. An explanation is that since the manager’s outside opportunity is held fixed, his share of first best firm value is smaller for the lower values Table 3 Characteristics of second best contract as a function of K, the exercise price of the real option K
Ω
q
V
Vfb
S*
∗ Sfb
γ −φ
85 90 95 100 105 110 115
.2780 .2734 .3084 .3690 .4497 .5552 .6900
2.954 2.146 2 2 2 2 2
10.165 7.514 5.565 3.976 2.706 1.709 .936
13.016 9.034 6.244 4.252 2.806 1.740 .943
107.44 113.65 117.75 121.73 126.28 131.32 136.72
100.77 106.69 112.62 118.55 124.48 130.40 136.33
1.69 1.77 1.26 .731 .374 .160 .051
is the equilibrium sensitivity of payment at S* , q is the manager’s expected discounted value and V is the net firm value ∗ are the values obtained in the first best case, and γ − φ is deadweight loss after expected payment to manager. Vfb and Sfb at exercise (not discounted) from perk consumption. All rows assume w (dollar value of outside payment) = 2, S0 (initial signal value) = 100, y (linear coefficient in asset diversion cost function) = 1, z (quadratic coefficient) = 1, β (function of discount rate and signal drift and diffusion) = 6.3912.
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Table 4 Characteristics of second best contract as a function of µ, the drift of the project payoff µ
β
Ω
q
V
Vfb
% Loss
S*
∗ Sfb
Med(τ)
−.14 −.12 −.10 −.08 −.06 −.04 −.02 0 .02 .04
8.30 7.34 6.39 5.46 4.55 3.68 2.87 2.16 1.58 1.16
.469 .419 .369 .319 .268 .218 .189 .161 .128 .080
2 2 2 2 2 2 2.4 3.1 4.0 4.8
2.59 3.20 3.98 5.01 6.44 8.52 11.8 17.7 29.0 55.4
2.72 3.38 4.25 5.43 7.11 9.62 13.6 20.5 33.4 61.0
.046 .054 .065 .078 .094 .114 .133 .140 .130 .091
115.3 118.0 121.7 127.2 135.6 150.0 171.8 215.0 330.5 1046.7
113.70 115.8 118.5 122.4 128.2 137.3 153.5 186.3 272.1 731.7
.48 .65 .91 1.35 2.14 3.79 7.21 17.2 78.5 83.3
Pr(Inv) .321 .314 .307 .301 .295 .296 .338 .465 1 1
Ω is the equilibrium sensitivity of payment at S* , q is the manager’s expected discounted value and V is the net firm value ∗ are the values obtained in the first best case. % Loss is percentage by after expected payment to manager. Vfb and Sfb which V is less than Vfb . All rows assume σ (volatility of project payoff) = .2, ρ (discount rate) = .05, w (dollar value of outside payment) = 2, S0 (initial signal value) = 100, K (required investment) = 100, y (linear coefficient in asset diversion cost function) = 1, and z (quadratic coefficient) = 1.
of K, leading to higher agency costs. However, unreported results show that, even if w is held at a constant fraction of first-best firm value, lower exercise prices lead to firm value deviating by a greater percentage of first-best value. In Table 4 the drift is varied. Other parameters are held fixed, including volatility of .2 and discount rate .05. The effect of increasing drift is somewhat similar to the effect of reducing exercise price, as the manager’s share of firm value falls as firm value rises. However, for high levels of drift, the percentage loss from first best firm value starts to drop. This is something of a surprise. Value lost does consistently rise with drift in absolute terms, however. The median time to investment and probability of eventual investment steadily rise with higher drift.12 In Table 5 we consider different values of β, the summary parameter for drift, diffusion, and discount rate. Recall that β is increasing in the discount rate and decreasing in the drift and volatility. Therefore, this exercise is similar to that portrayed in Table 4, except that we now also vary the manager’s reservation wage so as to keep it at a constant proportion of first best firm value. This may be representative of the variation we would see in the cross section. As β increases (that is, as the discount rate rises or the drift or volatility falls), agency costs (Vfb − V) stay approximately constant, therefore rising as a percentage of the falling first best value. So we have the contrast that projects that are long shots due to high required investment (relative to initial signal S0 ) experience low agency costs, while projects that are long shots due to high β have high agency costs. 8. Implementation of the optimal contract Once the characteristics of the optimal contract have been found, the firm must find a function that implements it. This function will have to fit the expected amount and sensitivity of the salary
Another possible experiment is raising the drift but holding β fixed, by lowering volatility or raising the discount rate. Given either type of adjustment, a higher drift leads to greater probability of eventual investment and longer median wait to investment. 12
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Table 5 Characteristics of second best contract as a function of β, holding w constant as a percentage of first best project value β
w
q
V
Vfb
Ω
S*
∗ Sfb
2.39 2.89 3.39 3.89 4.39 4.89 5.39 5.89 6.39 6.89 7.39 7.89
6.30 4.96 4.09 3.48 3.03 2.69 2.41 2.19 2 1.84 1.71 1.59
6.30 4.96 4.09 3.48 3.03 2.69 2.41 2.19 2 1.84 1.71 1.59
13.105 10.248 8.402 7.110 6.155 5.421 4.839 4.367 3.976 3.647 3.670 3.125
13.388 10.541 8.697 7.405 6.448 5.710 5.124 4.648 4.252 3.919 3.634 3.388
.3323 .3375 .3425 .3474 .3521 .3566 .3609 .3650 .3690 .3729 .3766 .3802
176.6 157.3 145.9 138.5 133.2 129.3 126.2 123.7 121.7 120.1 118.6 117.4
171.9 152.9 141.8 134.6 129.5 125.7 122.8 120.4 118.6 117.0 115.7 114.5
Ω is the equilibrium sensitivity of payment at S* , q is the manager’s expected discounted value and V is the net firm value ∗ are the values obtained in the first best case, and γ − φ is deadweight loss at after expected payment to manager. Vfb and Sfb exercise (not discounted) from perk consumption. All rows assume y (linear coefficient in asset diversion cost function) = 1, z (quadratic coefficient) = 1, K (required investment) = 100, and the initial signal S0 = 100. The table presents only results when diversion of all assets is not possible.
payment at the manager’s optimal S* and φ. That is, we use the results from above to find S* − γ(φ) and Q. Then we need to produce a contract such that E[M(ε(S ∗ − γ(φ)))] + φ = Q
and
∂ (E[M(ε(S ∗ − γ(φ)))]) = Ω. ∂S ∗
Now the distribution of the random variable ε comes into play, as the expected value of a function of ((S − γ)ε) depends on more than just E(ε). Also, we will require that the manager’s payoff at exercise be non-negative, at least in the case of residual uncertainty at the moment of investment. Any function that produces the desired managerial payoff value and sensitivity yields the same value for the firm. Here we consider contracts that take the form of call options on C. The manager’s salary M(C) is zero for C < C1 and is set equal to (C − C1 ) × C2 for C ≥ C1 .13 This solution is not unique, since there exist many functions that have a particular level and derivative at a certain point. But options are the least complex contract that will work, and option-based compensation is commonly observed in practice.14 Again consider the base case parameters {w, β, S, K, y, z} = {2, 6.1239, 100, 100, 1, 1}. The solution includes sensitivity Ω = .3690. We consider one example with residual uncertainty at the moment of investment and one without. (1) Truncated normal distribution: f(ε) = 0, ε < 0, ε ∼ N(.99999, .2763), ε > 0, Pr(ε = 0) = .0001. This distribution has mean 1. Using numerical methods, we find that {C1 ,C2 } = {134.81,
13
Earlier we assumed that the manager’s expected salary is differentiable with respect to his signal. The call option described here violates that condition if the manager faces no residual uncertainty, but the point of non-differentiability will be below the manager’s exercise level such that it is not a concern. 14 Of course, typical real-world employee options do not have payoffs that depend on cash flows of individual projects, nor is their exercise triggered by project investment.
32
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Fig. 1. Expected salary as a function of (S − γ) Plots display manager’s expected salary as a function of (S − γ), using S0 = K = 100, z = y = 1, β = 6.3912, w = 2, and either truncated normal distribution of ε or no residual uncertainty.
.8595}. If the manager invests at S* = 121.732 and is close to its mean, he will receive a salary of zero. (2) ε = 1 with certainty: Then the slope of the contract, C2 , is simply equal to Ω; {C1 ,C2 } = {103.42, .3690}. Notice that the example with residual uncertainty has a higher slope on the contract. In fact, other parameter choices suggest a positive relation between variance of ε and slope of contract, but that is likely sensitive to the risk neutral model we employ. For these two cases, Fig. 1 shows the manager’s expected salary as a function of S − γ, his signal net of the cost of perks. At S − γ = 120.146 (which occurs given the optimal contract),
Fig. 2. Strike price and slope of salary contract plots display strike price (divided by 100) and slope of manager’s contract in the case of no residual uncertainty and the parameter values S0 = K = 100, y = 1, β = 6.3912, w = 2.
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the two plots are tangent with slope Ω = .3690. Fig. 2 shows the slope and strike price of the manager’s contract under no residual uncertainty as we vary z, the quadratic component of perk cost. In Section 7 we mentioned that an increase in z leads to a reduction in the optimal Ω, which here translates into a lower slope of a call option. Roughly, a less severe agency problem implies a lower pay-for-performance sensitivity. Fig. 2 shows that this reduction in slope of the call option is accompanied by a lower strike price, so that the manager is willing to accept the contract. 9. Conclusion We have characterized optimal contracts in the presence of asset diversion and private signals of real option profitability. The solutions are based on the first order condition of the manager’s maximization, and while not in closed form they are easy to obtain numerically. The resulting compensation function is a call option itself. In our setting managers keep the real option alive too long, and are sometimes paid more than their reservation wages. A final thought on model implications pertains to the movement of share prices for firms with real options. Even if timing of investment is first best, uninformed outsiders view each instant without investment as bad news, since the trigger level of profitability evidently has not been reached. Therefore, if share price is set by outsiders, it will drift at less than the discount rate. When investment occurs, share price jumps. This share price jump will be larger when there is an agency conflict and delayed investment as in our model. There are many generalizations of the assumptions that could make the model more useful for application. In other versions of the paper, we have considered risk aversion, non-constant cost of investment, and differences in discount rates between manager and firm. Other possible extensions include contracts that depend on exercise time and managers who are overoptimistic or have an option to quit. It also might be more realistic to assume that a manager has multiple tasks with multiple payoffs, rather than just one real option signal to watch. These possibilities are left for future research. Acknowledgements I wish to thank two anonymous referees for several useful comments. Likewise, thanks are owed to Editor Sherrill Shaffer and Executive Editor Kenneth Kopecky. In addition, I appreciate the comments and suggestions of my advisor Jonathan Berk, Jeffrey Coles, Mintao Fan, Diego Garcia, Nils Hakansson, Hayne Leland, Spencer Martin, Mark Seasholes, Mark Taranto, Branko Uroˇsevi´c, Nancy Wallace, and others at Arizona State, KMV, U.C. Berkeley, Western Washington, and Wyoming for many helpful suggestions. All errors are mine. Appendix A Proof of Proposition 1. Under the first best, the incentive compatibility constraint is unnecessary. The problem for the firm is ∞ max ∗ e−τρ [S ∗ − K − Q]f (τ)dτ, M(C),S 0 ∞ (15) subject to e−τρ Qf (τ)dτ = q ≥ w 0
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f(τ,S* ) is just the first passage density for geometric Brownian motion. Karlin and Taylor (1975) show that the expected discount factor is15 ∗ −β ∞ S e−τρ f (τ)dτ = (16) S0 0 Under our assumption µ < ρ, β > 1. So the problem can be expressed as ∗ −β S ∗ (S − K − Q) × , max S ∗ ,Q S0 ∗ −β S subject to Q ≥w S0
(17)
The inequality will be binding as nothing is gained by giving the manager extra, and the first order condition of this problem with respect to S* is Eq. (6). Proof of Lemma 1. We can again employ the result (16) for the expected discount factor to write the manager’s problem as ∗ −β S [E(M(C)|S ∗ , φ) + φ] (18) max S ∗ ,φ S0 The first order condition for S* is ∂ β S0 −β(S ∗ )−β−1 Q + (S ∗ )−β ∗ [E(M(C)|S ∗ , φ)] = 0 ∂S
(19)
where we have inserted Q for the manager’s optimal [E(M(C)|S* ,φ) + φ]. Using the definition of ∗ −β Ω, along with q = Q SS0 , we can write (9).The first order condition for φ implies 0= or (10).
S∗ S0
−β
∗ −β ∂ S [E(M(C)|S ∗ , φ)] + 1 = [Ωγ (φ) + 1] ∂φ S0
Proof of Proposition 2. First, write the firm’s net value as Suppose φ = φfb . Under our assumption that ∂φ dV ∂V ∂S ∗ = ∂V fixed, dΩ ∂φ ∂Ω + ∂S ∗ ∂Ω .
γ (0) ≤ 1,
S∗ S0
−β
(20)
(S ∗ − K + φ − γ(φ)) − q.
this implies that Ω = 1. Now holding q
∂V ∂φ = 0. Notice that given a particular φ, the first best exercise level is such that −β−1 β ∗ (S ) S0 [−β(S ∗ − K + φ − γ(φ)) + S ∗ ] = 0.
But at φ = φfb , ∂V ∂S ∗
= If Ω = 1, and q is less than the first best value of the project, which has been a maintained assumption, S* exceeds this value (we saw from Eq. (9) that S* is strictly increasing in Ω q ), so that the value inside the brackets is negative and is not optimal.
15
∂V ∂S ∗
< 0. Since
Note that Karlin and Taylor’s µ equals what we have defined as µ −
∂S ∗ ∂Ω
σ2 2 .
> 0,
dV dΩ
< 0 and the contract
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∗ . This implies Ω < 1 and φ > φ . But the S* that maximizes firm value Now suppose S ∗ ≤ Sfb fb ∗ the optimal S* given φ . Therefore, by raising S* , given q and a choice of φ is greater than Sfb fb we can increase value by both improving timing and reducing φ.
A.1. Computational tools Matlab was used to find the strike price and slope of the call option in the presence of residual uncertainty. Matlab-provided quadrature functions were used to compute the expected payoff and its sensitivity to S* , and Matlab optimization procedures were used to minimize the differences between the computed values and the targeted values. References Antle, R., Bogetoft, P., & Stark, A. W. (2001). Incentive problems and investment timing options. Manchester Business School working paper. Arya, A., Glover, J., & Routledge, B. R. (2002). Project assignment rights and incentives for eliciting ideas. Management Science, 48(July (7)), 886–899. Comment, R., & Jarrell, G. (1995). Corporate focus and stock returns. Journal of Financial Economics, 37(January (1)), 67–87. Dixit, A., & Pindyck, R. (1994). Dynamic investment under uncertainty. Princeton University Press. Doukas, J., Kim, C., & Pantzalis, C. (2000). Security analysis, agency costs, and company characteristics. Financial Analysts Journal, 56(November–December (6)), 54–63. Grenadier, S. R., & Wang, N. E. (2005). Incentives and investment timing: real options in a principal-agent setting. Journal of Financial Economics, 75(March (3)), 493–533. Guay, W. (1999). The sensitivity of CEO wealth to equity risk: an analysis of the magnitude and determinants. Journal of Financial Economics, 53(July (1)), 43–71. Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes. New York: Academic Press. Kim, J., & Yoon, S. H. (2000). Investment timing choice, moral hazard and incentive compatibility, working paper. Hong Kong Polytechnic University. Maeland, J. (1999). Valuation of irreversible investments and agency problems, working paper.