Venture Capital Investment under Uncertainty and

  Asymmetric Beliefs: A Continuous-Time, Stochastic

                   ...
1    Introduction

Real-world productive activities are typically characterized by decentralized decision-making. The

age...
investments by the VC and human capital (effort) investments by the EN. We model the evolution

of the project’s terminatio...
EN’s pay-performance sensitivities, that is, the sensitivities of the change in the EN’s stake to

the change in the proje...
If the EN is optimistic, the VC can increase the performance-sensitive component of the EN’s

compensation because the EN ...
optimism lowers overall project value because the VC prolongs the project and over-invests to take

advantage of the EN’s ...
and financial contracting (DeMarzo and Fishman, 2006, Biais et al, 2007).

       We contribute to this literature by devel...
beliefs and agency conflicts play important roles.

    In summary, we contribute to the literature by developing and analy...
[t, t + dt], dVt , is the sum of a base output, a Gaussian process that is unaffected by the actions

of the VC and EN, and...
EN could be optimistic or pessimistic relative to the VC, that is, µEN could be greater or less than
                     ...
EN (see Holmstrom and Milgrom, 1987).

   Let {Ft } denote the information filtration generated by the history of terminati...
allow for all possible allocations of bargaining power that are indexed by different values of P0 .

    A contract (Pτ , c...
In light of Theorem 1, a contract is completely specified by the performance-invariant compen-

  sation and pay-performanc...
In (11), ∆t is the degree of asymmetry of beliefs at date t, defined in (4), and

                                        ...
– The EN’s performance-invariant compensation parameter is a∗ := at (b∗ , c∗ ), where at (·, ·)
                          ...
γ−β−αγ
    • Return on investment. The “return on investment” term,                           αγ   c(b),    reflects the VC...
Figure 1: Optimal investment function
                                          c(b)                             c (1) = 0...
the equilibrium values for the pay-performance sensitivity, investment and effort at each point in

time (conditional upon ...
(c) The EN’s pay-performance sensitivity b∗ exceeds 1 if t  t∗ , equals 1 at t = t∗ , and less than
                      ...
b∗
                 t
                    6
                    ..
                    ....
                      ....
   ...
c∗
                      t
                         6
               c(1)                           .........................
the pay-performance sensitivity and investment paths is ambiguous.4

Figure 4 demonstrates that the path of equilibrium in...
c∗
                       t
                          6
              c(1) ..................................................
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
Venture Capital Investment under Uncertainty and Asymmetric ...
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  1. 1. Venture Capital Investment under Uncertainty and Asymmetric Beliefs: A Continuous-Time, Stochastic Principal-Agent Model∗ Yahel Giat† , Steven T. Hackman‡ and Ajay Subramanian§ October 21, 2008 Abstract We develop a continuous-time, stochastic principal-agent model to investigate the effects of asymmetric beliefs and agency conflicts on the characteristics and valuation of venture capital projects. In our model, a venture capitalist (VC) and an entrepreneur (EN) have imperfect information and differing beliefs about the intrinsic quality of a project in addition to having asymmetric attitudes towards its risk. We characterize the equilibrium of the stochastic dynamic game in which the VC’s dynamic investments, the EN’s effort choices, the dynamic compensation contract between the VC and EN, and the project’s termination time are derived endogenously. Consistent with observed contractual structures, the equilibrium dynamic contracts feature both equity-like and debt-like components, the staging of investment by the VC, the progressive vesting of the EN’s stake, and the presence of inter-temporal performance targets or milestones that must be realized for the project to continue. We numerically implement the model and calibrate it to aggregate data on VC projects. Our analysis of the calibrated model shows that EN optimism significantly enhances the value that venture capitalists derive. Entrepreneurial optimism explains the discrepancy between the discount rates used by VCs (∼ 40%), which adjust for optimistic payoff projections by ENs, and the average expected return of VC projects (∼ 15%). Our results show how the “real option” value of venture capital investment is affected by the presence of agency conflicts and asymmetric beliefs. Key Words: Dynamic Principal-Agent Models, Stochastic Dynamic Games, Incentive Con- tracts, Imperfect Information, Heterogeneous Beliefs. ∗ We gratefully acknowledge financial support from the Kauffman Foundation under the “Roadmap for an En- trepreneurial Economy” initiative. We thank two anonymous referees and seminar audiences at the the 2007 Stan- ford Institute for Theoretical Economics (SITE) workshop on “Dynamic Financing and Investment”, the 2007 North American Summer Meeting of the Econometric Society (Duke University, Durham, NC), the 2007 Real Options Con- ference (Berkeley, CA), the 2008 Chicago-Minnesota Theory Conference (University of Chicago), the Fields Institute for Mathematical Sciences (Toronto, Canada), the University of Paris-Dauphine (Paris, France), and ESSEC (Paris, France) for valuable comments. The usual disclaimers apply. † Department of Industrial Engineering and Management, Jerusalem College of Technology, Jerusalem, Israel ‡ School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332 § Department of Risk Management and Insurance, J. Mack College of Business, Georgia State University, Atlanta, GA 30303
  2. 2. 1 Introduction Real-world productive activities are typically characterized by decentralized decision-making. The agents who control various aspects of production often have different objectives, access to informa- tion, and beliefs about project qualities (see Chen and Zenios, 2005, Gibbons, 2005). For example, entrepreneurs (ENs) are often more optimistic about the success of their start-up firms than the experienced venture capitalists (VCs) who provide capital (see Baker et al, 2005 for a recent sur- vey). Further, VCs are usually well-diversified and less exposed to firm-specific risk than the less diversified ENs who have significant human capital invested in their firms. As a result, VCs and ENs have differing attitudes towards the risks of projects, which leads to agency conflicts that affect the financing and operation of start-up firms. The interests of ENs (the “agents”) are aligned with those of VCs (the “principals”) through incentive contracts that are affected by the VC’s and EN’s heterogeneous beliefs about the outcomes of projects as well as their differing risk attitudes. We develop a dynamic, stochastic principal-agent model of venture capital investment to ex- amine the impact of asymmetric beliefs on the characteristics of venture capital projects—their values, the structures of dynamic contracts between VCs and ENs, the durations of VC projects, and the manner in which VC investment is staged over time. In our model, VCs and ENs have asymmetric beliefs about the intrinsic qualities of projects as well as asymmetric attitudes towards their risk. We characterize the equilibrium of the stochastic dynamic game in which the VC’s dynamic investments, the EN’s effort choices, the dynamic compensation contract for the EN, and the project’s termination time are derived endogenously. We calibrate the model parameters by matching the return distributions of VC projects predicted by the model to their observed values in the data. We show that the degree of EN optimism is significant enough to explain the discrepancy between the discount rates used by VCs to value projects (∼ 40%), which adjust for optimistic payoff projections by ENs, and the average expected return of VC projects (∼ 15%). EN optimism is a key determinant of the durations and economic values of VC relationships, and could explain features of observed contracts between VCs and ENs. Model Overview: In our continuous-time stochastic model, a cash-constrained EN with a project approaches a VC for funding. The project generates potential value through physical capital 1
  3. 3. investments by the VC and human capital (effort) investments by the EN. We model the evolution of the project’s termination payoff at each date, which is the total payoff (present value of future earnings) if the project is terminated at that date. The termination payoff evolves as a Gaussian process and is contractible. The variance of the termination payoff process is the project’s intrinsic risk, which remains invariant through time. The drift of the termination payoff process has two components: a fixed, non-discretionary component that represents the project’s intrinsic quality, and a discretionary component that is determined by the VC’s investment and the EN’s effort. The discretionary component is observable, but non-verifiable and, therefore, non-contractible. The VC and the EN have imperfect information about the project’s intrinsic quality and could have differing, normally distributed priors. Their respective beliefs are, however, common knowl- edge, that is, they “agree to disagree” about their respective mean assessments of project quality, the difference of which represents the degree of asymmetry in beliefs. We consider the general sce- nario in which the VC’s and EN’s mean assessments of project quality could differ from its true mean. Further, the EN could be either optimistic or pessimistic relative to the VC. The common variance of the VC’s and EN’s respective assessments of the project’s quality is the project’s tran- sient risk. The transient risk is resolved over time as the VC and EN update their assessments of the project’s quality based on observations of the project’s termination payoff. The VC has linear preferences whereas the EN is risk-averse with CARA preferences. The VC offers the EN a long-term contract that specifies her dynamic investment policy, the termination time (a stopping time) of the project, and the EN’s payoff. The EN dynamically chooses his effort to maximize his expected utility. The contractually specified payoffs of the VC and EN, the investment policy, the EN’s effort policy, and the termination time are derived endogenously in equilibrium of the dynamic game between the VC and EN. The Equilibrium Contracts: We derive the incentive efficient dynamic contracts between the VC and EN. Under an optimal contract, the change in the EN’s stake in the project or her promised payoff (his “certainty equivalent” expected future utility) evolves as an Ito process. The change in the EN’s stake has a performance-sensitive component that depends on the change in the project’s termination payoff and a performance-invariant component that does not. The key contractual parameters—the VC’s investments, the EN’s effort, and his compensation—are determined by the 2
  4. 4. EN’s pay-performance sensitivities, that is, the sensitivities of the change in the EN’s stake to the change in the project’s termination payoff. Conditional on the project’s continuation, the VC’s optimal investments and the EN’s pay-performance sensitivities are deterministic functions of time. The performance-invariant component of the change in the EN’s stake over the period is, however, stochastic and depends on the project’s termination payoff history through its effect on the VC’s and EN’s updated assessments of the project’s intrinsic quality. Consistent with observed contractual structures, (i) the VC’s payoff structure has “debt” and “equity” components; (ii) the VC optimally stages her investment; (iii) the EN’s stake in the project progressively vests over time; and (iv) the project is continued if and only if inter-temporal milestones or performance targets are realized (see Gompers, 1995, Kaplan and Stromberg, 2003). The Dynamics of Equilibrium Contracts: The time-paths of the VC’s investments and the EN’s pay-performance sensitivities depend on the relative magnitudes of the degree of asymmetry in beliefs and the costs of risk-sharing between the VC and EN. If the EN is pessimistic, then the pay-performance sensitivities and investments increase over time. If the EN is “reasonably optimistic,” i.e., the EN is optimistic, but the degree of his optimism is below a threshold relative to the costs of risk-sharing, then the pay-performance sensitivities and investments decrease over time. If, however, the EN is “exuberant,” i.e., the degree of EN optimism is above this threshold, then the pay-performance sensitivities decrease over time and investments increase in early periods and decrease in later periods. Hence, depending on the relative magnitudes of risk-sharing costs and asymmetry in beliefs, the VC’s investment policy could become more aggressive over time, less aggressive over time, or vary non-monotonically. The EN’s compensation could become either more or less sensitive to performance over time. The intuition for the above results hinges on the interplay among (i) the EN’s effort that is positively (negatively) affected by his optimism (pessimism); (ii) the costs of risk-sharing due to the EN’s risk aversion; and (iii) the complementary effects of the VC’s investment and the EN’s effort on output. The passage of time lowers the degrees of optimism (or pessimism) as successive project realizations cause the VC and the EN to revise their initial assessments of project quality. Hence, the beneficial (detrimental) effects of optimism (pessimism) in mitigating the agency costs of risk-sharing between the VC and the EN decline over time. 3
  5. 5. If the EN is optimistic, the VC can increase the performance-sensitive component of the EN’s compensation because the EN overvalues this component. Hence, the EN’s pay-performance sen- sitivity and effort are initially high. As the EN’s optimism declines over time, the positive effect of optimism on the power of incentives that can be provided to the EN declines so that the EN’s pay-performance sensitivity and effort decrease. (The opposite implications hold true when the EN is pessimistic.) If the EN is optimistic, but his degree of optimism is below a threshold, the VC’s investment also declines over time. Because investment and effort are complementary, the decline in the power of incentives to the EN and, therefore, his effort over time causes the VC to also lower her investment over time. If the EN’s optimism is above a threshold, the VC exploits the EN’s exuberance by initially increasing her investment to compensate for the decrease in effort of the EN. After a certain point in time when the VC’s investment attains its maximum, the decreasing effort of the EN makes it optimal for the VC to also lower her investments. The effects of risk on the investment path also depend on the degree of asymmetry in beliefs. If the EN is either pessimistic or reasonably optimistic, the time path of optimal investments decreases pointwise with the EN’s risk aversion, the project’s intrinsic risk, and its initial transient risk. If the EN is exuberant, however, the investment path is, in general, non-monotonic and increases with the EN’s risk aversion as well as the project’s intrinsic and transient risks in early periods, and decreases in later periods. In contrast with traditional real options models in which all decisions are made by monolithic agents (Dixit and Pindyck, 1994), the interaction between asymmetric beliefs and agency conflicts could lead to a positive or negative relation between risk and investment. Calibration and Numerical Analysis: We numerically implement the model and calibrate the baseline values of its parameters, which include the average intrinsic quality of VC projects, the degree of asymmetry in beliefs between VCs and ENs, the EN’s risk aversion, and his disutility of effort. We estimate these parameters by matching the predicted distributions of “round by round” returns of VC projects to their observed values reported in Cochrane (2005). Consistent with anecdotal evidence, our “indirect inference” approach shows that ENs are, indeed, significantly optimistic relative to VCs. (Recall that we do not assume that the EN is optimistic a priori.) Because the VC exploits EN optimism through the provision of more powerful incentives, EN optimism significantly enhances the value to the VC. Interestingly, however, EN 4
  6. 6. optimism lowers overall project value because the VC prolongs the project and over-invests to take advantage of the EN’s optimism and enhance her own value at the expense of project value. We examine how changes to the degree of EN optimism, the project’s intrinsic risk, and its transient risk affect the project value and the VC’s stake. The project’s intrinsic and transient risks have opposing effects on the “speed of learning” about project quality and, therefore, the rate at which the degree of EN optimism declines over time. As a result, they have differing effects on the project value and the VC’s stake: the project value and the VC’s stake decrease with the project’s intrinsic risk, but vary non-monotonically with its transient risk. Prior empirical and anecdotal literature documents that VCs use discount rates around 40% to value projects even though the average expected return of VC projects is approximately 15% (Cochrane, 2005). It has been suggested that higher discount rates could be a mechanism that VCs use to “adjust” optimistic projections by ENs. Previous research, however, has not ascertained whether EN optimism is, in fact, significant enough to generate such a large discrepancy between VC discount rates and the average expected returns of VC projects. We define the implied discount rate (IDR) as the rate at which the VC would discount the EN’s projections of the project’s payoffs to conform to her own valuation of the project’s payoffs. The IDRs for a wide range of parameter values predicted by the model lie between 30% and 50%, which is consistent with the range of VC discount rates reported in prior empirical research (e.g. Sahlman, 1990, Cochrane, 2005). Our study, therefore, confirms that entrepreneurial “optimism premia” are indeed high enough to justify the discount rates used by VCs in reality. Related Literature: Our study belongs to the growing body of literature that analyzes dynamic principal-agent models. In a seminal study, Holmstrom and Milgrom (1987) present a continuous-time principal-agent framework in which the principal and agent have CARA prefer- ences and payoffs are normally distributed. They show that the optimal contract for the agent is affine in the project’s performance. Schattler and Sung (1993) and Sung (1995) provide a rigorous development of the first-order approach to the analysis of continuous-time principal-agent prob- lems with exponential utility using martingale methods. Following Spear and Srivastava (1987), a significant stream of the literature applies dynamic principal-agent models to study executive compensation (Spear and Wang, 2005, Cvitanic et al, 2005, Cadenillas et al, 2006, Sannikov, 2007) 5
  7. 7. and financial contracting (DeMarzo and Fishman, 2006, Biais et al, 2007). We contribute to this literature by developing and analyzing a dynamic principal-agent model with imperfect public information and heterogeneous beliefs. The optimal dynamic contract reflects the effects of Bayesian learning and the resultant dynamic variation of the degree of asymmetry in beliefs in addition to the usual tradeoff between risk-sharing and incentives. Further, both the principal and the agent take productive actions in our model. In the specific context of venture capital, a strand of the literature investigates the importance of staging in the mitigation of VC-EN agency conflicts. Using a deterministic model, Neher (1999) shows that staging is essential to overcome the hold-up problem. As in Neher (1999), the manner in which VC investment is staged over time as well as the number of stages are determined en- dogenously in our framework. Our framework is, however, stochastic and incorporates asymmetric beliefs between the VC and EN.1 Another strand of the literature on venture capital analyzes the features of the optimal contracts that emerge in “double-sided” two-period moral hazard models in which the VC and EN exert effort (Casamatta, 2003, Cornelli and Yosha, 2003, Schmidt, 2003, Repullo and Suarez, 2004). We too develop a model in which the VC and EN take value-enhancing actions. Similar to these studies, the optimal contracts predicted by our analysis have “debt” and “equity” features consistent with observed contractual structures. Our study focuses on the effects of asymmetric beliefs on the characteristics of VC-EN relationships in a dynamic principal-agent model. The principal-agent paradigm is also applied to various operations management contexts. Fol- lowing the early work of Atkinson (1979), recent studies examine the inefficiencies arising from either hidden information (for example, Cachon and Lariviere, 2001, Ha, 2001) or hidden action (for example, Lal and Srinivasan, 1993, Plambeck and Zenios, 2000, 2003, Chen, 2005) in supply chain contracting. We contribute to this line of research by developing and analyzing a dynamic principal-agent model with heterogeneous beliefs, and in which both the principal and agent make value-enhancing decisions over time. Our framework could potentially be applied in supply chain contexts as well as scenarios such as venture capital investment and R&D in which heterogeneous 1 Kockesen and Ozerturk (2004) argue that some sort of EN “lock-in” is essential for staged financing to occur. Egli et al (2006) argue that staging can be used to build an EN’s credit rating. Berk et al (2004) develop an R&D model with a single, monolithic agent in which staging is exogenous. 6
  8. 8. beliefs and agency conflicts play important roles. In summary, we contribute to the literature by developing and analyzing a dynamic, stochastic principal-agent model of venture capital investment. The model is parsimonious, yet realistic enough to be taken to the data to yield quantitative assessments of the effects of the salient aspects of VC projects, namely, risky payoffs, agency conflicts, uncertainty about project quality and asymmetric beliefs. The tractability of the model, coupled with the fact that it is able to match disparate empirical data on the payoff distributions of projects as well as the discount rates used to value them, suggests that it could be useful as a tool to value risky ventures. 2 The Model We develop a continuous time framework with time horizon [0, T ]. At date zero, a cash-constrained entrepreneur (hereafter, EN) with a project approaches a venture capitalist (hereafter, VC) for funding. The project generates value through physical capital investments by the VC and human capital (effort) investments by the EN. Both the VC and the EN have imperfect information about the project and differ, in general, in their initial assessments of the project’s quality. If the VC agrees to invest in the project, she offers the EN a long-term contract that describes her subsequent investments in the project, the EN’s compensation, and the termination time of the relationship. The VC’s investments are made continuously over time. The termination time could be a random stopping time. The key state variable in the model is the project’s termination payoff Vt , which is the total payoff if the VC-EN relationship is terminated at date t. The termination payoff is the only economic variable that is contractible. For simplicity, we assume the project does not generate any intermediate cash flows so that all payoffs occur upon termination. 2.1 The Termination Payoff Process All stochastic processes are defined on an underlying probability space (Ω, F, P ) on which is defined a standard Brownian motion B. The initial termination payoff of the project is V0 . The incremental termination payoff, that is, the change in the termination payoff over the infinitesimal period 7
  9. 9. [t, t + dt], dVt , is the sum of a base output, a Gaussian process that is unaffected by the actions of the VC and EN, and a discretionary output, a deterministic component that depends on the physical capital investments by the VC and the human capital (effort) by the EN. It is given by base output discretionary output β dVt = (Θ − lt )dt + sdBt + Acα ηt dt. t (1) The first component, Θ, of the base output represents the project’s core output growth rate, which we hereafter refer to as the project’s intrinsic quality. The VC and EN have imperfect information about Θ and could also differ in their beliefs about its value. The second component of the base change, lt , represents “operating costs,” which could include wages to salaried employees, depreciation expenses, decline in revenues due to increased competition, fixed costs arising from increases in the scale of the project, etcetera. These costs are deterministic and increasing over time, which ensure that termination occurs in finite time almost surely. The third component of the base change, sdBt , where s > 0 is a constant, represents the “intrinsic” component of the project’s risk in period [t, t + dt]. It is the component of the project’s risk that remains invariant over time, and is independent of Θ. The discretionary output in period [t, t+dt] is a direct result of the VC’s capital investment rate ct and the EN’s effort ηt , and is described by a Cobb-Douglas production function. The discretionary output is observable to the VC and the EN. However, as in the literature on incomplete contracting (see Chapter 6 of Laffont and Martimort, 2002), the discretionary output is non-verifiable and, therefore, non-contractible. Because the discretionary output is non-contractible, the EN must be indirectly provided with appropriate incentives to exert effort through her explicit contract with the VC that can only be contingent on the termination payoff process. The uncertainty in the value of Θ is the project’s transient risk. The VC’s and EN’s initial priors on Θ are normally distributed with Θ ∼ N (µV C , σ0 ) and Θ ∼ N (µEN , σ0 ), respectively. Their 0 2 0 2 respective beliefs are, however, common knowledge, that is, they agree to disagree (see Morris, 1995, Allen and Gale, 1999). Because the equilibrium does not depend on how the EN’s and VC’s mean assessments of project quality relate to its true mean, we make no assumptions about the true mean of the project quality distribution. We consider the most general scenario in which the 8
  10. 10. EN could be optimistic or pessimistic relative to the VC, that is, µEN could be greater or less than 0 µV C . While the VC and EN disagree on the mean of the project’s intrinsic quality, they agree on 0 its variance, σ0 .2 2 The transient risk is resolved over time as the VC and the EN update their priors on Θ in a Bayesian manner based on observations of the project’s performance. Define dξt := dVt − (Φ(ct , ηt ) − lt )dt = Θdt + sdBt . (2) It follows from well-known formulae (Oksendal 2003) that the posterior distribution on Θ for each 2 date t ≥ 0 is N (µt , σt ), = V C, EN , where 2 s2 σ0 2 s2 µ0 + σ0 ξt 2 σt = 2 , µt = 2 , = V C, EN. (3) s2 + tσ0 s2 + tσ0 Note that the σt tend to zero. Let s2 ∆0 ∆0 2 ∆t := µEN − µV C = t t 2 + tσ 2 = 2 σt (4) s 0 σ0 denote the degree of asymmetry in beliefs at date t. It is resolved deterministically and monoton- ically over time, and its absolute value, | ∆t |, also declines over time. Consequently, if the EN is more optimistic (pessimistic) than the VC, the degree of optimism (pessimism) declines. 2.2 VC-EN Interaction The contract between the VC and the EN describes the VC’s capital investments over time, the EN’s effort, the termination date, and the EN’s payoff upon termination. The termination time is, in general, a random stopping time that is contingent on the project’s performance history. We follow the traditional principal-agent literature by having the contract also specify the EN’s effort and requiring that the contract be incentive compatible with respect to the specified effort of the 2 The literature on behavioral economics (see Baker et al, 2005) distinguishes between optimism and overconfidence. The EN is “optimistic” if his assessment of the mean (the first moment) of the project quality distribution is higher than that of the VC, while he is “overconfident” if his assessment of the variance (the second moment) of the project quality distribution is lower than that of the VC. In the terminology of the behavioral economics literature, therefore, the EN could be optimistic, but not overconfident in our framework. 9
  11. 11. EN (see Holmstrom and Milgrom, 1987). Let {Ft } denote the information filtration generated by the history of termination payoffs, the VC’s investments and the project’s discretionary outputs. A contract is described by the quadruple (Pτ , c, η, τ ), where c and η are {Ft }-adapted stochastic processes, τ is an {Ft }-stopping time, and Pτ is a nonnegative {Fτ }-measurable random variable. Pτ is the EN’s contractually promised payoff and Vτ − Pτ is the VC’s payoff at the termination time τ of the contractual relationship. In period [t, t + dt], the VC’s investment rate is ct and the EN’s effort is ηt . The VC offers the EN a long-term contract at date zero. The VC is risk-neutral whereas the EN is risk-averse with inter-temporal CARA preferences described by a negative exponential utility function. Their discount rates are equal and set to zero to simplify the notation. We extend the model to incorporate nonzero discount rates when we calibrate it to the data in Section 6. The EN’s expected utility at date zero from a contract (Pτ , c, η, τ ) is τ EN γ −E0 exp − λ Pτ − kηt dt . (5) t EN In (5), E0 denotes the expectation with respect to the EN’s beliefs at date zero and the parameter λ ≥ 0 characterizes the EN’s risk aversion. The EN’s disutility from effort in period [t, t + dt] is γ given by kηt dt with k > 0, γ > 0. For future reference in the derivation of the equilibrium, we follow Holmstrom and Milgrom (1987) by defining the EN’s certainty equivalent expected future utility, Pt , from the contract at any date t as τ EN γ exp(−λPt ) := Et exp − λ Pτ − kηu du , (6) t EN where the notation Et denotes the EN’s expectation conditioned on the information available at date t, that is, the σ-field Ft . Note that the EN’s certainty equivalent future expected utility at the contractual termination date τ is his contractually promised terminal payoff Pτ . For expositional convenience, we hereafter refer to the EN’s certainty equivalent expected future utility process {Pt , t ≥ 0} as his promised payoff process. The allocation of bargaining power between the VC and the EN is determined by the certainty equivalent reservation utility or promised payoff that the EN must be guaranteed at date zero. We 10
  12. 12. allow for all possible allocations of bargaining power that are indexed by different values of P0 . A contract (Pτ , c, η, τ ) is feasible if and only if it is incentive compatible for the EN with respect to his effort choices, that is, given the terminal payoff, Pτ , the VC’s investment policy, c, and the termination time τ , it is optimal for the EN to exert effort described by the process η. The risk-neutral VC’s optimal contract choice is a feasible contract that maximizes her expected payoff net of her investments, i.e., a feasible contract (Pτ , c, η, τ ) is optimal if and only if it solves τ (Pτ , c, η, τ ) = arg max E0 C Vτ − Pτ − V ct dt , (7) (Pτ ,c ,η ,τ ) t where E0 C denotes the expectation with respect to the VC’s beliefs at date zero and the maxi- V mization is over feasible contracts. 3 The Equilibrium We assume the following condition on the parameters for the remainder of the paper: Assumption 1 (1 − α)γ/β > 2. This condition implies that the EN faces decreasing returns to scale from the provision of effort. Further, the EN’s disutility from his effort is sufficiently pronounced relative to his positive contri- bution to output that an equilibrium contract between the VC and the EN exists. 3.1 Structure of Optimal Contract The following two theorems characterize the optimal contract. Proofs are provided in Appendix A. Theorem 1 (The EN’s Promised Payoff Process) The EN’s promised payoff evolves as dPt = at dt + bt dVt , where the contractual parameters at ∈ I bt ∈ I ++ are {Ft }-progressively measurable. R, R The parameter bt is the EN’s pay-performance sensitivity. It represents the sensitivity of the change in the promised payoff to performance during the infinitesimal period [t, t + dt]. The parameter at is the EN’s performance-invariant compensation. It determines the component of the change in the promised payoff that does not depend on performance during the infinitesimal period [t, t + dt]. 11
  13. 13. In light of Theorem 1, a contract is completely specified by the performance-invariant compen- sation and pay-performance sensitivity parameters, at , bt , the VC’s investment rate, ct , the EN’s effort, ηt , at each time t, and the termination time τ . 3.2 Existence and Characterization of Equilibrium We briefly outline the arguments involved in the derivation of the optimal contract, which is formally characterized in Theorem 2 below and proved in Appendix A. Fix date t ≥ 0. The derivation proceeds in four steps: Step 1. The EN’s incentive compatible effort. For a given EN’s pay-performance sensitivity bt and the VC’s investment rate ct at date t, we show that the EN’s incentive compatible effort is Aβcα bt t 1 γ−β η(bt , ct ) := . (8) γk Step 2. The EN’s performance-invariant compensation. The VC optimally chooses her investment rate ct and the EN’s pay-performance sensitivity bt incorporating the EN’s incentive compat- ible effort given by (8). The performance-invariant compensation parameter at is chosen to satisfy the “promise-keeping” constraint, that is, the EN’s promised payoff is actually deliv- ered by the contract. The promise-keeping constraint pins down the contactual parameter at as a function of the other two contract parameters bt and ct . In particular, at = at (bt , ct ) := 0.5λs2 b2 + kη(bt , ct )γ − bt Acα η(bt , ct )β − lt + µEN . t t t (9) Step 3. The optimal investment and pay performance sensitivity. Incorporating the EN’s incentive compatible effort (8) and the functional form for at in (9), the change in the VC’s continuation value (her expected future payoffs) at date t is CVt = Λt (bt , ct )dt, where (10) γ α γ−β Λt (bt , ct ) := ∆t b − 0.5λs2 b2 + φ(b)c − c + µV C − lt . (11) 12
  14. 14. In (11), ∆t is the degree of asymmetry of beliefs at date t, defined in (4), and  β β  γ−β  A γ 1 γ−β βb γ−β βb  k γ 1− γ , if 0 ≤ b ≤ γ/β, φ(b) := (12)    0, otherwise. The function Λt (bt , ct ) is the rate-of-change of the VC’s continuation value; hereafter, we shall refer to it simply as the continuation rate. The VC chooses the investment rate ct and the EN’s pay-performance sensitivity bt to maximize the continuation rate. Assumption 1 guarantees a unique solution b∗ , c∗ to this maximization problem. In particular, it implies t t that Λt (bt , ·) is strictly concave in ct , since the exponent on ct is less than one. Consequently, given the pay-performance sensitivity, there is a unique investment rate. We show that αγ γ−β γ−β c∗ = c(b∗ ) := t t (1−α)γ−β φ(bt ) (1−α)γ−β , (13) γ−β b∗ = arg max Λt (bt , c(bt )), where t (14) 0<bt γ − β − αγ Λt (bt , c(bt )) := ∆t bt − 0.5λs2 b2 + t c(bt ) + (µV C − lt ). t (15) αγ We refer to the function c(·) as the optimal investment function. We discuss the properties of the optimal investment function, which plays a central role in our analysis, in Section 3.4. Step 4. Determination of the optimal termination time. The optimal termination time of the contract is the solution to the optimal stopping problem τ τ ∗ = arg max E0 C V Λt (b∗ , c∗ )dt, t t (16) τ ≤T 0 where the maximization is over all Ft -stopping times τ ≤ T . Theorem 2 (Characterization of Equilibrium) (a) Conditional on the project not being terminated prior to date t ∈ [0, T ]: – The EN’s pay-performance sensitivity parameter, b∗ , solves (14). t – The VC’s equilibrium investment rate is c∗ = c(b∗ ), where c(·) is defined in (13). t t 13
  15. 15. – The EN’s performance-invariant compensation parameter is a∗ := at (b∗ , c∗ ), where at (·, ·) t t t is defined in (9). – The EN’s effort level is ηt := η(b∗ , c∗ ), where η(·, ·) is defined in (8). ∗ t t (b) The termination time of the relationship solves the optimal stopping problem (16). 3.3 The VC’s Controllable Rate Function Let γ − β − αγ Ft (b) := ∆t b − 0.5λs2 b2 + c(b) (17) αγ denote the “controllable” portion of the continuation rate; we hereafter refer to it as the VC’s controllable rate function. As summarized in Theorem 2, the equilibrium contract at date t is determined by b∗ , the solution to (14). An examination of (15) shows that b∗ is also the solution to t t b∗ = arg max Ft (b). t (18) 0<b By Theorem 2 and (18), the EN’s pay-performance sensitivity b∗ , the VC’s investment rate c∗ , t t ∗ and the EN’s effort ηt are deterministic functions of time (conditional on the project’s continuation). The performance-invariant compensation parameter a∗ is, however, stochastic and depends, in t particular, on the EN’s current mean assessment µEN of the project’s intrinsic quality. The proof t of the theorem shows that this parameter adjusts stochastically to ensure that the EN’s promise keeping constraints are satisfied at each date and state. The equilibrium contract critically depends on the VC’s controllable rate function, Ft (b). This function consists of three components: • Economic rent (cost) from the EN’s optimism (pessimism). When ∆t > 0, the term, ∆t b, reflects the rents that the VC extracts from the EN by exploiting his optimism about the project’s intrinsic quality. When ∆t < 0, ∆t b is the cost that the VC must bear to compensate the EN for his pessimism about the project’s intrinsic quality. 1 2 2 • Cost of risk. The term, 2 λs b , reflects the VC’s costs of risk-sharing with the risk-averse EN. We refer to λs2 as the price of risk ; hereafter, we denote it by p. 14
  16. 16. γ−β−αγ • Return on investment. The “return on investment” term, αγ c(b), reflects the VC’s expected return as a result of her investment and the EN’s effort. The interplay among these three “forces” determines the equilibrium dynamics. 3.4 Uniqueness and Stability of Equilibrium The characteristics of the contract depend on the optimal investment function c(·) given in (13). The following properties of c(·) play a central role in our subsequent analysis (see Figure 1). Proposition 1 γ γ (a) The function c(·) is strictly positive and strongly unimodal3 on [0, β ], satisfies c(0) = c( β ) = 0, and achieves its maximum at b = 1. γ γ (b) The function c(·) is strictly concave on [0, bM ] and strictly convex on [bM , β ], where bM ∈ (1, β ) is the unique minimum of the function c (·). Proof. The proof of this, and all subsequent results in the paper, are provided in Appendix B. The intuition for the non-monotonicity of the function is that an increase in the agent’s pay- performance sensitivity affects the principal’s investment in two distinct but opposite ways. On the positive side, the agent increases his effort. Because investment and effort are complementary, the increase in the agent’s effort provides an incentive for the principal to increase her investment. On the negative side, since the agent’s disutility of effort increases, the principal’s cost to maintain the agent’s participation also increases. For lower values of the pay-performance sensitivity, the complementarity of investment and effort causes the benefits of increased output to dominate. Hence, the principal finds it beneficial to increase her investment. However, beyond a threshold level of pay-performance sensitivity, the costs of inducing high effort from the agent are so high that the principal lowers her investment. In other words, it is optimal for the principal to allow output to be dominated by the agent’s effort. The ratio of the absolute value of the initial degree of asymmetry of beliefs to the price of risk, namely, | ∆0 | /p, provides an a priori bound on the equilibrium pay-performance sensitivity b∗ . t 3 A function f (·) is strongly unimodal on the interval [a, b], a < b, if there exists an x∗ ∈ (a, b) such that f (·) is increasing on [a, x∗ ] and f (·) is decreasing on [x∗ , b]. Obviously, the value x∗ maximizes f (·) on [a, b]. 15
  17. 17. Figure 1: Optimal investment function c(b) c (1) = 0 q .................. ..... ............. ... ... ... ... 6 .. .. .. .. ... .. .. .. .. .. .. . .. .. .. .. .. . .. .. ... . .. .. .. (b ) = 0 c .. .. . . . . . . . .. q convex - .. .. .. .. .. M . . .. .. . . .. .. .. .. .. .. .. .. .. . .. .. . . .. .. . . .. .. . . . . . . c o n c a v e- . ... .. ... ... .. .... .... . ..... . . . ..... -b .......... ......... γ γ c(0) = c( β ) = 0 1 bM β Proposition 2 An optimal solution to (18) is always less than or equal to max{ ∆0 , 1} if ∆0 ≥ 0 and is less than p 1 if ∆0 0. In our subsequent analysis we assume that the initial degree of asymmetry in beliefs, ∆0 , is below a threshold relative to the price of risk, p. The assumption ensures that the equilibrium is stable and the contractual parameters are continuous functions of the primitives of the model. Assumption 2 ∆0 /p ≤ bM . (The parameter bM above is defined in Proposition 1.) It follows immediately from Proposition 2 and Assumption 2 that a solution to (18) must lie in the interval [0, bM ). By Proposition 1, the optimal investment function c(·) is strictly concave on the interval [0, bM ]. It follows from (17) that the VC’s controllable rate function, Ft (b), is also strictly concave on [0, bM ] and hence strongly unimodal. Consequently, there exists a unique solution b∗ to (18). Moreover, it must also t be positive, since the proof of Proposition 1 shows that the marginal optimal investment c (0) is infinite. We summarize these observations with the following proposition. Proposition 3 Under Assumptions 1 and 2, the function Ft (·) is strictly concave on [0, bM ]. Further, the solution to (18) is strictly positive and less than bM . 4 Equilibrium Dynamics We investigate the dynamics of the EN’s compensation, his effort, and the VC’s investment condi- tional on continuation of the project. Since the degree of asymmetry in beliefs, ∆t , and variance, 2 σt , are deterministic functions of time (see (4)), it follows from Theorem 2 and Proposition 1 that 16
  18. 18. the equilibrium values for the pay-performance sensitivity, investment and effort at each point in time (conditional upon continuation) are also deterministic. The only component of the contract that is stochastic and is adjusted based on realizations of the termination payoff Vt of the project is the performance-invariant compensation parameter a∗ . t Let γ − β − αγ Ft (b) = F (b) := −0.5λs2 b2 + c(b) (19) αγ be the principal’s controllable rate function in the benchmark scenario in which beliefs are sym- metric. Since F (.) is time-independent, the agent’s equilibrium pay-performance sensitivities, the principal’s investments and the agent’s effort are all constant. Let b∗ , c∗ and ηp denote their values. p p ∗ It follows from (4) and (17) that the VC’s controllable rate function can be expressed as ∆0 2 Ft (b) = 2 σ b + F (b), (20) σ0 t where F (.) is defined in (19). Since σt → 0, it follows from Berge’s Theorem of the Maximum that b∗ → b∗ , and thus (c∗ , ηt ) → (c∗ , ηp ) by continuity where (b∗ , c∗ , ηp ) are the equilibrium pay- t p t ∗ p ∗ p p ∗ performance sensitivity, investment, and effort in the benchmark scenario with symmetric beliefs. We now describe the manner in which these economic variables converge to their asymptotic values. Theorem 3 (The Dynamics of the Equilibrium—Optimistic EN) Suppose that the EN is more optimistic than the VC so that ∆0 ≥ 0. (a) The EN’s pay-performance sensitivity b∗ decreases monotonically with t and approaches b∗ t p as t → ∞. (b) The value ∆0 s2 ∆0 s2 t∗ := ( − 1) 2 = 2 − 2 (21) p σ0 λσ0 σ0 is the point in time at which the EN’s pay-performance sensitivity and effort, and the VC’s investment rate equal their values in the “no agency” benchmark scenario. (This interpreta- tion of t∗ only applies if t∗ ≥ 0, which holds if and only if the initial degree of asymmetry of beliefs is at least as large as the price of risk.) 17
  19. 19. (c) The EN’s pay-performance sensitivity b∗ exceeds 1 if t t∗ , equals 1 at t = t∗ , and less than t 1 if t t∗ . (d) The VC’s investment rate c∗ increases until time t∗ and then decreases monotonically towards t c∗ as t → ∞. p (e) For t ≥ t∗ , ηt decreases monotonically towards ηp as t → ∞ . ∗ ∗ Note that if ∆0 ≤ p so that t∗ ≤ 0, then the EN’s pay-performance sensitivity, his effort, and the VC’s investment rate all decrease monotonically over time. Figures 2 and 3 illustrate the results of Theorem 3. They describe the trajectories of the EN’s pay-performance sensitivity and the VC’s investment for two different values of the initial degree of asymmetry in beliefs, ∆0 . The intuition for the results of Theorem 3 hinges on the interplay among the positive effect of optimism on the EN’s effort, the costs of risk-sharing due to the EN’s risk aversion that are negatively affected by the project’s intrinsic risk, and the complementary effects of investment and effort on output. The passage of time lowers the degree of optimism as successive project realizations cause the VC and the EN to revise their initial assessments of project quality. If the EN is optimistic, the VC can increase the performance-sensitive component of the EN’s compensation because the EN overvalues this component. Hence, the EN’s pay-performance sen- sitivity and effort are initially high. The negative effect of the evolution of time on the EN’s optimism, however, causes the EN’s pay-performance sensitivity and effort to decline over time. Due to the previously discussed non-monotonic relation between the VC’s investment and the EN’s pay-performance sensitivity, the VC’s investment initially increases to “compensate” for the de- crease in effort of the EN. After a certain point in time when investment attains its maximum, the decreasing effort of the EN makes it optimal for the VC to also lower her capital investments. Theorem 4 (The Dynamics of the Equilibrium—Pessimistic EN) Suppose that the EN is more pessimistic than the VC so that ∆0 0. (a) The EN’s pay-performance sensitivity b∗ increases monotonically with t and approaches b∗ as t p t → ∞. (b) The VC’s investment rate c∗ increases monotonically towards c∗ as t → ∞. t p 18
  20. 20. b∗ t 6 .. .... .... .... .. ... ... .. .... .... ..... ..... ...... ...... ........ ........ .......... .......... ∆ 0 ............... 0 ............... ........................ ........................ ............................................ ............................................. ∆0 = 0 ................................................. ................................................ b∗ p ........................................................................................ .................................................................................. ....................................... .................................... ................... .................. ........... ........... ........ ....... .. .. ..... ..... .... .... .... .... ∆0 0 - t 0 2 4 6 8 10 12 14 Figure 2: Possible equilibrium pay-performance sensitivity paths ∗ ∗ (c) The EN’s effort ηt increases monotonically towards ηp as t → ∞ . If the EN is pessimistic, he under-values the performance-sensitive portion of his compensation relative to the VC. Hence, the power of incentives that can be provided to the EN is initially low so that his pay-performance sensitivity and effort as well as the VC’s investment are initially low. With the evolution of time, the degree of pessimism declines, which has a positive effect on the power of incentives to the EN so that his pay-performance sensitivity, effort, and the VC’s investment all increase. Theorems 3 and 4 describe the paths of the EN’s pay-performance sensitivity and effort, and the VC’s investment rate conditional on the project’s continuation. Depending on the relationship between the degree of asymmetry in beliefs and the price of risk, it follows from the theorems that the VC’s investments until termination (these are the investments that are actually observed because there is no investment after termination) could either increase, decrease, or vary non- monotonically (initially increase and then decrease). 4.1 Sensitivity of Equilibrium Dynamics In light of Theorems 3 and 4, the manner in which the equilibrium dynamics are affected by changes in the underlying parameters critically depends on the initial value of the degree of asymmetry in beliefs ∆0 . In what follows the EN is said to be pessimistic if ∆0 0, reasonably optimistic if ∆0 ∈ [0, p) and exuberant if ∆0 ∈ (p, pbM ], where p = λs2 is the price of risk. (Assumption 2 guarantees that ∆0 ≤ pbM .) 19
  21. 21. c∗ t 6 c(1) ....................... .......................... ..... “no agency” investment level .... .. ........ ........ ... . ........ ........ ........ ........ . . ........ ........ . . . . . ∆ p ......... ......... 0... .......... .......... ........... . . ............ ............. ... . ... . ............. ............... .... ..... ............... ................. .... ...... .................. .................... . . ..... . ...................... .. ......... ...... ............ .... . . ....... ........ . ......... . . . . . ........... ............. ∆ p ................ 0 .................... ........................... . . ................................... .................................................... . .......................................................................... c ∗ .. . . ∆ =0 0 ......................................... p.. ..................................................................... ...................................................................... .............. ............ ............................................... .............................................. ....... ....... ................ ............... ....... ....... ..... ..... .. ... .... .... ∆ 0 0 .. ..... .. . - t 2 4 6 8 10 12 14 16 Figure 3: Possible equilibrium investment paths The following theorem characterizes the effects of the EN’s risk aversion, λ, the initial transient risk, σ0 , and the intrinsic risk, s2 , on the equilibrium dynamics. 2 Theorem 5 (Effects of Intrinsic Risk, Transient Risk and EN’s Risk Aversion) (a) If the EN is pessimistic, the paths of the EN’s pay-performance sensitivity and the VC’s investment are pointwise decreasing in the EN’s risk aversion, pointwise increasing in the initial transient risk, and pointwise decreasing in the intrinsic risk. (b) If the EN is optimistic, the path of the EN’s pay-performance sensitivity is pointwise decreas- ing in the EN’s risk aversion and the initial transient risk. (c) If the EN is reasonably optimistic, then the path of the VC’s equilibrium investment is pointwise decreasing in the EN’s risk aversion and the initial transient risk. (d) If the EN is exuberant, the VC’s investment path changes as depicted in Figure 4 as a result of a change in the EN’s risk aversion and the initial transient risk. More precisely, let λ1 λ2 and σ1 σ2 be two possible values of the EN’s risk aversion and the initial transient risk, respectively. There exist t∗ (λ1 , λ2 ) and t∗∗ (σ1 , σ2 ) such that the VC’s investments when the EN’s risk aversion is λ1 (the initial transient risk is σ1 ) are higher than her investments when the EN’s risk aversion is λ2 (the initial transient risk is σ2 ) for t t∗ (λ1 , λ2 ) (t t∗∗ (σ1 , σ2 )) and lower for t t∗ (λ1 , λ2 ) (t t∗∗ (σ1 , σ2 )). (e) If ∆0 ≤ 4p, parts (b)-(d) hold for the intrinsic risk. If ∆0 4p, the effects of intrinsic risk on 20
  22. 22. the pay-performance sensitivity and investment paths is ambiguous.4 Figure 4 demonstrates that the path of equilibrium investment converges to different limiting values depending on the EN’s risk aversion. The EN’s pay-performance sensitivity, b∗ , declines with his risk aversion because an increase in t the EN’s risk aversion increases the costs of risk-sharing. An increase in the transient risk lowers the degree of optimism or pessimism at each date because the “signal to noise ratio” is increased so that the VC and EN “learn faster.” An increase in the intrinsic risk increases the costs of risk- sharing and also increases the degree of optimism or pessimism at each date because the “signal to noise ratio” decreases so that the VC and EN “learn more slowly.” When the EN is pessimistic, the pointwise decline in the degree of pessimism with the transient risk increases the power of incentives to the EN so that the EN’s pay-performance sensitivity and the VC’s investment increase at each date. On the other hand, the pointwise increase in the degree of pessimism and the costs of risk-sharing with the intrinsic risk decreases the power of incentives to the EN so that his pay-performance sensitivity and the VC’s investment decrease at each date. When the EN is optimistic, the decline of the degree of optimism with the initial transient risk causes the economic rents to the VC in each period from the EN’s optimism to be lowered relative to the costs of risk-sharing. Hence, the EN’ pay-performance sensitivity declines. Intrinsic risk, however, has conflicting effects on the power of incentives to the EN. An increase in the intrinsic risk increases the degree of optimism at each date, which has a positive effect on the power of incentives. However, an increase in the intrinsic risk also increases the costs of risk-sharing, which has a negative effect on the power of incentives. When the EN is optimistic and ∆0 ≤ 4p, the costs of risk-sharing outweigh the benefits of the EN’s optimism so that the EN’s pay-performance sensitivity also decreases with intrinsic risk. When ∆0 4p, the conflicting effects of optimism and risk-sharing costs cause the effects of intrinsic risk to be ambiguous. The change in the VC’s investment path when the EN is optimistic critically depends on whether the EN is reasonably optimistic or exuberant. If the EN is reasonably optimistic, then the costs of risk-sharing outweigh the benefits of the EN’s optimism so that the VC’s investment path declines 4 The condition is trivially satisfied when ∆0 0. Under Assumption 2, the condition ∆0 ≤ 4p is automatically satisfied when bM ≤ 4. Since b∗ bM , this condition implies that the EN’s optimal pay-performance sensitivity t should be less than four, which is easily satisfied in reasonable parametrizations of the model. 21
  23. 23. c∗ t 6 c(1) ...................................................................................................................................“no agency” investment level .... ... ...... ..... ... . . ... .. ... ....... ....... ........... ........... ............ ............ ... .. ....... ....... ...... ...... ............ ............ .. ....... ....... ............. ............. .. .. ....... ....... .............. .............. ................ ................ .. . ........ ........ ........ ........ .................. .................. c (t)1 ... . ......... ......... .......... .......... .................... ..................... ..... ... . . ........... ........... . .. ............ ............ . . . .............. .............. .................. 2 ................ ................ . . . . . ..................c (t) .................... ................... .. . . . . . - t t∗ 2 t∗ 1 Figure 4: Sensitivity of the equilibrium investment path to a change in the EN’s risk aversion, the initial transient risk, or the intrinsic risk. Path c2 (·) corresponds to an increase in λ, σ0 or s2 . 2 with the EN’s risk aversion as well as the project’s intrinsic and transient risk. If the EN is exuberant, then an increase in intrinsic or transient risk increases the costs of risk-sharing, thereby partially offsetting the VC’s rents from the EN’s optimism. Early in the project, it is beneficial for the VC to compensate for the resulting decline in the EN’s effort by increasing investment. As time passes, however, the EN’s degree of optimism declines thereby reducing the rents to the VC. The costs of risk-sharing, therefore, dominate in later in the project so that an increase in risk results in a decline in the VC’s investment. In stark contrast with traditional real options models with monolithic agents (e.g. Dixit and Pindyck, 1992), the results of Theorem 5 show that the interactive effects of optimism and agency conflicts could lead to a positive or negative relation between risk and investment. Theorem 6 (Effects of Degree of Asymmetry in Beliefs) (a) The path of the EN’s equilibrium pay performance sensitivity is pointwise increasing in the initial degree of asymmetry in beliefs. (b) If the EN is pessimistic or reasonably optimistic, then the path of the VC’s equilibrium investment is pointwise increasing in the initial degree of asymmetry in beliefs. (c) If the EN is exuberant, then the path of equilibrium investment by the VC changes as in Figure 5 as a result of a change in the initial degree of asymmetry in beliefs—the time-path of investment shifts “to the right” if the initial degree of asymmetry increases. 22

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