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.
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.
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
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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.
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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
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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)
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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.
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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
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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
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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.
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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
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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].
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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.
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.
– 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.
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.
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.
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.
(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.
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.
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.
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.
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
24.
c∗
t
6
c(1) ......................................................................................................................................“no agency” investment level
.
... ..... .
... .......
.
..
. ..
.. ........
........ ...........
...........
...
... ........
........ ...........
............
..
.. ........
........
.........
............
............
.............
.............
..
..
.........
..........
.......... ..............
..............
..
. ...........
........... ...............
................
..................... 2
............
............ ..................
..................
..
.
..
. ..............
..............................
................
.................... c (t)
.......
....
.
. ...................
...................
.. .......................
.......................
.. .................
................
.
.
.
.
.
.
.
c1 (t)
.
.
.
..
.
- t
t∗
1 t∗
2
Figure 5: Sensitivity of equilibrium investment path to the initial degree of asymmetry in beliefs.
Path c2 (·) corresponds to an increase in ∆0 .
An increase in the initial degree of asymmetry in beliefs increases the power of incentives that
can be provided to the EN so that his pay-performance sensitivity increases at each date. When
the EN is pessimistic or reasonably optimistic, the VC increases her investment at each date. When
the EN is exuberant, however, the investment path is non-monotonic. The intuition for the eﬀects
of risk on investment discussed earlier is reversed so that the degree of asymmetry in beliefs aﬀects
the investment path as described in part (c) of the theorem.
5 Project Duration
The following proposition shows that there exists a trigger level of the project’s mean quality at
each date such that it is optimal for the VC to continue the project if and only if her current
assessment of the project’s quality exceeds the trigger.
Proposition 4 (The Optimal Termination Policy)
The optimal stopping policy for the VC is a trigger policy: there exist µ∗ such that the VC
t
terminates the project only if µV C < µ∗ .
t t
23
25.
Let Yt∗ dt := (c∗ α ηt β − lt )dt denote the equilibrium net discretionary output in period [t, t + dt].
t
∗
Since dVt = Yt∗ dt + ξt dt, it follows that
t t t
∗
Vt − V0 = dVu = Yu du + ξu du .
0 0 0
Given the formula for µt given in (3), we may conclude that µt ≥ µ∗ if and only if Vt ≥ Vt∗ , where
t
t
(s2 + tσ0 )µ∗ − s2 µ0
2
t
Vt∗ := V0 + ∗
Yu du + 2 .
0 σ0
The sequence of the Vt∗ may be thought of as the performance targets the project must reach at
each date to prevent termination.
An increase in the EN’s initial degree of optimism about project quality increases the rents to
the VC from the EN’s optimism thereby increasing her expected continuation value at each point
in time. Hence, an increase in the EN’s optimism prolongs the project’s duration. An increase
in the EN’s risk aversion or cost of eﬀort, however, increases the costs of risk-sharing for the VC,
thereby lowering her continuation value at each point in time.
Proposition 5 (Comparative Statics of Project Duration)
The project duration τ increases with the initial degree of asymmetry in beliefs, decreases with the
EN’s risk aversion, and decreases with the EN’s cost of eﬀort.
6 Numerical Analysis
We numerically explore further implications of the model using a discrete-time approximation of
the continuous-time model. We describe the details of the numerical implementation in the on-line
Appendix C. We directly model the evolution of the VC’s current assessment of project quality µV C
t
because, as explained in Section 5, it determines her continuation decision at any date t. In the
ﬁrst stage of the numerical implementation, we approximate the evolution of µV C using a discrete
t
lattice and derive the termination triggers µ∗ . In the second stage, given the triggers obtained from
t
the ﬁrst stage, we use Monte Carlo simulation to model the evolution of µV C and to obtain the
t
key output variables of interest. Gompers (1995) reports that the average length of a round of VC
24
26.
ﬁnancing is approximately one year. Accordingly, we set the time period between successive dates
in the discrete lattice to one year and assume that it corresponds to a single round of ﬁnancing.
6.1 Calibration
To obtain a reasonable set of “baseline” parameter values for our numerical analysis, we calibrate
the model to actual aggregate data on the distribution of round by round returns of venture
capital projects reported in Cochrane (2005). We classify the parameters of the model into two
groups: “direct” parameters whose baseline values can be set using guidance from previous empirical
research, and “indirect” parameters whose values are estimated by matching statistics predicted
by the model to their observed values in the data.
In our numerical implementation, we incorporate a nonzero discount rate for the VC (and EN).
Cochrane (2005) ﬁnds that the average expected return on venture capital investment in his sample
is 15%. Accordingly, we set the discount rate, Rb , to 15%. Further, we assume that the VC has
experience so that her prior assessment of the project quality distribution is correct. We assume a
production technology with constant returns to scale so that β = 1 − α. We assume a quadratic
form l(t) = l1 tl2 for the loss function. We normalize the initial seed capital V0 to one and set P0
to this value. We estimate the remaining parameters of the model (see Table 1) by matching its
predictions to data.
In our estimation, we use statistics on the round by round returns and standard deviations of
VC projects in each of the ﬁrst four rounds of ﬁnancing reported in Table 4 of Cochrane (2005).
As the length of a single round of ﬁnancing is set to one year, the round-by-round returns of a VC
Vt −Vt−1 −ct−1
project in the model are ct−1 , 1 ≤ t ≤ 4. Cochrane (2005) also reports the overall mean
and standard deviation of the VC project returns and the average number of rounds of ﬁnancing.
The statistics used for our estimation are displayed in the ﬁrst rows of the two panels of Table
2. We estimate the values of the indirect parameters of the model by matching the predicted
values of the statistics in Table 2 to their observed values. The standard errors of the estimates are
determined by parametric bootstrapping (see the on-line Appendix C).
As shown in Table 2, the model is able to closely match the observed statistics. The estimated
value of the degree of asymmetry in beliefs ∆0 = 0.504, while the VC’s assessment of the mean
25
27.
project quality µV C = 0.113. The data, therefore, suggest that the level of entrepreneurial optimism
0
is very signiﬁcant. The baseline values of the average intrinsic risk s and transient risk σ0 are high,
which conﬁrms anecdotal and empirical evidence that venture capital is risky and is characterized
by signiﬁcant uncertainty about project quality.
6.2 Numerical Results
We ﬁrst analyze the model when the parameters take their baseline values in Table 1. We then
explore various comparative static relationships by varying parameters about their baseline value.
In our numerical analyses, we compare the actual scenario in which there are asymmetric beliefs
and agency conﬂicts with two benchmark scenarios: the no agency scenario in which beliefs are
symmetric, and both the VC and EN have linear preferences; and the symmetric beliefs scenario
in which the VC and EN have symmetric beliefs, but the EN has CARA preferences.
We compute two output variables in each of the three scenarios. The
τ −1
P roject V alue := E0 C e−Rb τ Vτ −
V
e−Rb t ct . (22)
t=0
is the expected total payoﬀs to the ﬁrm less the capital investments discounted at the rate Rb . The
V C V alue := Project Value − E0 C e−Rb τ Pτ .
V
(23)
is the Project Value less the termination payoﬀ to the EN discounted at the rate Rb . The expecta-
tions in (22) and (23) are with respect to the VC’s beliefs about project quality, which are assumed
to be correct. The termination payoﬀ process evolves as in (1) with the contractual parameters,
(a∗ , b∗ , c∗ ), the EN’s eﬀort, η ∗ , and the performance targets, V ∗ , set to their equilibrium values
for the speciﬁc economic scenario (no agency, symmetric beliefs or actual) being analyzed.
6.2.1 Baseline Analysis
Table 3 reports the Project Value and VC Value in the actual scenario and the two benchmark
scenarios. The diﬀerence between the project values (VC values) in the no agency and symmetric
benchmark scenarios represent the deadweight agency costs of risk sharing between the VC and
26
28.
EN from the perspective of the ﬁrm (the VC fund). The diﬀerence between the project values (VC
values) in the actual and symmetric scenarios reﬂect the eﬀects of EN optimism on the project
value (VC value). We see that the VC signiﬁcantly beneﬁts from EN optimism. The project value,
however, is lower in the actual scenario than in the symmetric beliefs scenario. This is the due to
the fact that the VC exploits EN optimism by over-investing and prolonging the project’s duration
to increase her value at the expense of the overall value of the project.
Table 4 reports the EN’s pay-performance sensitivities and the investments for the ﬁrst four
rounds. Consistent with Theorem 3, the EN’s pay-performance sensitivity and the VC’s investments
decline over time. The EN’s pay-performance sensitivity decreases sharply across the four periods.
Successive capital infusions by the VC, therefore, rapidly reduce the EN’s stake in the ﬁrm.
6.2.2 Comparative Statics
Figure 6 (a) shows that the project value and VC value both vary non-monotonically with the
initial transient risk—they initially decrease and then increase. To understand the intuition for the
eﬀects of transient risk, note that, by (4), the degree of EN optimism at any date t declines with
the initial transient risk. This has a negative eﬀect on the power of incentives to the EN, his eﬀort,
and the economic rents the VC can extract from EN optimism. From (2) and (3), the standard
µ
deviation σt of the evolution of the mean assessment of project quality is
2
sσ0
µ
σt = 2. (24)
s2 + tσ0
From (24), an increase in the initial transient risk increases the standard deviation of the evolution
of the mean assessment of project quality and, therefore, the likelihood of both high and low
realizations. Since the VC can limit her downside by terminating the relationship if intermediate
signals of project quality are suﬃciently poor, the “real option value” of continuing the project
increases with the initial transient risk. The interaction between the negative eﬀects of transient
risk on the degree of EN optimism and its positive eﬀects on the real option value of continuation
causes the project value and VC Value to vary non-monotonically with the initial transient risk.
27
29.
Figure 6 (b) shows that the project value and VC value decline with the intrinsic risk, s.5 From
(24), the standard deviation of the evolution of the mean assessment of project quality decreases
with intrinsic risk above a threshold. Hence, the option value of continuing the relationship in any
period also declines. An increase in the intrinsic risk also increases the costs of risk-sharing, which
has a negative eﬀect on the EN’s eﬀort and the VC’s investment. The project value and VC value,
therefore, decline with intrinsic risk.
Figure 6 (c) shows that the VC value signiﬁcantly increases with the degree of asymmetry in
beliefs, which illustrates the beneﬁts to the VC from exploiting the EN’s optimism by providing
more powerful incentives. The project value, however, initially increases and then decreases with
the degree of asymmetry in beliefs. When the degree of EN optimism is low, the positive eﬀects
of increased optimism on the EN’s eﬀort, the VC’s investment and the project’s output cause the
project value to increase with optimism. When the degree of EN optimism is above a threshold,
however, the VC exploits the EN’s optimism by sub-optimally (from the standpoint of project
value) prolonging the project’s duration and over-investing in the project, thereby increasing her
value at the expense of project value.
6.2.3 Implied Discount Rates
There is considerable empirical and anecdotal evidence that VCs typically use high discount rates
in the range between 35% and 50% to value projects (see Sahlman, 1990, Gladstone and Gladstone,
2002). Sahlman (1990) suggests that high discount rates could be a mechanism that VCs use to
adjust optimistic projections by ENs. To the best of our knowledge, however, it has not been
ascertained whether optimism could indeed explain the high discount rates used by VCs. Do the
levels of EN optimism predicted by our model lead to the discount rates observed in reality?
We calculate the implied discount rate (IDR) of a project as the rate the VC would use to
discount the EN’s projections of the project’s payoﬀs to equal her own valuation of the project
deﬁned in (23). In other words, the IDR is the discount rate the VC would use to obtain her
valuation of the project if the project’s intrinsic quality were (hypothetically) distributed according
5
We allow for the discount rate to vary with the intrinsic risk as follows: Rb (s) := r + ((0.15 − r)/ˆ)s, where
s
r = 0.068 is the risk-free rate reported in Cochrane (2005) and s is the baseline value of s.
ˆ
28
30.
to the EN’s beliefs. The IDR βV C solves:
τ −1
−βV C τ
E0 e (Vτ − Pτ ) − e−βV C t ct Vt Θ ∼ N (µEN , σ0 )
0
2
= VC value. (25)
t=0
Figure 7 reports the IDR’s for varying values of α, σ0 , s, ∆0 . The range of IDR’s is consistent with
the discount rates that VC’s use to assess the value of a new venture (see Sahlman, 1990). Our
results, therefore, suggest that entrepreneurial optimism, indeed, explains the discount rates used
by VCs in reality.
7 Conclusions
We develop a continuous-time, stochastic principal-agent model to investigate the eﬀects of asym-
metric beliefs and agency conﬂicts on the characteristics and valuation of venture capital projects.
We characterize the equilibrium of the stochastic dynamic game in which the VC’s dynamic in-
vestments, 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 milestones or performance targets that must be realized for the project to continue.
We numerically implement the model and calibrate it to aggregate data on VC projects. Our
numerical analysis shows that EN optimism signiﬁcantly enhances the value that venture capitalists
derive. Entrepreneurial optimism explains the huge 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. Permanent and
transitory components of projects’ risks have diﬀering eﬀects on their values and durations.
29
31.
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Appendix A
Proof of Theorem 1
A rigorous proof of the theorem requires a precise interpretation of equation (1), which describes
the evolution of the termination payoﬀ process. As in the traditional principal-agent literature
(see Holmstrom and Milgrom, 1987), we consider the termination payoﬀ process V (·) to be a
given random process on a probability space with investment and eﬀort altering the probability
distribution of this process.
The Termination Payoﬀ Process
We consider an underlying probability space (Ω, F) with probability measures Q , ∈ {V C, EN },
representing the VC’s and EN’s beliefs. Θ is a normal random variable with variance σ0 and 2
mean µ0 under measure Q and B is a standard Brownian motion. The complete and augmented
ﬁltration of the probability space generated by the Brownian motion B(·) is denoted by {Ft }.
Consider the process V (·) = sB(·) where s2 is the intrinsic risk of the project. We will use the
Girsanov transformation (see Oksendal, 2003) to obtain new probability measures on (Ω, F) such
that the process V (·) evolves as in (1).
Suppose that η(·) and c(·) are strictly positive, square-integrable {Ft }-measurable stochastic
processes (under the measures QV C and QEN ) deﬁned on the time horizon [0, T ] describing the EN’s
choices of eﬀort and the VC’s choices of investments over time. Recall that l(·) is a deterministic
process describing the operating costs of the ﬁrm. Deﬁne the processes
t t
1
ζc,η (t) := exp[ (Θ + Ac(u)α η(u)β − l(u))s−1 dB(u) − (Θ + Ac(u)α η(u)β − l(u))2 s−2 du] (26)
0 2 0
t
Bc,η (t) := B(t) − (Θ + Ac(u)α η(u)β − l(u))s−1 du . (27)
0
The process ζc,η (·) is a positive, square-integrable martingale.6 Deﬁne the new measure Πc,η by
dΠc,η
= ζc,η (T ). (28)
dQ
By Girsanov’s theorem (see Oksendal, 2003), the process Bc,η (·) is a Brownian motion under the
6
The processes are assumed to satisfy the Novikov condition (see Oksendal, 2003):
T
1
E exp[ (Θ + Ac(u)α η(u)β − l(u))2 s−2 du] < ∞, ∈ {V C, EN } .
2 0
Because the equilibrium investment and eﬀort processes described in Theorem 1 are deterministic and Θ is a normal
random variable, the Novikov condition is satisﬁed by these processes.
In fact, we do not need to assume that feasible (not necessarily optimal) investment and eﬀort processes satisfy the
Novikov condition for our analysis to be valid; we only require that they be square-integrable. In this case, the process
ζc,η (·) is only guaranteed to be a local martingale and the measure Πc,η is a ﬁnite measure, but not necessarily a
probability measure. Our analysis, however, only requires that Πc,η be a ﬁnite measure. Since, as mentioned earlier,
the Novikov condition is satisﬁed by the equilibrium investment and eﬀort processes, the measure corresponding to
the equilibrium processes is a probability measure.
33
35.
measure Πc,η . Further, under this measure, the process V (·) evolves as
dV (t) = [Θ + Ac(t)α η(t)β − l(t)]dt + sdBc,η (t) . (29)
Equation (29) describes the evolution of the termination payoﬀ process and is identical to equation
(1), but with the Brownian motion and the probability measures representing the VC’s and EN’s
beliefs depending on the investment and eﬀort processes. It is important to keep in mind that V (·)
is a ﬁxed process whose sample paths are not aﬀected by investment and eﬀort. Investment and
eﬀort, however, alter the probability distribution of the sample paths of V (·).
For future reference, we make an important observation. The process
dWc,η (t) := s−1 [dV (t) − (Ac(t)α η(t)β − l(t))dt − µt dt] (30)
is an {Ft }-Brownian motion with respect to the probability measure Πc,η . Moreover, the complete
and augmented ﬁltration generated by this Brownian motion is {Ft }. The EN’s and VC’s mean
assessments of project quality Θ at date t, µEN , µV C are given by (2) and (??).
t t
Utility Related Processes
Let τ ≤ T be an {Ft }-stopping time denoting the termination time of the VC-EN relationship.
Let c(·), η(·) and η(·) be strictly positive {Ft }-adapted square-integrable processes on [0, τ ].7 A
contract is represented by the quadruple (Pτ , c(·), η(·), τ ). We now deﬁne some processes that are
used frequently in the sequel.
• Cumulative value process of the EN. This is the conditional expected future utility to the EN at
any date including the sunk disutilities of prior eﬀort, from a given contract (Pτ , c(·), η(·), τ ).
Formally,
τ
EN
U P,c,η,τ (t) := Ec,η [− exp(−λ[P (τ ) − kη(u)γ du]) | Ft ] . (31)
0
Here, Ec,η [· | Ft ]; ∈ {P r, Ag} denotes conditional expectation at date t under the probability
measure Πc,η deﬁned in (28). For future reference, we note that the cumulative value process
of the EN is a square-integrable {Ft }-martingale under the measure ΠEN . c,η
• Promised payoﬀ process for the EN. The EN’s promised payoﬀ process corresponding to a
given contract (Pτ , c(·), η(·), τ ) is deﬁned by (6).
• Adjusted cumulative value process of the EN. This process represents the cumulative value
process of the EN from a contract where his eﬀort is η(s); s ≤ t and eﬀort η(s); s ≥ t. Formally,
t τ
EN
YP,c,τ (η(·); t; η(·)) := Ec,η [− exp(−λ[P (τ ) − kη(u)γ du − kη(u)γ du]) | Ft ] . (32)
0 t
• EN’s maximum conditional expected utility process. This process represents the EN’s maxi-
mum conditional expected utility at date t given that he has exerted eﬀort η(s); s ≤ t and his
7
These processes are assumed to satisfy the Novikov condition—see footnote 6.
34
36.
terminal payoﬀ, the VC’s investment process, and the termination time are (Pτ , c(·), and τ ),
respectively. Formally, we deﬁne
ZPτ ,c,η,τ (t) := supη(·) YP,c,τ (η(·); t; η(·)) . (33)
To simplify the subsequent notation, we drop the subscripts denoting the dependence of the pro-
cesses deﬁned in (31)-(33) on the contract wherever there is no danger of confusion.
Structure of Incentive Compatible Contracts
A contract (Pτ , c(·), η(·), τ ) is incentive compatible with respect to the EN’s eﬀort if and only if,
given the terminal payoﬀ, Pτ , the VC’s investment process, c(·), and the termination time, τ ,
the EN’s optimal eﬀort choices are η(·). The following lemma, from which the theorem follows,
characterizes incentive compatible contracts.
Lemma 1 (Incentive Compatible Contracts)
A contract (Pτ , c(·), η(·), τ ) is incentive compatible only if the EN’s promised payoﬀ process P (·)
satisﬁes the following stochastic diﬀerential equation:
dP (t) = a(t)dt + b(t)dV (t) (34)
where
γk
b(t) = η(t)γ−β (35)
Aβc(t)α
and
λ
a(t) := b(t)2 s2 + kη ∗ (t)γ − b(t) Ac(t)α η(t)β − l(t) + µEN .
t (36)
2
Proof. Deﬁne the process η (.) as follows:
˜
η (s) = η(s) for s = t, η (t) = η (t),
˜ ˜ (37)
where η (t) is any candidate (possibly sub-optimal) eﬀort choice of the EN at date t. By the
principle of optimality of dynamic programming (Oksendal, 2003), the eﬀort η(t) is optimal for the
EN at date t when his prior eﬀort choices are η(·) only if
EN
η(t) = argmaxη (t) Ec,˜ [Z(˜(·); t + dt) − Z(η(·); t) | Ft ] = argmaxη (t) Ec,˜[dZ(˜(·); t) | Ft ] . (38)
η η η η
In what follows, we derive the inﬁnitesimal change dZ(˜(·); t) and then use (38) to establish the
η
statements of the Lemma. It follows from (32), (33), and (37) that
dZ(˜(·); t) = dZ(η(·); t) + Z(η(·); t)kλ(η (t)γ − η(t)γ )dt .
η (39)
35
37.
Since the process η(·) represents the EN’s optimal eﬀort choices by hypothesis, it follows that
Z(η(·), t) = U (η(·), t), (40)
and hence the process Z(η(·), ·) is a square-integrable {Ft }-martingale under the measure ΠEN .
c,η
By the martingale representation theorem (see Oksendal, 2003), there exists a square-integrable,
{Ft }-adapted process ω(·) such that8
dZ(η(·); t) = ω(t)dWc,η (t) = ω(t)s−1 [dV (t) − (Ac(t)α η(t)β − l(t))dt − µEN dt] .
t (41)
Since the expectation in the dynamic programming equation (38) is taken under the measure ΠEN ,
c,˜
η
it follows from (30) and (41) that Z(η(·); t) evolves under this measure as
dZ(η(·); t) = ω(t)s−1 Ac(t)α (η (t)β − η(t)β )dt + ω(t)dWc,˜(t) .
η (42)
Substituting (42) in (39) yields
dZ(˜(·); t) = ω(t)s−1 Ac(t)α (η (t)β − η(t)β ) + kλZ(η(·); t)(η (t)γ − η(t)γ ) dt
η
+ω(t)dWc,˜(t) .
η (43)
Having derived the requisite expression for dZ(˜(·); t), we substitute it in (38) to obtain
η
η(t) = argmaxη (t) [ω(t)s−1 Ac(t)α η (t)β + kλZ(η(·); t)η (t)γ ] . (44)
It follows that the eﬀort η(t) is optimal over the interval [t, t + dt] if and only if
ω(t) kλs γ
=− η(t)γ−β . (45)
Z(η(·); t) Ac(t)α β
From the deﬁnition of the promised payoﬀ process in (6), and using (40), we have
t
logZ(η(·); t)
P (t) = − + kη(u)γ du . (46)
λ 0
Using Ito’s Lemma and (41), we obtain
ω(t)s−1 [dV (t) − (Ac(t)α η(t)β − l(t))dt − µEN dt]
t ω(t)2
dP (t) = − + dt + kη(t)γ dt. (47)
λZ(η(·); t) 2λZ(η(·); t)2
ω(t)
Now substituting the expression (45) for the quantity Z(η(·);t) in (47) yields
dP (t) = a(t)dt + b(t)dV (t), (48)
8
Identity (41) is an almost sure relation that holds under all equivalent probability measures on the probability
space. It is only under the measure ΠEN deﬁned in (28) that the process [dV (t) − (c(t)α η(t)β − l(t))dt − µEN dt] is
c,η t
the increment of a Brownian motion.
36
38.
where b(t) and a(t) are given by (35) and (36), respectively, as claimed. This completes the proof
of the lemma.
Because an incentive compatible contract must have the aﬃne form by Lemma 1, it immediately
follows that the optimal contract must also have the same aﬃne form. This completes the proof of
Theorem 1.
Proof of Theorem 2
By (35) and (36), a candidate optimal contract is completely described by the investment process
c(·), the processes a(·), b(·) describing the performance-invariant and performance-dependent com-
ponents of the EN’s promised payoﬀ process, and the termination time τ . By (35), the contract is
incentive compatible if and only if the EN’s eﬀort at date t is
Aβcα bt
t
1
γ−β
ηt = η(bt , ct ) := . (49)
γk
Deﬁne τ
VC
Ma,b,c,τ (0) = Ec,η (V (τ ) − P (τ ) − c(s)ds) (50)
0
as the VC’s expected future payoﬀ at date 0 if she chooses a contract (Pτ , c(·), η(·), τ ) ≡ (a(·), b(·), c(·), τ ),
where η(.) is given by (49). The VC’s contract choice problem is then the following:
(a∗ (·), b∗ (·), c∗ (·), τ ∗ ) = argmax(a,b,c,τ ) Ma,b,c,τ (0). (51)
Let the “state” of the system at any date t be described by the ordered pair (t, µV C ). We
t
ﬁrst restrict consideration to Markov controls where a(t), b(t), c(t) and the decision to terminate
the relationship only depend on the current state (t, µV C ). We derive the optimal Markov control
t
policy. We then appeal to the veriﬁcation theorem of dynamic programming (see Theorem 11.2.3 of
Oksendal, 2003) to conclude that the optimal Markov control policy is, in fact, the optimal control
policy over the entire space of admissible {Ft }-adapted controls.
We note from (35) and (36) that the control a(·) is determined by the controls b(·), c(·) and
the state of the system. Hence, a Markov control policy is completely described by (b(·), c(·), τ ).
For simplicity, we abuse notation by denoting the VC’s continuation value in state (t, µV C ) from
t
adopting the Markov control policy (b(·), c(·), τ ) by
τ
Mb,c,τ (t, µV C ) = Et;c,η (V (τ ) − V (t)) − (P (τ ) − P (t)) −
t
VC
c(s)ds , (52)
t
where η(·) is determined by (49). Let M ∗ (t, µV C ) be the optimal continuation value within the
t
space of Markov controls and (b∗ (·), c∗ (·), τ ∗ ) be the optimal Markov control policy (we derive this
policy in the following).
Suppose that the VC deviates from the optimal policy over the inﬁnitesimal time interval
[t, t + dt] by choosing the controls (b(t), c(t)), Let M (t, µV C ) denote the VC’s continuation value at
t
37
39.
date t under this deviated policy. It follows from Lemma 1 and (52) that
M (t, µV C ) = Et;c,η − a(t)dt + (1 − b(t))dV (t) − c(t)dt + M ∗ (t + dt, µV C ) .
t
VC
t+dt (53)
By (29), we have
M (t, µV C ) = Et;c,η −a(t)dt+(1− b(t))[µV C +Ac(t)α η(t)β −l(t)]dt−c(t)dt+M ∗ (t+dt, µV C ) , (54)
t
VC
t t+dt
where η(t) is given by (49) with b(t) replacing b(t). Since the VC’s investment and EN’s eﬀort are
observable, the VC’s assessment µV C of project quality at date t + dt is independent of the choices
t+dt
of controls (b(t), c(t)). Hence, the function M ∗ (t + dt, µV C ) is also independent of these choices.
t+dt
By the principle of optimality of dynamic programming (see Oksendal, 2003), the optimal controls
(b∗ (t), c∗ (t)) at date t must maximize the “ﬂow” term in (54), that is,
(b∗ (t), c∗ (t)) = argmaxb(t),c(t) − a(t)dt + (1 − b(t))[µV C + Ac(t)α η(t)β − l(t)]dt − c(t)dt
t (55)
By (36) and (49), we can show (after some algebra) that
(b∗ (t), c∗ (t)) = arg max Λt (b(t), c(t))dt, (56)
b(t),c(t)
where
γ
α γ−β
Λt (b, c) := (∆t b − 0.5λs2 b2 + φ(b)c − c + µV C − lt )
t (57)
and
β β
γ 1 γ−β βb γ−β βb
φ(b) := A γ−β 1− . (58)
k γ γ
We ﬁrst determine the VC’s optimal investment rate c(b) as a function of the EN’s pay-
performance sensitivity b and then simultaneously derive the optimal investment rate and pay-
performance sensitivity. By (57) and (58), the optimal investment rate is zero if b ≥ γ/β. For
b ∈ (0, γ/β), Assumption 1 guarantees that the function Λt (b, ·) is strictly concave in the invest-
ment rate c (the exponent on c is guaranteed to be less than 1). As a consequence, setting the
partial derivative of Λt (b, ·) with respect to c equal to zero implies that the optimal investment as
a function of the pay-performance sensitivity b is given by (13). Substitution in (56), we see that
the VC chooses the pay-performance sensitivity at date t to solve (14).
By the above arguments, the VC receives a “ﬂow” payoﬀ Λt (b∗ , c∗ )dt in each inﬁnitesimal
t t
time period [t, t + dt]. It immediately follows that the risk-neutral VC chooses to terminate the
relationship at the stopping time that solves (16).
The Markov control policy derived above trivially satisﬁes the conditions of the dynamic pro-
gramming veriﬁcation theorem (see Section 11 of Oksendal, 2003). Hence, it is, in fact, the optimal
control policy among the space of all square-integrable {Ft }-adapted controls. This completes the
proof of Theorem 2.
38
40.
Appendix B: Proofs of Remaining Results
For each model parameter “Π” (e.g. σ0 , s2 , λ, ∆0 , k) we let bt (π) denote the solution to (18)
2
at time t, deﬁne ct (π) := c(bt (π)), and let b(π) and c(π) denote the corresponding time paths
when the parameter Π’s value equals π. We write Ft (b, π) when we wish to explicitly indicate the
functional dependence of the derivative of Ft on the parameter value π. For subsequent reference,
the derivative of the VC’s controllable rate function (17) is given by
γ − β − αγ s2 γ − β − αγ
Ft (b) = ∆t − pb + c (b) = 2 2 ∆0 − pb + c (b). (59)
αγ s + tσ0 αγ
Proof of Proposition 1.
The marginal optimal investment is given by
1
1 γ
c (b) ∝ (1−α) −1
β br1 (γ − b)r2 (1 − b) (60)
k
where
γ γ
2 − (1 − α) β αβ
r1 := γ and r2 := γ ,
(1 − α) β − 1 (1 − α) β − 1
and where the symbol ∝ means “equal up to a positive multiplicative constant”. Under Assumption
γ
2, the parameter r2 is positive and the parameter r1 is negative. Since β > 1 (Assumption 1), the
strong unimodality of c(·) easily follows from (60). Since c(0) = c(γ) = 0 and c (0) = +∞, it also
follows from (60) that c(·) achieves its maximum at b = 1. Part (a) has been established.
To establish part (b), we note that the second derivative of the optimal investment function is
γ γ γ
c (b) ∝ br1 −1 ( − b)r2 −1 [r1 ( − b)(1 − b) − r2 b(1 − b) − b( − b)].
β β β
The expression inside the brackets is a strictly convex quadratic function whose value at 1 is nega-
γ
tive, whose value at β > 1 is positive, and whose value at 0 is negative since r1 < 0. Consequently,
γ
there is exactly one root bM of the quadratic in the interval (1, β ) such that c (bM ) = 0. At bM the
marginal investment is at its minimum. Moreover, since c (·) is negative on [0, bM ) and is positive
γ γ
on (bM , β ), the function is strictly concave on [0, bM ] and strictly convex on [bM , β ].
Proof of Proposition 2.
Suppose ∆0 ≥ 0. It directly follows from (59) that
γ − β − αγ
Ft (b) ≤ ∆0 − pb + c (b),
αγ
γ
since ∆t ≤ ∆0 for all t. Figure 1 (p. 16) shows that c (1) = 0 and c (b) < 0 for all b ∈ (1, β ). It is
straightforward then to check that Ft (b) < 0 for all b > max{ ∆0 , 1}, which proves the claim when
p
39
41.
∆0 ≥ 0. Suppose ∆0 < 0 so that ∆t < 0 for all t. Then
γ − β − αγ
Ft (b) < −pb + c (b),
αγ
By the previous arguments, the right hand side above is less than zero for b ≥ 1 so that b < 1.
Proof of Theorem 3.
s 2
a) By (59), for any ﬁxed b > 0, Ft (b) decreases with t because ∆t = s2 +tσ2 ∆0 decreases with t for
0
∆0 > 0. By deﬁnition, Ft (b∗ ) = 0. Hence, Fs (b∗ ) < 0 for s > t. Since Fs (b∗ ) = 0 by deﬁnition,
t t s
we must have b∗ < b∗ by the strong unimodality of Fs (·). The fact that b∗ → b∗ as t → ∞ follows
s t t p
from the fact that ∆t → 0 as t → ∞ and the Theorem of the Maximum. This establishes part (a).
s2
b) The derivative Ft (1) = s2 +tσ2 ∆0 − p is zero when t = t∗ and t∗ ≥ 0. Thus, bt∗ = 1 and ct∗ is at
0
its maximum, which coincides with the “no agency” benchmark case, as required for part (b).
c) Since Ft (1) > 0 if t < t∗ and Ft (1) < 0 if t > t∗ , part (c) follows from the strong unimodality of
each Ft (·).
d) Part (a) establishes that the b∗ decrease with time, and so part (d) follows from part (c) and
t
the fact that the optimal investment function c(·) increases on [0, 1] and decreases on [1, bM ].
e) Since b∗ and c∗ both decrease with time on [t∗ , ∞), part (e) follows immediately from the
t t
functional form (8).
Proof of Theorem 4.
s 2
a) By (59), for any ﬁxed b > 0, Ft (b) increases with t because ∆t = s2 +tσ2 ∆0 increases with t for
0
∆0 < 0. By deﬁnition, Ft (b∗ ) = 0. Hence, Fs (b∗ ) > 0 for s > t. Since Fs (b∗ ) = 0 by deﬁnition,
t t s
we must have b∗ > b∗ by the strong unimodality of Fs (·). The fact that b∗ → b∗ as t → ∞ follows
s t t p
from the fact that ∆t → 0 as t → ∞ and the Theorem of the Maximum. This establishes part (a).
b) By Proposition 2, 0 < b∗ < 1 when ∆0 < 0. By Proposition 1, the optimal investment function
t
c(·) is strictly increasing on (0, 1). It follows that, because b∗ increases with t, c∗ = c(b∗ ) also
t t t
increases. This establishes part (b).
c) Since ηt = η(b∗ , c∗ ), part (c) follows directly from (8) and the fact that both b∗ and c∗ increase
∗
t t t t
with t.
The following Lemma will be used repeatedly in the proofs to follow.
Lemma 2
If Ft (b, π) is an increasing (decreasing) function of π, then bt (π) is an increasing (decreasing)
function of π.
Proof. Let π 1 < π 2 . Suppose ﬁrst that Ft (b, π) is an increasing function of π. By deﬁnition,
0 = Ft (bt (π 2 ), π 2 ) = Ft (bt (π 1 ), π 1 ) < Ft (bt (π 1 ), π 2 ),
40
42.
which immediately implies bt (π 1 ) < bt (π 2 ) by the strong unimodality of Ft . The proof in the
decreasing case is analogous.
Proof of Theorem 5.
a) If ∆0 < 0, Ft (b) decreases with λ, decreases with s, and increases with σ0 by (59). By Lemma 2,
b∗ decreases with λ, decreases with s and increases with σ0 . Since b∗ ∈ (0, 1) by Proposition 2, and
t t
c(·) is increasing in (0, 1) by Proposition 1 that c∗ = c(b∗ ) also decreases with λ, decreases with s
t t
and increases with σ0 . This establishes part (a).
b) Part (b) follows using similar arguments by observing that Ft (b) decreases with λ and decreases
with σ0 .
c) Since Ft (1, π) = ∆t − p < ∆0 − p < 0, it follows from the strong unimodality of Ft (·, π) that
bt (π) ∈ (0, 1) for all t. Since the function c(·) increases on [0, 1], part (c) follows from part (b).
d) We turn to part (d). (Please refer to Figure 4.) Suppose π 1 < π 2 where π = λ or σ0 . First 2
we note that by part (a) the path b(π 1 ) lies strictly above the path b(π 2 ). Let t∗ , j = 1, 2, denote
j
the value of t∗ in (21) corresponding to π j . Clearly, t∗ > t∗ . By Theorem 3(c), in the interval
1 2
[0, t∗ ) both paths b(π 1 ) and b(π 2 ) lie above one. Since b(π 1 ) > b(π 2 ) and since the function c(·)
2
decreases on [1, bM ], it follows that c(π 1 ) < c(π 2 ) in this interval. Analogously, by Theorem 3(c),
in the interval (t∗ , ∞) both paths b(π 1 ) and b(π 2 ) lie below one. Since b(π 1 ) > b(π 2 ) and since
1
the function c(·) increases on [1, bM ], it follows that c(π 1 ) > c(π 2 ) in this interval. By Theorem
3(d), we know that in the interval [t∗ , t∗ ] the path c(π 1 ) increases whereas the path c(π 2 ) decreases.
2 1
Moreover, by deﬁnition of t∗ we have that ct∗ (π j ) = c(1), j = 1, 2, and so
j j
ct∗ (π 1 ) = c(1) > ct∗ (π 2 ) and ct∗ (π 1 ) < ct∗ (π 2 ) = c(1).
1 1 2 2
Thus, the trajectories c(π 1 ) and c(π 2 ) cross exactly once in this interval, as claimed. Part (d) has
been established.
e) We turn our attention to s2 . Since the proofs of parts (c) and (d) when π = λ or σ0 are based
2
on part (b) (and Proposition 1 and Theorem 3), it is suﬃcient to establish part (b) when π = s2 .
∂F (bt (π),π)
To this end, we shall prove that for each t and π, t ∂π < 0, from which the result will follow
by direct application of Lemma 1. First, we examine the case when bt (π) ≤ 1. Since
∆0 π γ − β − αγ
Ft (bt (π), π) = 2 − λbt (π)π + c (bt (π)) = 0, (61)
π + tσ0 αγ
it follows that
∂Ft (bt (π), π) 2
∆0 tσ0 π
π = 2 − λbt (π)π
∂π (π + tσ0 )2
2
∆0 tσ0 π ∆0 π γ − β − αγ
= 2 )2 − π + tσ 2 + c (bt (π)) by (61)
(π + tσ0 0 αγ
∆0 π 2 γ − β − αγ
= − 2 + c (bt (π))
(π + tσ0 )2 αγ
41
43.
which is less than zero since bt (π) ≤ 1 implies that c (bt (π)) ≥ 0. (The condition ∆0 ≤ 4p is not
required in this case.) Now consider the case when bt (π) ≥ 1. Here,
∂Ft (bt (π), π) 2
∆0 tσ0 2
4λtσ0 π
= 2 − λbt (π) < 2 − λ since bt (π) ≥ 1 and ∆0 ≤ 4p = 4λπ
∂π (π + tσ0 )2 (π + tσ0 )2
(π − tσ0 )2
2
= −λ 2 ≤ 0.
(π + tσ0 )2
The proof is complete.
Proof of Theorem 6.
The proofs of parts (a) and (b) are identical to the proofs of Theorem 5(a, b). The proof of part (c)
follows the same arguments given in the proof of Theorem 5(d), except that here part (a) implies
that the path b(π 1 ) lies strictly below the path b(π 2 ).
Proof of Proposition 4.
Let
τ
φ(t, µt , τ ) := Et C
V ∗
(Fv − lv + Θ)dv (62)
t
denote the VC’s continuation value function at date t given her current project assessment µt and
∗
a given (possibly sub-optimal) stopping time τ . In the above, Fv satisﬁes (18). The VC’s optimal
continuation value function is
φ∗ (t, µt ) := sup φ(t, µt , τ ), (63)
τ ≥t
By standard dynamic programming arguments, the optimal termination time τ ∗ (if it exists) must
solve (63) for any t ∈ [0, T ] and µt ∈ (−∞, ∞). Further, the VC continues the project at any date t
and project assessment µt if and only if φ∗ (t, µt ) > 0. The proof proceeds by showing that φ∗ (t, ·) is
monotonic (non-decreasing) and lower semi-continuous. It then follows that at each date t ∈ [0, T ]
there exists a trigger µ∗ such that the VC continues the project if and only if µt > µ∗ .
t t
We prove the monotonicity and lower semi-continuity of φ∗ (t, ·) by considering the sequence of
discrete stopping time problems in which for each ﬁxed positive integer N the VC is constrained to
N
terminate the project only at the discrete set of times {0, 2T , . . . , (2 2−1)T , T }. We show that the
N N
VC’s optimal value functions φ∗ (t, ·) in the discrete problems are continuous and monotonic. We
N
then use a convergence argument to show that φ∗ (t, ·) is lower semi-continuous and monotonic.
Pick a positive integer N . We use backward induction to show continuity and monotonicity of
φ∗ (t, ·). To establish continuity we further show there exist positive constants κ1 , κ2 such that
N t t
φ∗ (t, µt ) ≤ κ1 + κ2 max{µt , 0}.
N t t (64)
42
44.
The assertions of continuity, monotonicity and (64) are trivial at date T since φ∗ (T, ·) ≡ 0. Suppose
N
that the assertion is true for t ∈ [t + 21 , . . . , T ]. We will establish that the assertion is true for
N
t ∈ [t , t + 21 ). Consider ﬁrst any t ∈ (t , t + 21 ]. By the dynamic programming principle,
N N
1
t+
2N 1
φ∗ (t, µt )
N = Et (Fv − lv + Θ)dv + φ∗ (t +
∗
N , µt + 1 ) . (65)
t 2N 2N
∗
Since Fv is bounded and deterministic by (18), (4), (58) and (13) and Θ is normally distributed,
T
E0 C
V
((Fv )2 + (lv )2 + (Θ)2 )dv < ∞.
∗
(66)
0
We can therefore apply Fubini’s theorem to conclude that
1
t+
2N 1
φ∗ (t, µt )
N = (Fv − lv + µt )dv + Et φ∗ (t +
∗
N , µt + 1 ) . (67)
t 2N 2N
We ﬁrst establish monotonicity of φ∗ (t, ·). The integral on the right-hand side of (67) obvi-
N
ously increases with µt ; it remains to show the expectation on the right-hand side of (67) is also
monotonic in µt . A bit of algebra applied to (2) and (3) shows that µt + 1 ∼ N (µt , σ 2 ) is normally
ˆ
2N
distributed. Further, µt + 1 may be expressed in the form ft (µt , Z) where Z ∼ N (0, 1) and ft (·, ·)
2N
is an increasing, linear function of its arguments. The monotonicity of Et φ∗ (t + 21 , ·) now follows
N N
from
1 1
Et φ∗ (t + N , µt + 1 ) = Eφ∗ (t + N , ft (µt , Z)),
N N (68)
2 2N 2
since the expectation on the right-hand side of (68) is taken with respect to the standard normal
density, which is independent of the problem parameters, and since both ft (·, z) and φ∗ (t + 21 , ·)
N N
are monotonic in µt (the latter by the inductive assumption).
Next we show continuity of φ∗ (t, ·). Once again, this property obviously holds for the integral
N
on the right-hand side of (67); it remains to show the expectation on the right-hand side of (67)
is also continuous in µt . This result will follow from identity (68) if the limit and expectation
operators may be interchanged, since both ft (·, z) and φ∗ (t + 21 , ·) are continuous in µt (the
N N
latter by the inductive assumption). By the inductive assumption (64) the function φ∗ (t + 21 , ·)
N N
is bounded above by a positive function whose expectation
σt −1/2( µt )2
ˆ µt
E κ1 +
t 1 + κ2 +
t 1 max{µt + 1 , 0} = κ1 +
t 1 + κ2 +
t 1 {√ e σt
ˆ + µt P (Z > − )} (69)
2N 2N 2N 2N 2N 2π σt
ˆ
is ﬁnite, and thus the interchange is justiﬁed by the dominated convergence theorem.
To complete the inductive argument we must show that (64) holds for t. The integral on the
right-hand side of (67) is bounded above by (t + 21 − t)(F0 + max(µt , 0))—recall the Ft∗ decrease
N
∗
with t. Since (64) holds for t = t + 21 , the inequality (69) shows that the expectation on the right-
N
σt
ˆ
hand side of (67) is bounded above by (κ1 + 1 + κ2 + 1 √2π ) + κ2 + 1 max(µt , 0). It is therefore
t t t
2N 2N 2N
possible to deﬁne positive constants κ1 , κ2 such that (64) holds for t, as required.
t t
43
45.
Finally, we must establish the inductive step for t = t . We have
1
t+
2N 1
φ∗ (t, µt )
N = max 0, (Fv − lv + µt )dv + Et φ∗ (t +
∗
N , µt + 1 ) , (70)
t 2N 2N
where (70) diﬀers from (67) because the VC can terminate at date t . It should be clear that the
previous arguments still apply, and hence the inductive step is established.
N N −1)T
Because {0, 21 , . . . , (2 2−1)T , T } ⊂ {0, 2N , . . . , (2 2N
N N
1
, T } for all N < N , it follows that
∗ (t, µ ) ≤ φ∗ (t, µ ). For each (t, µ ) ∈ [0, T ] × (−∞, ∞) we may therefore deﬁne
φN t N t t
φ(t, µt ) := lim φ∗ (t, µt ) .
N (71)
N →∞
We claim that φ = φ∗ . Fix (t, µt ) ∈ [0, T ] × (−∞, ∞). Since φ∗ (t, µt ) ≥ φ∗ (t, µt ) for all N ,
N
φ∗ (t, µt ) ≥ φ(t, µt ). Suppose that φ∗ (t, µt ) > φ(t, µt ). Choose any < (φ∗ (t, µt ) − φ(t, µt ))/2.
There exists a stopping time τ such that φ(t, µt ) < φ∗ (t, µt ) − < φ(t, µt , τ ) where φ(t, µt , τ ) is
deﬁned in (62). Deﬁne the stopping time τN = 2i 1{ i <τ < i+1 } . There exists N suﬃciently large
N
2N 2N
such that φ(t, µt , τN ) > φ(t, µt , τ ) − . It follows that φ(t, µt ) < φ(t, µt , τN ). By deﬁnition of the
function φ∗ (t, µt ), however, φ(t, µt , τN ) ≤ φ∗ (t, µt ) ≤ φ(t, µt ), which is a contradiction. Hence,
N N
φ(t, µt ) = φ∗ (t, µt ). The monotonicity of φ∗ (t, ·) easily follows from the monotonicity of φ∗ (t, ·)
N
and the fact that φ∗ (t, µt ) = limN →∞ φ∗ (t, µt ). The lower semi-continuity of φ∗ (t, ·) follows from
N
the fact that the supremum of continuous functions is lower semi-continuous.
Proof of Proposition 5.
The controllable rate function Ft (·) (17) is an increasing function of ∆0 , which implies that Ft∗
is also a increasing function of ∆0 . One may proceed exactly as in the proof of Proposition 4 to
establish that each CVt (·) is a pointwise increasing function of ∆0 . Consequently, the trigger values
will decrease. Since a change in ∆0 has no eﬀect on the sample paths, part (a) follows. The proof
of (b) is the same, except that each Ft (·) is now a decreasing function of either λ or k, and thus
the trigger values will increase.
44
46.
On-Line Appendix C
Numerical Implementation of the Model
By Proposition 4, the relationship is terminated at date i if and only if µV C is less than a trigger µ∗ .
i i
In our numerical implementation, therefore, we focus on directly modeling the evolution of µi V C,
which is hereafter denoted by µi to simplify the notation. In the ﬁrst stage of our implementation,
we approximate the evolution of µi using a discrete lattice and derive the triggers µ∗ that determine
i
whether the relationship is continued or terminated in each period. In the second stage, given the
triggers obtained from the ﬁrst step, we use Monte Carlo simulation to model the evolution of µi
and derive the key output variables of interest, e.g., the duration of the relationship, the “rational
expectations” value of the ﬁrm, and the continuation value of the VC.
Lattice design
We use a lattice to approximate the possible evolution of µi , i = 0, 1, 2, . . . , T − 1, which is given
by (3). Gompers (1995) ﬁnds that the average time between investments for diﬀerent investment
stages is 1.09 years. We, therefore, set the length of each period δt to one year. Sahlman (1990)
provides empirical evidence there are at most 8 investment stages. We, therefore, choose a ﬁnite
time horizon T = 10 in our numerical implementation.
At date 0 the project quality is given by µ0 . Let n(i) denote the number of states at date i > 0
and let µi,j denote the ﬁrm’s quality at the j th state at date i, j = 1, . . . , n(i). The lattice is designed
so that the minimal and maximal states at date i, µi,1 and µi,n(i) , are κ standard deviations below
and above the minimal and maximal states at date i − 1, respectively. More precisely,
µ
µi,n(i) = µi−1,n(i−1) + κσi−1 (72)
and
µ
µi,1 = µi−1,1 − κσi−1 (73)
µ
In (72) and (73), σi−1 is the standard deviation of the evolution of the assessment of project quality
over the period (i − 1, i). This can be derived from equation (3) as
2
σi−1
µ
σi−1 = . (74)
2
s2 + σi−1
The values for the remaining n(i)−2 states are equally spaced between the minimum and maximum
states. The number of states in the lattice increases linearly from period to period. That is,
n(i) = M i for i > 0. The value of M is set to 25 and the value of κ is set to 2.5.9
9
We found that an increase in M or κ or both did not change the optimal trigger values (to within a 3% tolerance),
and that the probability of the ﬁrm surviving to the 10th period was zero.
45
47.
The VC’s Continuation Value and the Termination Triggers
Let CVi,j denote the VC’s continuation value at state µi,j . At date T − 1 the continuation value is
∗
independent of the future and is given by CVT −1,j = µT −1,j + FT −1 − lT −1 for each state µT −1,j ,
j = 1, 2, . . . , n(T − 1). At earlier dates the continuation values are given by:
n(i+1)
CVi,j = µi,j + Fi∗ − li + pi+1,k max(CVi+1,k , 0) .
i,j (75)
k=1
i+1,k
In the above, pi,j denotes the probability that the assessment of project quality will transition
from state µi,j at date i to state µi+1,k at date i + 1.10
Starting from the last investment period T and working backwards through time we use dy-
namic programming to compute the continuation values for all states and dates. Since the true
continuation value function is continuous and increasing, we complete the approximation to CVi (·)
by linear interpolation. We then determine the optimal trigger µ∗ , which solves CVi (µ∗ ) = 0.
i i
In the second stage, given the termination triggers, {µ∗ }, we directly model the evolution of
i
(3) using Monte Carlo simulation to compute the various economic statistics of interest. We run
50, 000 simulations; the key economic statistics that we derive do not change by more than 1% if
the number of simulation runs is increased beyond 50, 000.
Estimation of Indirect Parameters
We use the simulated method of moments to estimate the indirect (or the “deep” structural)
parameters of the model. Let O denote the vector of the 11 aggregate statistics (see Table 2) we
are trying to match.
Phase I: Estimation of Indirect Parameter Values
We simulate a large number N of ﬁrms and ﬁx this simulated sample. The number N is chosen
large enough to minimize simulation errors. The simulation error is negligible when N = 50, 000.
For a given candidate indirect parameter vector π, we compute the vector V of simulated values of
the 11 statistics. Let di (Vi , Oi ) = Vi − Oi denote the diﬀerence between the simulated and observed
values of ith statistic and d denote the vector of diﬀerences. Deﬁne
f (π) := dT Σd. (76)
10 µ
If µi+1,k is within ±κσi from µi,j , we set
1 1 1 1
pi+1,k := Φ ( (µi+1,k + µi+1,k+1 ) − µi ) µ − Φ ( (µi+1,k + µi+1,k−1 ) − µi ) µ
i,j ,
2 σi 2 σi
where Φ(·) denotes the cdf of the standard normal distribution. Otherwise, the transition probability is zero.
46
48.
In (76) the matrix Σ is diagonal. The vector π ∗ of parameter estimates solves
π ∗ = arg min f (π). (77)
π
Phase II: Bootstrapping to Determine Conﬁdence Intervals for π ∗
We use parametric bootstrapping to determine the conﬁdence intervals for the estimated parameters
(see Davison and Hinkley, 1997). This method consists of two steps:
Step 1: Generation of bootstrapped statistics.
We ﬁx the parameter vector π = π ∗ . We generate X samples of NC ﬁrms denoted as (S1 , . . . , SX ).
We set NC = 7765 to incorporate the fact that the 11 Cochrane statistics are calculated from a
sample of 7765 ﬁrms. We use samples Si to compute the ith vector of the 11 statistics, respec-
tively. In this manner, we obtain a set of X vectors of “bootstrapped” Cochrane statistics Vj ,
j = 1, 2, . . . , X.
Step 2: Repeat Phase I for each bootstrapped vector of statistics
We replace the vector O of actual values of the statistics of Phase I with the vector Vj of
bootstrapped values. We solve (76) and (77) to obtain a set of X “bootstrapped” estimates of
∗ ∗
the indirect parameter vector (π1 , . . . , πX ). We use these vectors to obtain standard errors for the
estimated parameters π ∗ .
Phase III: Computation of Output Statistics and Conﬁdence Intervals
We use π ∗ as the baseline vector in our numerical analysis. To compute the conﬁdence intervals for
all output statistics of interest when parameters are set to their baseline values, we compute the
values of the statistics for each of the X bootstrapped samples described above with the parameter
∗
vector set to the corresponding vector πi , i = 1, 2, . . . , X, determined above. In our “comparative
statics” analyses, where one or more parameters are varied while keeping the other parameters
at their baseline values, we apply the same percentage changes to the parameter values in the
∗
bootstrapped collection πi , i = 1, 2, . . . , X, to compute conﬁdence intervals. For example, if we
analyze the eﬀect of varying the degree of asymmetry in beliefs ∆0 by 10% from its baseline value
on some output statistic X, we determine the conﬁdence interval for X by computing its values for
the bootstrapped samples where ∆0 is varied by 10% from its corresponding baseline value for the
bootstrapped sample.
47
49.
Table 1: Baseline Parameter Values
The table displays the baseline parameter values obtained by the model calibration. The point estimates and their
standard errors are shown in the second and third rows, respectively. The standard errors are obtained by generating
bootstrapped samples using the model as described in Appendix C.
Technology Parameters Belief Parameters Preference Parameters
A α l1 l2 µV C
0 ∆0 s σ0 λ k γ
0.6834 0.3212 0.0434 2.1726 0.1128 0.5039 0.7308 0.4401 1.5518 0.0480 4.9614
(0.001) (0.005) (0.001) (0.021) (0.002) (0.015) (0.113) (0.083) (0.034) (0.001) (0.077)
Table 2: Predicted and Observed Statistics
The table displays the observed values of the statistics used to calibrate the model, their predicted values from the
model, and the standard deviations of the statistics obtained by generating bootstrapped samples using the model
as described in Appendix C. The ﬁrst row shows the round by round returns (RTRR) and standard deviations of
VC projects, the mean overall return (Mean All), standard deviation (Stdev All), and mean duration (Duration All)
of VC projects reported in Cochrane (2005, Table 4). The second row shows the model’s predictions, and the third
shows the corresponding standard errors.
Standard Deviation of RTRR Standard Deviation of RTRR Mean StdDev Duration
Rd 1 Rd 2 Rd 3 Rd 4 Rd 1 Rd 2 Rd 3 Rd 4 All All
0.26 0.20 0.15 0.09 0.90 0.83 0.77 0.84 0.20 0.71 2.10
0.26 0.19 0.15 0.09 0.86 0.84 0.82 0.81 0.21 0.71 2.08
(0.008) (0.008) (0.010) (0.020) (0.012) (0.012) (0.013) (0.022) (0.014) (0.006) (0.012)
Table 3: Project Value, VC Value, Duration, and Survival Probabilities
The table shows the Project Value, VC Value, Expected Total Investment (Inv), Expected Duration, and the proba-
bilities of termination in successive round (p∗ , p∗ , p∗ , p∗ ) in the Actual Scenario with asymmetric beliefs and agency
1 2 3 4
conﬂicts, the Symmetric benchmark scenario in which beliefs are symmetric, and the No Agency benchmark scenario
in which beliefs are risk attitudes are symmetric.
Agency Project VC Duration p∗
1 p∗
2 p∗
3 p∗
4
Scenario Value Value
Actual 0.954 0.385 2.080 0.250 0.455 0.259 0.035
Symmetric 0.960 0.177 1.834 0.401 0.388 0.188 0.023
No Agency 1.014 0.288 1.961 0.330 0.411 0.226 0.032
Table 4: Contract Parameter Values (ﬁrst four periods)
The table shows the contractual parameters—the EN’s pay-performance sensitivities and the VC’s investments—in
the ﬁrst four rounds in the Actual Scenario with asymmetric beliefs and agency conﬂicts, the Symmetric benchmark
scenario in which beliefs are symmetric, and the No Agency benchmark scenario in which beliefs are risk attitudes
are symmetric.
Agency b∗
1 b∗
2 b∗
3 b∗
4 c∗
1 c∗
2 c∗
3 c∗
4
Scenario
Actual 0.477 0.397 0.348 0.315 0.097 0.095 0.093 0.092
Symmetric 0.141 0.147 0.150 0.152 0.083 0.083 0.083 0.084
No Agency 1.000 1.000 1.000 1.000 0.101 0.101 0.101 0.101
48
50.
Figure 6: Variations of Project Value and VC Value with Transient Risk, Intrinsic Risk, and
Degree of Asymmetry in Beliefs
a) Effects of Transient Risk
σ0
1.2
1
0.8
0.6
0.4
0.2
V-Project V-VC
0
-100 -80 -60 -40 -20 0 20 40 60 80 100
Percent Deviation
b) Effects of Intrinsic Risk
s
1.6
1.4 V-Project
1.2
V-VC
1
0.8
0.6
0.4
0.2
0
-100 -80 -60 -40 -20 0 20 40 60 80 100
Percent Deviation
c) Effects of Degree of Asymmetry in Beliefs
Δ0
1
0.8
0.6
0.4
0.2
V-Project V-VC
0
-100 -80 -60 -40 -20 0 20 40 60 80 100
Percent Deviation
Figure 7: Variations of Implied Discount Rates (IDRs) with Transient Risk, Intrinsic Risk,
and Degree of Asymmetry in Beliefs
Implied Discount Rate
0.8
sigma s_risk
0.6 Delta_0
0.4
0.2
0
-100 -80 -60 -40 -20 0 20 40 60 80 100
Percent Deviation from Baseline Values
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