Venture Capital Contracting with Ambiguity Sharing and Effort Complementarity Effect

In this paper, we established a continuous-time agency model in which an ambiguity-averse venture capitalist (VC) employs an ambiguity-neutral entrepreneur (EN) to manage an innovative project. We analyzed the connection between ambiguity sharing and incentives under double moral hazard. Applying a stochastic dynamic programming approach, we solved the VC’s maximization problem and obtained the Hamilton–Jacobi–Bellman (HJB) equation under a special form of the value function. We showed that the optimal pay-performance sensitivity was a fixed point of a nonlinear equation. The model ambiguity on the probability measure induced a tradeoff between ambiguity sharing and the incentive compensation that improved the EN’s pay-performance sensitivity level. Besides, we simulated the model and showed that when two efforts were complementary, the VC’s effort did not monotonically decrease with respect to the pay-performance sensitivity, while the EN’s effort did not monotonically increase in the pay-performance sensitivity level. More importantly, we found that as efforts tended to be more complementary, the optimal pay-performance sensitivity tended to approach those that maximized the efforts exerted by the EN and the VC.


Introduction
The venture capital market plays an important role in financing and nurturing innovative start-ups. Many highly successful companies, such as Google, Facebook, Amazon, and Alibaba, receive venture capital funding in their early stages of development. According to a report released by Crunchbase in January 2019, more than 56% of private technology companies complete financing through large-scale venture capital in 2018. There is a typical principal-agent relationship between the venture capitalist (VC) and the entrepreneur (EN). Signing an investment contract is an important sign for the two parties to reach a formal agency relationship.
Traditionally, financial contract models rely on the assumption that partners in two-party contracting problems have the same beliefs on the uncertainty output [1][2][3][4]. However, the lack of information in start-up enterprises may lead to the ambiguity of individual knowledge about future enterprise performance; that is, there are multiple possible distributions on firm value. The famous Ellsberg Paradox [5] shows that people treat ambiguity and risk from different perspectives. Nowadays, it is well-accepted that economic future outcomes can be subject to 'risk' and 'ambiguity/Knightian' uncertainties. Risk refers to the situation in which the true probability distribution of the uncertain outcome is known, whereas ambiguity refers to the case in which the true probability distribution is unknown. literature assume that VC knows the true probability distribution of project cash flow; hence there is no ambiguity uncertainty.
The novelty of this paper focused on the effect of the efforts complementarity and the optimal pay-performance sensitivity that the VC would allocate to the EN. Following [14,19,20], we assumed the VC's investment in the project is endogenous, and the VC plays a leadership role in the game relationship. Our model took a similar angle with [18] on considering the complementarity effect but departed from it in three ways. First, we established the model and solved the optimization problem, where the VC holds bargaining power. Second, we extended the contract analysis to a continuous-time situation, in which cash flow and the compensation awarded to the EN are dynamic. Third, we used a project revenue function proposed by [17] to simulate the model, instead of the constant elasticity of production function in [18].
Another paper closely related to our study is that of Wu et al. [9], who introduced probability measure ambiguity to analyze the continuous-time contract and focus on the research of relative performance evaluation. They found ambiguity induces a tradeoff between ambiguity sharing and incentives. However, many important models in finance and economics do not consider synergy between partners in the project [9,[21][22][23]. In contrast, we analyzed how complementarity of efforts affects the optimal pay-performance sensitivity level that the VC is willing to award to the EN. We simulated the model in the scenario that two efforts are complementary and found that the synergy improves the project's output and increases the pay-performance sensitivity allocated to the EN.
The rest of this paper is structured as follows. Section 2 characterizes the key elements of contracting and deduces the evolution process of the value function. In Section 3, we have solved the contracting problem and provide several applications. Finally, Section 4 concludes the paper.

Model Setup and Optimal Contracting
In this section, firstly, we studied the contract designing problem with the cash flow process. Second, we dealt with the belief distortion on model uncertainty by employing the discounted relative entropy. Third, we derived the Hamilton-Jacobi-Bellman (HJB) equation that characterizes the optimal contract.

General Model
We studied a continuous-time contracting problem. A risk-neutral VC (she) hires a risk-averse EN (he) to operate an innovative project. Both of their actions (efforts) are crucial to the projected revenue. The project produces cash flow Y t per unit of time, where Y t follows the process Here, we interpreted a t and e t as the EN's and the VC's effort choice at the time t, respectively. As an agent, the EN has the motivation to hide his action (effort), we assumed the EN's effort is unobservable and unverifiable to the VC. ζ is a constant nonnegative scale adjusted parameter, which describes the efficiency of the VC's effort in the project's output. α measures efforts complementarity, with α ∈ [0,α) andα < 2. De Bettignies [20] pointed out that when α = 0, both efforts are perfect substituted, but two efforts become more complementary as α increases. σ > 0 is the volatility of cash flow. The last parameter ρ measures the elasticity (impact) of the common shock. Similar to most project management literature, we adopted the quadratic cost function for both a t and e t in the forms of g(e t ) = δ 1 e 2 t /2 and g(a t ) = δ 2 a 2 t /2, δ 1 and δ 2 are positive constants, which measure the effects of being more or less efficient in the delivery of efforts. B t denotes common shock, while Z t denotes idiosyncratic shock. In addition, we assumed that (B, Z) is a two-dimensional independent standard Brownian motion under probability measure P.
Suppose, following [1,9,24], that in addition to project performance, compensation contracts can be written on another variable that is correlated with the noise component of project performance. This variable is assumed to be uninformative about the EN's action. An example would be the performance of other start-ups in the given industry or in the market as a whole under the assumption that the actions of an EN do not affect the performance of other firms in his industry. Similar to [9], we introduced the industry average performance into a contract designing problem, which is not affected by efforts exerted by the EN and the VC. For a given start-up, the industry average performance refers to the average performance of other start-ups engaged in a similar business within a particular industry. Denote M t as the industry's average performance, which follows a martingale process in the following form dM t = σdB t .
Both the EN and the VC discount future cash flow at the market interest rate r. Following [1], we assume that the EN has a constant absolute risk aversion (CARA) utility function where γ > 0 is the EN's absolute risk aversion coefficient, and c t is the EN's wage at the time t.
The VC provides a long-term compensation contract Π(c t , a t ) to the EN based on the past rates of project return. Π(c t , a t ) specifies the EN's wage policy {c t } and the recommended effort process {a t }. In order to avoid confusion, we used {ĉ t ,â t } that indicates EN's actual contract policies.
For simplicity, we assumed the EN's initial wealth as S 0 = 0. Then, the contract problem faced by the EN is subject to The first constraint represents the actual cash flows faced by the EN when he exerts actual effort {â t }. The second constraint states that, the change of the EN's saving dS t is the interest accrual rS t dt plus the wage deposit c t dt and minus the consumption withdrawalĉ t dt. To save, the EN can set his consumptionĉ t strictly below the wage c t .
The problem faced by the VC is expressed as subject to The value V 0 could be interpreted as the reservation utility that the EN would achieve in the best alternative offer he has. Following Holmström and Milgrom [1], under the CARA assumption framework, the constraint (9) is bind.

Discounted Relative Entropy
Due to the model ambiguity, we considered belief distortion on the possibility measure. The VC does not trust the probability measure P and considers alternative models under probability measure P to protect him from probability measure ambiguity. Defining two real-valued density generators {h t } and g t , satisfying Novikov-condition exp{ 1 2 t 0 (h 2 t + g 2 t )dt} < ∞ for all t > 0, we had Radon-Nikodym derivative with respect to P.
where ξ 0 = 1. According to the Cameron-Martin-Girsanov Theorem, we knew d P dP = ξ t and, two processes B h t and Z g t are defined by where B h t and Z g t are standard Brownian motions under the probability measure P. Under probability measure P, the cash flow and the industry average performance could be rewritten as Similar to [25,26], we employed the discounted relative entropy to measure the discrepancy between P and P, To incorporate concerns on the robustness of probability measure ambiguity, in our paper, the VC's optimization problem could be written as The parameter θ > 0 could be interpreted as the ambiguity aversion coefficient. A large value of θ implies a high degree of probability measure ambiguity or a large degree of concern for robustness. When θ converges to zero, the VC's optimization problem is reduced to be the case without probability measure ambiguity.

Contracting Problem
That is, the EN would choose the recommended contract Π(c t , a t ), as this choice would bring him the highest revenue, that is, (ĉ t ,â t ) ∈ Π (c t , a t ). Thus, t 0 (c s −ĉ s )ds = 0. In other words, when the contract is incentive-compatible, there would be no savings.
Given contract Π(c t , a t ), the EN's continuation value at time t is defined as Under Equation (17), Lemma 1 summarizes the relationship between the EN's expected time utility and his continuation value. This result is documented in [21]. Lemma 1. (He [21]). When the EN has a CARA utility function, then.
Lemma 1 indicates that the EN's utility function can be regarded as the time discount form with respect to continuation value. This property is peculiar to the exponential utility and would be convenient to deal with the process V t .
By Martingale Representation Theorem [27], there would be two measurable processes φ t and φ t which are relevant to continuation value V t such that where the second equation is obtained with the help of Lemma 1. In order to derive the specific optimal contract rather than providing the HJB equation of the VC's value function, we assumed simple forms We interpreted the two real variables β t and β t as pay-performance sensitivity level and relative performance sensitivity level, which are paid to the EN. While, relative performance evaluation occurs if the optimal contract lists a negative value β t to the industry average performance.
According to Equations (8) and (19), it could be rewritten as Now, we turned to deal with the EN's incentive compatibility. It is not difficult to find that the EN's effortâ t not only affects his instantaneous U(c t ,â t ) but also his continuation value, thus, the EN's optimal effort a t satisfies Taking the EN's first-order condition of his effort decision, we have Together with Lemma 1, the above equation implies that a t = β t (1 + αe t )/(γrδ 2 ).
Proposition 1. The contract Π(c t , a t ) is incentive-compatible with respect to the EN if and only if his effort satisfies. a t = β t (1 + αe t )/(γrδ 2 ).
It is worth noting that, due to the complementarity between the EN and the VC, the EN's effort depends not only on the pay-performance sensitivity level β t but also VC's effort. In this section, we assume that the EN can observe and verify the VC's effort e t . Thus, this is a single moral hazard problem.
By Proposition 1, the VC's objective function (16) Based on Equation (3) and Lemma 1, we have c t = δ 2 2 a 2 t − 1 γ ln(−γrV t ). Formally, we defined the EN's continuation value at time t as here V t evolves as Using Equations (25) and (26), under the measure P, the HJB equation for the VC's problem (24) is

Model Solution
To make further research on the optimal contract terms in addition to the HJB equation that characters the optimal contracting problem, we studied a special form of the value function J(V).
Thanks to the CARA preference, it plays a key role in solving the optimal contract. Following [9], we conjecture that where A is the constant part of J(V t ). Moreover, it can be easily verified as

Plugging them back into the HJB Equation (27) yields
As c t = δ 2 2 a 2 t − 1 γ ln(−γrV t ), the equation above yields Let then, applying the Envelope Theorem, we obtained the First-Order-Condition (FOC) of H 1 with respect to e t as ζ + 2αβ t (1 + αe t )/(γrδ 2 ) − δ 1 e t − α(β t ) 2 (1 + αe t )/(γr) 2 δ 2 = 0. Evaluating the equation above, we had Based on (23) and (30), we could obtain Taking the FOCs of H 1 for (h t , g t , β t , β t ) and re-ordering, we had By Equations (35) and (36), we could derive the optimal pay-performance sensitivity and the optimal relative performance sensitivity stated in the following proposition, which characterizes the robust contract items.

Proposition 2.
Suppose that the effort exerted by the EN is unobservable while he can observe and verify the VC's effort. With model uncertainty, the optimal pay-performance sensitivity β * t given to the EN is non-linear and at a fixed point takes the form of β * t = l(β * t ), where: The optimal relative performance sensitivity β * t is given by It follows from the above proposition that when the project's cash flow is deterministic (σ = 0) or when the project's performance is perfectly correlated with industry average performance (ρ 2 = 1), the EN would obtain the largest pay-performance sensitivity. Meanwhile, the probability measure ambiguity has no impact on the EN's pay-performance sensitivity.

Applications
In applications, we assumed the project's cash flow is subject to an industry-level shock.

Comparative Statics on Ambiguity
To better understand the impact of the probability measure ambiguity on the optimal contract, we made a robust analysis of ambiguity.

Proposition 3.
For any α ∈ [0, 2), we haddβ * t /dθ > 0. As we had Taking the derivative of both sides of the equation above with respect to θ and arranging, it follows that Graphically, Figure 1a,b illustrate the probability measure ambiguity effect for the pay-performance sensitivity and the relative performance sensitivity using the parameters in Table 1. Figure 1a shows the changes between the pay-performance sensitivity and ambiguity aversion, we could find that the pay-performance sensitivity increases in θ, coinciding with Proposition 3. As shown in Figure 1b, it depicts the relative performance sensitivity dynamics at a different level of ambiguity aversion θ. When θ > 0, V t evolves related to the industry average performance. In other words, the adoption of the relative performance evaluation improves the EN's compensation.
To better understand the impact of the probability measure ambiguity on the optimal contract, we made a robust analysis of ambiguity.

Proposition 3. For any
[0, 2) a Î , we had 0 Proof. Define we had Taking the derivative of both sides of the equation above with respect toθ and arranging, it follows that □ Graphically, Figure 1a,b illustrate the probability measure ambiguity effect for the pay-performance sensitivity and the relative performance sensitivity using the parameters in Table  1. Figure 1a shows the changes between the pay-performance sensitivity and ambiguity aversion, we could find that the pay-performance sensitivity increases in q , coinciding with Proposition 3.
As shown in Figure 1b

Lack of Relative Performance Evaluation
According to the standard principal-agent theory, relative performance evaluation should be employed to improve the efficiency of compensation contracts [10]. However, in the empirical literature, many studies document a lack of relative performance evaluation in compensation contracts [9,12]. Our model provided an explanation for this executive compensation puzzle. As shown in Proposition 2, the adoption of the relative performance evaluation allowed the VC to weaken the effect of the common shock on the project's cash flow.
When there is no model ambiguity on probability measure, that is, θ = 0, we had β * t = −ρβ * t , then V t follows Intuitively, V t varies without the common shock B t , that is, the common shock has no effect on V t . In the field of investment, an investor would take a short position in order to hedge the market risk. This finding is consistent with a number of studies, where they consider idiosyncratic shock only.
However, with probability measure ambiguity, θ > 0, according to Proposition 2, we had which means that the use of the relative performance evaluation is reduced due to probability measure ambiguity. Thus, based on (38), we can rewrite V t as It is worth noting that the EN's continuation value V t is related to the common shock B t . Moreover, we can find that V t increases in response to common shocks to performance beyond the EN's control. This result may be related to the empirical evidence of "reward for luck" by Bertrand and Mullainathan [13].

Effort Dynamic and Complementarity Effect
The efforts complementarity plays an important role in allocating compensation equity between the VC and the EN. The revenue function with the complementarity parameter is extensively used in the production field of microeconomics research. As shown in (31), (32), and (37), α is indeed related to the effort dynamics and the optimal contract, thus, we discussed the effect that the degree of complementarity has on the efforts and the pay-performance sensitivity. Figure 2 depicts the effort dynamics of each partner at a different pay-performance sensitivity level using parameters in Table 1. The aim of this simulation is to observe the complementarity effect that the parameter α has on the dynamics of the two efforts exerted by the VC and the EN. In case 1, the two efforts are perfectly substituted; it can be shown that when the EN obtains the highest pay-performance sensitivity level, β t = 1, he would exert maximum effort. On the contrary, the VC would deploy a fixed effort, which we could interpret as capital assets put in this project.
In case 2-case 6, different complementarity levels are assumed. Note that as the degree of effort complementarity increases, for instance, case 6: α = 1.9, both the entrepreneur and the VC deploy low efforts if β t = 0 or β t = 1. A large degree of complementarity means that in order for the project to be successful, both partners must put their efforts at work at the same time, as only one partner making effort would be worthless.
Furthermore, it could be seen that two efforts become concave on the pay-performance sensitivity degree β t . That is, there would be a pay-performance sensitivity β EN t assigned to the EN that maximizes his effort. The same holds for the VC. This finding is established in Proposition 4. In addition, the distance between β EN (a) There is a pay-performance sensitivity level β VC t allocated to the VC that maximizes his effort in the form of β VC t = min γr, 1 . (b) There is a pay-performance sensitivity level β EN t allocated to the EN that maximizes his effort in the form of β EN t = min (γr) Proof. Proof of (a): According to (31), we had As α ≥ 0, δ 1 > 0, δ 2 > 0 and γr > 0, then, if γr > 1, ∀β t ∈ [0, 1], we had de t /dβ t ≥ 0, β t = 1 is the maximum point of e t ; if 0 < γ r < 1, β t ∈ [0, γr], de t /dβ t ≥ 0, while β t ∈ (γr, 1], de t /dβ t ≤ 0, then e t reaches a maximum at β t = γr. On the whole, β VC t = min γr, 1 assigned to the EN maximizes the VC's effort.  In case 2-case 6, different complementarity levels are assumed. Note that as the degree of effort complementarity increases, for instance, case 6: 1.9 a = , both the entrepreneur and the VC deploy low efforts if A large degree of complementarity means that in order for the project to be successful, both partners must put their efforts at work at the same time, as only one partner making effort would be worthless. Furthermore, it could be seen that two efforts become concave on the pay-performance sensitivity degree t b . That is, there would be a pay-performance sensitivity EN t b assigned to the EN that maximizes his effort. The same holds for the VC. This finding is established in Proposition 4. In addition, the distance between Proof. Proof of (a): According to (31), we had Proof of (b): Taking the first derivative of (23) with respect to β t , it yields da t dβ t The FOC of a t with respect to β t is a quadratic equation (γr)(δ 1 + ζα) δ 1 δ 2 (γr) 2 − α 2 (β t ) 2 δ 1 δ 2 (γr) 2 − α 2 β t (2γr − β t ) 2 = 0.