Financier Search and Boundaries of the Angel and VC Markets

Abstract

This paper studies how critical entrepreneurial finance outcomes such as the investment return and equity division are shaped by venture characteristics, financier risk preferences, and competitive searching. Our analysis uses a double-hazard agency model in which financiers determine the equity division to maximize the expected utility of their investment return while entrepreneurs search for the best deal. Model results provide new theoretical insights on the venture funding cycle, the coexistence of angels/venture capitalists (VCs) with heterogeneous risk aversion, and risk separation in the entrepreneurial finance market. The model predicts that financiers with higher funding capacity and advisory capabilities (e.g., VC firms) will prefer to fund at later stages as their expected investment return rises with the venture’s initial value and financier productivity. Competitive searching by entrepreneurs enables financiers with a diverse set of risk preferences to coexist profitably by reducing the advantage (disadvantage) of lower (higher) risk aversion financiers and making investment returns more similar. Further, the model shows the emergence of a risk separation cutoff beyond which only angels/VCs with lower levels of risk aversion can profitably fund riskier ventures.

Keywords

angel/VC risk preferences competitive search double hazard expected investment returns equity division

New start-ups and ventures formed by entrepreneurs typically obtain financing from private investors such as angels and venture capitalists (VCs) instead of traditional sources such as banks or public stock markets. Studies report that the size of the angel finance market approximately equals that of the VC market and can be larger during certain periods (e.g., Organization for Economic Cooperation and Development [OECD] report by Wilson, 2011; Sohl, 2015). Sohl (2015) reports that 71,110 entrepreneurial ventures received angel funding in 2015 from 204,930 investors and the total investment was $24.6 billion. Pure equity deals are common in angel finance in the United States and Canada where the contract stipulates a simple division of future venture value between the entrepreneur and financier (e.g., Cumming, 2005; Wong, Bhatia, & Freeman, 2009). In contrast, venture capital deals involve the use of preferred or preferred convertible securities (Kaplan & Stromberg, 2004). Recent empirical research also suggests that angel investors and VCs can act as “dynamic substitutes” in that companies that obtain VC funding are less likely to obtain subsequent angel funding and vice versa (Hellmann, Schure, & Vo, 2013).

Given that VCs can potentially provide funding in earlier or later stages, the theoretical venture finance literature has attempted to rationalize the concentration of angel funding at early stages and VC funding at later stages (e.g., Chemmanur & Chen, 2006). Further, prior to settling on a particular deal, it is in the interest of entrepreneurs to search for the best funding deal from several potential financiers (angels/VCs). This paper provides additional insights into the role played by heterogeneity in financier risk preferences and competitive searching on critical entrepreneurial finance outcomes such as the expected investment return and equity division.

Our analysis uses a double-hazard agency model in which financiers determine the equity division that maximizes the expected utility of their investment return while entrepreneurs search for the best deal. Model results provide new insights on how interactions between venture characteristics (venture risk, financier risk aversion, initial project value, funding size, and financier/entrepreneur productivities) can rationalize the venture funding cycle, the coexistence of angels and VCs with heterogeneous risk aversions, and the emergence of a risk separation in the entrepreneurial finance market. Our theoretical results can also provide valuable insights to angels, VCs, entrepreneurs, and practitioners in designing and analyzing venture deals.

Consistent with the pattern of funding activity by angels/VCs observed in the entrepreneurial finance industry, the model shows analytically that financiers with access to larger investment funds (e.g., VC firms) will benefit by funding ventures at the later stage of development because their expected investment return increases with the venture’s initial value and financier productivity. At the later stage, there is an increase in both the ventures’ initial value and VCs’ ability to add value by providing advice and management services (Shane, 2009). As the venture matures and the productivity of the VC rises (relative to the entrepreneur), the model predicts that this will lead to an increase in the expected investment return and a lower equity share for the entrepreneur.

The model further shows that competitive searching by entrepreneurs enables financiers with heterogeneous risk preferences to coexist in a market that otherwise would favor financiers with lower risk aversions. Without search, lower risk aversion financiers have an advantage in the venture finance market: their expected investment return is higher, the equity share given to entrepreneurs is lower, and they can more profitably finance higher risk projects. In addition to the expected return, high risk–averse investors are also concerned about venture risk (which is present in their expected utility). To improve risk sharing and mitigate the negative impact of uncertainty on entrepreneur effort, they provide more equity to entrepreneurs, which reduces their expected return. In the presence of competitive search by entrepreneurs, an equilibrium emerges that allows a diverse set of financiers with heterogeneous risk preferences to operate profitably by reducing the advantage (disadvantage) of lower (higher) risk aversion financiers and makes returns similar. This result is agnostic to the assumption of whether VC firms are more or less risk averse than angel investors (see Section Competitive Search and Coexistence of Angels/VCs).

Finally, a risk separation emerges in the entrepreneurial finance market beyond which higher risk ventures can only be feasibly funded by angels/VCs with lower levels of risk aversion. This arises because while the expected investment return and equity division is unaffected by venture risk for risk-neutral financiers, the expected return declines with risk and risk aversion for risk-averse financiers. To raise declining entrepreneur effort in the face of higher venture uncertainty and reduce their risk exposure, risk-averse utility-maximizing financiers respond by increasing the entrepreneur’s equity share and the net result is a lower expected investment return. The paper also introduces a number of new theoretical features in venture finance agency modeling that are important to generating analytical results that connect the expected investment return to financier risk aversion and other venture characteristics. These include generalizing the financier to be risk averse, modeling the venture’s future value as a continuous random variable that depends functionally on a rich set of inputs (e.g., initial venture value, funding size, entrepreneur/financier productivities and efforts), and using the investment return in the financier’s objective function. Many venture finance agency models assume risk-neutral financiers and outcomes take binary values (with effort affecting the probability of the higher outcome). This is also among the first papers in entrepreneurial finance that incorporates equilibrium search by entrepreneurs over financiers with heterogeneous risk preferences.

A broad theoretical literature seeks to explain the nature of venture capital contracts and explores the implications for entrepreneurs, VCs, and policy makers. As well as theories about why start-ups may find equity preferable to debt contracts (Chemmanur & Chen, 2006; De Bettignies & Brander, 2007; Ueda, 2004) and the nature of projects financed by equity (Winton & Yerramilli, 2008), researchers have attempted to offer theoretical rationales for a range of specific equity contracts. These include convertible debt–equity contracts (Berglöf, 1994; Casamatta, 2003; Hellmann,1998, 2006; Kaplan & Strömberg, 2003)); staged financing whereby VCs release funds in discrete tranches (Bergemann & Hege, 1998; Li, 2008); and vesting and control rights (Gompers & Lerner, 1999). Other theoretical work explores strategic interactions between entrepreneurs and financiers, including the possibility of opportunistic behavior (Cornelli & Yosha, 2003; Elitzur & Gavious, 2003; Fairchild, 2011; Schmidt, 2003; Schure, 2006). This work provides a rich theoretical landscape that helps us to comprehend a wide variety of equity contract forms and their role in financing entrepreneurial activity. We discuss some relevant aspects of this literature below and relate it to the model and analysis of this paper.

One of these streams explores the boundaries between VCs and bank financing. Ueda (2004) assumes that VCs have greater capabilities to screen entrepreneurial opportunities than banks and, therefore, need to rely less on collateral to finance entrepreneurs. Their model predicts that VCs are more likely than banks to finance companies with high return and risk profiles and entrepreneurs who possess relatively little collateral will be more likely to seek VC financing. De Bettignies and Brander (2007) point to financier effort as a major distinction between VCs and banks and show that equity contracts best balance the incentives for both contracting parties when the marginal value of VC effort is sufficiently high. In a similar direction, this paper extends our understanding of the coexistence of financiers with heterogeneous risk preferences and differing funding capacity (e.g., angels, VCs) in a context of searching by entrepreneurs for the best deal.

The literature has also examined strategic issues at the exit stage of the venture. Berglöf (1994) analyzes the potential conflict of interests between entrepreneurs and VCs that arises from a future sale of the company to a third party. The entrepreneur values private benefits of control that increase with firm value and is afraid of a sale that compensates him insufficiently for their loss while the VC is afraid of a premature “cheap exit” that leads to poor investment returns. Hellmann (1998) studies why entrepreneurs give up control rights to VCs who expend costly effort to search for a superior management team. This exposes entrepreneurs to the risk of being fired before their shares are fully vested. In subsequent work, Hellmann (2006) shows why VCs use contracts with convertible preferred equity to mitigate concerns regarding exit decisions as they allocate different cash flow rights for acquisitions and initial public offerings (IPOs). With a similar focus on exit, Elitzur and Gavious (2003) analyze a sequential investment game with moral hazard comprising entrepreneurs, angels, and VCs where angels invest before VCs and are concerned of value transfer to VCs; all agents are risk neutral.¹ Inderst & Müller, 2004 adopt a bargaining perspective and argue that bargaining power between entrepreneurs and VCs, as reflected in equity shares inter alia, varies with technological and economic conditions. Fairchild (2011) allows for potential “stealing” by either the entrepreneur or the financier at the IPO stage and shows how entrepreneur/angel empathy relative to the VC’s value-adding abilities affects financing choices.

Hellmann and Thiele (2015) use a costly search model on deal flows between the angel and VC markets to obtain an equilibrium relationship that endogenously characterizes market sizes, valuation levels, and exit rates. Our analysis incorporates double-sided hazard but abstracts over issues related to the mode of exit as our primary focus is on understanding how heterogeneity in financier risk preferences interacts with important venture characteristics (e.g., project risk, initial value, financier/entrepreneur productivities) to shape funding outcomes under competitive search by entrepreneurs.

Cumming (2005) finds that pure equity deals remain in widespread use around the world and are especially common in the context of angel finance. In our analysis, they provide a common benchmark for comparing the expected investment return and equity division in contracts designed by financiers with differing funding capacities and risk preferences (e.g., angels, VCs). A number of studies rationalize the use of convertible securities in VC financing contracts (Baiman, Bar-Yosef, & Sarath, 2008; Casamatta, 2003; Schmidt, 2003). Schmidt (2003) analyzes a double moral hazard model where the entrepreneur provides effort first and the VC second. With conversion, the VC has stronger equity-like incentives and the entrepreneur works hard so that the VC sees enough upside potential to convert and become a common equity holder. Casamatta (2003) also considers the possibility that an outsider can provide value-adding advice in addition to effort by the entrepreneur and shows that the provision of advice creates information rents and that convertible preferred equity (CPE) is the optimal contracting security for sufficiently large investments. Baldenius and Meng (2010) consider how entrepreneurs can signal their firm’s value to investors who have the expertise to contribute actively to the start-up company’s operations and firm decision-making.

In a different vein, Kanniainen and Keuschnigg (2003, 2004) study how a risk-neutral VC decides on the size of his portfolio of start-ups with costly advisory effort. The VC faces a trade-off in that advice could be spread too thinly, thereby reducing the survival chances of all firms in the portfolio. As projects become riskier, the VC must cede a higher profit share to entrepreneurs to secure their effort that is critical for survival. With the corresponding erosion in the VC’s own profit share, the VC eventually finds it unattractive to expand the portfolio further. Keuschnigg and Nielsen (2004) show that by failing to internalize effort costs of the other party, equity share divisions are generally inefficient and require a corrective tax to restore efficiency. Sørensen (2007) considers a matching model in which more experienced VCs add more value and bring companies public at a higher rate.

The remainder of the paper is organized as follows. The Model section presents the model and its solution. Discussion of model results and implications follow in the Discussion and Implications section. The Summary and Conclusion section concludes the paper.

The Model

The model reflects the strategic interaction between an entrepreneur E (agent) and potential financiers F (principal) willing to provide venture financing in return for an equity position. Financiers are heterogeneous in their risk aversion $α_{F} \in [0, α_{M a x}]$ and their distribution over this support is given by the probability measure $P (α) = P (α_{F} \leq α)$ . The entrepreneur does not know the type of the financier $F (α_{F})$ , $α_{F} \in [0, α_{M a x}]$ approached for funding. The timing of actions in the model are as follows. First, the entrepreneur E searches over financiers of type $F (α_{F})$ , $α_{F} \in [0, α_{M a x}]$ , to obtain the best equity financing deal. Second, each financier $F (α_{F})$ offers an equity contract that maximizes the expected utility from the investment. Third, both E and $F (α_{F})$ simultaneously choose their levels of effort given the equity share. Fourth, the future value of the venture $V$ is realized and payoffs based on the equity deal are realized.

Project Value and Financing

An entrepreneur E owns a project with initial value $V_{0}$ and seeks additional financing $f$ from potential financiers at the start of the period $[0, t]$ . Next, with efforts $a$ and $b$ provided by the entrepreneur and financier, respectively, the value of the venture at the end of the period at time $t$ is

V = (V_{0} + f) (1 + β a + γ b + ε), ε \sim N (0, σ_{}^{2})

(1)

where

β

represents the productivity of the entrepreneur’s effort;

γ

is the productivity of the financier effort; and

ε

reflects the return uncertainty in the future value of the venture (assumed to be normally distributed). While various market scenarios can be imagined regarding the realization of future value

V

(e.g., IPO, acquisition, private equity buyout), we view

V

as simply the terminal value of the venture at time

t

Next, E searches for the best financing deal from financiers of types $F (α_{F})$ , $α_{F} \in [0, α_{M a x}]$ . We describe the entrepreneur’s search problem below (see Section Optimal Equity Contract and Efforts) after first solving for the optimal contract designed by a financier of type $F (α_{F})$ . In exchange for funding f, the financier $F (α_{F})$ offers the entrepreneur an equity share $x (α_{F}) \in [0, 1]$ in the new venture with the remaining value $(1 - x (α_{F})) V$ retained by the financier $F (α_{F})$ . Given the equity share $x (α_{F})$ , the entrepreneur determines his effort level to maximize his expected utility. Principal agency models commonly use the exponential utility specification with constant absolute risk aversion (CARA) since it yields tractable expressions for incentives and effort that are independent of the agent’s initial wealth (e.g., Gao, 2010; Garen, 1994; Grossman & Hart, 1983; Jensen & Murphy, 1990). We adopt this specification as well and, accordingly, the entrepreneur determines her best effort response from maximizing the utility

U_{E} (x, a, b) = - \exp (- α_{E} (w - \frac{1}{2} g a^{2}))

(2)

where

α_{E}

is the entrepreneur’s coefficient of risk aversion, w = xV is the entrepreneur’s payoff and the parameter

g

reflects the entrepreneur’s cost of exerting effort.

The Financier’s Investment Problem

The financier $F (α_{F})$ ’s investment return is given by

r_{F} = \frac{(1 - x) V - f}{f} = \frac{(1 - x) (V_{0} + f) (1 + β a + γ b) - f}{f}

(3)

The utility of the financier from the venture’s investment return is given by

U_{F (α_{F})} (x, a, b) = - \exp (- α_{F} (r_{F} - \frac{1}{2} h b^{2}))

(4)

where

α_{F}

is the financier’s coefficient of risk aversion and the parameter

h

represents the financier’s personal cost of exerting effort.

The financier $F (α_{F})$ maximizes the expected utility of his investment return $\ E U_{F (α_{F})} (x, a, b)$ by offering the equity contract $w = x V$ while taking into account the best effort response of the entrepreneur and his own effort while ensuring entrepreneur’s participation. The entrepreneur has an outside alternative opportunity with monetary value $\bar{w}$ (in addition to the sale price of the current venture with initial value $V_{0}$ ). The timing of actions in the model is similar to De Bettignies and Brander (2007) and there are two stages in each interaction between F and E. The equity division is determined at the first stage and the two efforts are determined at the second stage. The model is solved backward. First, at the second stage, the entrepreneur and financier simultaneously determine their best effort responses $a$ and $b$ , respectively, given the equity contract $w = x V$ . This requires solving the first-order condition of E’s expected utility (6) with respect to $a$ and the F’s expected utility (5) with respect to $b$ (see (A.1) and (A.2) Appendix A). Both these first-order conditions do not depend on the risk aversion parameters of E and F. Then, at the first stage, the financier determines the optimal utility-maximizing equity division $x$ in (5) taking into account the best effort responses $a$ and $b$ from the second stage.

The financing problem of $F (α_{F})$ is as follows:²

\begin{array}{ll} \underset{{x}}{m a x} E U_{F (α_{F})} (x, a, b) & = \underset{{x}}{m a x} \frac{(1 - x) (V_{0} + f) (1 + β a + γ b) - f}{f} - \frac{1}{2} h b^{2} \\ - α_{F}^{} \frac{{(1 - x)}^{2} {(V_{0} + f)}^{2} σ^{2}}{f^{2}} \end{array}

(5)

subject to

max_{{a}} E U_{E} (x, a, b) = \underset{{a}}{m a x} (x (V_{0} + f) (1 + β a + γ b) - V_{0} - \frac{1}{2} α_{E} x^{2} {(V_{0} + f)}^{2} σ^{2} - \frac{1}{2} g a^{2})

(6)

max_{{b}} E U_{F (α_{F})} (x, a, b)

(7)

E U_{E} (x, a, b) \geq U (\bar{w} + V_{0})

(8)

where

U (\bar{w} + V_{0}) = - \exp (- α_{E} (\bar{w} + V_{0}))

. For the discussion below, let

x^{*} (α_{F})

generically denote the optimal equity share offered by the financier of type

F (α_{F})

by solving (5) subject to (6) and (7). Further, let

a^{*} (α_{F})

and

b^{*} (α_{F})

represent the resulting optimal efforts, and from (1),

V^{*} (a^{*} (α_{F}), b^{*} (α_{F})) = (V_{0} + f) (1 + β a^{*} (α_{F}) + γ b^{*} (α_{F}) + ε)

is the terminal venture value.

Note that the financier’s utility in (5) is based on F’s investment return while the same for the entrepreneur in (6) is E’s value from the venture. This is because the venture is a mutually exclusive project for the entrepreneur, while it is not for the financier (VC or angel). Entrepreneurs will typically be fully engaged in their ventures while VCs and angels will invest in a number of different opportunities concurrently. Corporate finance theory suggests that the decision rule for non-mutually exclusive projects should be the profitability index (PI) constructed by dividing each project’s net present value (NPV) by the scare resource. In the venture context, the relevant scarce resource is the funding capital f so that $P I = (1 - x) V / f$ and the financier’s investment return defined in (3) is a simple transformation of this: $r_{F} = P I - 1$ . When projects are mutually exclusive, finance theory suggests that the project with the highest NPV be selected. Therefore, the entrepreneur’s expected utility in (6) is based on E’s total project value $x V$ .³

The weak inequality in (8) represents two possible situations. The first is that the constraint holds with equality, which implies that entrepreneurs receive the lowest equity valuation that makes them indifferent to choosing between pursuing the entrepreneurial venture and their best alternative monetary opportunity. It also implies that the equity share $x$ will be determined purely by the entrepreneur’s reservation wage, making the financier’s problem (5) irrelevant.⁴ Furthermore, forcing equality in (8) also assumes that the financier knows the best alternative opportunity $\bar{w}$ of the entrepreneur (who is not his employee) with a fair degree of precision. Given these concerns, the more realistic and palatable assumption is to assume slackness in the inequality (8). Accordingly, the financier determines the equity division $x^{*} (α_{F})$ that maximizes the expected utility of his investment return (5) while taking into account the best effort responses of both the entrepreneur and the financier from (6) and (7), respectively. The entrepreneur then accepts the equity contracts only if the expected utility gain exceeds the one derived from his (privately known) outside opportunity as described in Definition 1.

Definition 1. The Optimal Feasible Contract

An optimal contract $x^{*} (α_{F})$ is feasible if it satisfies the following conditions:

The equity stake $x^{*} (α_{F})$ maximizes financier $F (α_{F})$ ’s investment return utility (5) subject to best effort responses (6) and (7).

The entrepreneur’s participation inequality (8) holds, meaning that the contract $x^{*} (α_{F})$ is selected if it offers a larger utility gain over the best alternative opportunity $\bar{w}$ . $E U_{E} (x^{*}, a^{*}, b^{*}) - U (\bar{w} + V_{0}) > 0$ .⁵

Our interest is naturally on feasible deals that the entrepreneur will accept (i.e., satisfy Definition 1). The expressions for $x^{*} (α_{F})$ , $a^{*} (α_{F})$ and $b^{*} (α_{F})$ in Proposition 1 and 2 below provide a direct way for the entrepreneur to evaluate condition ii) in Definition 1 or (8).

Optimal Equity Contract and Efforts

The problem (5–7) for financier $F (α_{F})$ is solved backward and requires first obtaining the best effort responses. The first-order conditions for (6) and (7) lead to the following best effort responses for the entrepreneur and financier given the contract $w = x V$ (see Appendix A):

a = \frac{x (V_{0} + f) β}{g}

(9)

b = \frac{(1 - x) (V_{0} + f) γ}{f h}

(10)

Both efforts increase with productivity, incentive share, initial project value, and funding size, while they decrease with the agent’s cost of effort. Incorporating the best effort response (9) and (10) into (5), the first-order condition for the equity share leads to the equity division in Proposition 1 (see Appendix A for details).

Proposition 1. Optimal Equity Division—Entrepreneur and Financier

The optimal equity share for the entrepreneur that maximizes $F (α_{F})$ ’s expected utility (4) from the investment return $r_{F}$ is

x^{*} (α_{F}) = \frac{(V_{0} + f) (β_{}^{2} f h + α_{F} σ_{}^{2} g h - γ_{}^{2} g) - f g h}{(V_{0} + f) (2 β_{}^{2} f h + α_{F} σ_{}^{2} g h - γ_{}^{2} g)}

where

γ_{}^{2} < 2 f β_{}^{2} (h / g) + α_{F} σ_{}^{2} h

Financier $F (α_{F})$ ’s optimal equity share is

1 - x^{*} (α_{F}) = \frac{f (β_{}^{2} (V_{0} + f) + g) h}{(V_{0} + f) (2 β_{}^{2} f h + α_{F} σ_{}^{2} g h - γ_{}^{2} g)}

(12)

The optimal efforts for the entrepreneur and financier in Proposition 2 follow from using the equity shares (11) and (12) in (9) and (10).

Proposition 2. Return-Maximizing Efforts

The optimal effort levels of the entrepreneur and financier $F (α_{F})$ are

a^{*} (α_{F}) = \frac{(β_{}^{2} f (V_{0} + f) h + α_{F} (V_{0} + f) σ_{}^{2} g h - γ_{}^{2} (V_{0} + f) g - f g h) β}{(α_{F} σ_{}^{2} g h + 2 β_{}^{2} f h - γ_{}^{2} g) g}

(13)

b^{*} (α_{F}) = \frac{(β_{}^{2} f (V_{0} + f) h + g) γ}{α_{F} σ_{}^{2} g h + 2 β_{}^{2} f h - γ_{}^{2} g}

(14)

Finally, Proposition 3 reports the optimal expected investment return for the financier. The expression follows directly from substituting the financier’s equity share (12) and efforts (13) and (14) into the financier’s investment return (3) and simplifying.

Proposition 3. Financier’s Optimal Investment Return

Financier $F (α_{F})$ ’s optimal expected investment return is

E r_{F} (α_{F}) = \frac{(β_{}^{2} (V_{0} + f) + g) h}{(2 β_{}^{2} f h + α_{F} σ_{}^{2} g h - γ_{}^{2} g)^{2}} [\begin{matrix} 1 + β^{2} (\begin{matrix} β_{}^{2} f (V_{0} + f) h + α_{F} (V_{0} + f) σ_{}^{2} g h \\ - γ_{}^{2} (V_{0} + f) g - f g h \end{matrix}) \\ + γ^{2} (β_{}^{2} f (V_{0} + f) h + g) \end{matrix}] - 1

(15)

The Entrepreneur’s Equilibrium Search Contract

Prior to accepting a venture funding deal, it is in the interest of any entrepreneur to search for the best deal from several potential financiers (e.g., angels, VCs). These financiers are likely to have differing risk preferences $F (α_{F})$ , $α_{F} \in [0, α_{M a x}]$ , which are not known to the entrepreneur. We identify the entrepreneur’s equilibrium contract that arises from optimally searching for the best financing deal available from financiers of different risk aversions.

Let $W (x (α))$ be the maximized expected value to the entrepreneur of either accepting the equity share $x (α_{F})$ currently offered by a financier $F (α_{F})$ or performing another search next period for a better deal across financiers of different types. $W (x (α))$ solves the following problem:

W (x (α_{F})) = m a x (x (α_{F}) E V (a (α_{F}), b (α_{F})), ϕ \int_{0}^{α_{M a x}} W (x (α_{F})) d P (α))

(16)

where

x (α_{F}) E V (a (α_{F}), b (α_{F}))

is the expected value of the current equity contract

x (α_{F})

to the entrepreneur,

\int_{0}^{α_{M a x}} W (x (α_{F})) d P (α)

represents the expected value from continuing the searching for a better deal, and

ϕ < 1

is the discount factor (time value of money to entrepreneur in waiting for the next search). The offer

x (α_{F})

from the financier satisfies equation (11) of Proposition 1 and must also be a feasible contract for the entrepreneur — that is, it satisfies Definition 1. Otherwise, the entrepreneur will reject contract

x (α_{F})

because the alternative outside opportunity provides higher expected utility. We assume that if

x (α_{F})

is not a feasible contract, the entrepreneur will end the search and pursue the best outside option.

Using standard arguments, the feasible contract $x (\bar{α})$ with $\bar{α} \in [0, α_{M a x}]$ that makes the entrepreneur indifferent to choosing between accepting the current deal $x (\bar{α})$ and performing another search is given by the condition

x (\bar{α}) E V (a (\bar{α}), b (\bar{α})) = ϕ \int_{0}^{α_{M a x}} x (α) E V (a (α), b (α)) d P (α)

(17)

Equation (17) implicitly defines the critical level of financier risk aversion $\bar{α} \in [0, α_{M a x}]$ and corresponding financier contract $x (\bar{α})$ that solves the search problem of finding the best deal available from various types of financiers $F (α)$ of probability measure $P (α)$ .

In solving (17), Assumption 1 below ensures that the optimal solution for the equity division in Proposition (1) holds during competitive search by the entrepreneur across financiers of different types. It reflects the realism that funds available from financiers are a much scarcer resource than entrepreneurial projects seeking funding. It excludes perfect competition among potential financiers of the same type $F (α_{F})$ when the entrepreneur is searching for the best deal. This ensures that financiers with the same risk aversion $α_{F} \in [0, α_{M a x}]$ will not undercut each other and deviate from the optimal feasible contract for that type of financier $F (α_{F})$ in Proposition 1 and Definition 1.

Assumption 1. Excess Funding Demand

The total demand for venture funds by entrepreneurs exceeds their supply available from each type of financier $F (α_{F})$ .

Under Assumption 1, the financier would simply wait for one of many entrepreneurs to approach and provide funding that yields the expected return of Proposition 3. Excess demand for venture funds ensures that their investment capital will be fully deployed. If there was excess supply of venture funds, perfect competition between financiers will lead them to increase the entrepreneur’s equity stake beyond the optimal level in Proposition 1 until their expected investment utility in (5) goes to zero.⁶ It is also useful to note that Assumption 1 is implicitly made in other venture finance models where the analysis is based on a contract designed by a risk-neutral financier ( $α_{F} = 0$ , single type).

The expected value of the equity deal offered to the entrepreneur $x (α) E V (a (α), b (α))$ by the financier $F (α)$ in (17) is highly nonlinear in $α$ . This makes the integration on the right-hand side of (17) in a general setting intractable. We, therefore, apply a first-order approximation to (17) to find an analytical solution under the special case where financier types $F (α)$ are uniformly distributed over $α \in [0, α_{M a x}]$ so that $d P (α) = 1 / α_{M a x}$ (see Appendix B). Proposition 4 characterizes the equilibrium search contract under a particular assumption regarding the distribution of financier types $P (α)$ .

Proposition 4. Equilibrium Search Contract

If financiers $F (α)$ are uniformly distributed over $α \in [0, α_{M a x}]$ , then the equilibrium search contract solving (18) is given by the feasible contract $x^{*} (\bar{α})$ in (11) with

\bar{α} \approx \frac{ϕ}{2} α_{M a x} - (1 - ϕ) \frac{x (0) E V (a (0), b (0))}{\frac{\partial {x (α) E V (a (α), b (α))}}{\partial α} |_{α = 0}}

(18)

where

x (0) E V (a (0), b (0)) = x (0) (V_{0} + f) (1 + β a (0) + γ b (0))

and

\begin{array}{ll} \frac{\partial {x (α) E V (a (α), b (α))}}{\partial α} |_{α = 0} = & {x^{/} (0) (V_{0} + f) (1 + β a (0) + γ b (0)) \\ + x (0) (V_{0} + f) (β a^{/} (0) + γ b^{/} (0))} α \end{array}

If the feasible equilibrium search contract $x^{*} (\bar{α})$ defined by (11) is offered to the entrepreneur, she will accept the funding deal because the expected value from another search would give the same expected value. Although more complex distributions of financiers $P (α)$ will lead to somewhat differing results on the positioning of $\bar{α}$ (and the contract $x^{*} (\bar{α})$ ), these solutions will generally satisfy $\bar{α} \in (0, α_{M a x})$ . Hence, our discussion and insights in the Discussion and Implications section on how the equilibrium search contract affects the dynamics of expected investment returns among financiers of different types are largely robust to specific assumptions regarding the probability measure $P (α)$ .

Comparative Statics and Numerical Results

The model’s comparative statics are summarized in Table 1. The signs of the partial derivatives for the entrepreneur’s equity share, expected investment return, and entrepreneur/financier effort (columns) are reported with respect to various model parameters (rows). To aid in the analysis, the signs are reported in separate columns for both a risk-neutral financier ( $α_{F} = 0$ ) and a risk-averse financier ( $α_{F} > 0$ ). The model’s unambiguous results are in the shaded cells of Table 1 (parameters $V_{0}$ , $γ$ , $α_{F}$ and $σ$ ). For these parameters, the response under the case $α_{F} > α_{F}^{*}$ holds with $α_{F}^{*} = 0$ (see (1) in Table 1). The intuition behind many of these results and their important implications are discussed in the Discussion and Implications section.

Table 1.

Comparative Statics.

	Entrepreneur’s equity share		Financier’s expected investment return		Entrepreneur’s effort		Financier’s effort
	$x$		$E r_{F}$		$a$		$b$
$α_{F} = 0$	$α_{F} > α_{F}^{*}$	$α_{F} = 0$	$α_{F} > α_{F}^{*}$	$α_{F} = 0$	$α_{F} > α_{F}^{*}$	$α_{F} = 0$	$α_{F} > α_{F}^{*}$
$α_{F}$	+	+	─	─	+	+	─	─
$σ$	0	+	0	─	0	+^(a)	0	─^(a)
$β$	+	─^(b)	─	?	+	?^(a)	─	+^(b)
$γ$	─	─	+	+	─	─	+	+
$V$	+	+	+	+	+	+	+	+
$f$	+	─^(c)	─	?	+	?^(a)	─	+^(d)

^a Holds for

α_{F}^{*} = 0

^b Holds for

α_{F}^{*} = \frac{γ_{}^{2} (V_{0} + f) + 2 f h}{(V_{0} + f) σ_{}^{2} h}

^c Holds for

α_{F}^{*} = \frac{γ_{}^{2} (β_{}^{2} {(V_{0} + f)}^{2} + V_{0} g) + 2 β_{}^{2} f^{2} h}{(β_{}^{2} {(V_{0} + f)}^{2} + V_{0} g) σ_{}^{2} h}

^d Holds for

α_{F}^{*} = \frac{β_{}^{2} g + 2 (β_{}^{2} V_{0} + g) h}{σ_{}^{2} g h}

The table displays the sign of the partial derivatives for the entrepreneur’s equity share, expected investment return, entrepreneur effort, and financier effort (columns) with respect to various model parameters (rows). When $α_{F} > α_{F}^{*}$ holds, the sign of the partial derivative is as is given in the cell; the sign switches to the opposite sign when $α_{F} < α_{F}^{*}$ .

Figures 1–3 report three-dimensional graphs for the entrepreneur’s optimal equity share and angel/VC expected investment returns. They depict how these key deal outcomes respond to combinations of financier risk aversion with venture risk, financier/entrepreneur productivities, and funding size.

Figure 1.

Venture uncertainty and risk aversion. (a) Entrepreneur’s equity share and (b) angel/venture capitalist expected return.

Figure 2.

Productivity ratio and risk aversion. (a) Entrepreneur’s equity share and (b) angel/venture capitalist expected return.

Figure 3.

Funding size and risk aversion. (a) Entrepreneur’s equity share and (b) angel/venture capitalist expected return.

The numerical results are generated by setting model parameters to the following values: $V_{0} = 1$ , $α_{F} \in [0, 5]$ , $σ \in [0, 200 %]$ , $β = 1$ , $γ \in β \times [0, 1.6]$ , $g = h = 1$ , $f \in V \times [0, 5]$ . We follow De Bettignies and Brander (2007) and normalize the initial value of the project to $V_{0} = 1$ and calibrate the model to generate similar equity shares under risk neutrality (this leads to the baseline parameter values $β = 1$ , $γ = β / 2$ , $g = h = 1$ , $σ = 100 %$ , $f = V_{0} \times 5$ ). These settings view the entrepreneur as being twice as productive as the financier and assume that the effort disutility is the same. The six graphs in Figures 1–3 are produced by varying the parameters for financier risk aversion ( $α_{F} \in [0, 5]$ ), return volatility ( $σ \in [0, 200 %]$ ), productivity ratio ( $γ / β \in [0, 1.6]$ ), and funding ratio ( $f / V_{0} \in [1, 5]$ ). The annualized expected investment return in reported assumes a 5-year horizon for the venture.

The graphs below show that risk aversion of the angel/VC and venture risk plays an important role in determining their expected investment return and the corresponding entrepreneur’s equity share. When F is risk neutral or venture risk is zero, the equity share and expected return remain constant. When F is risk averse, the equity share rises with venture return volatility, but the expected return declines and E’s equity share rises uniformly with risk aversion. There is also positive interaction effect where the equity share rises with risk aversion and venture return volatility; on the other hand, an increase in both these parameters leads to a fall in the expected return.

A rise in venture uncertainty discourages entrepreneur effort, as outcomes become more random. To elicit the utility-maximizing level of effort, risk-averse financiers respond by offering a higher equity share to entrepreneurs ( $\frac{\partial x}{\partial σ} > 0$ ). This also reduces the financier’s risk exposure and the overall effect is that the financier’s expected investment return declines with project risk ( $\frac{\partial E r_{F}}{\partial σ} < 0$ if $α_{F} > 0$ ). Although the model includes additional parameters (e.g., risk aversion, funding size, venture risk, and initial venture value), it generates a broad range of equity shares corresponding to the numerical results of De Bettignies and Brander (2007).

It is interesting that the financier’s effort and expected return decreases with $α_{F}$ while the entrepreneur’s equity share and effort increases with F’s risk aversion (see Figure 1 and top row of Table 1). More risk-averse financiers provide higher incentives to E ( $\frac{\partial x}{\partial α_{F}} > 0$ ), which increases E’s effort ( $\frac{\partial a}{\partial α_{F}} > 0$ ), but this also lowers F’s effort ( $\frac{\partial b}{\partial α_{F}} < 0$ ) as F’s equity share ( $1 - x$ ) decreases. The variance component in F’s expected utility also declines in this scenario and has a positive utility effect. These actions maximize F’s expected utility from the investment return; however, the overall impact is a reduced expected investment return ( $\frac{\partial E r_{F}}{\partial α_{F}} < 0$ ). This occurs even when F is more productive than E (see Figure 2), suggesting that E’s incentive effect and F’s risk reduction effect dominates F’s negative incentive effect when α_F rises.

The graphs in Figure 2 show that as the angel/VC becomes more productive relative to the entrepreneur (productivity ratio γ/β rises), the financier’s expected investment return increases and less equity is offered to the entrepreneur ( $\frac{\partial E r_{F}}{\partial γ} > 0$ , $\frac{\partial x}{\partial γ} < 0$ ). Since VCs are able to add more value through management advice as the venture matures (Gompers & Lerner, 1999; Shane, 2009), these model results suggest that higher financier productivity at later stages of venture development will generate a higher return for VCs and a decline in entrepreneurs’ equity share.

Funding size has an interesting effect on the entrepreneur’s equity share that depends on the level of financier’s risk aversion (Figure 3a). When F is risk neutral, the equity share rises with the funding multiple ( $f / V_{0}$ ); however, at higher levels of risk aversion, the equity share declines slowly with funding size. This decline is not very sharp because a financier committing larger amounts of capital to the venture also needs to induce greater effort from the entrepreneur.

The financier’s expected return generally rises with funding size; however, there is an interesting reversal pattern for lowest risk–averse angels/VCs. The return initially declines with funding size and then continues to increase. Overall, Figure 3b exhibits a funding scale effect, suggesting that financiers with access to larger pools of funding capital will benefit by investing more in fewer similar projects than investing smaller amounts in more projects.

Discussion and Implications

We now discuss the major implications of model results on how critical entrepreneurial finance outcomes (e.g., expected investment return and equity division) are influenced by venture characteristics, heterogeneity in financier risk preferences, and competitive searching by entrepreneurs. More specifically, we show how interactions between venture risk, financier risk aversion, initial project value, funding size, and financier/entrepreneur productivities rationalize the venture funding cycle, the coexistence of angels/VCs with heterogeneous risk aversions, and the emergence of risk separation in the entrepreneurial finance market. It is also useful to point out that the implications and discussion below are based entirely on the unambiguous results from the model—these are summarized in the shaded cells of Table 1 (for parameters $V_{0}$ , $γ$ , $α_{F}$ and $σ$ ).

The Venture Funding Cycle

Model results show that the venture’s initial value $V_{0}$ and financier productivity $γ$ play an important role in differentiating the profitability of financiers (Table 1). This has direct implications for the participation of angels and VCs at different stages of venture development.

First, note that the model shows analytically that the financier’s expected investment return rises with initial project value and financier productivity: $\frac{\partial E r_{F}}{\partial V_{0}} > 0$ and $\frac{\partial E r_{F}}{\partial γ} > 0$ (Table 1, Appendix D). This means that it is in the interest of venture investors to finance projects with high initial $V_{0}$ where the financier can also add greater value to the venture after funding (high financier productivity $γ$ ). The model’s contract provides greater incentives to entrepreneurs of projects with higher initial value ( $\frac{\partial x}{\partial V_{0}} > 0$ ) and this increase is greater for financiers who are less risk averse ( $\frac{\partial^{2} x}{\partial V_{0} \partial α_{F}} < 0$ ; (C.6) in Appendix C). The financier will accordingly seek greater effort from the entrepreneur when initial value $V_{0}$ is higher ( $\frac{\partial a}{\partial V_{0}} > 0$ ) by providing stronger equity incentives ( $\frac{\partial x}{\partial V_{0}} > 0$ ).

Further, higher financier productivity reduces the equity offered to entrepreneurs ( $\frac{\partial x}{\partial γ} < 0$ ), and a higher share will be retained by the financier. This incentivizes financiers to make greater efforts to add value to the venture when they have stronger capabilities to add value (e.g., advice, management services).

Model results on project size and financier productivity support the emergence of a venture capital cycle (e.g., Gompers & Lerner, 1999; Hellmann et al., 2013; Shane, 2009). In this cycle, new start-ups often raise relatively modest sums initially and then seek larger funding in later rounds when the VC is in a position to add more value. Note that the model (5 – 8) with equilibrium search (17) captures the strategic interaction in any independent funding round between the entrepreneur and potential financiers (angel or VC). At the start of each such round, the initial value of the venture $V_{0}$ will change. Since the initial value of the start-up $V_{0}$ increases as the venture moves to later stages of development, the model predicts that the financiers’ expected investment return will increase: $\frac{\partial E r_{F}}{\partial V_{0}} > 0$ . Therefore, financiers with larger pools of available funds (e.g., institutional VC firms) will prefer to participate in later stages because their expected investment returns will be larger due to the higher $V_{0}$ , as predicted by the model. VCs are also in a better position at later stages of venture development to provide value-adding management advice (Shane, 2009). The model predicts that the increase in VC productivity (relative to the entrepreneur) as the venture matures will further raise VCs’ expected return ( $\frac{\partial E r_{F}}{\partial γ} > 0$ ) and reduce the equity share of entrepreneurs ( $\frac{\partial x}{\partial γ} < 0$ ).

Proposition 5. VC Timing and Financing Rounds

Since $\frac{\partial E r_{F}}{\partial V_{0}} > 0$ and $\frac{\partial E r_{F}}{\partial γ} > 0$ , financiers with access to larger investment funds (e.g., VCs) can increase their expected investment return by investing in larger ventures (higher $V_{0}$ ) at later stages when they can also provide more value-adding advice and management services (higher $γ$ ).

Finally, the numerical results in Comparative Statics and Numerical Results show a funding scale effect where the financier’s expected return generally increases with funding size (Figure 3b). This means that angels/VCs with larger amounts of funding capital can raise their expected return by investing more in fewer similar projects than investing smaller amounts in more projects.

Risk Separation in the Entrepreneurial Finance Market

An interesting risk separation emerges due to the interaction between financier risk aversion ( $α_{F}$ ) and venture risk ( $σ$ ). It has the implication that higher risk ventures can only be profitably financed by angels/VCs with sufficiently low levels of risk aversion.

Model results show that the expected investment return and equity division of risk-neutral financiers is unaffected by venture risk ( $\frac{\partial E r_{F}}{\partial σ} = 0$ , $\frac{\partial x}{\partial α_{F}} = 0$ if $α_{F} = 0$ , Table 1). Meanwhile, for risk-averse financiers, the expected return declines with venture risk and risk aversion ( $\frac{\partial E r_{F}}{\partial σ} < 0$ , $\frac{\partial E r_{F}}{\partial α_{F}} < 0$ if $α_{F} > 0$ ) and the entrepreneur’s equity share rises with risk aversion ( $\frac{\partial x}{\partial α_{F}} > 0$ if $α_{F} > 0$ ). As project risk increases, entrepreneur effort is discouraged as venture outcomes become more random. To elicit the utility-maximizing level of effort, risk-averse financiers respond by offering a higher equity share to entrepreneurs ( $\frac{\partial a}{\partial σ} > 0$ ) and this also improves risk sharing by reducing their exposure to venture risk. The overall effect is that the financier’s expected investment return declines with project risk ( $\frac{\partial E r_{F}}{\partial σ} < 0$ if $α_{F} > 0$ ) and a risk cutoff emerges beyond which financing risky projects becomes unprofitable for higher risk aversion financiers. These insights are visually reflected in the expected return surface for $E r_{F}$ in Figure 1b where the financier’s expected investment return declines as F becomes more risk averse and this decline becomes very sharp at higher levels of risk aversion. Therefore, at high enough combinations of venture uncertainty and finance risk aversion, venture projects yield much lower expected returns to angels and VCs.

While the preceding analytical results are consistent with intuitive reasoning, it is useful to note that they arise from the model’s new theoretical features. This includes allowing the generality of risk-averse financiers and defining their utility function in terms of the investment return. Venture finance models in the literature typically assume risk-neutral financiers and their objective function are based on venture profits (which are dependent on investment scale).

Competitive Search and Coexistence of Angels/VCs

We next show how competitive searching by entrepreneurs enables a broader spectrum of financiers with differing risk preferences to operate profitably in the venture finance market. Searching reduces the comparative advantage that lower risk aversion financiers have over higher risk aversion financiers.

First, note that in the absence of search, venture financiers with lower levels of risk aversion benefit as they have higher expected investment returns. This follows directly from the negative relationship between the financier’s expected investment return and risk aversion: $\frac{\partial E r_{F}}{\partial α_{F}} < 0$ . Under the optimal equity contract, higher risk-averse financiers experience lower expected returns and also provide a larger equity stake to entrepreneurs ( $\frac{\partial x}{\partial α_{F}} > 0$ ). This encourages greater effort from the entrepreneur ( $\frac{\partial a}{\partial α_{F}} > 0$ ) that positively impacts the size of the total venture “pie” ( $V$ ), but also reduces the financier’s share and the financier’s exposure to venture risk (second term of (5)). The risk reduction effect is larger for more risk-averse financiers and increases their expected utility. Therefore, it is utility enhancing for higher risk aversion financiers to offer a higher equity share to entrepreneurs, but this comes at a cost of lower expected investment returns.

Next, with competitive searching by entrepreneurs, the model shows that the equilibrium outcome allows financiers with a broader range of risk preferences to coexist. Here, the equilibrium search contract $x^{*} (\bar{α})$ of Proposition 4 (e.g., $\bar{α} = \frac{1}{2} α_{M a x}$ in the special case $ϕ = 1$ ) reduces the higher expected return of lower risk aversion financiers while improving the return of higher risk aversion financiers. Hence, the search for the best venture deal by entrepreneurs leads to an equilibrium that enables financiers with a broader range of risk preferences to coexist in the venture finance market by making their expected returns similar.

These insights from the model are visible in Figures 1b, 2b, and 3b of the numerical study (Comparative Statics and Numerical Results) which show that the expected investment return of the financier declines secularly with F’s risk aversion at differing levels of venture risk, financier/entrepreneur productivities, and funding size. Since the equilibrium search contract is interior to the range, a financier’s deal will be accepted by the entrepreneur if it is close to the one defined by (11). This means that under the equilibrium search contract, the expected return of higher risk aversion financiers increases as their return moves up the return surface, while the same for the lower risk–averse financiers decreases (they move down the return surface).

Proposition 6. Financier Expected Returns—Equilibrium Search

The equilibrium search contract $x^{*} (\bar{α})$ of Proposition 4 (e.g., with $\bar{α} = \frac{1}{2} α_{M a x}$ if $ϕ = 1$ ) enables a broader set of financiers with heterogeneous risk preferences to operate profitably in the venture market by making their expected investment returns similar. The equilibrium contract $x^{*} (\bar{α})$ :

Reduces the higher expected investment return of lower risk aversion financiers $α_{F} \in [0, \bar{α}]$ and increases the equity share of entrepreneurs: $E r_{F} (\bar{α}) < E r_{F} (α_{F})$ , $x (\bar{α}) > x (α_{F})$ .

Raises the lower expected investment return of higher risk aversion financiers $α_{F} \in [\bar{α}, α_{M a x}]$ and decreases the equity share of entrepreneurs: $E r_{F} (α_{F}) < E r_{F} (\bar{α})$ , $x (\bar{α}) < x (α_{F})$ .

Note that the model result on the coexistence of financiers with differing risk profiles under competitive search by entrepreneurs is agnostic to the assumption of whether VCs are more or less risk averse than angel investors. The equilibrium dynamics under search means that a larger set of angels and VCs clustered around the equilibrium level of risk aversion $\bar{α}$ can continue to operate with similar investment performance in the venture finance industry.

Empirical Implications

Given the preceding discussion of model results, we summarize below some testable empirical implications from the model.

The venture funding cycle—role of initial value and VC productivity . The model predicts that venture firms with greater access to funding capital and value-adding capacity will prefer to wait and fund at a later stage as this leads to a higher expected investment return. This follows from the positive relationship between the expected return and initial project value and financier productivity: $\frac{\partial E r_{F}}{\partial V_{0}} > 0$ , $\frac{\partial E r_{F}}{\partial γ} > 0$ (Table 1). This broadens the result of Chemmanur and Chen (2006) and De Bettignies and Brander (2007) whose models predict that VCs will finance ventures where they can add more value (higher $γ$ ) and in industries where the potential for doing so is greatest (e.g., technology firms), while angels and banks will tend to fund firms where this potential is limited. The model further shows that the equity share of entrepreneurs will increase with initial value ( $\frac{\partial x}{\partial V_{0}} > 0$ ) and that a rise in VC productivity as the venture matures will positively affect their expected investment return ( $\frac{\partial E r_{F}}{\partial γ} > 0$ ).

Role of financier risk aversion—return and equity division. In a market with more risk-averse investors, the model predicts a lower expected investment return and a higher equity share for entrepreneurs ( $\frac{\partial E r_{F}}{\partial α_{F}} < 0$ , $\frac{\partial x}{\partial α_{F}} > 0$ ; Figures 2 and 3). These investors are also concerned about venture risk and the negative impact of uncertainty on entrepreneur effort. They mitigate this problem by providing more equity to entrepreneurs and this also improves risk sharing. Given that the model extends previous analysis to risk-averse financiers, these are new hypotheses open to empirical exploration.

Risk separation—entrepreneurial finance market . Another interesting hypothesis from the model is that only angels/VCs with sufficiently low levels of risk aversion can profitably finance high-risk ventures. This follows from the model result that the expected investment return declines with venture risk for risk-averse financiers ( $\frac{\partial E r_{F}}{\partial σ} < 0$ if $α_{F} > 0;$ Figure 1). Further, to elicit higher levels of effort and improve risk sharing, risk-averse financiers respond by offering a higher equity share to entrepreneurs when venture risk rises ( $\frac{\partial x}{\partial σ} > 0$ ).

Funding size . The model also suggests that risk-averse investors with access to larger pools of funding capital will benefit by investing more in fewer projects than investing smaller amounts in more projects. Meanwhile, very low risk aversion investors will prefer to make smaller or larger investments. This is illustrated by Figure 3, which shows that the expected investment return rises with relative funding size ( $f$ ) for higher risk aversion investors. For lower risk aversion angels/VCs, however, the return declines and then rises with relative funding size.

Summary and Conclusion

This paper expands our theoretical understanding of how critical entrepreneurial finance outcomes such as the expected investment return and equity division are shaped by important venture characteristics (e.g., project risk, financier risk aversion, initial project value, funding size, financier and entrepreneur productivities). Our analysis uses a double-hazard agency framework where financiers maximize the utility of their investment return, both sides can contribute to the future value of the venture, and entrepreneurs search for the best deal. Model results provide a number of insights on the role played by funding capacity, heterogeneity in financier risk preferences and competitive searching on the venture funding cycle, the coexistence of angels/VCs, and risk separation in the entrepreneurial finance market.

The model provides theoretical support for the pattern of funding activity observed in the venture finance industry with angels funding start-ups in their earlier stage and VCs engaging at the later stage when the venture’s initial value is higher and when they have greater capability to provide value-adding advice and management services. Here, the model shows analytically that the expected investment return increases with the venture’s initial value and financier productivity. Therefore, financiers with access to larger investment funds (e.g., VC firms) will benefit by waiting to fund larger ventures at later stages of their development. Moreover, as the venture matures, the productivity of the financier rises relative to the entrepreneur (Gompers & Lerner, 1999; Shane, 2009). The model predicts that this will have a positive impact on the expected investment return of VCs at later stages while reducing entrepreneurs’ equity share. The tendency of VC firms to concentrate their investment portfolio on specific sectors (e.g., biotech, IT, social media) is also consistent with the model. This allows them to develop more focused capabilities that increase financier productivity, leading to higher expected returns.

The model further shows that competitive searching by entrepreneurs leads to outcomes where financiers with a diverse range of risk preferences can coexist in the venture finance market by making their investment returns similar. Without search, lower risk aversion financiers would have the advantage as their expected investment return is higher, the equity share given to entrepreneurs is lower, and they can profitably fund higher risk projects. With search, an equilibrium arises that reduces the advantage (disadvantage) of lower (higher) risk aversion financiers. The lower expected investment return of higher risk aversion financiers rises and the equity share given to entrepreneurs declines. Similarly, there is a decline in the higher expected return of lower risk aversion financiers and the equity share given to entrepreneurs increases.

Lastly, a risk separation emerges in the entrepreneurial finance market where higher risk ventures can only be profitably funded by angels/VCs with lower levels of risk aversion. The model shows that while the expected investment return and equity division of risk-neutral financiers is unaffected by venture risk, the expected return declines with venture risk and risk aversion. In maximizing the utility of their investment, risk-averse financiers respond to lower entrepreneurial effort (due by venture risk) by offering a higher equity share and this also improves risk sharing by reducing their exposure to venture uncertainty.

The paper also contributes to the entrepreneurial finance literature by introducing some new modeling features. The model allows the generality of a risk-averse financier and our analysis is based on the financier’s investment return, which is a primary measure of performance in the venture finance industry. Further, the venture’s future value is modeled as a continuous random function that incorporates a rich set of inputs (initial venture value, funding size, entrepreneur/financier productivities and efforts). Finally, the analysis incorporates equilibrium search by entrepreneurs over financiers with heterogeneous risk preferences. Therefore, model results regarding expected investment returns and equity division for risk-averse financiers are new contributions to the entrepreneurial finance literature. These results show analytically how financier risk aversion, venture risk, initial venture value, and funding size impact these important entrepreneurial finance quantities.

There is considerable potential to extend the model and analysis in a number of promising directions. First, the model provides a rich set of predictions and hypotheses for future empirical studies (see Table 1). For example, studies could investigate if equity shares and realized investment returns from financing deals are in line with model predictions. Second, model results can provide useful analytical guidance on the design of venture finance deals by relating the optimal equity division and expected investment return to a number of venture characteristics. Further, VC deals can include convertible features. Although this will involve some technical complexity in the extend framework (e.g., risk-averse financiers, modeling future venture value as a continuous function of a rich set of inputs), incorporating such option-like features into the model’s modeling framework with search (see (5 – 8) and (17)) can provide additional insights on the role of asymmetric risks and incentives.

Finally, although we do not explicitly explore implications for public policy in this paper, the model and its extensions can shed useful light in this direction. For example, the model shows that the positive impact of initial project value on financier expected returns and the negative impact of project risk are a source of disadvantage to investors with lower access to funding (e.g., angels) and higher risk aversion. Targeted subsidies, preferential tax treatment, and provision of partial insurance may further enhance financing opportunities in the entrepreneurial finance market for entrepreneurs with smaller and/or risker ventures.

Footnotes

The Entrepreneur’s Optimal Equity Stake

The best effort response of the entrepreneur (E) to the equity stake $w = x V$ offered by the financier (F) follows from differentiating (5) with respect to $a$ , which yields (A.1)

x (V_{0} + f) β - g a = 0

This leads to the entrepreneur’s best effort response (A.2)

a = \frac{β (V_{0} + f) x}{g}

Similarly, differentiating (7) with respect to $b$ leads to the financier’s best effort response (A.3)

b = \frac{γ (V_{0} + f) (1 - x)}{f h}

Next, to determine the optimal equity incentive share, the financier $F (α_{F})$ maximizes his expected utility (6)

max_{{x}} E U_{F (α_{F})} (x, a, b) = max_{{x}} \exp (\begin{matrix} α_{F} (\frac{(1 - x) (V_{0} + f) (1 + β a + γ b) - f}{f} - \frac{1}{2} h b^{2}) \\ - α_{F}^{2} \frac{{(1 - x)}^{2} {(V_{0} + f)}^{2} σ^{2}}{f^{2}} \end{matrix})

while taking into account the best effort responses by the entrepreneur and financier in (A.2) and (A.3).

The corresponding first-order condition is (A.4)

\begin{matrix} - \frac{(V_{0} + f) (1 + β a + γ b)}{f} + \frac{(1 - x) (V_{0} + f) (1 + β \frac{\partial a}{\partial x} + γ \frac{\partial b}{\partial x})}{f} - h b (\frac{\partial b}{\partial x}) \\ - α_{F}^{} \frac{(1 - x) {(V_{0} + f)}^{2} σ^{2}}{f^{2}} = 0 \end{matrix}

where (A.5)

\frac{\partial b}{\partial x} = - \frac{β (V_{0} + f)}{f h}

It is easy to show using (A.5) in (A.4) that the optimal equity share for the entrepreneur is (A.6)

x (α_{F}) = \frac{β_{}^{2} f (V_{0} + f) h + α_{F} (V_{0} + f) σ_{}^{2} g h - γ_{}^{2} (V_{0} + f) g - f g h}{(V_{0} + f) (2 β_{}^{2} f h + α_{F} σ_{}^{2} g h - γ_{}^{2} g)}

where

γ_{}^{2} < 2 f β_{}^{2} (h / g) + α_{F} σ_{}^{2} h

for all plausible values of entrepreneur and financier productivities

β

and

γ

.⁷

The optimal efforts for the entrepreneur and financier follow from using the equity share (A.6) in (A.2) and (A.3) and simplifying: (A.7)

a (α_{F}) = \frac{(β_{}^{2} f (V_{0} + f) h + α_{F} (V_{0} + f) σ_{}^{2} g h - γ_{}^{2} (V_{0} + f) g - f g h) β}{α_{F} σ_{}^{2} g h + 2 β_{}^{2} f h - γ_{}^{2} g}

(A.8)

b (α_{F}) = \frac{(β_{}^{2} f (V_{0} + f) h + g) γ}{α_{F} σ_{}^{2} g h + 2 β_{}^{2} f h - γ_{}^{2} g}

Entrepreneur’s Equilibrium Search Contract

As noted earlier, the key to solving the entrepreneur’s equilibrium search contract $x (\bar{α})$ with $\bar{α} \in [0, α_{M a x}]$ in (16) is to identify the contract that makes the entrepreneur indifferent to choosing between accepting the current deal $x (\bar{α})$ and carrying out another search: (17)

x (\bar{α}) E V (a (\bar{α}), b (\bar{α})) = ϕ \int_{0}^{α_{M a x}} x (α) E V (a (α), b (α)) d P (α)

Equation (17) implicitly defines the critical level of risk aversion $\bar{α} \in [0, α_{M a x}]$ and corresponding financier contract $x (\bar{α})$ that solves the search problem (16) of finding the best financing deal available from various types of financiers $F (α)$ with measure $P (α)$ .

The expected value of the equity deal to the entrepreneur $x (α) E V (a (α), b (α))$ by the financier $F (α)$ is highly nonlinear in $α$ . This makes the integration on the right-hand side of (17) technically complicated. We, therefore, settle for a first-order approximation to (17). This approximation for $x (α) E V (a (α), b (α)) = x (α) (V_{0} + f) (1 + β a (α) + γ b (α))$ is given by (B.1)

x (α) E V (a (α), b (α)) \approx x (0) E V (a (0), b (0)) + \frac{\partial {x (α) E V (a (α), b (α))}}{\partial α} |_{α = 0} α

where (B.2)

\begin{matrix} \frac{\partial {x (α) E V (a (α), b (α))}}{\partial α} = x^{/} (α) E V (a (α), b (α)) + x (α) E V^{/} (a (α), b (α)) \\ = x^{/} (α) (V_{0} + f) (1 + β a (α) + γ b (α)) + x (α) (V_{0} + f) (β a^{/} (α) + γ b^{/} (α)) \end{matrix}

The partial derivatives $x^{/} (α) > 0$ , $a^{/} (α) > 0$ and $b^{/} (α) < 0$ are obtained in Appendices C and E. Using these expressions, a sufficient condition for $\frac{\partial {x (α) E V (a (α), b (α))}}{\partial α} > 0$ is that $γ$ . It is easy to show that this holds when the following condition on funding size holds: (B.3)

f > \frac{γ^{2} g}{β^{2} h}

This condition is easily satisfied because even if the financier is 10 times more productive than the entrepreneur ( $γ^{2} / β^{2} = 100$ ) and the effort aversion of the entrepreneur is 10 times larger than that of the financier $(g / h = 10)$ , condition (B.3) is easily satisfied as $f > 100 (10) = 1000$ in almost any realistic venture finance deal. In reality, the discrepancy in these parameters will not be so extreme. Therefore, $\frac{\partial {x (α) E V (a (α), b (α))}}{\partial α} > 0$ holds in realistic funding situations.

To obtain a tractable result, we assume that financier types $F (α)$ are uniformly distributed over $α \in [0, α_{M a x}]$ so that $d P (α) = 1 / α_{M a x}$ . Using (B.2) on both sides of the equilibrium search condition (17) leads to the following result for the equilibrium search contract: (B.3)

\bar{α} = \frac{ϕ}{2} α_{M a x} - (1 - ϕ) \frac{x (0) E V (a (0), b (0))}{\frac{\partial {x (α) E V (a (α), b (α))}}{\partial α} |_{α = 0}}

where

x (0) E V (a (0), b (0)) = x (0) (V_{0} + f) (1 + β a (0) + γ b (0))

and

\begin{array}{ll} \frac{\partial {x (α) E V (a (α), b (α))}}{\partial α} |_{α = 0} & = {x^{/} (0) (V_{0} + f) (1 + β a (0) + γ b (0))) \\ + x (0) (V_{0} + f) (β a^{/} (0) + γ b^{/} (0))} α \end{array}

The equilibrium search contract is given by using (B.3) in Proposition 1 leading to

x^{*} (\bar{α}) = \frac{(V_{0} + f) (β_{}^{2} f h + \bar{α} σ_{}^{2} g h - γ_{}^{2} g) - f g h}{(V_{0} + f) (2 β_{}^{2} f h + \bar{α} σ_{}^{2} g h - γ_{}^{2} g)}

Sensitivity of Entrepreneur’s Equity Stake

It is easy to show using (A.6) that the partial derivatives of the entrepreneur’s equity stake with respect to various model parameters as given below.

Sensitivities of Financier’s Expected Investment Return

The partial derivatives of the financier’s investment return (15) with respect to various –project–financier–entrepreneur characteristics are reported below (for complicated expressions, only the sign is reported; the complete expression is available from the authors).

Optimal Effort Sensitivities

Further, from the optimal efforts (A.7) and (A.8), the sensitivities of the entrepreneur’s optimal effort level to various model parameters are provide below (for complex expressions with a large number of terms, only the sign is reported; complete expressions are available from the authors).

Acknowledgment

The authors would like to thank the Editor, Professor Sophie Manigart, and two anonymous Reviewers for providing very detailed and useful comments on this research.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by a grant from the Canadian Social Science & Humanities Research Council (SSHRC).

Notes

Author Biographies

Gurupdesh Pandher is professor of finance at University of Windsor’s Odette School of Business. He completed his PhD studies at Cornell University and has held faculty positions at the University of British Columbia and DePaul University (Chicago) and visiting positions at the University of Maryland and University of Hong Kong. His research interests include venture finance, banking risk management, innovation in organizations and executive compensation and his research has been published in journals including Econometric Theory, Journal of Banking & Finance, Journal of Derivatives, Journal of Economic Theory, International Statistical Review, Real Estate Economics, Strategic Entrepreneurship Journal, and Strategic Management Journal. He served as Senior Associate Dean (2012-17) and has professional and consulting experience with a number of organizations including Sunlife Insurance Company, Treasury Board, Statistics Canada, IMF, Arthur Andersen and Deloitte.

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