# The Two-Stage Model of Entrepreneurs Financing Based on the Entry/Exit Decision

Authors: Kaihong Wang; Yin Mingyang; Zuo Jiani
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

Normally entrepreneur would raise fund from angel investors during the initial round. If the venture program was by then successful, the entrepreneur would then continue the fund-raising process from venture capitalist. By adopting the convertible preferred stock, we managed to construct the two-stage angel investment decision process. This research reveals the following: (1) The probability of the first stage’s success has negative relationships with levels of priority dividend in both first and second stages, as well as with the venture capitalist’s proportion of shares. (2) The probability of the second stage’s success has negative relationships with the venture capitalist’s proportion of shares and the dividend level of both first and second stage funding. (3) There has been a threshold of dividend distribution, which belongs to angel investor. While the level of angel investor’s shares is higher than the threshold, AN would decide to join the second phase of the program; otherwise, AN would exit the project at the end of the first stage.


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## Body

## 1. Introduction

Venture entrepreneur (EN), due to a lack of internal financial sources at early stages, depends crucially on outside financing and two potential sources include (1) angel investor (AN) and (2) venture capitalist (VC). Among the two sources of external financing, the AN is a perfect match for start-ups. This is because the VCs are simply not available for ENs at early stages. Li et al. [1] claimed that AN investment, as a relatively mature informal supporting scheme for start-ups, was the dominant capital source for the ENs at the seed-stage and the start-stage and effectively improved the development of the start-ups. Currently, the ANs market is gathering momentum in China. For instance, compared to 2014, Chinese angel institutions absorbed 124 new funds with disclosed amount of 20.357 billion RMB and invested in 2,075 cases with the disclosed amount of more than 10.188 billion RMB, showing a remarkable speed of development of Chinese angel investment market (Researching Center of Zero2IPO Group, 2016). These findings further illustrate that angel investment is a substantial source of financing for the start-ups.Recent literature on angel investments is based on the following two aspects:Firstly, the characteristics of AN. Shi [2] analyzed the characteristics, motivations, transaction structures, investing process, and postinvestment management of angel investment in China and proposed anticipation of evolution of AN. Wang et al. [3] analyzed the Chinese angel investment market in terms of development trend and countermeasures, based on first-hand research data and secondary sources. They found the following: (1) Chinese angel investment is still developing rapidly. (2) Problems still exist in investor recognition, standardization, and codification. Rodriguez [4] pointed out that, because the advantages and attractiveness of the projects cannot be easily revealed at the development phase and start-up phase, some ENs start from getting private loans from their friends or relatives, which is the original form of angel investments. Based on empirical results on 121 AN in 1038 new-type ventures, Wiltbank et al. [5] found the impacts of AN’s predictive control strategies on the results of ventures. Through empirical studies, Hoberg et al. [6] concluded that EN prefers VC rather than the combination of AN and VC and will obtain better return in this way.Secondly, moral hazard of EN. Kerr et al. [7] stated the fact that ventures funded by two successful angel groups experience superior outcomes to rejected ventures: they have higher survival rate, exits, employment, patenting, Web traffic, and financing. Kerr et al. [7] confirmed the positive effects for venture operations, with qualitative support for a higher likelihood of successful exits. Chemmanur and Chen [8] develop a theoretical analysis of an entrepreneur’s choice between venture capital and angel financing at various stages in a private firm’s life and characterize the dynamic evolution of the firm’s contract with its financier (VC or AN). Johnson and Sohl [9] recommended that angel investors and ENs carry out different stages of phased investment when faced with ENs with completely external financing plans. Elitzur and Gavious [10] studied the relationship among ENs, AN, and VC by analyzing the connections and the relevant contract designs. They also took moral hazard into consideration and argued that an EN without a powerful helmsman tends to have an inefficient operation. Guo and Wang [11] introduced a signaling model to analyze the role of AN in screening of high-quality projects.The two aspects above mainly focus on characteristics of angel financing and provisions of contract before investment realization. However, the process from entry to exit should be complete, and thus we should not ignore the exit of angel capital. The approaches that the capital financed from AN exits are related to financial instrument agreements when the angel capital enters. Convertible preferred stock is the preferred stock that allows owner to convert it to another type of stock under certain conditions. Yao [12] studied efficient exit of VC through convertible preferred stocks. He showed that the time of preferred stock conversion influences VC’s exit decision. When incomes obtained from corporation’s growth are greater than a specified interest, VC has an incentive to convert preferred shares into ordinary shares in order to gain benefits from the firm’s growth. Generally speaking, VC’s preferred stock is automatically converted into common shares when the company goes public. Therefore, the flexibility of convertible preferred stocks allows VC to exit more efficiently. Yao [12] also showed that the allocation of control power between the two parties is a problem that needs to be solved in the process of investment. Through conversion of preferred stock, EN and VC reallocate income and control so that the VC can have an optimal exit and the social benefit can be maximized. Zhang and Yang [13] studied the optimal exit decision with partaking convertible preferred stocks under certain conditions; VC can always achieve the optimal exit by using partaking convertible preferred stocks. Moreover, financier’s gains from distribution of EN’s surplus value, reputation, and corporate private interests also impact the control power of exit of VC. Cumming and Johan [14] studied the relation between venture capital contracts and exit. Flix and Gulamhussen [15] analyzed the exit decision in the European venture capital market and studied when to exit and how to interact with the exit form. The study of Medin [16] suggested that PE-backed firms demonstrate superior operating performance after exit, which to a large extent is driven by superior performance of VC-backed firms. BO-backed firms, however, did not demonstrate superior postexit operating performance. Zheng, et al. [17] thought that the strategies of VC and EN in the venture capital market may form an exit dilemma; therefore, the analysis of their strategy selection and entanglement using quantum game is given.These studies mainly paid attention to the exit of venture capital but rarely considered the impact of convertible preferred stock owned by angel. Carpentier and Suret [18] analyzed exit-related perceptions of the members of a large, well-structured Canadian angel group; these angels should consider the initial public offering (IPO) as a possible exit mode. Securities regulation was also perceived as a major impediment to exit onto the stock market. Mahapatra [19] discussed the exit phase of the individual angels, angel syndicates, and corporate angels. His work tried to overcome this gap and examined only the exit phase of the angel syndicates and the corporate investors. The exit phase of the individual investors was reexamined in the light of their reported professionalism and evolution. Hellmann and Thiele [20] developed a theory of how angel and venture capital markets interact.In current literature, most scholars mainly studied optimal exit of VC. As an approach of venture financing, besides “love fund” and VC, AN is relatively rarely discussed with VC and the incentive mechanism, and exit mechanism of AN is rarely explained from multistage dynamic angles. AN differs from the traditional sense of VC in that the former will choose the optimal time to exit. When start-up is mature or bankrupt, and AN will find ways to sell equity for funds; then the exit phase is particularly important for AN. These are the differences between this article and other studies.This paper, in the aspect of AN, considers convertible preferred stock as an incentive mechanism of venture capital investment and analyzes the mechanism of angel investment from the multistage dynamic investment process.The innovation is as follows: the operation of the project is divided into two stages: EN finances from AN in the first stage, EN finances from AN and VC in the second stage; we study how to design the contract and when to quit in the cases, with and without AN. The exit decision is given by AN numerical calculation.


## 2. Model Assumption and Establishment

EN has a project while its own capital is zero. In such case, EN needs external financing. Considering the uncertainty and riskiness of innovative project, AN finances the project with the following method. At the first stage of project, AN provides capitalI1(I1>0). After the end of first phase of the project, if the project performs badly and fails, then AN liquidates and exists. On the other hand, if the project has a good operation condition, AN transmits signal to the outside world and begins the second round of financing. At this time, VC provides capital I2(I2>0). Due to the asymmetry of information, EN as an agent of this project will try to maximize his own benefit. In order to enhance the probability of project’s success, AN not only needs to invest capital I1(I2) but also would provide value-added services (like participation in decision making and management of strategy, providing consulting services, assisting in public relation, doing product promotion, and increasing market share). Considering these factors, the profit of the project not only relates to EN’s effort e11(e11>0) but also relates to AN’s effort e21(e21>0). That is the same case in the second phase of the project. EN and AN choose effort, respectively, to maximize their own benefit. The same for VC, he would provide the project with effort e32(e32>0).The success probability of the first phase is related to the effort of AN and EN in the first stage. Thus we assume success probability of the first phase is(1)p1=r1e11+r2e21,0≤r1e11+r2e21≤1.The success probability of the second phase is related to AN and EN’s second stage effort; at the same time the effort of first stage will also continue affecting the success probability of the second phase. Thus we assume the success probability of second phase is(2)p2=θ1e12+θ2e22+θ3e32+ρe11+e21,0≤θ1e12+θ2e22+θ3e32+ρe11+e21≤1.ρ ( ρ < θ 1 , θ 2 , θ 3 ) is the coefficient representing the effort of the first stage, which affects the second stage’s probability. The magnitude of ρ is lower than θ1, θ2, and θ3, because the first stage effort has little influence on the probability of the second stage.This article assumes the success probability of project is a linear function of participanti’s effort. This means that the level of each participant’s effort is substitute rather than complementary. The reasons for this assumption are as follows.Firstly, Ma [21] points out that there is symmetry information between EN and VC. EN has lots of uncertainties, for example, the uncertainty of research and technology, the uncertainty of management ability, and the uncertainty of employees. It is hard to separate their contribution of effort. In such case, it is meaningful; VC as a principle motivates EN to work harder.Secondly, Casamatta [22] states EN and VC face a double moral hazard. The consulting services they provide to project are indivisible. In order to encourage each other to work hard, EN and VC must have the right to share the benefit from project. In addition, in this article, the efforts of both sides are unobservable. EN provides the project with technical and innovative services while VC provides the project with management service.Supposing the cost of effort can be measured by money and cost function satisfiesc′(·)>0, c′′(·)>0, we assume the cost of various stages with EN’s effort is c(e11)=(1/2)e112, c(e12)=(1/2)e122.And the cost of various stages with AN’s effort isc(e21)=(1/2)e212, c(e22)=(1/2)e222,The cost of VC’s effort isc(e32)=(1/2)e322.The income of project is a random variableR. The input of EN decreases marginally with the increase of input. Bascha and Walz [23] assume the density function of random income R is g(R)=λe-λR.In order to simplify the computation, this paper constructs a new density functiong(R)=2a22/R3 ( Figure 1 shows the image of the two density functions).Figure 1
Comparison of two density function.Newly constructed density function has consistent characteristic of density function of Bascha and Walz [23], so the density this paper uses is feasible.The variables and symbols are summarized as Table1.Table 1
The variables and symbols.
Symbol Specification e i j The effort ofi at the j -stage, i=EN,AN,VC, j=1,2 c(eij)=(1/2)eij2 The effort cost ofi at the stage j, i=EN,AN,VC, j=1,2 Π E N 1, ΠAN1 The expected income of EN and AN in the first stage Π E N 2, ΠAN2, ΠVC2 The expected income of EN, AN, and VC at the second stage I 1 The financing amount of AN at the first stage I 2 The financing amount of VC at the second stage δ Discount factor of income of project a 1 The project residual at the first stage a 2 The project residual at the second stage with the participation of AN a ~ 2 The residual of project at the second stage without the participation of AN φ 1 In the second stage, VC transfers the shares to the EN, when AN can participate φ ~ 1 In the second stage, VC transfers the shares to the EN, when AN does not participate φ 2 In the second stage, VC transfers the shares to the AN, when AN can participate φ ~ 2 In the second stage, VC transfers the shares to the AN, when AN does not participate γ In the second stage, a share of AN from the restp2R-D2 of the income, when AN can participate γ ~ In the second stage, a share of AN from the restp2R-D2 of the income, when AN does not participate y In the first stage, AN transfers the shares to the EN D 2 In the second stage, VC’s bond yields, when AN can participate D ~ 2 In the second stage, VC’s bond yields, when AN does not participate D 1 In the first stage, AN’s bond yields, when AN can participate D ~ 1 In the first stage, AN’s bond yields, when AN does not participate k 1 Risk free interest rate of capitalI1 k 2 Risk free interest rate of capitalI2


## 3. Two-Stage Game Model of AN Continued Investment

In this section, we consider the second stage investment which AN can do. The game order is as follows:Phase 0: AN analyzes the quality of project and chooses the high-quality project to invest.Phase 1: As the equity holder, AN will design the contract to get the best return. AN will not only supply funds but also assist in the management which the firm needs.Phase 2: If the project succeeds, in the second round of investment, VC is introduced to control project. As a result, VC becomes the equity holder of the project. If AN does not exit, EN and AN will become the agents. As the principal, VC will design the contract to get the best return and will supply funds as well as assisting in management.Then, we design the contract according to backward induction.


### 3.1. The Second Stage Game Model of Contract

In the second stage, VC invests in the project. EN/VC provides a convertible preferred stock contract; see Figure2.Figure 2
AN participates, the two-stage contract design.Explanation of Figure2. In the first stage, if the project is successful and AN can execute the convertible preferred stock, then y is a share of AN and 1-y is a share of EN. If AN does not execute the convertible preferred stock, then D1 is bond yield, γ is a share of AN from the residue p2R-D1 of the income, and 1-γ is a share of EN from the residue p2R-D2 of the income. In the second stage, if the project is successful, if VC can execute the convertible preferred stock, then φ1 is a share of VC, φ2 is a share of AN, and the share of EN is 1-φ1-φ2, if VC does not execute the convertible preferred stock, then D2 is bond yield.According to the analysis above, the optimal financing problem of VC is(3)maxD2,φ1,φ2,γΠVC2=maxD2,φ1,φ2,φ3,γ⁡∫a2D21-p2R2a22R3dR+∫D2D2/φ1p2D22a22R3dR+∫D2/φ1+∞φ1p2R2a22R3dR-12e322-I2.(4)st:e12∗∈arg⁡maxe12⁡ΠEN2=maxe12⁡∫D2D2/φ11-γp2R-D22a22R3dR+∫D2/φ1+∞1-φ1-φ2p2R2a22R3dR-12e122(5)e22∗∈arg⁡maxe22⁡ΠAN2=maxe22⁡∫D2D2/φ1p2R-D22a22R3dR+∫D2/φ1+∞φ2p2R2a22R3dR-12e222(6)e32∗∈arg⁡maxe32⁡ΠVC2=maxe32⁡∫a2D21-p2R2a22R3dR+∫D2D2/φ1p2D22a22R3dR+∫D2/φ1+∞φ1p2R2a22R3dR-12e322-I2.Equation (3) is VC’s objective function. VC gets the best return by determining the D2,φ1,φ2,γ in the contract. ∫D2D2/φ1D2(2a22/R3)dR is the profit of VC when the project fails and the possibility is 1-p2. ∫D2D2/φ1D2(2a22/R3)dR, ∫D2/φ1+∞φ1R(2a22/R3)dR are the profit of VC in two conditions: does not convert or convert. And the possibility is p2.Equation (4) means the incentive compatible constraint of EN. EN gets the best profit by determining the effort level. When the project fails, the profit of EN is 0, and ∫D2D2/φ11-γR-D2(2a22/R3)dR and ∫D2/φ1+∞(1-φ1-φ2)p2R(2a22/R3)dR are the profit of EN in the two different conditions: the preferred stock does not convert or convert. The possibility is also p2.Equation (5) means the incentive compatible constraint of AN. AN gets the best profit by determining the effort level. When the project fails, the profit of AN is 0, and ∫D2D2/φ1p2(R-D2)(2a22/R3)dR and ∫D2/φ1+∞φ2p2R(2a22/R3)dR are the profit of EN in the two different conditions: the preferred stock does not convert or convert. The possibility is still p2.Equation (6) means the incentive compatible constraint of VC. VC gets the best profit by determining the effort level. Solving the first-order of (4), (5), and (6), the optimal effort of EN, AN, and CV are as follows: (7)e12∗=1-γθ1a22D2-2a22φ1D2+2a22φ12D2+1-φ1-φ2θ12a22φ1D2.e22∗=γθ2a22D2-2a22φ1D2+2a22φ12D2+φ2θ22a22φ1D2.e32∗=θ3-2a2-3a22D2+2a22φ12D2.


### 3.2. The First Stage Game Model of Contract

In the first stage of the project, AN finances the project; according to the assumption in Section1, the success probability of the first stage is given as p2=r1e11+r2e21, and the income of the project is stochastic, whose density function is given as g(R)=2a12/R3.AN and EN make a contract of the convertible preferred stock and the mechanism of the contract; see Figure2.AN and EN need to make sure that their efforts in the first stage maximize their total income in the first and second stage; optimal financing problem is as follows:(8)maxD1,φΠAN=ΠAN1e11,e21+δΠAN2∗e11,e21,e12∗,e21∗,e32∗(9)S.T.maxe11⁡ΠEN=ΠEN1e11,e21+δΠEN2∗e11,e21,e12∗,e21∗,e32∗(10)maxe21ΠAN=ΠAN1e11,e21+δΠAN2∗e11,e21,e12∗,e21∗,e32∗,of which(11)ΠEN1=∫D1D1/φr1e11+r2e21R-D2a12R3dR+1-φ∫D1/φ+∞r1e11+r2e21R-D2a12R3dR-12e112.(12)ΠEN2∗=∫D2D2/φ11-γθ1e12∗+θ2e22∗+θ3e32∗+ρe11+e21R-D22a22R3dR+1-φ1-φ2∫D2/φ1+∞1-φ1-φ2θ1e12∗+θ2e22∗+θ3e32∗+ρe11+e21R2a22R3dR-12e122.(13)ΠAN1∗=∫a1D11-r1e11-r2e21R2a12R3dR∫D1D1/φD12a22R3dR+φ∫D1/φ1+∞r1e11+r2e21R2a12R3dR-12e212-I1.(14)ΠAN2∗=∫D2D2/φ1γθ1e12∗+θ2e22∗+θ3e32∗+ρe11+e21R-D22a22R3dR+φ2∫D2/φ1+∞θ1e12∗+θ2e22∗+θ3e32∗+ρe11+e21R2a22R3dR-12e222.Formula (9) is AN’s object function; AN decides D1,φ in the contract to maximize his income.Formula (10) is EN’s incentive compatibility constrain; EN decides the effort to maximize his income.Formula (15) is AN’s incentive compatibility constrains; AN decides the effort to maximize his income.According to the above-mentioned model, we solve the optimal effort of AN and EN in the first stage:(15)e11∗=r1a12D1-a12φ2D1+δρ1-γa22D2-2a22φ1D2+2a22φ1D2+1-φ1-φ22a22φ1D2.(16)e21∗=r2-2a1+3a12D1+a12φ2D1+δργa22D2-2a22φ1D2+2a22φ1D2+φ22a22φ1D2.According to the first stage optimal effort level, we can get the optimal income of AN in the second stage of the project:(17)ΠAN∗=ΠAN1e11∗,e21∗+δΠAN2∗e11∗,e21∗,e12∗,e21∗,e32∗.


### 3.3. The Optimal Contract and Its Characteristics

According to (8), (9), (10), (15), and (16), we take the second stage optimal effort into the second stage success probability:(18)p2∗=θ1e12∗+θ2e22∗+θ3e32∗+ρe11∗+e21∗=θ1e12∗+δρ2θ1e12∗+θ2e22∗+δρ2θ2e22∗+θ3e32∗+ρr1a12D1-a12φ2D1+r2-2a+3a12D1+a12φ2D1.To simplify the model, we assumeθ1=θ2=θ3=θ, r1=r2; thus(19)p2∗=δρ2a22D2-a22φ12D2+θ2-2a2+4a22D2+rρ-2a2+4a12D1.VC’s optimal utility is as follows:(20)ΠVC2∗=2a2-2a22D2+δρ2a22D2-a22φ12D2+θ2-2a2+4a22D2+rρ-2a1+4a12D1×-2a2+3a22D2+a22φ12D2-12θ12-2a2+3a22D2+a22φ12D2-I2.In (20), we calculate the first-order derivatives of φ1 and get (21)∂ΠVC2∗∂φ1=δρ2-2a22φ1D2-2a2-3a22D2+a22φ12D2+2a22φ12D2+θ2-2a2+4a22D2+rρ-2a1+4a12D1-θ12-2a2+3a22D2+a22φ12D22a22φ1D2=0.According to formula (21), there is (22)φ1∗2=2δρ2a2-2rρa1+4rρa12/D12δρ2a22+θ12a22D2+-2δρ2a22+θ12a222δρ2a22+θ12a22.Guo and Zeng [24] thought that the income of VC from the initial contract should be no less than the external opportunity income; thus we assume that if the project is not liquidated, the income of VC should be no less than his external opportunity income. Risk-free income of venture capital I2 is k2I2, among which, k2 is risk-free rate of interest; thus we can determine the preferred stock coupon, VC gets D2=k2I2; thus optimal contract of the second stage is (23)D2=k2I2φ1∗=2δρ2a2-2rρa1+4rρa12/D12δρ2a22+θ12a22D2∗+-2δρ2a22+θ12a222δρ2a22+θ12a22.According to (21) and (22), there is Conclusion 1.Conclusion 1.
The second stage success probabilityp2 is inversely related to VC’s share φ1, is inversely related to the second stage convertible preferred stock coupon D2, and is inversely related to the first stage preferred stock coupon D1.Practical significance of Conclusion1 is that the success probability of the second stage is related to the contract designed by VC; lowering φ1,D2 can incentivize VC and EN to work harder, thus decreasing the agency cost and improving the success probability of the project. This is because modestly lowering φ1,D2 can increase the share of profit of AN and EN. Besides, the success probability p2 of the second stage is related to the first stage convertible point D1; lowering the convertible point can make EN and AN share the profit, thus giving both sides incentive to work hard.Conclusion 2.
The VC’s share is a linear function of his preferred stock coupon, and whenD1>2ra12/(ra1-δρa2), φ12 is an increasing function of D2; when D1<2ra12/(ra1-δρa2), φ12 is an decreasing function of D2.Conclusion2 shows that the linear relationship between φ12 and D2 is related to the preferred stock coupon in the first stage; according to Conclusion 1, a lower D1 can incentivize EN and AN to work hard in the second stage, although increasing D2 can adversely influence the hard work of EN and AN; a good outcome of the first stage can motivate EN and AN to go on working hard, and the success probability and the income of the project are guaranteed; thus VC can moderately decrease the share φ1 to motivate EN and AN to go on working hard. Similarly, when D1 is higher, EN and AN can slug in the second stage, while increasing D2 will adversely affect the hard work of EN and AN; these two factors affect the project’s success probability and income; thus VC can moderately increase share φ1 to guarantee his own income. In reality, AN’s preferred stock coupon should guarantee his investment capital; that is, D1>I1, and 2ra12/(ra1-δρa2)>2ra12/ra1=2a1; in other words, only if the preferred stock coupon of AN satisfies the condition D1>2ra12/(ra1-δρa2)>2ra12/ra1=2a1, AN will make sure he will not lose.Conclusion 3.
The share of VC is the decreasing function of the residual value with the projecta2; when a1>0.25D1, VC’s share is an increasing function of a1; when a1<0.25D1, VC’s share is a decreasing function of a1.Practical significance of Conclusion3 is that the greater the residual value of the project in the second stage, the greater the income of VC when the project fails; thus VC can decrease his share in a high income scenario and incentivize AN and EN to work hard. The share of VC is also related to the residual value in the first stage; when a1<0.25D1, if D1 is smaller, EN and AN will work hard, and the success probability of the project will rise.According to (15) and (16), we substitute the optimal efforts into the probability of success in the first period; there is (24)p1∗=r1e11+r2e21∗=δρrθ1e12∗+δρrθ1e22∗+r12a12D1-a12φ2D1+r22-2a1+3a12D1+a12φ2D1=δρrθ1e12∗+δρrθ1e22∗+r2-2a1+4a12D1=δρra22D2-a22φ12D2+r2-2a1+4a12D1so the optimal utility of AN in the first period is(25)ΠAN1∗=2a1-2a12D1+δρ2ra22D2-a22φ12D2+r2-2a1+4a12D1-2a1+3a12D1+a12φ2D1-12r1-2a1+3a12D1+a12φ2D1+δρθ1e22∗2-I1.The partial derivation of (25) with respect to φ is (26)∂ΠAN1∗∂φ=δρra22D2-a22φ12D2+r2-2a1+4a12D12a12φD1-r1r1-2a1+3a12D1+a12φ2D1+δρθ1e22∗2a12φD1=0.From (25) and (26), we can get(27)φ∗2=r1δρe12a12r12D1-1.On the basis of the assumptions of Guo and Zeng [24], if the project is not liquidated, the least return that the AN gains is equal to AN’s external opportunity. If k1I1 is AN’s return of external risk-free investments, the dividends that AN gains on prefer stocks must be D1=k1I1.In the first period, the contract designed by AN and EN should be(28)D1∗=k2I2φ1∗=r1δρe12a12r12D1-1,=r1δρ1-γθ1a22/D2-2a22φ1/D2+a22φ12/D2+1-φ1-φ2θ12a22φ1/D2a12r12D1∗-1.From (24) (27) we have following conclusions.Conclusion 4.
The success probabilityp1 in the first period is inversely proportional to the dividends D1, D2 converted on preferred stocks and inversely proportional to VC’s shares φ1.Conclusion 5.
The AN’ stock share strictly monotonically increases with the increment of the dividendsD1 that AN gains on preferred stocks.Practical significance of Conclusion5 is that when D1 goes up, EN’s return will goes down if the project has a poor quality. As a result, the smaller the success probability of the project, the lower the profits that will come up when EN becomes slake and indolent. In this case, the only way for AN is to increase dividend φ. This suggests that a higher D1 gives rise to a higher φ.The formulae22∗=γθ1(a22/D2-2a22φ1/D2+a22φ12/D2)+φ2θ1(2a22φ1/D2) indicates that AN’s efforts in the second period are a function of φ1,D2,γ.On the basis ofD1∗,φ∗,D2∗,φ1∗, the optimal total utility of AN is as follows: (29)ΠAN∗=ΠAN1∗+δΠAN2∗=2a1-2a12D1∗+δρ2ra22D2∗-a22φ12D2∗+r2-2a1+4a12D1∗-2a1+3a12D1∗+a12φ2D1∗-12r1-2a1+3a12D1∗+a12φ2D1∗+δρθ1e22∗2-I1+δγδρ2a22D2∗-a22φ1∗2D2∗+θ2-2a2+4a22D2∗+rρ-2a1+4a12D1∗×a22D2∗-a22φ1∗2D2∗+a22φ1∗2D2∗+φ2δρ2a22D2∗-a22φ1∗2D2∗+θ2-2a2+4a22D2∗+rρ-2a1+4a12D1∗2a22φ1∗2D2∗-12e22∗2.


## 3.1. The Second Stage Game Model of Contract

In the second stage, VC invests in the project. EN/VC provides a convertible preferred stock contract; see Figure2.Figure 2
AN participates, the two-stage contract design.Explanation of Figure2. In the first stage, if the project is successful and AN can execute the convertible preferred stock, then y is a share of AN and 1-y is a share of EN. If AN does not execute the convertible preferred stock, then D1 is bond yield, γ is a share of AN from the residue p2R-D1 of the income, and 1-γ is a share of EN from the residue p2R-D2 of the income. In the second stage, if the project is successful, if VC can execute the convertible preferred stock, then φ1 is a share of VC, φ2 is a share of AN, and the share of EN is 1-φ1-φ2, if VC does not execute the convertible preferred stock, then D2 is bond yield.According to the analysis above, the optimal financing problem of VC is(3)maxD2,φ1,φ2,γΠVC2=maxD2,φ1,φ2,φ3,γ⁡∫a2D21-p2R2a22R3dR+∫D2D2/φ1p2D22a22R3dR+∫D2/φ1+∞φ1p2R2a22R3dR-12e322-I2.(4)st:e12∗∈arg⁡maxe12⁡ΠEN2=maxe12⁡∫D2D2/φ11-γp2R-D22a22R3dR+∫D2/φ1+∞1-φ1-φ2p2R2a22R3dR-12e122(5)e22∗∈arg⁡maxe22⁡ΠAN2=maxe22⁡∫D2D2/φ1p2R-D22a22R3dR+∫D2/φ1+∞φ2p2R2a22R3dR-12e222(6)e32∗∈arg⁡maxe32⁡ΠVC2=maxe32⁡∫a2D21-p2R2a22R3dR+∫D2D2/φ1p2D22a22R3dR+∫D2/φ1+∞φ1p2R2a22R3dR-12e322-I2.Equation (3) is VC’s objective function. VC gets the best return by determining the D2,φ1,φ2,γ in the contract. ∫D2D2/φ1D2(2a22/R3)dR is the profit of VC when the project fails and the possibility is 1-p2. ∫D2D2/φ1D2(2a22/R3)dR, ∫D2/φ1+∞φ1R(2a22/R3)dR are the profit of VC in two conditions: does not convert or convert. And the possibility is p2.Equation (4) means the incentive compatible constraint of EN. EN gets the best profit by determining the effort level. When the project fails, the profit of EN is 0, and ∫D2D2/φ11-γR-D2(2a22/R3)dR and ∫D2/φ1+∞(1-φ1-φ2)p2R(2a22/R3)dR are the profit of EN in the two different conditions: the preferred stock does not convert or convert. The possibility is also p2.Equation (5) means the incentive compatible constraint of AN. AN gets the best profit by determining the effort level. When the project fails, the profit of AN is 0, and ∫D2D2/φ1p2(R-D2)(2a22/R3)dR and ∫D2/φ1+∞φ2p2R(2a22/R3)dR are the profit of EN in the two different conditions: the preferred stock does not convert or convert. The possibility is still p2.Equation (6) means the incentive compatible constraint of VC. VC gets the best profit by determining the effort level. Solving the first-order of (4), (5), and (6), the optimal effort of EN, AN, and CV are as follows: (7)e12∗=1-γθ1a22D2-2a22φ1D2+2a22φ12D2+1-φ1-φ2θ12a22φ1D2.e22∗=γθ2a22D2-2a22φ1D2+2a22φ12D2+φ2θ22a22φ1D2.e32∗=θ3-2a2-3a22D2+2a22φ12D2.


## 3.2. The First Stage Game Model of Contract

In the first stage of the project, AN finances the project; according to the assumption in Section1, the success probability of the first stage is given as p2=r1e11+r2e21, and the income of the project is stochastic, whose density function is given as g(R)=2a12/R3.AN and EN make a contract of the convertible preferred stock and the mechanism of the contract; see Figure2.AN and EN need to make sure that their efforts in the first stage maximize their total income in the first and second stage; optimal financing problem is as follows:(8)maxD1,φΠAN=ΠAN1e11,e21+δΠAN2∗e11,e21,e12∗,e21∗,e32∗(9)S.T.maxe11⁡ΠEN=ΠEN1e11,e21+δΠEN2∗e11,e21,e12∗,e21∗,e32∗(10)maxe21ΠAN=ΠAN1e11,e21+δΠAN2∗e11,e21,e12∗,e21∗,e32∗,of which(11)ΠEN1=∫D1D1/φr1e11+r2e21R-D2a12R3dR+1-φ∫D1/φ+∞r1e11+r2e21R-D2a12R3dR-12e112.(12)ΠEN2∗=∫D2D2/φ11-γθ1e12∗+θ2e22∗+θ3e32∗+ρe11+e21R-D22a22R3dR+1-φ1-φ2∫D2/φ1+∞1-φ1-φ2θ1e12∗+θ2e22∗+θ3e32∗+ρe11+e21R2a22R3dR-12e122.(13)ΠAN1∗=∫a1D11-r1e11-r2e21R2a12R3dR∫D1D1/φD12a22R3dR+φ∫D1/φ1+∞r1e11+r2e21R2a12R3dR-12e212-I1.(14)ΠAN2∗=∫D2D2/φ1γθ1e12∗+θ2e22∗+θ3e32∗+ρe11+e21R-D22a22R3dR+φ2∫D2/φ1+∞θ1e12∗+θ2e22∗+θ3e32∗+ρe11+e21R2a22R3dR-12e222.Formula (9) is AN’s object function; AN decides D1,φ in the contract to maximize his income.Formula (10) is EN’s incentive compatibility constrain; EN decides the effort to maximize his income.Formula (15) is AN’s incentive compatibility constrains; AN decides the effort to maximize his income.According to the above-mentioned model, we solve the optimal effort of AN and EN in the first stage:(15)e11∗=r1a12D1-a12φ2D1+δρ1-γa22D2-2a22φ1D2+2a22φ1D2+1-φ1-φ22a22φ1D2.(16)e21∗=r2-2a1+3a12D1+a12φ2D1+δργa22D2-2a22φ1D2+2a22φ1D2+φ22a22φ1D2.According to the first stage optimal effort level, we can get the optimal income of AN in the second stage of the project:(17)ΠAN∗=ΠAN1e11∗,e21∗+δΠAN2∗e11∗,e21∗,e12∗,e21∗,e32∗.


## 3.3. The Optimal Contract and Its Characteristics

According to (8), (9), (10), (15), and (16), we take the second stage optimal effort into the second stage success probability:(18)p2∗=θ1e12∗+θ2e22∗+θ3e32∗+ρe11∗+e21∗=θ1e12∗+δρ2θ1e12∗+θ2e22∗+δρ2θ2e22∗+θ3e32∗+ρr1a12D1-a12φ2D1+r2-2a+3a12D1+a12φ2D1.To simplify the model, we assumeθ1=θ2=θ3=θ, r1=r2; thus(19)p2∗=δρ2a22D2-a22φ12D2+θ2-2a2+4a22D2+rρ-2a2+4a12D1.VC’s optimal utility is as follows:(20)ΠVC2∗=2a2-2a22D2+δρ2a22D2-a22φ12D2+θ2-2a2+4a22D2+rρ-2a1+4a12D1×-2a2+3a22D2+a22φ12D2-12θ12-2a2+3a22D2+a22φ12D2-I2.In (20), we calculate the first-order derivatives of φ1 and get (21)∂ΠVC2∗∂φ1=δρ2-2a22φ1D2-2a2-3a22D2+a22φ12D2+2a22φ12D2+θ2-2a2+4a22D2+rρ-2a1+4a12D1-θ12-2a2+3a22D2+a22φ12D22a22φ1D2=0.According to formula (21), there is (22)φ1∗2=2δρ2a2-2rρa1+4rρa12/D12δρ2a22+θ12a22D2+-2δρ2a22+θ12a222δρ2a22+θ12a22.Guo and Zeng [24] thought that the income of VC from the initial contract should be no less than the external opportunity income; thus we assume that if the project is not liquidated, the income of VC should be no less than his external opportunity income. Risk-free income of venture capital I2 is k2I2, among which, k2 is risk-free rate of interest; thus we can determine the preferred stock coupon, VC gets D2=k2I2; thus optimal contract of the second stage is (23)D2=k2I2φ1∗=2δρ2a2-2rρa1+4rρa12/D12δρ2a22+θ12a22D2∗+-2δρ2a22+θ12a222δρ2a22+θ12a22.According to (21) and (22), there is Conclusion 1.Conclusion 1.
The second stage success probabilityp2 is inversely related to VC’s share φ1, is inversely related to the second stage convertible preferred stock coupon D2, and is inversely related to the first stage preferred stock coupon D1.Practical significance of Conclusion1 is that the success probability of the second stage is related to the contract designed by VC; lowering φ1,D2 can incentivize VC and EN to work harder, thus decreasing the agency cost and improving the success probability of the project. This is because modestly lowering φ1,D2 can increase the share of profit of AN and EN. Besides, the success probability p2 of the second stage is related to the first stage convertible point D1; lowering the convertible point can make EN and AN share the profit, thus giving both sides incentive to work hard.Conclusion 2.
The VC’s share is a linear function of his preferred stock coupon, and whenD1>2ra12/(ra1-δρa2), φ12 is an increasing function of D2; when D1<2ra12/(ra1-δρa2), φ12 is an decreasing function of D2.Conclusion2 shows that the linear relationship between φ12 and D2 is related to the preferred stock coupon in the first stage; according to Conclusion 1, a lower D1 can incentivize EN and AN to work hard in the second stage, although increasing D2 can adversely influence the hard work of EN and AN; a good outcome of the first stage can motivate EN and AN to go on working hard, and the success probability and the income of the project are guaranteed; thus VC can moderately decrease the share φ1 to motivate EN and AN to go on working hard. Similarly, when D1 is higher, EN and AN can slug in the second stage, while increasing D2 will adversely affect the hard work of EN and AN; these two factors affect the project’s success probability and income; thus VC can moderately increase share φ1 to guarantee his own income. In reality, AN’s preferred stock coupon should guarantee his investment capital; that is, D1>I1, and 2ra12/(ra1-δρa2)>2ra12/ra1=2a1; in other words, only if the preferred stock coupon of AN satisfies the condition D1>2ra12/(ra1-δρa2)>2ra12/ra1=2a1, AN will make sure he will not lose.Conclusion 3.
The share of VC is the decreasing function of the residual value with the projecta2; when a1>0.25D1, VC’s share is an increasing function of a1; when a1<0.25D1, VC’s share is a decreasing function of a1.Practical significance of Conclusion3 is that the greater the residual value of the project in the second stage, the greater the income of VC when the project fails; thus VC can decrease his share in a high income scenario and incentivize AN and EN to work hard. The share of VC is also related to the residual value in the first stage; when a1<0.25D1, if D1 is smaller, EN and AN will work hard, and the success probability of the project will rise.According to (15) and (16), we substitute the optimal efforts into the probability of success in the first period; there is (24)p1∗=r1e11+r2e21∗=δρrθ1e12∗+δρrθ1e22∗+r12a12D1-a12φ2D1+r22-2a1+3a12D1+a12φ2D1=δρrθ1e12∗+δρrθ1e22∗+r2-2a1+4a12D1=δρra22D2-a22φ12D2+r2-2a1+4a12D1so the optimal utility of AN in the first period is(25)ΠAN1∗=2a1-2a12D1+δρ2ra22D2-a22φ12D2+r2-2a1+4a12D1-2a1+3a12D1+a12φ2D1-12r1-2a1+3a12D1+a12φ2D1+δρθ1e22∗2-I1.The partial derivation of (25) with respect to φ is (26)∂ΠAN1∗∂φ=δρra22D2-a22φ12D2+r2-2a1+4a12D12a12φD1-r1r1-2a1+3a12D1+a12φ2D1+δρθ1e22∗2a12φD1=0.From (25) and (26), we can get(27)φ∗2=r1δρe12a12r12D1-1.On the basis of the assumptions of Guo and Zeng [24], if the project is not liquidated, the least return that the AN gains is equal to AN’s external opportunity. If k1I1 is AN’s return of external risk-free investments, the dividends that AN gains on prefer stocks must be D1=k1I1.In the first period, the contract designed by AN and EN should be(28)D1∗=k2I2φ1∗=r1δρe12a12r12D1-1,=r1δρ1-γθ1a22/D2-2a22φ1/D2+a22φ12/D2+1-φ1-φ2θ12a22φ1/D2a12r12D1∗-1.From (24) (27) we have following conclusions.Conclusion 4.
The success probabilityp1 in the first period is inversely proportional to the dividends D1, D2 converted on preferred stocks and inversely proportional to VC’s shares φ1.Conclusion 5.
The AN’ stock share strictly monotonically increases with the increment of the dividendsD1 that AN gains on preferred stocks.Practical significance of Conclusion5 is that when D1 goes up, EN’s return will goes down if the project has a poor quality. As a result, the smaller the success probability of the project, the lower the profits that will come up when EN becomes slake and indolent. In this case, the only way for AN is to increase dividend φ. This suggests that a higher D1 gives rise to a higher φ.The formulae22∗=γθ1(a22/D2-2a22φ1/D2+a22φ12/D2)+φ2θ1(2a22φ1/D2) indicates that AN’s efforts in the second period are a function of φ1,D2,γ.On the basis ofD1∗,φ∗,D2∗,φ1∗, the optimal total utility of AN is as follows: (29)ΠAN∗=ΠAN1∗+δΠAN2∗=2a1-2a12D1∗+δρ2ra22D2∗-a22φ12D2∗+r2-2a1+4a12D1∗-2a1+3a12D1∗+a12φ2D1∗-12r1-2a1+3a12D1∗+a12φ2D1∗+δρθ1e22∗2-I1+δγδρ2a22D2∗-a22φ1∗2D2∗+θ2-2a2+4a22D2∗+rρ-2a1+4a12D1∗×a22D2∗-a22φ1∗2D2∗+a22φ1∗2D2∗+φ2δρ2a22D2∗-a22φ1∗2D2∗+θ2-2a2+4a22D2∗+rρ-2a1+4a12D1∗2a22φ1∗2D2∗-12e22∗2.


## 4. The Game Model that AN Does Not Invest in the Second Period

We still use backward induction method to find the optimal contracts.


### 4.1. The Optimal Decision in the Second Period

If AN quits in the second stage and VC continues to invest in the project, now the success probability of project isp2=θ~1e12+θ~3e32+ρ(e11+e21). The profits of the project are random and the probability density is g(R)=2a~2/R3.In this case, VC and EN sign a convertible preferred stock contract and the mechanism of the contract; see Figure3.Figure 3
AN does not invest project in the second period.Explanation of Figure3: In the first stage, if the project is successful and AN executes the convertible preferred stock, then y is AN’s a share and 1-y is a share of EN; if AN does not execute the convertible preferred stock, then D~1 is bond yield, γ~ is a share of AN from the rest p2R-D1 of the income, and 1-γ~ is a share of EN from the rest p2R-D~1 of the income. In the second stage, if the project is successful, if VC can execute the convertible preferred stock, then ϕ~1 is a share of VC and the share of EN is 1-ϕ~1; if VC does not execute the convertible preferred stock, then D~2 is bond yield.The following is the optimal problem of VC:(30)maxD~2,φ1Π~VC2=maxD~2,φ~1⁡∫a~2D~21-p2R2a~22R3dR+∫D~2+∞φ~1p2R2a~22R3dR-12e322-I2.(31)S.T.maxe12⁡Π~EN2=maxe12⁡∫a~2D~21-p2R2a~12R3dR+∫D~2/φ~1+∞1-φ~2p2R2a~22R3dR-12e122.(32)maxe32Π~VC2=maxe32⁡∫a~2D~21-p2R2a~12R3dR+∫D~2D~2/φ~1p2D~22a~22R3dR+∫D~2/φ~1+∞φ~1p2R2a~22R3dR-12e322-I2.Formula (30) is the target function of VC. VC determines the D~2,φ~1 in the contracts to maximize his or her own returns.Formula (31) is the incentive compatibility constraint of EN. EN determines his or her efforts to maximize his or her own returns.Formula (32) is the incentive compatibility constraint of AN. AN determines his or her efforts to maximize his or her own returns.From formula (31) and (32), the optimal efforts of VC and EN are as follows:(33)e~12∗=θ~1a~22D~2-2a~22φ~12D~2+a~22φ~12D~2+1-φ~1θ~12a~22φ~12D~2.(34)e~32∗=θ~3-2a~2+3a~22D~2+a~22φ~12D~2.


### 4.2. The Optimal Decision in the First Period

In the first period, the success probability of project isp1=r1e11+r2e21. The profits of the project are still random and the probability density is g(R)=2a~1/R3.The following is the optimal problem of AN:(35)maxD~1,φ~Π~AN=maxD~1,φ~Πa1e11,e(21).(36)S.T.maxe21Π~a=maxe21ΠAN1e11,e21.(37)maxe11Π~EN=maxe11Π~EN1e11,e21+δΠ~EN2e11,e21,e~12∗,e~32∗.(38)Π~AN1=∫a~1D11-r1e11-r2e21R2a~12R3dR+∫D~1D~1/φ~r1e11+r2e21D~1R2a~12R3dR+φ~∫D~1/φ~+∞r1e11+r2e21R2a~12R3dR-12e212-K1.Π~EN1=∫D~1D~1/φ~r1e11+r2e21R-D~1R2a~12R3dR+1-φ~∫D~1/φ~+∞r1e11+r2e21R2a~12R3dR-12e112.Formula (35) is the target function of VC. VC determines the D~1,φ~ in the contracts to maximize his or her own returns.Formula (36) is the incentive compatibility constraint of EN. EN determines his or her efforts to maximize his or her own returns.Formula (37) is the incentive compatibility constraint of AN. AN determines his or her efforts to maximize his or her own returns.From formula (36) and (37), the optimal efforts are(39)e~11∗=r1a~12D~1-2a~12D~12+δρa~22D~22-a~22φ~12D~22.(40)e~21∗=r1-2a~1+3a~12D~1+2a~12φ~2D~1.From the optimal efforts (39) and (40) in the first period, we can get the optimal utility of AN if he or she participates in the project:(41)Π~AN∗∗=Π~AN1e~11∗,e~21∗+δΠ~AN2e~11∗,e~21∗,e~12∗,e~32∗.


### 4.3. Optimal Contract and Contract Characteristics

In Sections4.1 and 4.2, the optimal decision in the first and second stage is obtained. In this section, we need to find the optimal contract and further analyze the characteristics of the contract.Substituting (33), (34), (39), and (40) to successful probability of second stage, we get (42)p~1=θ~1e~12∗+θ~3e~32∗+ρe~11∗+e~21∗=θ~1e~12∗+δρ2θ~1+θ~3e~32∗+ρ-2a~1+4a~1D~1.To simplify model, we assumeθ~1=θ~2=θ~3=θ~, r~1=r~2=r~. Then, from (41), we get(43)p~1=δρ22a~2D~2-2a~2φ12D~2-+ρr-2a1+4a1D~1+θ~2-2a~2+4a~2D~2.Substituting (33), (34), (39), and (40) to the second stage utility function of VC, there is (44)Π~VC2∗=2a~2-2a~22D~2+δρ2a~22D~2-a~22φ~12D~2+θ2-2a~2+4a~22D~2+rρ-2a1+4a12D~1×-2a2+3a~22D~2+a~22φ~12D~2-12θ~12-2a~2+3a~22D~2+a~22φ~12D~2-I2.From (44), there is the first-order equation(45)∂Π~VC∗∂φ~1=δρ2-2a~22φ~1D~2-2a~2-3a~22D~2+a~22φ~12D~2+2a~22φ~12D~2δρ2a~22D~2-a~22φ~12D~2+θ~2-2a~2+4a~22D~2+rρ-2a~1+4a~12D~1-θ~12-2a~2+3a22D~2+a22φ12D~22a~22φ~1D~2=0.Then we get(46)φ~1∗2=2δρ2a~2-2rρa1+4rρa12/D~12δρ2a~22+θ~12a~22D~2∗+-2δρ2a~22+θ~12a~222δρ2a~22+θ~12a~22.Similarly, VC’s lowest profit is ask2 times of investment, so in the second stage, the contract is as follows: (47)D~2∗=k2I2.φ~1∗=2δρ2a~2-2rρa1+4rρa12/D~12δρ2a~22+θ~12a~22D~2∗+-2δρ2a~22+θ~12a~222δρ2a~22+θ~12a~22.Substituting (39) and (40) to successful probability of first stage, there is (48)p~1=r1e~11∗+r2e~21∗=δρr1θ~1e~12∗+ρr-2a~2+4a~22D~2=δρ2r12a~2D~2-2a~2φ12D~2-+ρr-2a1+4a1D~1.Substituting (39), (40), and (48) to first stage utility function of AN, there is (49)Π~AN1∗=2a1-2a12D1+δρra~22D~2-a~22φ~12D~2+r2-2a1+4a12D~1-2a1+3a12D~1+a12φ~2D~1-12r1-2a1+3a12D~1+a12φ2D~1+δρθ~1e22∗2-I1.Calculating derivative of (49), we have (50)φ~2=5+r1δρe12/θ~-4a1r12a12r12D~1.Similarly, assuming that AN’s minimum value is times of the investment amount, in the first stage, the contract is as follows:(51)D~1∗=k1I1,φ∗=r1δρe12/θ~a12r12D~1∗-1.According to the equations above, total utility of AN is as follows:(52)Π~AN1∗=2a1-2a12D1∗+δρra~22D~2∗-a~22φ~12∗D~2∗+r2-2a1+4a12D~1∗-2a1+3a12D~1∗+a12φ~2∗D~1∗-12r1-2a1+3a12D~1∗+a12φ2∗D~1∗+δρθ~1e22∗2-I1.


## 4.1. The Optimal Decision in the Second Period

If AN quits in the second stage and VC continues to invest in the project, now the success probability of project isp2=θ~1e12+θ~3e32+ρ(e11+e21). The profits of the project are random and the probability density is g(R)=2a~2/R3.In this case, VC and EN sign a convertible preferred stock contract and the mechanism of the contract; see Figure3.Figure 3
AN does not invest project in the second period.Explanation of Figure3: In the first stage, if the project is successful and AN executes the convertible preferred stock, then y is AN’s a share and 1-y is a share of EN; if AN does not execute the convertible preferred stock, then D~1 is bond yield, γ~ is a share of AN from the rest p2R-D1 of the income, and 1-γ~ is a share of EN from the rest p2R-D~1 of the income. In the second stage, if the project is successful, if VC can execute the convertible preferred stock, then ϕ~1 is a share of VC and the share of EN is 1-ϕ~1; if VC does not execute the convertible preferred stock, then D~2 is bond yield.The following is the optimal problem of VC:(30)maxD~2,φ1Π~VC2=maxD~2,φ~1⁡∫a~2D~21-p2R2a~22R3dR+∫D~2+∞φ~1p2R2a~22R3dR-12e322-I2.(31)S.T.maxe12⁡Π~EN2=maxe12⁡∫a~2D~21-p2R2a~12R3dR+∫D~2/φ~1+∞1-φ~2p2R2a~22R3dR-12e122.(32)maxe32Π~VC2=maxe32⁡∫a~2D~21-p2R2a~12R3dR+∫D~2D~2/φ~1p2D~22a~22R3dR+∫D~2/φ~1+∞φ~1p2R2a~22R3dR-12e322-I2.Formula (30) is the target function of VC. VC determines the D~2,φ~1 in the contracts to maximize his or her own returns.Formula (31) is the incentive compatibility constraint of EN. EN determines his or her efforts to maximize his or her own returns.Formula (32) is the incentive compatibility constraint of AN. AN determines his or her efforts to maximize his or her own returns.From formula (31) and (32), the optimal efforts of VC and EN are as follows:(33)e~12∗=θ~1a~22D~2-2a~22φ~12D~2+a~22φ~12D~2+1-φ~1θ~12a~22φ~12D~2.(34)e~32∗=θ~3-2a~2+3a~22D~2+a~22φ~12D~2.


## 4.2. The Optimal Decision in the First Period

In the first period, the success probability of project isp1=r1e11+r2e21. The profits of the project are still random and the probability density is g(R)=2a~1/R3.The following is the optimal problem of AN:(35)maxD~1,φ~Π~AN=maxD~1,φ~Πa1e11,e(21).(36)S.T.maxe21Π~a=maxe21ΠAN1e11,e21.(37)maxe11Π~EN=maxe11Π~EN1e11,e21+δΠ~EN2e11,e21,e~12∗,e~32∗.(38)Π~AN1=∫a~1D11-r1e11-r2e21R2a~12R3dR+∫D~1D~1/φ~r1e11+r2e21D~1R2a~12R3dR+φ~∫D~1/φ~+∞r1e11+r2e21R2a~12R3dR-12e212-K1.Π~EN1=∫D~1D~1/φ~r1e11+r2e21R-D~1R2a~12R3dR+1-φ~∫D~1/φ~+∞r1e11+r2e21R2a~12R3dR-12e112.Formula (35) is the target function of VC. VC determines the D~1,φ~ in the contracts to maximize his or her own returns.Formula (36) is the incentive compatibility constraint of EN. EN determines his or her efforts to maximize his or her own returns.Formula (37) is the incentive compatibility constraint of AN. AN determines his or her efforts to maximize his or her own returns.From formula (36) and (37), the optimal efforts are(39)e~11∗=r1a~12D~1-2a~12D~12+δρa~22D~22-a~22φ~12D~22.(40)e~21∗=r1-2a~1+3a~12D~1+2a~12φ~2D~1.From the optimal efforts (39) and (40) in the first period, we can get the optimal utility of AN if he or she participates in the project:(41)Π~AN∗∗=Π~AN1e~11∗,e~21∗+δΠ~AN2e~11∗,e~21∗,e~12∗,e~32∗.


## 4.3. Optimal Contract and Contract Characteristics

In Sections4.1 and 4.2, the optimal decision in the first and second stage is obtained. In this section, we need to find the optimal contract and further analyze the characteristics of the contract.Substituting (33), (34), (39), and (40) to successful probability of second stage, we get (42)p~1=θ~1e~12∗+θ~3e~32∗+ρe~11∗+e~21∗=θ~1e~12∗+δρ2θ~1+θ~3e~32∗+ρ-2a~1+4a~1D~1.To simplify model, we assumeθ~1=θ~2=θ~3=θ~, r~1=r~2=r~. Then, from (41), we get(43)p~1=δρ22a~2D~2-2a~2φ12D~2-+ρr-2a1+4a1D~1+θ~2-2a~2+4a~2D~2.Substituting (33), (34), (39), and (40) to the second stage utility function of VC, there is (44)Π~VC2∗=2a~2-2a~22D~2+δρ2a~22D~2-a~22φ~12D~2+θ2-2a~2+4a~22D~2+rρ-2a1+4a12D~1×-2a2+3a~22D~2+a~22φ~12D~2-12θ~12-2a~2+3a~22D~2+a~22φ~12D~2-I2.From (44), there is the first-order equation(45)∂Π~VC∗∂φ~1=δρ2-2a~22φ~1D~2-2a~2-3a~22D~2+a~22φ~12D~2+2a~22φ~12D~2δρ2a~22D~2-a~22φ~12D~2+θ~2-2a~2+4a~22D~2+rρ-2a~1+4a~12D~1-θ~12-2a~2+3a22D~2+a22φ12D~22a~22φ~1D~2=0.Then we get(46)φ~1∗2=2δρ2a~2-2rρa1+4rρa12/D~12δρ2a~22+θ~12a~22D~2∗+-2δρ2a~22+θ~12a~222δρ2a~22+θ~12a~22.Similarly, VC’s lowest profit is ask2 times of investment, so in the second stage, the contract is as follows: (47)D~2∗=k2I2.φ~1∗=2δρ2a~2-2rρa1+4rρa12/D~12δρ2a~22+θ~12a~22D~2∗+-2δρ2a~22+θ~12a~222δρ2a~22+θ~12a~22.Substituting (39) and (40) to successful probability of first stage, there is (48)p~1=r1e~11∗+r2e~21∗=δρr1θ~1e~12∗+ρr-2a~2+4a~22D~2=δρ2r12a~2D~2-2a~2φ12D~2-+ρr-2a1+4a1D~1.Substituting (39), (40), and (48) to first stage utility function of AN, there is (49)Π~AN1∗=2a1-2a12D1+δρra~22D~2-a~22φ~12D~2+r2-2a1+4a12D~1-2a1+3a12D~1+a12φ~2D~1-12r1-2a1+3a12D~1+a12φ2D~1+δρθ~1e22∗2-I1.Calculating derivative of (49), we have (50)φ~2=5+r1δρe12/θ~-4a1r12a12r12D~1.Similarly, assuming that AN’s minimum value is times of the investment amount, in the first stage, the contract is as follows:(51)D~1∗=k1I1,φ∗=r1δρe12/θ~a12r12D~1∗-1.According to the equations above, total utility of AN is as follows:(52)Π~AN1∗=2a1-2a12D1∗+δρra~22D~2∗-a~22φ~12∗D~2∗+r2-2a1+4a12D~1∗-2a1+3a12D~1∗+a12φ~2∗D~1∗-12r1-2a1+3a12D~1∗+a12φ2∗D~1∗+δρθ~1e22∗2-I1.


## 5. Numerical Simulation

To test and verify the conclusions which we get, this section uses simulation method to study relationships among them.


### 5.1. Simulation of the Second Stage Investment in Which AN Participates

Let values of parametersI,a1,a2,k1,k2,δ,ρ,θ,r be as in Table 2.Table 2
Parameter value ofI, a1, a2, k1, k2, δ, ρ, θ, r.
Parameter I a 1 a 2 k 1 k 2 δ ρ θ r  Value 10 1 4 1.2 1.5 0.6 0.1 0.267 0.5Substitute parameter values of Table2 to optimal solution D2∗,φ1∗,D1∗,φ∗ of Section 3, so (53)D2∗=15,φ1∗=0.55,D1∗=12,φ∗=0.39+0.40γ+0.45φ2.According to theD2∗,φ1∗,D1∗,φ∗ above, two-stage total utility of AN is as follows: (54)ΠAN∗=ΠAN1∗+δΠAN2∗=2.517+0.0128γ+0.125φ2-0.5-0.826+0.063γ+0.126φ222-0.50.058γ+0.313φ22-I1.


### 5.2. Simulation of Second Stage Investment That AN Does Not Participate

Let values of parametersI,a1,a2,k1,k2,δ,ρ,θ,r be as in Table 3.Table 3
Parameter value ofI, a1, a2, k1, k2, δ, ρ, θ, r.
Parameter I a 1 a 2 k 1 k 2 δ ρ θ r  Value 10 1 4 1.2 1.5 0.5 0.1 0.45 0.5Substituting parameter values of Table3 to optimal solution D~2∗,φ~1∗,D~1∗,φ~∗ of Section 4, we get(55)D~2∗=15,φ~1∗=0.78,D~1∗=12,φ~∗=0.57.In the same way, substituting parameter values of Table3 to the two-stage total utility of AN, there is (56)ΠAN∗=2.25-I1.To analyze the exit selection of AN, we compare the value (57), (58) of ΠAN∗ and Π~AN∗:(57)ΠAN∗==2.517+0.0128γ+0.125φ2-0.5-0.826+0.063γ+0.126φ222-0.50.058γ+0.313φ22-I1.(58)Π~AN∗=2.25-I1.Further, setφ2=0.2 and assume γ is uniformly distributed in [0,1]. See Table 4.Table 4
Value of parameterγ.
0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 0.1400 0.1600 0.1800 0.2000  0.2200 0.2400 0.2600 0.2800 0.3000 0.3200 0.3400 0.3600 0.3800 0.4000  0.4200 0.4400 0.4600 0.4800 0.5000 0.5200 0.5400 0.5600 0.5800 0.6000  0.6200 0.6400 0.6600 0.6800 0.7000 0.7200 0.7400 0.7600 0.7800 0.8000  0.8200 0.8400 0.8600 0.8800 0.9000 0.9200 0.9400 0.9600 0.9800 1.0000Substituting these 50 parameter’s values into (57), results are shown in Table 5.Table 5
Second stage utility of AN when AN participates.
γ 0.0200 0.0400 0.0600 0.0800 0.1000  AN’s profit 2.2154 2.2166 2.2179 2.2191 2.2204  γ 0.1200 0.1400 0.1600 0.1800 0.2000  AN’s profit 2.2216 2.2229 2.2242 2.2254 2.2267  γ 0.2200 0.2400 0.2600 0.2800 0.3000  AN’s profit 2.2279 2.2292 2.2304 2.2316 2.2329  γ 0.3200 0.3400 0.3600 0.3800 0.4000  AN’s profit 2.2341 2.2354 2.2366 2.2378 2.2391  γ 0.4200 0.4400 0.4600 0.4800 0.5000  AN’s profit 2.2403 2.2415 2.2428 2.2440 2.2452  γ 0.5200 0.5400 0.5600 0.5800 0.6000  AN’s profit 2.2464 2.2477 2.2489 2.2501 2.2513  γ 0.6200 0.6400 0.6600 0.6800 0.7000  AN’s profit 2.2525 2.2538 2.2550 2.2562 2.2574  γ 0.7200 0.7400 0.7600 0.7800 0.8000  AN’s profit 2.2586 2.2598 2.2610 2.2622 2.2634  γ 0.8200 0.8400 0.8600 0.8800 0.9000  AN’s profit 2.2646 2.2658 2.2670 2.2682 2.2694According to (58) and Table 5, we have Figure 4.Figure 4
Comparison of AN utility.According to Table4 and Figure 4, we get Conclusion 6.Conclusion 6 (numerical conclusion).
There is aγ∗; when γ>γ∗, AN chooses to participate in second stage, and when γ<γ∗, AN chooses to exit after the end of first stage.


## 5.1. Simulation of the Second Stage Investment in Which AN Participates

Let values of parametersI,a1,a2,k1,k2,δ,ρ,θ,r be as in Table 2.Table 2
Parameter value ofI, a1, a2, k1, k2, δ, ρ, θ, r.
Parameter I a 1 a 2 k 1 k 2 δ ρ θ r  Value 10 1 4 1.2 1.5 0.6 0.1 0.267 0.5Substitute parameter values of Table2 to optimal solution D2∗,φ1∗,D1∗,φ∗ of Section 3, so (53)D2∗=15,φ1∗=0.55,D1∗=12,φ∗=0.39+0.40γ+0.45φ2.According to theD2∗,φ1∗,D1∗,φ∗ above, two-stage total utility of AN is as follows: (54)ΠAN∗=ΠAN1∗+δΠAN2∗=2.517+0.0128γ+0.125φ2-0.5-0.826+0.063γ+0.126φ222-0.50.058γ+0.313φ22-I1.


## 5.2. Simulation of Second Stage Investment That AN Does Not Participate

Let values of parametersI,a1,a2,k1,k2,δ,ρ,θ,r be as in Table 3.Table 3
Parameter value ofI, a1, a2, k1, k2, δ, ρ, θ, r.
Parameter I a 1 a 2 k 1 k 2 δ ρ θ r  Value 10 1 4 1.2 1.5 0.5 0.1 0.45 0.5Substituting parameter values of Table3 to optimal solution D~2∗,φ~1∗,D~1∗,φ~∗ of Section 4, we get(55)D~2∗=15,φ~1∗=0.78,D~1∗=12,φ~∗=0.57.In the same way, substituting parameter values of Table3 to the two-stage total utility of AN, there is (56)ΠAN∗=2.25-I1.To analyze the exit selection of AN, we compare the value (57), (58) of ΠAN∗ and Π~AN∗:(57)ΠAN∗==2.517+0.0128γ+0.125φ2-0.5-0.826+0.063γ+0.126φ222-0.50.058γ+0.313φ22-I1.(58)Π~AN∗=2.25-I1.Further, setφ2=0.2 and assume γ is uniformly distributed in [0,1]. See Table 4.Table 4
Value of parameterγ.
0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 0.1400 0.1600 0.1800 0.2000  0.2200 0.2400 0.2600 0.2800 0.3000 0.3200 0.3400 0.3600 0.3800 0.4000  0.4200 0.4400 0.4600 0.4800 0.5000 0.5200 0.5400 0.5600 0.5800 0.6000  0.6200 0.6400 0.6600 0.6800 0.7000 0.7200 0.7400 0.7600 0.7800 0.8000  0.8200 0.8400 0.8600 0.8800 0.9000 0.9200 0.9400 0.9600 0.9800 1.0000Substituting these 50 parameter’s values into (57), results are shown in Table 5.Table 5
Second stage utility of AN when AN participates.
γ 0.0200 0.0400 0.0600 0.0800 0.1000  AN’s profit 2.2154 2.2166 2.2179 2.2191 2.2204  γ 0.1200 0.1400 0.1600 0.1800 0.2000  AN’s profit 2.2216 2.2229 2.2242 2.2254 2.2267  γ 0.2200 0.2400 0.2600 0.2800 0.3000  AN’s profit 2.2279 2.2292 2.2304 2.2316 2.2329  γ 0.3200 0.3400 0.3600 0.3800 0.4000  AN’s profit 2.2341 2.2354 2.2366 2.2378 2.2391  γ 0.4200 0.4400 0.4600 0.4800 0.5000  AN’s profit 2.2403 2.2415 2.2428 2.2440 2.2452  γ 0.5200 0.5400 0.5600 0.5800 0.6000  AN’s profit 2.2464 2.2477 2.2489 2.2501 2.2513  γ 0.6200 0.6400 0.6600 0.6800 0.7000  AN’s profit 2.2525 2.2538 2.2550 2.2562 2.2574  γ 0.7200 0.7400 0.7600 0.7800 0.8000  AN’s profit 2.2586 2.2598 2.2610 2.2622 2.2634  γ 0.8200 0.8400 0.8600 0.8800 0.9000  AN’s profit 2.2646 2.2658 2.2670 2.2682 2.2694According to (58) and Table 5, we have Figure 4.Figure 4
Comparison of AN utility.According to Table4 and Figure 4, we get Conclusion 6.Conclusion 6 (numerical conclusion).
There is aγ∗; when γ>γ∗, AN chooses to participate in second stage, and when γ<γ∗, AN chooses to exit after the end of first stage.


## 6. Conclusion

In venture investment projects, AN usually uses common stocks as financial instrument, while VC uses more sophisticated tools such as combination of common stocks and bonds, convertible preferred stocks, and participating convertible preferred stock. As the initial stage during the process of venture investment, it is necessary to improve the awareness of self-protection for AN. In current venture investment projects, AN is motivated to innovate financial instruments, among which convertible preferred stock is the most common and simple choice. This article regards convertible preferred stock as an incentive to analyze behaviors of AN.Capital exit is the main target of investors. Proper exit mechanism enables maximum profit not only for investors but also for ENs. Therefore, capital appreciation and investors’ exit are the most essential parts in venture capital operation. As the financier of venture investment, AN usually participates in the beginning stage of projects instead of continuing until the last stage.Based on previous line of reasoning, this article improves exit mechanism of venture capital and analyzes exit mechanism of AN. This article assumes that the level of effort from the former stage influences the success probability of latter stage project, so the design of two-stage contract will interact with and influence the success probability for two-stage project. This article assumes that the choice of AN to exit in first stage is related to their own profits. If AN decides to exit, the final profits that are the difference between the profits and the cost of effort and investment in first stage are negative. If AN decides to stay, then the final profits that are the difference between the profits and the cost of effort and investment in first stage are positive. In the second stage contract, the more the profits VC gives to AN, the smaller the probability of exiting in the first stage.



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