# American Institute of Mathematical Sciences

• Previous Article
Pricing and remanufacturing decisions for two substitutable products with a common retailer
• JIMO Home
• This Issue
• Next Article
Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction
April  2017, 13(2): 1105-1123. doi: 10.3934/jimo.2016064

## The optimal exit of staged investment when consider the posterior probability

 School of Economic Mathematics, Southwestern University of Finance and Economics, Cheng Du 610074, China

Received  February 2015 Revised  August 2016 Published  October 2016

The current main method to analyze the staged venture investment is some game models, which finally get the optimal contract between the venture entrepreneurs and the venture capitalists by constructing the participation constraint and the incentive constraint. But this method only considers the probability of the success of the project, and ignores whether the project itself is enforceable or not. This paper introduces the concept of the posterior probability, extends the Bergemann and Hege model from the single period to the multi period. Then by using the posterior probability and the successful chance of the project, this paper analyzes the numerous factors which influence the optimal design of the contract under three conditions, such as the fixed and the floating investment in multi-stage and the time when the successful result is related to the current investment quota. What's more, it dose not only give the optimal stop point but compares it in case of the information symmetry and the contrary condition in the floating multi-stage investment. At the same time, it pays attention to the importance of the posterior probability in the present multi-stage venture investment researches. Last but not the least, it provides a reference for the related researches and makes great significance to the venture investment practice.

Citation: Meng Wu, Jiefeng Yang. The optimal exit of staged investment when consider the posterior probability. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1105-1123. doi: 10.3934/jimo.2016064
##### References:

show all references

##### References:
The research framework and time line of decision
Variable Definitions
 Notation Definition $p_t$ The probability of the project become successful when the project is "good" $\alpha_t$ The probability that VC concluded a project was "good" through its previous performance $V_t$ The expected return of VCs in the t period $R$ The Total return of the project $C$ The cost of VC $\theta$ The discount rate $I$ The fixed investment amount under fixed investment $S$ The return of EN when the project become successful $e_k$ Represents the effort of EN, $e_d$ indicates the minimum effort, $e_h$ indicates the maximum effort $h_k$ Represents the effort of VC, $h_d$ indicates the minimum effort, $h_h$ indicates the maximum effort $K, K_T$ Represents an investment stage $E$ The cost of supervision by VC $\beta$ The share ratio of EN $D$ The parameter which can represents the relationship between return and investment $M$ The parameter which can represents the relationship between the amount of investment and the posterior probability of the current period
 Notation Definition $p_t$ The probability of the project become successful when the project is "good" $\alpha_t$ The probability that VC concluded a project was "good" through its previous performance $V_t$ The expected return of VCs in the t period $R$ The Total return of the project $C$ The cost of VC $\theta$ The discount rate $I$ The fixed investment amount under fixed investment $S$ The return of EN when the project become successful $e_k$ Represents the effort of EN, $e_d$ indicates the minimum effort, $e_h$ indicates the maximum effort $h_k$ Represents the effort of VC, $h_d$ indicates the minimum effort, $h_h$ indicates the maximum effort $K, K_T$ Represents an investment stage $E$ The cost of supervision by VC $\beta$ The share ratio of EN $D$ The parameter which can represents the relationship between return and investment $M$ The parameter which can represents the relationship between the amount of investment and the posterior probability of the current period
Some Specific Examples
 p A D C The stop point 0.5 100000 3 40000 0.666671 0.5 90000 8 40000 0.400008 0.5 70000 7 40000 0.285722 0.6 100000 5 40000 0.333337 0.6 90000 6 40000 0.277783 0.7 100000 9 40000 0.158734 0.7 90000 9 40000 0.158735 0.7 100000 8 40000 0.178575 0.7 90000 8 40000 0.178576 0.7 100000 7 40000 0.204086 0.7 90000 7 40000 0.204087 0.7 100000 6 40000 0.238099 0.7 90000 6 40000 0.238101 0.7 100000 5 40000 0.285718 0.7 90000 5 40000 0.285719 0.7 90000 4 40000 0.357148 0.7 80000 4 40000 0.357149 0.7 70000 4 40000 0.357151 0.7 100000 3 40000 0.476194 0.7 90000 3 40000 0.476195 0.7 100000 2 40000 0.714289 0.7 90000 2 40000 0.714291 0.8 100000 8 40000 0.156254 0.8 70000 8 40000 0.156258 0.9 100000 7 40000 0.158734 0.9 90000 3 40000 0.370375 0.9 80000 4 40000 0.277783
 p A D C The stop point 0.5 100000 3 40000 0.666671 0.5 90000 8 40000 0.400008 0.5 70000 7 40000 0.285722 0.6 100000 5 40000 0.333337 0.6 90000 6 40000 0.277783 0.7 100000 9 40000 0.158734 0.7 90000 9 40000 0.158735 0.7 100000 8 40000 0.178575 0.7 90000 8 40000 0.178576 0.7 100000 7 40000 0.204086 0.7 90000 7 40000 0.204087 0.7 100000 6 40000 0.238099 0.7 90000 6 40000 0.238101 0.7 100000 5 40000 0.285718 0.7 90000 5 40000 0.285719 0.7 90000 4 40000 0.357148 0.7 80000 4 40000 0.357149 0.7 70000 4 40000 0.357151 0.7 100000 3 40000 0.476194 0.7 90000 3 40000 0.476195 0.7 100000 2 40000 0.714289 0.7 90000 2 40000 0.714291 0.8 100000 8 40000 0.156254 0.8 70000 8 40000 0.156258 0.9 100000 7 40000 0.158734 0.9 90000 3 40000 0.370375 0.9 80000 4 40000 0.277783
 [1] Jinghuan Li, Shuhua Zhang, Yu Li. Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020154 [2] Sun-Ho Choi, Hyowon Seo, Minha Yoo. A multi-stage SIR model for rumor spreading. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2351-2372. doi: 10.3934/dcdsb.2020124 [3] Li Deng, Wenjie Bi, Haiying Liu, Kok Lay Teo. A multi-stage method for joint pricing and inventory model with promotion constrains. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1653-1682. doi: 10.3934/dcdss.2020097 [4] Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109 [5] Ming-Jong Yao, Tien-Cheng Hsu. An efficient search algorithm for obtaining the optimal replenishment strategies in multi-stage just-in-time supply chain systems. Journal of Industrial & Management Optimization, 2009, 5 (1) : 11-32. doi: 10.3934/jimo.2009.5.11 [6] Pengyu Yan, Shi Qiang Liu, Cheng-Hu Yang, Mahmoud Masoud. A comparative study on three graph-based constructive algorithms for multi-stage scheduling with blocking. Journal of Industrial & Management Optimization, 2019, 15 (1) : 221-233. doi: 10.3934/jimo.2018040 [7] Jinghuan Li, Yu Li, Shuhua Zhang. Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2065-2086. doi: 10.3934/jimo.2019043 [8] William Chad Young, Adrian E. Raftery, Ka Yee Yeung. A posterior probability approach for gene regulatory network inference in genetic perturbation data. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1241-1251. doi: 10.3934/mbe.2016041 [9] Xue Dong He, Roy Kouwenberg, Xun Yu Zhou. Inverse S-shaped probability weighting and its impact on investment. Mathematical Control & Related Fields, 2018, 8 (3&4) : 679-706. doi: 10.3934/mcrf.2018029 [10] Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021003 [11] Xia Han, Zhibin Liang, Yu Yuan, Caibin Zhang. Optimal per-loss reinsurance and investment to minimize the probability of drawdown. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021145 [12] Tianyuan Xu, Shanming Ji, Chunhua Jin, Ming Mei, Jingxue Yin. Early and late stage profiles for a chemotaxis model with density-dependent jump probability. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1345-1385. doi: 10.3934/mbe.2018062 [13] Chuangqiang Hu, Shudi Yang. Multi-point codes from the GGS curves. Advances in Mathematics of Communications, 2020, 14 (2) : 279-299. doi: 10.3934/amc.2020020 [14] Rongfei Liu, Dingcheng Wang, Jiangyan Peng. Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Journal of Industrial & Management Optimization, 2017, 13 (2) : 995-1007. doi: 10.3934/jimo.2016058 [15] Michael Herty, Reinhard Illner. On Stop-and-Go waves in dense traffic. Kinetic & Related Models, 2008, 1 (3) : 437-452. doi: 10.3934/krm.2008.1.437 [16] Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118 [17] John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273 [18] Lingju Kong, Qingkai Kong. Existence of nodal solutions of multi-point boundary value problems. Conference Publications, 2009, 2009 (Special) : 457-465. doi: 10.3934/proc.2009.2009.457 [19] Lan Yi, Zhongfei Li, Duan Li. Multi-period portfolio selection for asset-liability management with uncertain investment horizon. Journal of Industrial & Management Optimization, 2008, 4 (3) : 535-552. doi: 10.3934/jimo.2008.4.535 [20] Xianping Wu, Xun Li, Zhongfei Li. A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints. Journal of Industrial & Management Optimization, 2018, 14 (1) : 249-265. doi: 10.3934/jimo.2017045

2020 Impact Factor: 1.801