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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
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##### 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
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