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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 and Management Optimization, 2017, 13 (2) : 1105-1123. doi: 10.3934/jimo.2016064
References:
 [1] D. Bergemann and U. Hege, Venture capital financing, moral hazard, and learning, Journal of Banking and Finance, 22 (1998), 703-735.  doi: 10.1016/S0378-4266(98)00017-X. [2] D. A. Blum, Factors contributing to independent venture capital successful exits, Journal of Business & Economics Research, 13 (2015), 1-6.  doi: 10.19030/jber.v13i1.9074. [3] D. J. Cumming and S. A. Johan, Venture capital and private equity contracting (Second Edition), Elsevier Science Academic Press, 2014 (2014), 633-675. [4] S. Dahiya and K. Ray, Staged investments in entrepreneurial financing, Journal of Corporate Finance, 18 (2011), 1193-1216. [5] P. Dietmar and J. Leisen, Staged venture capital contracting with ratchets and liquidation rights, Review of Financial Economics, 21 (2012), 21-30. [6] R. Elitzur and A. Gavious, A multi-period game theoretic model of venture capitalists and entrepreneurs, European Journal of Operational Research, 144 (2002), 440-453.  doi: 10.1016/S0377-2217(02)00144-3. [7] S. Espenlaub, A. Khurshed and A. Mohamed, Venture capital exits in domestic and cross-border investments, Journal of Banking & Finance, 53 (2015), 215-232.  doi: 10.1016/j.jbankfin.2014.11.014. [8] E.G. S. Félix, C. P. Pires and M. A. Gulamhussen, The exit decision in the European venture capital market, Quantitative Finance, 14 (2014), 1115-1130.  doi: 10.1080/14697688.2012.714903. [9] P. Gompers and J. Lerner, The Venture Capital Cycle, MIT Press, Cambridge Mas, 1999. [10] V. Gerasymenko and J. D. Arthurs, New insights into venture capitalists' activity: IPO and time-to-exit forecast as antecedents of their post-investment involvement, Journal of Business Venturing, 29 (2014), 405-420.  doi: 10.1016/j.jbusvent.2013.06.003. [11] L. J. Hu and D. H. Pan, Decision-making on stage financing based on markov process, Systems Engineering, 2 (2003), 12-16. [12] H. Z. Huang and C. H. Xu, Financial syndication and R & D, Economics Letters, 80 (2003), 141-146.  doi: 10.1016/S0165-1765(03)00087-9. [13] Y. H. Jin, Y. Q. Xi and Z. X. Ye, Study of multi-period dynamic financial model of venture capital considering reputation, Systems Engineering Theory and Practice, 5 (2003), 76-80. [14] Z. L. Liu and W. X. Xu, Decision-making on Stage Financing, Chinese Journal of Management, 9 (2002), 106-111. [15] P. Maik and S. Bernhard, Exit control, capital structure and financial contracts in venture capital -a comment on BerglöF 1994, Social Science Research Network, (2006), 1-20.  doi: 10.2139/ssrn.889549. [16] D. L. Muriel and P. Sophie, Venture capital syndication and the financing of innovation: Financial versus expertise motives, Economics Letters, 106 (2010), 75-77.  doi: 10.1016/j.econlet.2009.10.004. [17] J. Medin, Post Exit Operating Performance of PE-backed Firms: Evidence from Sweden, http://hdl.handle.net/2077/36445, (2014). [18] D. Neher, Staged financing: An agency perspective, Review of Economic Studies, 66 (1999), 255-274.  doi: 10.1111/1467-937X.00087. [19] E. Ramy and G. Arieh, A multi-period game theoretic model of venture capitalists and entrepreneurs, European Journal of Operational Research, 144 (2003), 440-453.  doi: 10.1016/S0377-2217(02)00144-3. [20] R. Repullo and J. Suarez, Venture capital finance: A security design approach, Review of Finance, 8 (2004), 75-108. [21] Y. Suzuki, Commitment Problem, Optimal Incentive Schemes, and Relational Contracts in Agency with Bilateral Moral Hazard, Economics Society European Meeting, Stockholm, 2003. [22] A. Schwienbacher, Innovation and venture capital exits, The Economic Journal, 118 (2008), 1888-1916.  doi: 10.1111/j.1468-0297.2008.02195.x. [23] M. L. Sang and B. Lee, Entrepreneur characteristics and the success of venture exit: An analysis of single-founder start-ups in the U.S, International Entrepreneurship and Management Journal, 11 (2015), 891-905. [24] Z. M. Wang and X. J. Chen, The Contextual Factors and Multi-Stage Risk Assessment in Simulated Investment Decisions, Psychological Science, 25 (2002), 7-9. [25] S. S. Wang and H. L. Zhou, Staged financing in venture capital: Moral hazard and risks, Journal of Corporate Finance, 10 (2004), 131-155. [26] Q. Yang and L. S. Yin, The decision model of multi-stage venture capital based on theory of option pricing, The theory and the advancement of science in management, 5 (2004), 95-97. [27] J. J. Zheng, X. Tan and W. T. Fan, An incentive model with stocks based on principal-agent theory, Journal of management sciences in china, 8 (2005), 24-29. [28] S. D. Zhang and D. X. Wei, Analysis on multi-stage investment game of double moral hazard in venture capital, Nankai Economics Studies, 54 (2008), 142-150. [29] X. L. Zhang and B. Huo, A single contract of optimization of multi-period venture capital and its optimal exit, Mathematics in economics, 252 (2008), 148-155. [30] J. J. Zheng, P. Zhang, W. L. Jiang and C. J. Rao, Venture capital exit dilemma and quantum equilibrium in presence of heterogeneous bidders, Journal of Management Sciences in China, 18 (2015), 62-71.

show all references

References:
 [1] D. Bergemann and U. Hege, Venture capital financing, moral hazard, and learning, Journal of Banking and Finance, 22 (1998), 703-735.  doi: 10.1016/S0378-4266(98)00017-X. [2] D. A. Blum, Factors contributing to independent venture capital successful exits, Journal of Business & Economics Research, 13 (2015), 1-6.  doi: 10.19030/jber.v13i1.9074. [3] D. J. Cumming and S. A. Johan, Venture capital and private equity contracting (Second Edition), Elsevier Science Academic Press, 2014 (2014), 633-675. [4] S. Dahiya and K. Ray, Staged investments in entrepreneurial financing, Journal of Corporate Finance, 18 (2011), 1193-1216. [5] P. Dietmar and J. Leisen, Staged venture capital contracting with ratchets and liquidation rights, Review of Financial Economics, 21 (2012), 21-30. [6] R. Elitzur and A. Gavious, A multi-period game theoretic model of venture capitalists and entrepreneurs, European Journal of Operational Research, 144 (2002), 440-453.  doi: 10.1016/S0377-2217(02)00144-3. [7] S. Espenlaub, A. Khurshed and A. Mohamed, Venture capital exits in domestic and cross-border investments, Journal of Banking & Finance, 53 (2015), 215-232.  doi: 10.1016/j.jbankfin.2014.11.014. [8] E.G. S. Félix, C. P. Pires and M. A. Gulamhussen, The exit decision in the European venture capital market, Quantitative Finance, 14 (2014), 1115-1130.  doi: 10.1080/14697688.2012.714903. [9] P. Gompers and J. Lerner, The Venture Capital Cycle, MIT Press, Cambridge Mas, 1999. [10] V. Gerasymenko and J. D. Arthurs, New insights into venture capitalists' activity: IPO and time-to-exit forecast as antecedents of their post-investment involvement, Journal of Business Venturing, 29 (2014), 405-420.  doi: 10.1016/j.jbusvent.2013.06.003. [11] L. J. Hu and D. H. Pan, Decision-making on stage financing based on markov process, Systems Engineering, 2 (2003), 12-16. [12] H. Z. Huang and C. H. Xu, Financial syndication and R & D, Economics Letters, 80 (2003), 141-146.  doi: 10.1016/S0165-1765(03)00087-9. [13] Y. H. Jin, Y. Q. Xi and Z. X. Ye, Study of multi-period dynamic financial model of venture capital considering reputation, Systems Engineering Theory and Practice, 5 (2003), 76-80. [14] Z. L. Liu and W. X. Xu, Decision-making on Stage Financing, Chinese Journal of Management, 9 (2002), 106-111. [15] P. Maik and S. Bernhard, Exit control, capital structure and financial contracts in venture capital -a comment on BerglöF 1994, Social Science Research Network, (2006), 1-20.  doi: 10.2139/ssrn.889549. [16] D. L. Muriel and P. Sophie, Venture capital syndication and the financing of innovation: Financial versus expertise motives, Economics Letters, 106 (2010), 75-77.  doi: 10.1016/j.econlet.2009.10.004. [17] J. Medin, Post Exit Operating Performance of PE-backed Firms: Evidence from Sweden, http://hdl.handle.net/2077/36445, (2014). [18] D. Neher, Staged financing: An agency perspective, Review of Economic Studies, 66 (1999), 255-274.  doi: 10.1111/1467-937X.00087. [19] E. Ramy and G. Arieh, A multi-period game theoretic model of venture capitalists and entrepreneurs, European Journal of Operational Research, 144 (2003), 440-453.  doi: 10.1016/S0377-2217(02)00144-3. [20] R. Repullo and J. Suarez, Venture capital finance: A security design approach, Review of Finance, 8 (2004), 75-108. [21] Y. Suzuki, Commitment Problem, Optimal Incentive Schemes, and Relational Contracts in Agency with Bilateral Moral Hazard, Economics Society European Meeting, Stockholm, 2003. [22] A. Schwienbacher, Innovation and venture capital exits, The Economic Journal, 118 (2008), 1888-1916.  doi: 10.1111/j.1468-0297.2008.02195.x. [23] M. L. Sang and B. Lee, Entrepreneur characteristics and the success of venture exit: An analysis of single-founder start-ups in the U.S, International Entrepreneurship and Management Journal, 11 (2015), 891-905. [24] Z. M. Wang and X. J. Chen, The Contextual Factors and Multi-Stage Risk Assessment in Simulated Investment Decisions, Psychological Science, 25 (2002), 7-9. [25] S. S. Wang and H. L. Zhou, Staged financing in venture capital: Moral hazard and risks, Journal of Corporate Finance, 10 (2004), 131-155. [26] Q. Yang and L. S. Yin, The decision model of multi-stage venture capital based on theory of option pricing, The theory and the advancement of science in management, 5 (2004), 95-97. [27] J. J. Zheng, X. Tan and W. T. Fan, An incentive model with stocks based on principal-agent theory, Journal of management sciences in china, 8 (2005), 24-29. [28] S. D. Zhang and D. X. Wei, Analysis on multi-stage investment game of double moral hazard in venture capital, Nankai Economics Studies, 54 (2008), 142-150. [29] X. L. Zhang and B. Huo, A single contract of optimization of multi-period venture capital and its optimal exit, Mathematics in economics, 252 (2008), 148-155. [30] J. J. Zheng, P. Zhang, W. L. Jiang and C. J. Rao, Venture capital exit dilemma and quantum equilibrium in presence of heterogeneous bidders, Journal of Management Sciences in China, 18 (2015), 62-71.
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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