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doi: 10.3934/jimo.2019043

## Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies

 1 Department of Mathematics, Tianjin University of Commerce, Tianjin 300134, China 2 Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China

* Corresponding author: Jinghuan Li

Received  September 2018 Revised  January 2019 Published  May 2019

Fund Project: This project was supported in part by the the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322, 71471132, 71573189), and Tianjin Education Commission Scientific Research Plan(2017SK076, 2017KJ236)

The topic of investment timing in multi-stage public-private partnership (PPP) projects has not been received much attention so far. This study investigates optimal expansion timing decisions in multi-stage PPP projects under an uncertain demand and where the first-stage greenfield project involving a dedicated asset is immediately implemented as the PPP contract is closed, whereas the timing of the later expansion is flexibly decided according to the demand. In this setting, the optimal multiple stopping timing theory is adopted to model the expansion framework. Furthermore, we integrate a government subsidy, including an investment subsidy and revenue subsidy, into the expansion timing decisions. Through a hypothetical three-stage investment plan for a sanitary sewerage project, the optimal expansion strategies and the value of the multi-stage project before and after the subsidy are provided using a least squares Monte Carlo simulation. Also, the influences of a dedicated asset on the expansion strategies and project value are illustrated. In addition, we compare the incremental value before and after the subsidy and earlier expansion derived from two types of subsidies. The comparisons show that there is more incremental value for the revenue subsidy, and that the investment subsidy brings an earlier expansion.

Citation: 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, doi: 10.3934/jimo.2019043
##### References:
 [1] M. J. R. Armada, P. J. Pereira and A. Rodrigues, Optimal subsidies and guarantees in public-private partnerships, Eur. J. Financ., 18 (2012), 469-495. Google Scholar [2] L. E. T. Brandao and E. Saraiva, The option value of government guarantees in infrastructure projects, Constr. Manag. Econ., 26 (2008), 1171-1180. Google Scholar [3] G. Cortazar, M. Gravet and J. Urzua, The valuation of multidimensional American real options using the LSM simulation method, Comput. Oper. Res., 35 (2008), 113-129. doi: 10.1016/j.cor.2006.02.016. Google Scholar [4] E. Dahlgren and T. Leung, An optimal multiple stopping approach to infrastructure investment decisions, J. Econ. Dynam. Contr., 53 (2015), 251-267. doi: 10.1016/j.jedc.2015.02.001. Google Scholar [5] T. Dangl, Investment and capacity choice under uncertain demand, Eur. J. Oper. Res., 117 (1999), 415-428. doi: 10.1016/S0377-2217(98)00274-4. Google Scholar [6] P. Doan and K. Menyah, Impact of irreversibility and uncertainty on the timing of infrastructure projects, J. Constr. Eng. Manage., 139 (2013), 331-338. doi: 10.1061/(ASCE)CO.1943-7862.0000615. Google Scholar [7] U. Dörr, Valuation of Swing Options and Examination of Exercise Strategies by Monte Carlo Techniques, Master thesis, University of Oxford, 2003.Google Scholar [8] E. Engel, R. Fischer and A. Galetovic, The basic public finance of public-private partnerships, J. Euro. Econ. Assoc., 11 (2013), 83-111. doi: 10.3386/w13284. Google Scholar [9] B. C. Esty, Modern Project Finance: A Casebook, Princeton, Wiley, 2003.Google Scholar [10] B. Flyvbjerg, M. Holm and S. Buhl, How common and how large are cost overruns in transport infrastructure projects?, Transp. Rev., 23 (2003), 71-88. doi: 10.1080/01441640309904. Google Scholar [11] C. C. Gkochari, Optimal investment timing in the dry bulk shipping sector, Transp. Res. Part E, 79 (2015), 102-109. doi: 10.1016/j.tre.2015.02.018. Google Scholar [12] H. B. Herath and C. Park, Mult-stage capital unvestment opportunities as compound real options, Eng. Econ., 47 (2002), 1-27. Google Scholar [13] Y. Huang, Project and Policy Analysis of Build-Operate-Transfer Infrastructure Developments, Ph.D thesis, University of California at Berkeley, 1995.Google Scholar [14] Y. L. Huang and S. P. Chou, Valuation of the minimum revenue guarantee and the option to abandon in BOT infrastructure projects, Constr. Manag. Econ., 24 (2006), 379-389. doi: 10.1080/01446190500434997. Google Scholar [15] Y. Huang and C. Pi, Valuation of multi-stage BOT projects involving dedicated asset investments: a sequential compound option approach, Constr. Manag. Econ., 27 (2009), 653-666. doi: 10.1080/01446190903002789. Google Scholar [16] B. Klein and K. B. Leffler, The role of market forces in assuring contractual performance, J. Polit. Econ., 89 (1981), 615-641. doi: 10.1086/260996. Google Scholar [17] Y. Kwak, Y. Chih and C. I. William, Towards a comprehensive understanding of public-private partnerships for infrastructure development, California Manag. Rev., 51 (2009), 51-78. doi: 10.2307/41166480. Google Scholar [18] S. Li and H. B. Cai, Government incentive impacts on private investment behaviors under demand uncertainty, Transp. Res. Part E, 101 (2017), 115-129. doi: 10.1016/j.tre.2017.03.007. Google Scholar [19] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, J. Ind. Manag. Optim., 9 (2013), 365-389. doi: 10.3934/jimo.2013.9.365. Google Scholar [20] F. A. Longstaff and E. S. Schwartz, Valuing American options by simulation: A simple least squares approach, Rev. Financ. Stud., 14 (2001), 113-147. doi: 10.1093/rfs/14.1.113. Google Scholar [21] L. J. Maseda, Real Options Analysis of Flexibility in a Hospital Emergency Department Expansion Project: A Systems Approach, Master thesis, MIT, 2008.Google Scholar [22] N. Meinshausen and B. M. Hambly, Monte Carlo methods for the valuation of multiple exercise options, Math. Financ., 14 (2004), 557-583. doi: 10.1111/j.0960-1627.2004.00205.x. Google Scholar [23] S. Nadarajah, F. Margot and N. Secomandi, Comparison of least squares Monte Carlo methods with applications to energy real options, Eur. J. Oper. Res., 256 (2017), 196-204. doi: 10.1016/j.ejor.2016.06.020. Google Scholar [24] R. Neufville, Y. S. Lee and S. Scholtes, Flexibility in hospital infrastructure design, Working Paper, MIT, 2008.Google Scholar [25] E. Pennings, Optimal pricing and quality choice when investment in quality is irreverible, J. Ind. Econ., 52 (2004), 569-589. Google Scholar [26] M. Skamris and B. Flyvbjerg, Inaccuracy of traffic forecasts and cost estimates on large transport projects, Transp. Policy, 4 (1997), 141-146. doi: 10.1016/S0967-070X(97)00007-3. Google Scholar [27] S. Szymanski, The optimal timing of infrastructure investment, J. Transp. Econ. Policy, 25 (1991), 247-258. Google Scholar [28] Z. Tan and H. Yang, Flexible build-operate-transfer contracts for road franchising under demand uncertainty, Transp. Res. Part B, 46 (2012), 1419-1439. doi: 10.1016/j.trb.2012.07.001. Google Scholar [29] O. E. Williamson, The Economic Institutions of Capitalism, The Free Press, New York, 1985. Google Scholar [30] Y. Xenidis and D. Angelides, The financial risks in build-operate-transfer projects, Constr. Manag. Econ., 23 (2005), 431-441. doi: 10.1080/01446190500041552. Google Scholar [31] X. Zhang, Financial viability analysis and capital structure optimization in privatised public infrastructure projects, J. Constr. Eng. Manag., 131 (2005), 656-668. Google Scholar

show all references

##### References:
 [1] M. J. R. Armada, P. J. Pereira and A. Rodrigues, Optimal subsidies and guarantees in public-private partnerships, Eur. J. Financ., 18 (2012), 469-495. Google Scholar [2] L. E. T. Brandao and E. Saraiva, The option value of government guarantees in infrastructure projects, Constr. Manag. Econ., 26 (2008), 1171-1180. Google Scholar [3] G. Cortazar, M. Gravet and J. Urzua, The valuation of multidimensional American real options using the LSM simulation method, Comput. Oper. Res., 35 (2008), 113-129. doi: 10.1016/j.cor.2006.02.016. Google Scholar [4] E. Dahlgren and T. Leung, An optimal multiple stopping approach to infrastructure investment decisions, J. Econ. Dynam. Contr., 53 (2015), 251-267. doi: 10.1016/j.jedc.2015.02.001. Google Scholar [5] T. Dangl, Investment and capacity choice under uncertain demand, Eur. J. Oper. Res., 117 (1999), 415-428. doi: 10.1016/S0377-2217(98)00274-4. Google Scholar [6] P. Doan and K. Menyah, Impact of irreversibility and uncertainty on the timing of infrastructure projects, J. Constr. Eng. Manage., 139 (2013), 331-338. doi: 10.1061/(ASCE)CO.1943-7862.0000615. Google Scholar [7] U. Dörr, Valuation of Swing Options and Examination of Exercise Strategies by Monte Carlo Techniques, Master thesis, University of Oxford, 2003.Google Scholar [8] E. Engel, R. Fischer and A. Galetovic, The basic public finance of public-private partnerships, J. Euro. Econ. Assoc., 11 (2013), 83-111. doi: 10.3386/w13284. Google Scholar [9] B. C. Esty, Modern Project Finance: A Casebook, Princeton, Wiley, 2003.Google Scholar [10] B. Flyvbjerg, M. Holm and S. Buhl, How common and how large are cost overruns in transport infrastructure projects?, Transp. Rev., 23 (2003), 71-88. doi: 10.1080/01441640309904. Google Scholar [11] C. C. Gkochari, Optimal investment timing in the dry bulk shipping sector, Transp. Res. Part E, 79 (2015), 102-109. doi: 10.1016/j.tre.2015.02.018. Google Scholar [12] H. B. Herath and C. Park, Mult-stage capital unvestment opportunities as compound real options, Eng. Econ., 47 (2002), 1-27. Google Scholar [13] Y. Huang, Project and Policy Analysis of Build-Operate-Transfer Infrastructure Developments, Ph.D thesis, University of California at Berkeley, 1995.Google Scholar [14] Y. L. Huang and S. P. Chou, Valuation of the minimum revenue guarantee and the option to abandon in BOT infrastructure projects, Constr. Manag. Econ., 24 (2006), 379-389. doi: 10.1080/01446190500434997. Google Scholar [15] Y. Huang and C. Pi, Valuation of multi-stage BOT projects involving dedicated asset investments: a sequential compound option approach, Constr. Manag. Econ., 27 (2009), 653-666. doi: 10.1080/01446190903002789. Google Scholar [16] B. Klein and K. B. Leffler, The role of market forces in assuring contractual performance, J. Polit. Econ., 89 (1981), 615-641. doi: 10.1086/260996. Google Scholar [17] Y. Kwak, Y. Chih and C. I. William, Towards a comprehensive understanding of public-private partnerships for infrastructure development, California Manag. Rev., 51 (2009), 51-78. doi: 10.2307/41166480. Google Scholar [18] S. Li and H. B. Cai, Government incentive impacts on private investment behaviors under demand uncertainty, Transp. Res. Part E, 101 (2017), 115-129. doi: 10.1016/j.tre.2017.03.007. Google Scholar [19] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, J. Ind. Manag. Optim., 9 (2013), 365-389. doi: 10.3934/jimo.2013.9.365. Google Scholar [20] F. A. Longstaff and E. S. Schwartz, Valuing American options by simulation: A simple least squares approach, Rev. Financ. Stud., 14 (2001), 113-147. doi: 10.1093/rfs/14.1.113. Google Scholar [21] L. J. Maseda, Real Options Analysis of Flexibility in a Hospital Emergency Department Expansion Project: A Systems Approach, Master thesis, MIT, 2008.Google Scholar [22] N. Meinshausen and B. M. Hambly, Monte Carlo methods for the valuation of multiple exercise options, Math. Financ., 14 (2004), 557-583. doi: 10.1111/j.0960-1627.2004.00205.x. Google Scholar [23] S. Nadarajah, F. Margot and N. Secomandi, Comparison of least squares Monte Carlo methods with applications to energy real options, Eur. J. Oper. Res., 256 (2017), 196-204. doi: 10.1016/j.ejor.2016.06.020. Google Scholar [24] R. Neufville, Y. S. Lee and S. Scholtes, Flexibility in hospital infrastructure design, Working Paper, MIT, 2008.Google Scholar [25] E. Pennings, Optimal pricing and quality choice when investment in quality is irreverible, J. Ind. Econ., 52 (2004), 569-589. Google Scholar [26] M. Skamris and B. Flyvbjerg, Inaccuracy of traffic forecasts and cost estimates on large transport projects, Transp. Policy, 4 (1997), 141-146. doi: 10.1016/S0967-070X(97)00007-3. Google Scholar [27] S. Szymanski, The optimal timing of infrastructure investment, J. Transp. Econ. Policy, 25 (1991), 247-258. Google Scholar [28] Z. Tan and H. Yang, Flexible build-operate-transfer contracts for road franchising under demand uncertainty, Transp. Res. Part B, 46 (2012), 1419-1439. doi: 10.1016/j.trb.2012.07.001. Google Scholar [29] O. E. Williamson, The Economic Institutions of Capitalism, The Free Press, New York, 1985. Google Scholar [30] Y. Xenidis and D. Angelides, The financial risks in build-operate-transfer projects, Constr. Manag. Econ., 23 (2005), 431-441. doi: 10.1080/01446190500041552. Google Scholar [31] X. Zhang, Financial viability analysis and capital structure optimization in privatised public infrastructure projects, J. Constr. Eng. Manag., 131 (2005), 656-668. Google Scholar
The schematic diagram of a three-stage PPP project
The project value for different demand levels
The optimal exercise boundaries for the i-th (i = 1, 2) expansions
The influences of the dedicated asset ratio
The project value and subsidy amount under different demands
The influences of the investment subsidy proportion
Revenue subsidy at different demand levels
The influences of the revenue subsidy price
The comparison of the subsidy amount
The comparison of the incremental value
The comparison of the exercise boundary under the same subsidy amount
Default parameters used in the calculations
 Constant Symbol Value Unit Concession Period $T_{c}$ 30 Year Investment period $T$ 10 Year Planned investment times $N$ 3 time Construction period $\nu$ 1 Year Refraction time $\delta$ 2 Year Capacity of i-th stage $m_{i}$ 40,000 $m^3$/day Unit price $p$ 1.8 CNY/$m^3$ Unit operational cost $c$ 0.8 CNY/$m^3$ Construction cost parameter $b$ 2917.8 Construction cost parameter $\gamma$ 0.9427 Drift $\alpha$ 6% Volatility rate $\sigma$ 15% Discount rate $\rho$ 8% Dedicated asset ratio $\eta$ 10%
 Constant Symbol Value Unit Concession Period $T_{c}$ 30 Year Investment period $T$ 10 Year Planned investment times $N$ 3 time Construction period $\nu$ 1 Year Refraction time $\delta$ 2 Year Capacity of i-th stage $m_{i}$ 40,000 $m^3$/day Unit price $p$ 1.8 CNY/$m^3$ Unit operational cost $c$ 0.8 CNY/$m^3$ Construction cost parameter $b$ 2917.8 Construction cost parameter $\gamma$ 0.9427 Drift $\alpha$ 6% Volatility rate $\sigma$ 15% Discount rate $\rho$ 8% Dedicated asset ratio $\eta$ 10%
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