# Effects of Short-Term Uncertainties on the Revenue Estimation of PPP Sewage Treatment Projects

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Sewage Revenue in Practice

^{th}year of the operation period. r is the benchmark discount rate of industry.

^{th}year of the operation period.

^{th}year has a strong correlation with that of the t

^{th}year. This stochastic evolution process can be expressed in Equation (7):

#### 2.2. Short-Term Uncertainty and Ornstein-Uhlenbeck Process

## 3. Case Study

#### 3.1. Illustrative Case Example

#### 3.2. Results and Discussion

^{7}CNY without the short-term uncertainty. The NPV increased to 6.31 × 10

^{7}CNY when the mean value of the short-term uncertainty is positive ($\mathrm{u}=1$). Additionally, as the mean value of the short-term uncertainty changed to −1, the NPV decreased from 5.86 × 10

^{7}CNY to 4.78 × 10

^{7}CNY. From the above statistic, we can see that the revenue increases by about 7.6% when the mean value of the short-term uncertainty increased to 1, compared with the revenue without the short-term uncertainty. Furthermore, the revenue decreases by about 18.4% when the mean value of the short-term uncertainty decreased to −1, compared with the revenue without the short-term uncertainty. The results show that short-term uncertainty can have both positive and negative effects on the revenue. The historical project experiencing negative noise shows the reduction in the expected revenue of the new project, while the positive noise depicts an increment in the expected revenue in a new project. This is consistent with the study by Hong et al. [44], which states that the effects of short-term uncertainties on the real option value of the project could be positive or negative. Empirical evidence demonstrated that the demand deviation was about 20% in public works projects [45]. The deviations in revenue estimated in this paper are consistent with this empirical evidence.

^{7}CNY without short-term uncertainty. The subsidy increased to 7.51 × 10

^{7}CNY when the mean value of the short-term uncertainty is positive ($\mathrm{u}=1$). Additionally, as the mean value of the short-term uncertainty changed from 0 to −1, the subsidy decreased from 3.49 × 10

^{7}CNY to −2.99 × 10

^{7}CNY. An increase in subsidy could lead to high-performance costs for the government and prevents the PPP contract from being successfully completed [46]. Over-valuation caused by not taking into account short-term uncertainty may explain unprofitable projects [27]. From the above statistic, we can see that the government subsidy increased about 115% when the mean value of short-term uncertainty increased to 1, compared with the revenue without the short-term uncertainty. Additionally, the government subsidy decreased about 185% when the mean value of the short-term uncertainty decreased to −1, compared with the revenue without the short-term uncertainty.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The distribution of the private revenue without and with noise. (The x-axis shows the NPV of the private sector. The y-axis shows the frequency. Subplot

**a**: Revenue without short-term uncertainty. Subplot

**b**,

**c**,

**d**explicitly show the revenue with short-term uncertainty, and the mean reversion values are 1, 0, and −1 respectively).

**Figure 4.**The distribution of the government subsidy without and with noise. (The x-axis shows the government subsidy. The y-axis shows the frequency. Subplot

**a**: Subsidy without short-term uncertainty. Subplot

**b**,

**c**,

**d**explicitly show the subsidy with short-term uncertainty, and the mean reversion values are 1, 0, and −1 respectively).

**Figure 6.**Time series simulation of revenues (The x-axis shows the timeline of the operation period. The y-axis shows the revenue. Subplot

**a**: Base sample. Subplot

**b**: High rate of the mean reversion, $\mathrm{k}$. Subplot

**c**: High rate of the mean value $\mathrm{u}$. Subplot

**d**: High rate of the noise volatility term, ${\sigma}_{y}$. This figure indicates that the time series with an alternative parameter are different from each other. The blue lines X represent the estimated revenue, the red lines Z represent the actual revenue, while the yellow lines Y represent the basis revenue. Subplots

**a**,

**b**,

**c**and

**d**represent the time trends of three scenarios. Note the different scales in the y-axis).

Project Parameters | Symbol | Value |
---|---|---|

Construction cost | $\mathrm{I}$ | 101.18 million |

Operation and maintain costs | ${\mathrm{C}}_{\mathrm{t}}$ | 2 million |

Construction period | ${\mathrm{T}}_{\mathrm{c}}$ | 2 years |

Operation period | ${\mathrm{T}}_{\mathrm{o}}$ | 20 years |

Toll Rate | $\mathrm{P}$ | 0.8285 yuan |

Design capacity | $\mathrm{Q}$ | 80,000 T |

Discount rate | $\mathrm{r}$ | 6.90% |

Upper threshold | ${\mathsf{\theta}}_{\mathrm{max}}$ | 110% |

Lower threshold | ${\mathsf{\theta}}_{\mathrm{min}}$ | 70% |

Annual growth rate | ${\mathsf{\mu}}_{\mathrm{X}}$ | 8.40% |

Long-term volatility | ${\mathsf{\sigma}}_{\mathrm{X}}$ | 6.20% |

Mean reverting rate | $\mathrm{k}$ | 0.1 |

Short-term volatility | ${\mathsf{\sigma}}_{\mathrm{Y}}$ | 0.2 |

Mean reversion value | $\mathrm{u}$ | –1 |

Parameters | Subplot a | Subplot b | Subplot c | Subplot d |
---|---|---|---|---|

Reverting speed ($\mathrm{k}$) | 0.1 | 1 | 0.1 | 0.1 |

$\mathrm{Mean}\text{}\mathrm{value}\text{}\left(\mathrm{u}\right)$ | −1 | −1 | 1 | −1 |

Volatility (${\mathsf{\sigma}}_{\mathrm{y}}$) | 0.5 | 0.5 | 0.5 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Q.; Liao, Z.; Guo, Q.; Degefu, D.M.; Wang, S.; Jian, F. Effects of Short-Term Uncertainties on the Revenue Estimation of PPP Sewage Treatment Projects. *Water* **2019**, *11*, 1203.
https://doi.org/10.3390/w11061203

**AMA Style**

Liu Q, Liao Z, Guo Q, Degefu DM, Wang S, Jian F. Effects of Short-Term Uncertainties on the Revenue Estimation of PPP Sewage Treatment Projects. *Water*. 2019; 11(6):1203.
https://doi.org/10.3390/w11061203

**Chicago/Turabian Style**

Liu, Qian, Zaiyi Liao, Qi Guo, Dagmawi Mulugeta Degefu, Song Wang, and Feihong Jian. 2019. "Effects of Short-Term Uncertainties on the Revenue Estimation of PPP Sewage Treatment Projects" *Water* 11, no. 6: 1203.
https://doi.org/10.3390/w11061203