Determining concession periods and minimum revenue guarantees in public-private-partnership agreements

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Determining concession periods and minimum revenue guarantees in public-private-partnership agreements Mr Hongyu Jin , Miss Shijing Liu , Jide Sun , Chunlu Liu PII: DOI: Reference:

S0377-2217(19)31014-8 https://doi.org/10.1016/j.ejor.2019.12.013 EOR 16221

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European Journal of Operational Research

Received date: Accepted date:

30 January 2019 2 December 2019

Please cite this article as: Mr Hongyu Jin , Miss Shijing Liu , Jide Sun , Chunlu Liu , Determining concession periods and minimum revenue guarantees in public-private-partnership agreements, European Journal of Operational Research (2019), doi: https://doi.org/10.1016/j.ejor.2019.12.013

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Highlights 

A synthetic measure to determine concession period and minimum revenue guarantee



Imperfect information bargaining for the equilibrium return rate on investment



Effects of the probability of achieving the equilibrium return rate on investment



The concession period decision range is sensitive to change in the concession price

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Determining concession periods and minimum revenue guarantees in public-private-partnership agreements

All authors: Mr Hongyu Jin School of Architecture and Built Environment, Deakin University, 1 Gheringhap Street, Geelong, VIC 3220, Australia Miss Shijing Liu School of Architecture and Built Environment, Deakin University, 1 Gheringhap Street, Geelong, VIC 3220, Australia Jide Sun, Corresponding Author School of Economics and Management, Tongji University, Shanghai, China Chunlu Liu, Corresponding Author School of Architecture and Built Environment, Deakin University, Geelong, Australia Corresponding authors: Associate Professor Jide Sun School of Economics and Management, Tongji University, Shanghai 200092, China Tel: +86-137-61008158 Email: [email protected] Associate Professor Chunlu Liu School of Architecture and Built Environment, Deakin University, Geelong, VIC 3220, Australia Tel: +61-3-52278306 Email: [email protected]

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Abstract Public–private partnership (PPP) schemes show strong capability in delivering infrastructure projects. One challenge in designing PPP contracts is optimising the length of the concession period and level of the minimum revenue guarantee (MRG) to satisfy both public and private parties’ interests. Existing research excludes interaction between the concession period and MRG, but a method that can determine their values simultaneously is needed. This study fills the research gap by proposing a synthetic measure to determine the values of the concession period and MRG. An imperfect information bargaining model is created to find the equilibrium return rate on investment. To achieve the equilibrium of the bargaining game, the required length of the concession period and level of the MRG are calculated based on Monte Carlo simulation and real option analysis. Project QJ is created as a numerical example to verify the applicability of the proposed method. The outcome shows the proposed determination process identifies the optimal length of the concession period and level of the MRG. The length of the concession period is inversely proportional to the level of the MRG and this correlation is influenced by the probability of achieving the equilibrium return rate on investment. When this probability equals 70%, an MRG is not required once the concession period exceeds 24 years. The results also show the concession period decision range is sensitive to change in the concession price. Keywords: game theory; concession period; imperfect information bargaining; minimum revenue guarantee; public–private partnership

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1. Introduction Over the past decade, public–private partnerships (PPPs) have gained popularity in delivering public infrastructure. Daube et al. (2008) defined a PPP project as “a long-term contractual arrangement between the public and private sector to realise public infrastructure and services more cost-effectively and efficiently than under conventional procurement”. However, the implementation of PPPs is not without its challenges over a long operational period. The uncertainty of the annual revenue is one of the critical risk factors that may lead to financial failure for governments or private investors (Demirel et al., 2017). To share this revenue risk and deliver value for money in PPP projects, governments and private investors have attempted to determine the optimal values of the contract parameters, which can greatly influence the revenues for both parties. Concession periods and government guarantees have been recognised as two of the critical parameters that are capable of balancing the financial interests for project parties (Song et al., 2015). Xu et al. (2012) claimed that a reasonable concession price is also beneficial in creating a win–win situation for governments and private investors. Nevertheless, for the type of user-pays PPPs, the decision on the concession price is usually a commercial one. Since with an increase in the price the number of users may decrease, a stochastic price will make the prediction of the project profits even more difficult. In practice, the concession price for a user-pays project can be predetermined by governments on the basis of historical data of similar projects or the outcome of public hearings that provide information about the affordable price level of public users (Ng et al., 2007).

Concession period design is considered an effective way of balancing the financial interests between project parties (Hanaoka & Palapus, 2012; Medda, 2007; Shen et al., 2007). Net present value (NPV) analysis, Monte Carlo simulation and barging game theory are often adopted as methods to find the optimal length of the concession period. In addition to a reasonable length for the concession period, there is a variety of government financial supports that can be provided to private investors to ensure the profitability of the project for them (Chen et al., 2012). The minimum revenue guarantee (MRG) is one of the most common government supports in real-life PPP contracting, aiming to alleviate financial risks for private investors. If the annual net income is lower than the minimum threshold regulated by the MRG scheme, governments will compensate private investors for the deficit. The MRG is usually recognised as a European put option whose value has been evaluated in many 4

research studies (Ashuri et al., 2011; Huang & Chou, 2006; Wibowo, 2004). However, these studies determined the level of the MRG based on the assumption that the length of the concession period has been predetermined. In practice, as mentioned above, the concession period is one of the most important contract parameters in PPP agreements and its value also needs to be carefully examined. The practice of predetermining the concession period without plausible reasoning can lead to an inefficient selection of concessionaires (Zhang, 2009) and encourage renegotiation of concession contracts (Nombela & de Rus, 2004). Particularly in some countries like Australia, it is regulated that the concession period has to be decided by governments before issuing requests for proposals (Department of Infrastructure, Transport and Regional Economics, 2015). Thus, developing a methodology that can find the optimal combined values for the concession period and MRG in the absence of information from private investors contributes to overcoming these issues.

In recognising the importance of finding a synthetic measure to determine the values of the concession period and MRG in PPP contracts, this paper contributes to the body of knowledge by proposing such a determination process, aiming to find the optimal combined values for the concession period and MRG to protect financial interests for both private investors and governments. The remainder of the paper is structured as follows: a detailed literature review is given in Section 2, which helps to identify the knowledge gap in concession period and MRG calculations, followed by a summary of the determination process. Sections 3 to 5 elaborate the steps of the determination process and Section 6 is a numerical study to verify the applicability of the proposed method. Section 7 discusses the research findings and the conclusion in Section 8 ends the paper.

2. Existing Models for Concession Period and MRG Calculations The length of the concession period has been recognised to be influenced by many uncertain project variables, which has driven both researchers and practitioners to attempt to identify the critical factors that influence the length of the concession period (Ullah et al., 2016). Different models have been proposed for identifying a reasonable length for the concession period. NPV analysis is used to construct the core of some of these models. For example, Wu et al. (2011) calculated the length of the concession period to ensure positive NPVs for both governments and private investors. PPP studies have also highlighted that the long-term uncertainties in PPPs should not be neglected, since they can 5

significantly influence the payoffs for project parties (Cruz & Marques, 2013; Wang et al., 2012) and Monte Carlo simulation is considered the appropriate method to evaluate the financial risk profile for PPPs (Arnold & Yildiz, 2015; Ng et al., 2007). Hanaoka and Palapus (2012) proposed a simulation-based method for calculating concession period intervals for build–operate–transfer projects. Yu and Lam (2013) investigated the influencing factors via Monte Carlo simulation, based on which the length of the concession period was proposed. Carbonara et al. (2014a) constructed a win–win model based on NPV and Monte Carlo analyses for determining the length of the concession period. Game-theoretical models are also used to explore the bargaining behaviour of governments and private investors by considering PPP a social game (Feng et al., 2019; Scharle, 2002). In addition, some studies have evaluated the option value of a flexible concession period in PPP contracts (Qiu & Wang, 2011; Xiong & Zhang, 2016).

The majority of studies on MRG calculation have adopted real option theory to assess the option value of the MRG (e.g. Blank et al., 2016; Jun, 2010). Only the model proposed by Carbonara et al. (2014b) is capable of determining the specific level of the MRG. Table 1 classifies the existing models used for concession period determination concerning two aspects: whether the model evaluates the concession period as an instant of time, a time interval or a flexible value; and whether the calculated MRG is a specific level or option value. It can be seen from Table 1 that there is a lack of research exploring the interaction between the concession period and MRG, and no single research work optimises the length of the concession period and the level of the MRG in combination. This study fills the knowledge gap by proposing a synthetic measure to determine the optimal combined values of the concession period and MRG which follows a step-by-step process. As shown in Figure 1, the uncertain and decisive variables for PPP projects are first recognised. Second, an imperfect information bargaining model is developed to calculate the equilibrium return rate on investment. A simulation analysis is conducted in the third step to evaluate the NPVs of the operating incomes for private investors and governments respectively. The ranges of the concession period and MRG are then determined according to the simulation outcomes. Based on the equilibrium return rate and the decision ranges of the concession period and MRG, the optimal combinations of the concession period and MRG will be finally determined. Each of these steps will be elaborated in the following.

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Model

BOT concession model Simulation model

Concession period calculation Instant Time Flexible of time interval concession ✓ ✓

MRG calculation Specific Option level value

✓ Fuzzy multi-objective model Simulation–critical path method Fair risk allocation model

✓ ✓ ✓ ✓

Option game model Pareto-optimal model Bargaining game model Concession period and price model Discrete stochastic model Concession extension model Concession renegotiation model Real option model

✓ ✓ ✓ ✓ ✓

Least-present-value of revenue PPP franchising mechanism Flexible contract model Risk-neutral pricing model

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Table 1. Models for concession period and MRG calculations 7

Source

Wu et al., 2011; Zhang et al., 2016 Hanaoka & Palapus, 2012; Wang et al., 2009; Yu & Lam, 2013 Wibowo, 2004 Ng et al., 2007 Zhang, 2009 Carbonara et al., 2014a; Jin et al., 2019 Carbonara et al., 2014b Lv et al., 2014; Wang et al., 2015 Niu & Zhang, 2013 Feng et al., 2019; Shen et al., 2007 Ma et al., 2018 Xu et al., 2016 Qiu & Wang, 2011; Xiong & Zhang, 2014 Ho, 2006; Xiong & Zhang, 2014; Xiong & Zhang, 2016 Blank et al., 2016; Brandao & Saraiva, 2008; Huang & Chou, 2006; Jun, 2010 Liu & Cheah, 2017 Engel et al., 2001; Gómez-Lobo & Hinojosa, 2000; Vassallo, 2006 Nombela & de Rus, 2004 Tan & Yang, 2012 Ashuri et al., 2011

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Fig. 1. Concession period and MRG determination process

3. Uncertain and Decisive Variables for PPP Projects 3.1 Uncertain variables and their distributions The uncertain variables recognised in a PPP project involve the total amount of capital investment, the percentage of the capital investment shared by private investors, the operation and maintenance costs, and the user volume. The amount of capital investment mainly depends on the forecasted value of the construction cost. Construction cost includes direct or indirect cost spent on construction works. The study of Wing Chau (1995) stated that the probability distribution of the construction cost shows positive skewness, so the symmetric distributions are not an appropriate choice. In this study, considering the capability of evaluating positive skewness, beta distribution is chosen to describe the distribution of construction cost. The percentage of capital investment shared by private investors depends on the local policy. For example, in Australia, transportation PPPs are normally privately funded (English, 2006), while in China private investors have to contribute at least 50% of the total capital investment (Ministry of Finance, 2014).

The operation and maintenance costs are paid to ensure a smooth operation of the project. Due to the long lifespans of PPP projects, the operation and maintenance costs are reckoned as long-term costs whose values are mainly influenced by the inflation rate (Jin et al., 2019). It is feasible to accurately

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forecast the inflation rate in a short period (Fisher et al., 2002) but no simulation can generate inflation with high accuracy over a long time (Yu & Lam, 2013). Hence, the inflation rate is not considered as the appropriate indicator of evaluating the operation and maintenance costs. In this research, the annual operation and maintenance costs are forecasted as a proportion of the annual gross income.

The annual user volume often follows a geometric Brownian motion, which has been justified in many studies (e.g. Soumaré, 2016; Wang et al., 2015). The user volume in one year can be expressed as a function of the previous year (Dixit et al., 1994): ⁄

where rate and

is the user volume in the following year.

(1)

is the expected value of the annual growth

gives the annual volatility of the user volume.

is a random variable that follows a

standardised normal distribution.

3.2 Decisive variables Decisive variables recognised in this research include the concession price, the minimum return rate on investment required by private investors, the maximum return rate on investment regulated by the government, and the discount rate. As mentioned in the Introduction, the concession price for a user-pays PPP project can be predetermined by governments based on historical data gathered from similar projects or the outcomes of public hearings which provide information about the tariff preferences of public users.

The return rate on investment gained at the end of the concession equals the NPV of the total operating income divided by the initial capital investment. The minimum value of the return rate is often proposed by the private investors, who can estimate this value on the basis of the return rate of a similar project. The maximum value is often regulated by the government to avoid an overly lucrative condition for private investors.

The modern methodology on economic capital identifies the discount rate as a sum of the risk-free rate and the risk premium required by private investors. However, in this research, the risk premium is not considered for the following reasons. First, the imperfect information bargaining model and the real 10

option method that are used here for pricing the value of the MRG are based on the assumption that the private investors are risk-neutral. This assumption has been used in many studies when constructing the bargaining process and pricing the real options (Ashuri et al., 2011; Bao et al., 2015; Buyukyoran & Gundes, 2018). In a risk-neutral world, the risk-free rate should be used as the discount rate. Second, the risk premium for an infrastructure project is difficult to find since the market portfolio could be hard to define (Eriksen & Jensen, 2010). For instance, to find a portfolio for transport infrastructure projects, one possible solution is to find a portfolio of assets that has the same systematic risk as the transport infrastructure, such as stocks of transport companies. Nevertheless, this value cannot be used directly when operating in a regulated market (Eriksen & Jensen, 2010). Third, Monte Carlo simulation is the method adopted in this research to evaluate the influence of the uncertain variables. Brealey et al. (2012) stated that the risk-free discount rate should be used in a Monte Carlo simulation because market risks have already been taken into consideration through assigning the probability distributions of the uncertain variables.

4. Imperfect Information Bargaining Model 4.1 Model assumptions Many studies adopted game theory to assess the bargaining behaviours between project parties (e.g. Lukas & Thiergart, 2019; Scandizzo & Ventura, 2010). An imperfect information bargaining model is proposed here to find the equilibrium return rate on investment so that a successful partnership can be established without a sacrifice of either private investors’ or governments’ interest. The bargaining behaviour of governments and private investors in PPP contracting will be fully evaluated when the information in relation to private investors is limited. Several assumptions need to be clarified in constructing the model. First, as rational behaviours, the involved parties endeavour to achieve the maximisation of their payoffs in a bargaining process. Second, the government and private investor are risk-neutral. They may hold other types of risk preferences, i.e. risk-averse and risk-loving, in real-life PPP projects and it is difficult to find a universal risk preference in PPP contracting (Bao et al., 2015). The utilities for project parties can be adjusted according to their risk preference by defining their payoff utility functions. Project parties that are risk-averse are represented by a concave utility function to represent the risk premium, while for those that are risk-loving, a convex utility function can reveal their utilities. If the project parties are risk-neutral, their payoff utility function should be 11

linear (Watson, 2002). Third, the private investor has private information about its opportunity cost. Opportunity cost describes the returns that could have been earned if the money had been invested in another instrument (Arrow & Kruz, 2013). Therefore, it can be expected that a high opportunity cost investor will require a higher return rate on investment in a bargaining process compared with an investor with low opportunity cost. The government can judge the type of private investor based on its prior beliefs, which will be updated according to the newly acquired information.

4.2 Bargaining process The modelled bargaining game consists of two bargaining stages and ends when the offer proposed by one of the bargainers is accepted. As can be seen in Figure 2, at the beginning of the bargaining game, the nature (N) classifies the type of the private investor into low opportunity cost investor (L) and high opportunity cost investor (H). It is expected that the first-round offer proposed by an L investor is and the offer proposed by an H investor in the first round equals

(

.

The L investor may behave speculatively which is expressed as a dotted line from L to G in Figure 2. If an L investor is a speculator, it will offer

instead of

for a higher payoff. The

bargaining game will end in the first round if the government decides to accept the offer proposed by the private investor. Otherwise, the government needs to propose a counteroffer which will be for an L investor and

for an H investor.

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Fig. 2. Bargaining towards the return rate on investment

4.3 Payoffs for the private investor and government The payoff for the private investor equals the difference between the actual money earned and its opportunity cost. Providing that for an L investor, the project is worthwhile to invest only if the return rate on investment exceeds the value of

and the minimum threshold is

for an H investor,

then in the first-round bargaining stage, the payoff for an L investor can be calculated via the function . Similarly, for an H investor, the payoff function is where

stands for the present value of the capital investment contributed by the private investor.

If the government rejects the first-round offer of the private investor, the bargaining game will step into the next round where the government will propose a counteroffer. If the private investor accepts the counteroffer, the payoffs will be

for an L investor and

for an H investor or speculator. Otherwise, the bargaining game will end in the second round, in which case both of the government and private investor will earn a zero payoff.

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For the maximisation of social welfare, the government usually limits the maximum return rate on investment

for the private investor. If the private investor proposes a return rate on

investment that is higher than the value of

, the proposal will be rejected by the government in

the first round to avoid an overly lucrative condition for the private investor. Based on this common knowledge, an L investor will always offer a return rate on investment within the range [

while an H investor will negotiate within the range [

. Although the

government cannot distinguish the type of the private investor at the initial stage of the game, it can infer that the payoff of taking part in the project will be at least

where

denotes the NPV of the project. If the government receives an offer investor, its minimum payoff will rise to

from an L

. Hence, when the offer

is

accepted in the first round, the government’s payoff can be calculated as: (2)

Similarly, the government will earn a payoff of

if the offer

is

accepted in the first round. Following the same reasoning process, if the counteroffer is accepted by the private investor, the government will gain a payoff whose value equals or

depending on the first-round offer of the private investor.

4.4 Equilibrium return rate on investment The method of backward induction is employed to find the equilibrium of this bargaining game. As shown in the right-bottom nodes of Figure 2, the government’s payoff can be maximised via offering a counteroffer

to make it no difference for the private investor to accept or reject the

counteroffer. If the private investor is an L-type investor or speculator, the government will propose a counteroffer

in the second stage of the bargaining process to maximise its payoff. To

reflect the patience level of the game player, an impatience factor will be included whose value changes from zero to one if the game player changes from being extremely impatient to extremely patient (Shen et al., 2007). The payoffs for the project parties will be devalued with the progress of the bargaining game and it is assumed in this research that the government and private investor are equally patience throughout the bargaining game. Therefore, to make it no difference for the

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government to accept or reject the offer in the first bargaining stage, the following equations should be satisfied: (3) (4) where

denotes the value of the impatience factor. Eqs. (3) and (4) indicate the principle that to

achieve the equilibrium of the game, the private investor should provide the government with a payoff in the first round that is equivalent to the amount obtained in the second round. Thus, the equilibrium values of

and

can be expressed as: (5) (6)

The optimal strategies for the government can be discovered through the following reasoning process. First, the offer

will always be accepted since the government know it can only be offered by

an L investor. Second, since the type of the private investor is unclear when the offer

is

proposed, the government will make a judgement based on the acquired information about the private investor. The government's prior belief regarding the type of the investor will be updated after receiving the private investor’s first-round offer. In this research, of the private investor is L,

stands for the event that the type

represents the event that the type of the private investor is H, and B

stands for the event that the offer

is proposed in the first bargaining stage. The offer

will be accepted only when the government believes the private investor is not a speculator based on the acquired information. Hence, the government will accept

|

under which case the expected utility value of the government is . Third, the offer

will be rejected with a probability of

government will propose a counteroffer

|

|

and the

to maximise its payoff. Under this case, the |

government’s utility value will be [ rule,

|

with a probability of

. Based on Bayes’

can be expressed as: |

|

⁄

|

|

Given |

| |

⁄

(7)

, Eq. (7) can be simplified as: (8)

15

Based on the common knowledge, the private investor knows the government will update the prior belief according to the observed action in the first round bargaining. As a result, the private investor should make an offer with which the government will gain the equivalent utility value no matter whether to accepts the offer or not. Based on this logic, the following equation is obtained: |

|

[

(9)

The following equation is derived after inserting Eq. (6) and Eq. (8) into Eq. (9),: ⁄

(10)

Then Eq. (8) can be rewritten as: |

⁄

(11)

To sum up, the equilibrium return rate on investment could be two-stage bargaining game. The offer government

will

|

or

for the given

will always be accepted by the government. The

accept

with

⁄

a

probability

of

and reject the offer with a probability of

|

⁄

.

5. Determining Concession Periods and Minimum Revenue Guarantees 5.1 Simulation on NPVs of operating incomes In order to run the simulation on the NPVs of the operating incomes, the NPV functions for the private investor and government need to be clarified. For a PPP project with no MRG provided, the NPV of the operating income for private investors (

can be calculated by:

∑ where

⁄

denotes the value of the concession price.

length of the concession period.

(12)

is the independent variable indicating the

demonstrates the operation and maintenance cost in year .

stands for the risk-free rate.

indicates the NPV of the operating income for private investors under guaranteed contracts whose value can be calculated based on the following equation: (

∑

)

16

⁄

(13)

where { indicates the value of the net cash inflow in year .

and

are the independent

variables.

denotes the NPV of the operating income for the government when there is no guarantee provided. The value of

is calculated by: ∑

where

⁄

stands for the project lifespan.

(14)

denotes the present value of the capital investment shared

by the government. By assigning the values of

and

as well as the distributions of the

uncertain variables recognised in Section 3, the distributions of

,

and

can be

derived after running multiple simulations.

5.2 Decision range of concession period The decision range of the concession period can be found based on the simulation outcomes. The length of the concession period is closely related to the amount of operating income earned by private investors. Thus, the length of the concession period cannot be given without an upper limit, since an overlong concession period brings overly lucrative conditions for private investors. Based on this logic, the maximal length of the concession period ( {

) can be calculated via: (

)

}

(15)

On the other hand, the concession period should be long enough to guarantee an expected minimum return for private investors. It is expected that the higher the level of the MRG, the shorter the concession period required to recover private investors’ investment (Wang et al., 2015). Therefore, under a guaranteed contract, the minimum length of the concession period (

can be derived

based on two conditions. First, the minimum length of the concession period should be one during which the expected minimum return for private investors is achieved. Second, governments should provide the maximal affordable level of the MRG. Even though governments are usually not profit-oriented when contracting PPP projects (Gu et al., 2019), they limit the amount of the guarantee 17

provided annually so that the liquidity of the fiscal expenditure can be protected. For instance, in order to guarantee financial support for public services, the Chinese Government regulates that the annual investment on all PPP projects cannot exceed 10% of the annual fiscal expenditure (Ministry of Finance, 2015). In this research, the minimum annual net income during the operational period (

needs

to be found first to help locating the maximal level of the MRG. Then the governments have to evaluate the maximum amount of money they can afford annually for the proposed PPP project ( Mathematically, the value of { where

.

can be expressed as: (

)

}

gives the minimum return rate on investment whose value is

(16)

for an L investor or

for an H investor.

5.3 Decision range of minimum revenue guarantee After clarifying the decision range of the concession period, i.e. [

], the maximum level

of the MRG can be further limited by: {

̅̅̅̅̅̅̅

where ̅̅̅̅̅̅̅ denotes the option value of the

(

)}

(17)

. To prevent governments from suffering risk

spillover, Eq. (17) regulates that the option value of the MRG cannot exceed the NPV which governments earn through operating the project to prevent themselves from suffering a financial deficit. Also, as mentioned in Eq. (16), the level of the MRG cannot be given beyond the annual financial capabilities of governments. As a result, the maximum level of the MRG

can be

found via: (18) Since governments can choose not to provide an MRG if the concession period is already long enough for private investors to recover their investment share, the level of the MRG should range between zero and

.

Eq. (19) is proposed to find the optimal combinations of the concession period and MRG: (

)

{(

)

18

(

)

}

(19)

The value of the equilibrium return rate on investment information bargaining process and its value can be

is derived from the imperfect or

. The optimal combinations

should contribute to achieving the equilibrium return on investment for private investors so that a win– win situation can be achieved for governments and private investors.

6. Numerical Example 6.1 The Data In reference to a real highway PPP project in China, Project QJ is now used as a numerical case to verify the developed determination process. Conducted as a highway PPP, the pre-construction stage of Project QJ will last one year, during which the feasibility study and tendering process will be completed. The construction work is scheduled to be completed within two years and the service life of the project is 60 years. As summarised in Table 2, the values of the uncertain variables are as follows. The required capital investment ranges from 2.9 billion to 3.3 billion US dollars. The private investor contributes 60– 70% of the total capital investment, while the government takes responsibility for the remaining part. The annual operation and maintenance costs are assumed to be 30% of the annual operating income. The traffic volume in the first operational year is estimated at

. The value of

is 5.1% and

equals 27% according to historical data from a similar project.

Table 2. Statistical distributions of uncertain variables Uncertain variable

Probability distribution

Parameters

Total capital investment

Beta distribution

Capital investment shared by private investors Operation and maintenance cost Traffic volume

Even distribution

Minimum=2.9 (billion USD) Maximum=3.3 (billion USD) =2 =5 Minimum=60% Maximum=70% 30% of operating income =5.1% (2015–2020) =27%

– Geometric motion

Brownian

The decisive variables are given as follows. The maximum amount of money that the government can afford annually for the proposed PPP project is 10 million US dollars. The risk-free rate is 3.8%, taking the interest rate of 10-year government bonds as a proxy. The toll scheme of the project is shown in

19

Table 3. The minimum return rate on investment is assumed as 10% for an L investor and 15% for an H investor. The maximum return rate is given as 20% and the government’s prior belief on private investor’s probability of taking low opportunity cost is 50%. Finally, it is assumed that the private investor and government are equally patient and they barely care when they will achieve their payoffs, in which case the impatience factor is given as 0.98.

Table 3. Predetermined toll regime Vehicle type

Tariff (cents USD)

7 seats 8–19 seats 20–39 seats 40 seats Truck

8/km 12/km 18/km 24/km 1.2/ton/km

Average toll rate per trip (USD)

3.1

6.2 Equilibrium return rate on investment The equilibrium return rate on investment can now be calculated by the proposed imperfect information bargaining model. Based on the given data, an L investor will negotiate a return rate on investment within the interval [10%, 20%]. No return rate lower than 10% will be accepted by L investor since it will not invest in a project when the expected return is lower than the opportunity cost. No return rate higher than 20% will be accepted by the government to avoid the private investor from earning excess profits. Similarly, the negotiation range for an H investor is [15%, 20%]. By applying the project parameters to Eqs. (5) and (6), the equilibrium return rates on investment are calculated as 10.2% and 15.1%. Using Eq. (11), the value of

|

is calculated as 67%. In summary, to achieve the

equilibrium of the bargaining game, the L investor has to propose a 10.2% return rate on investment and the government will always accept the offer. The speculator or H investor will propose a 15.1% return rate on investment, in which case the government will accept the offer with a probability of 67% and reject the offer with a probability of 33%.

6.3 Decision range of the concession period Before calculating the decision range of the concession period, the minimum annual net income throughout the operational period needs to be valued. To evaluate the influence of the uncertainty of the

20

traffic volume, the minimum annual net income is simulated using the method of Monte Carlo analysis. It can be seen from Figure 3 that the value of the minimum annual net income with the highest frequency equals

US dollars.

Fig. 3. 1000-times simulation of annual net income

Figure 4 is plotted based on 1000-times Monte Carlo simulations of the NPVs for both the private investor and government. The risk situations of the project parties can also be observed via the Monte Carlo analysis. As shown in Figure 4, with an increase in the length of the concession period, the probability of achieving the minimum return rate on investment rises more sharply than that of the maximum return rate on investment. It can be inferred that for Project QJ, the private investor’s financial risk, i.e. the probability of earning less than the minimum return rate on investment, can be easily alleviated by a reasonable length of the concession period, while it is not easy for the private investor to achieve the highest return rate on investment allowed by the government. Assuming the maximum acceptable incidence of the financial risk for the private investor is 20%, the minimal lengths of the concession period are 12 years for an L investor and 14 years for an H investor. Similarly, given that the government will not allow the risk of an overly lucrative condition for the private investor to be higher than 20%, the maximum length of the concession period is found to be 34 years. Therefore, for an L investor, the decision range of the concession period should be [12, 34] while for an H investor the range should be [14, 34]. It can also be expected that the decision range will narrow if the private investor

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requires more stringent financial risk control or the government’s control over the private investor’s profit becomes more rigorous.

Fig. 4. Probabilities of achieving

and

with different values of concession period

6.4 Decision range of minimum revenue guarantee As the minimum annual net income has been calculated as

and the maximum amount of

money the government can afford annually for the proposed PPP project is 10 million US dollars, the value of

should be

US dollars from Eq. (16). To find the value of

the government’s NPV is first evaluated via a 1000-times Monte Carlo simulation. Then a real option analysis is conducted to find the maximum level of the MRG that the government can accept. Table 4 summarises the highest frequency value of

and the value of

corresponding to each

given concession period in the decision range. It can be observed that all the values of higher than

and therefore the maximum level of the MRG is kept at

are US dollars

no matter the which type of the private investor is observed. Since the government can choose not to provide an MRG if the concession period is already long enough for the private investor to recover their investment, the decision range of the MRG is decided as [0,

22

].

Table 4. Values of Concession period (years) 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

and (USD 7.18 7.23 7.30 7.10 6.50 6.25 6.02 5.92 5.82 5.78 5.38 5.33 5.46 5.39 5.13 5.02 4.59 4.36 4.22 4.32 4.05 3.98 3.79

with different lengths of concession periods )

(USD 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51

)

(USD 11.09 10.73 9.66 9.30 8.13 7.33 7.22 7.01 6.21 6.20 5.89 5.30 5.35 5.29 4.85 4.57 4.24 4.06 3.92 4.02 3.73 3.52 3.40

)

(USD 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51 2.51

)

6.5 Optimal combinations of concession period and minimum revenue guarantee After clarifying the decision ranges of the concession period and MRG, the final step is to find the optimal combinations of them. The optimal combinations are calculated based on the principle that the equilibrium return on investment for the private investors should be achieved with the granted values of the concession period and MRG so that a win–win situation for both government and private investor can be achieved. Assuming the government holds the belief that the potential private investor has a low opportunity cost, then 10.2% should be the value of the equilibrium return rate on investment. Figure 5 draws a probability surface demonstrating the probabilities of achieving the equilibrium return rate on investment with different combination values of the concession period and MRG. The optimal combination values are the coordinate values of the intersection line of the probability and PER surfaces, where PER surface gives the expected probability of achieving the equilibrium return rate on investment. For example, when the decision-maker wants to achieve the equilibrium return rate on investment with 80% probability, the optimal combinations of the concession period and MRG can be found as:

23

{

The unit of the optimal MRG is USD

}

. The result shows that when the concession period is longer

than 34 years, the government should not provide an MRG since the concession period is already long enough for the private investor to recover its investment.

Fig. 5. Probabilities of achieving

with different values of concession period and MRG

Figure 6 further depicts the relationship between the optimal lengths of concession periods and the optimal levels of MRGs with different values of PERs. It can be noted that the length of the concession period is inversely proportional to the level of the MRG. In addition, when the decision-maker decides to increase the PER to 90%, an MRG is always required by the private investor. This is because even though the length of the concession period reaches its maximum value, the value of PER will still be under 90% if no MRG is provided. On the contrary, when the PER equals 70%, an MRG is not required once the concession period exceeds 24 years.

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Fig. 6. Optimal combinations of concession period and MRG with changes of PER

A sensitivity analysis of the decisive variables is also conducted here to evaluate the impact of the selected decisive variables on the outcomes of the data analysis. By keeping the other decisive variables constant, Figure 7 plots the optimal combinations of the concession period and MRG with different values of the risk-free rates. Following the same calculation process, the range of the concession period is calculated as [11, 34] when the risk-free rate decreases by 1% while the range of the concession period changes to [13, 34] when the risk-free rate increases by 1%. The decision range of the MRG remains constant with changes in the risk-free rate. There is a small change in the optimal combinations of the concession period and MRG with changes in the risk-free rate.

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Fig. 7. Optimal combinations of concession period and MRG with changes in the risk-free rate

Figure 8 demonstrates the optimal combinations of the concession period and MRG with changes in the concession price. The ranges of the concession period are [11, 29] when the concession price decreases by one dollar and [20, 38] when the concession price increases by one dollar. The decision range of the MRG remains unchanged. Comparing Figure 8 with Figure 7, it can be observed that the values of the optimal combinations are more sensitive to change in the concession price. However, this influence can be mitigated since the user volume also varies with a change in the concession price, which is beyond the scope of this research.

Fig. 8. Optimal combinations of concession period and MRG with changes in concession price 26

7. Discussion Recognising the importance of the design of the critical contract parameters in PPP agreements, this research proposes a determination process for optimising the values of the concession period and MRG. The incomplete information bargaining model is proposed first to find the equilibrium return rate on investment. After the first move of the private investor in the bargaining process, the government can infer the type of private investor (low or high opportunity cost) based on its updated information. Although negotiation may not be conducted in some cases, the equilibrium return rate on investment calculated by a simulated bargaining model provides a good reference for the government to balance the financial interests between the private investor and itself. The bargaining results demonstrate that in order to derive the equilibrium return rate on investment, the government needs some prior information on the level of the opportunity cost of the private investor. As the government faces the problem of incomplete information when designing the contract in the pre-tender stage, it may have to choose one of the following two strategies derived from the data analysis outcomes. First, if the government wants to attract social capital to its full extent, a 15.1% return rate on investment should be adopted since both L and H investors will be motivated. However, speculators will enjoy an excess profit in this case. Second, if the government wants to prevent the emergence of speculative behaviour or it has enough prior information indicating the private investor is an L investor, a 10.2% return rate on investment should be set by the government.

There are also some lessons to be learned from the outcomes of the data analysis. First, as shown in Figure 5, the longer the concession period, the lower the level of the MRG. However, with an increase in the value of PER, the downward trend of the curve becomes more and more gentle, which means the length of the concession period will have less influence on the level of the MRG. Hence, for the proposed determination process, the expected probability of achieving the equilibrium return rate (PER) is an important parameter that needs to be evaluated by the decision-makers from the government side, since the value of PER will impact on the level of correlation between the length of the concession period and the level of the MRG. Second, as shown in Table 1, the research proposed by Carbonara et al. (2014b) is capable of determining the specific level of the MRG. In their research, the data analysis outcomes showed that with a predetermined concession period, a specific length of the MRG can be determined when the public and private risks are evenly shared at around 75%. This result seems 27

irrational since with a 75% probability of suffering financial risk, both government and private investor may choose to abandon the project. In this research, as shown in Figure 5, the financial risk for the private investor is constrained within 20% and the data outcomes suggest that when the concession period is 12 years, the level of the MRG should be

USD, in which case the probability of

suffering financial risk for the government is calculated as 22.4%. The data analysis outcomes are more reasonable in this research and, as demonstrated in Figure 6, the value of PER can be adjusted depending on which party is the main risk-taker. Third, although it is a common practice for governments to continue charging after the end of the concession period at the toll level decided by consultants to avoid a financial deficit (Department for Transport, 2017), cases show that some governments eliminate the toll for highway PPPs to improve social welfare. Thus, the second constraint on the maximal level of MRG, i.e. Eq. (17), could be optional for decision-makers. As shown in Table 4, the second constraint has no influence over the outcomes for the given numerical example. However, the scenario may change for other projects.

8. Conclusions The length of the concession period plays a vitally important role in deciding the profitability of a PPP project for private investors. The longer the concession period, the easier it is for private investors to recover their investment share. However, in real-life projects, some PPPs with long concession periods are still full of challenges since the market surroundings keep changing. As a result, private investors often require the provision of government guarantees to alleviate their financial risk. The MRG, which regulates a minimum revenue threshold, is one common guarantee provided by governments in PPP contracting. In this research, the concession period and MRG have been recognised as two of the critical parameters that are capable of balancing the financial interests between governments and private investors. The majority of the existing literature focuses on the determination method for the concession period and the real option valuation of the MRG. There is a lack of research exploring the interaction between the concession period and MRG, and no single research work optimises the length of the concession period and the level of the MRG simultaneously. This research fills the research gap by proposing a determination process which determines the optimal combinations of the concession period and MRG. An imperfect information bargaining model is constructed to find the equilibrium return rate on investment, which provides a reference for governments to balance financial interests 28

between private investors and themselves. Project QJ is given as a numerical example to validate the proposed determination process. The data outcomes show that the optimal combinations of the concession period and MRG can be successfully determined following the proposed determination process. An MRG is not required by private investors when the concession period is already long enough to recover their investment.

The proposed determination process is not without limitations. First, as the NPVs for the project parties are forecasted by a simulation process in this study, the determination process may be inapplicable to social infrastructure PPPs where the income stream is less predictable. Second, several uncertain variables are recognised in this research to reflect the NPV for project parties under uncertainty. The accuracy of the analysis outcomes relies on the accuracy of the prediction on the probability distributions of the uncertain variables. However, it is difficult for decision-makers without statistical backgrounds to select the most appropriate distributions for the identified uncertain variables. In addition, probability distributions for certain variables are still under debate (e.g. the distribution for the inflation rate). Future research work could focus on the improvement of the accuracy of probability distributions for the key uncertain variables in PPP projects, which would contribute to improving the effectiveness of simulation-based models.

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