Compound deferrable options for the valuation of multi-stage infrastructure investment projects

Abstract Multi-stage planning is common for expanding infrastructure complexes or networks. Previous real-option studies developed a sequential compound call option (SCCO) for evaluating multi-stage infrastructure investment projects, including evaluating abandonment options during individual pre-scheduled investment stages. In practice, however, investment schedules involve risk and uncertainties forcing investors to change plans, raising an important issue regarding investors having more options at each commencement date. This study introduces a new model, the sequential compound deferrable call option (SCDCO), incorporating deferment options for each fold in an n-fold setting and generalizing the exercise of each deferment option into mn periods. A closed-form solution to the valuation of SCDCO is derived accordingly. A real-world case demonstrates that introducing deferment options increases project value, but the marginal benefit of increasing deferment time diminishes. The presence of dedicated assets for the usage of future expansions can also increase project value, but this increase is restricted by deferment options. Furthermore, the investment cost growth with an increase of deferment time rapidly decreases project value and vice versa. Inflation or deflation is therefore important to consider in deferment decisions. Finally, although deferment options only produce limited project value, they have the flexibility to manage risks, changes, and budgetary constraints.


Introduction
Multi-stage planning is a common approach for the expansion of infrastructure complexes or networks. By staged investment, infrastructure facilities are built, commissioned, and delivered in stages. When the first stage of investment is delivered, the investor further decides if the next stage can be carried on and when. This planning approach allows the investor to maintain flexibility and adaptability during project execution. To maintain flexibility and adaptability, or as indicated by Priemus (2010), "to keep open as many options as possible" is key to managing risks, uncertainties, and changes in megaprojects. In addition, major infrastructure investments often face financial, budgetary, and other resource constraints. A staged investment plan allows the investor to overcome these constraints by buying time for future expansions. However, a staged investment plan will produce a valuation problem when the investor is given a list of options at the beginning of each stage of investment. The valuation of multi-stage plans has become an important research topic in the field of real-option theories, and Machiels et al. (2021) and Martins et al. (2015) have surveyed some of the recent developments.
The sequential compound call option (SCCO) is an emerging approach used in the real-option literature. An SCCO is an option on options, and the value of each option is determined by the valuation of its subsequent options. In theory, a multi-stage investment plan can be modeled as an SCCO. The valuation of the SCCO can be solved by numerical methods (e.g. Herath andPark 2002, Kr€ uger 2012) or by closed-form solutions (e.g. Cassimon et al. 2004, Huang andPi 2009). Currently, the SCCO provides two types of options at the beginning of each stage of investment, namely, to execute or to abandon. This approach can be extended to cover the valuation of dedicated assets, which are up-front investments intended for the usage of future expansions (Huang and Pi 2011). The SCCO model also has potential applications in other areas, such as pharmaceutical R&D (Cassimon et al. 2011a), mobile payments (Cassimon et al. 2011b), production and inventory (Cortazar andSchwartz 1993, Dixit andPindyck 1994), general capital investment (Herath and Park 2002), information technology (Panayi andTrigeorgis 1998, Pendharkar 2010), bank expansion (Panayi and Trigeorgis 1998), and construction (Huang and Pi 2009).
In the SCCO context, it is assumed that the staged investment dates are pre-scheduled according to the staged investment plan. The investor can further decide to execute or abandon at the beginning of these staged investment periods. In practice, however, investors are usually confronted with the issue of uncertain demand (e.g. Bernanke 1983, Zhao and Tseng 2003, Arboleda 2006, Doan and Menyah 2013 and other risks, such as the introduction of new regulation (Hoffmann et al. 2009). To provide the investor with more flexibility in the sequentially staged decision-making process, the investor should have the option to defer the staged investment dates (McDonald andSiegel 1986, Ingersoll andRoss 1992).
This study introduces an option to defer into each investment stage of the SCCO model. The resulting model, a sequential compound deferrable call option (SCDCO), has n folds, and each fold has m n options to defer before execution. In this model, the number of deferment options is generalized into a discrete-time framework that differs from the continuous-time American option, as reviewed in the next section. In the SCDCO, the investor has three mutually exclusive options at the beginning of each investment stage: to execute the expansion, to abandon, and to defer. Therefore, the flexibility to keep open more options is enhanced, more exhaustive investment timing and risk management scenarios are covered, and a more comprehensive valuation of flexibility is achieved.
In the next section of the paper, we provide a brief background and literature review. The sections that follow first describe the SCDCO valuation problem, outline a closed-form solution to the problem, apply the solution to the valuation of a real-world project, and discuss the implication of deferment rights. The final section summarizes and concludes this paper.

Background and literature review
Long-term infrastructure investment projects with high sunk costs face substantial risks and uncertainties concerning demand, capital costs, and construction delays (Cruz and Marques 2013). The flexibility to manage risks and uncertainties and to adapt to future changes is, therefore, widely recognized. As Martins et al. (2015) summarized, seven types of flexibility are embedded in infrastructure projects: deferment, staged investment, altering operating scale, abandonment, switching usage, growth or expansion, and the combination of these means.
Real-options theories can be applied to the valuation of flexibility, especially when the projects in question involve irreversible investments and faces uncertain environments (e.g. Dixit andPindyck 1994, De Neufville andScholtes 2011). Real-option theories can also be applied to the valuation of credit enhancement and contract mechanisms. For example, Huang and Chou (2006) proposed a real-option model to evaluate voluntary abandonment and minimum revenue guarantees in BOT projects. Zapata Quimbayo et al. (2019) used a mean-reverting process instead of geometric Brownian motion to evaluate minimum revenue guarantees. Brandao and Saraiva (2008) developed a real-option model for the valuation of minimum traffic guarantee. Huang and Pi (2014) used a compound option approach to examine the value of performance bonds in BOT contracts. Finally, the general application of real-option theories for the planning and valuation of megaprojects has received wider attention (e.g. Machiels et al. 2021).
For major infrastructure investments, real-option theories provide promising tools for the valuation of staged investment plans. Staged investment plans allow the investor to have a list of options at the beginning of each stage to manage risks and uncertainties and adapt to changes. Staged planning can also overcome investors' budgetary and resource constraints.
To avoid the risk of oversimplification, two general real-option approaches are observed and summarized below for option valuation in staged investments. The first approach is derived from McDonald and Siegel (1986). This approach focuses on investment timing and treats it as a standard first passage problem to obtain optimal investment time and the value of waiting. In this model, the optimal investment time is endogenous, and the investor should wait until the benefit outweighs twice the cost of investment. In this context, Garvin and Cheah (2004) used a one-step binomial tree to examine the value of waiting. Tahon et al. (2014) further examined the value of waiting in telecommunication infrastructure projects. Penizzotto et al. (2019) examined the value of waiting to invest in irreversible photovoltaics assets. Van den Boomen et al. (2019) developed a decision tree approach to examine the option value of waiting. J. Li et al. (2020a) used Monte Carlo simulation to solve the optimal expansion timing problem. Finally, J. Li et al. (2022) used a multi-exercise American option to solve the multi-stage optimal investment timing problem.
The second approach is derived from Dixit and Pindyck (1994). Dixit and Pindyck treated a staged investment plan as a series of discrete-time sequential investment decisions. They proposed a dynamic programming model for the valuation problem. Berk et al. (2004) extended their work by introducing a systematic risk component in stochastic cash flow. Smit (2003) introduced a real-option game model to evaluate airport infrastructure expansion. Adetunji and Owolabi (2016) combined the growth option with a time-to-build option to evaluate a two-stage railroad expansion project. Spiegel et al. (2020) developed a scenario tree implemented by Monte Carlo simulation for staged farm-level investments wherein the number of investment stages is not limited. Torres-Rinc� on et al. (2021) introduced Monte Carlo sampling and a K-Medoids clustering algorithm to solve a multi-stage stochastic programming problem for infrastructure expansion projects.
In the context of discrete-time sequential investments, and on the basis of Geske (1979), Huang and Pi (2009) developed a sequential compound call option (SCCO) approach for the valuation of staged infrastructure projects. They treated irreversible dedicated-asset investments explicitly. Huang and Pi (2011) further used the SCCO model to examine the impact of demand risk and technological obsolescence on project value. Polat and Battal (2021) used a compound option model to examine the value of expansion options in airport investments. Finally, compound options were used by Yoon and Kim (2016) to evaluate a coal bed methane development project and by Y. Li et al. (2020b) to evaluate hydrogen-fuel infrastructure investments.
The scope of application of the compound option theory has been extended into other areas. Herath and Park (2002) used Monte Carlo simulation to solve the sequential compound option valuation problem for R&D projects. Panayi and Trigeorgis (1998) applied the theory to the valuation of information technology and bank expansion. Cassimon et al. (2004Cassimon et al. ( , 2011 applied the theory to the valuation of pharmaceutical R&D projects. Overall, it is widely recognized that the option to abandon and the option to exercise expansion rights in staged investments can increase project value (Cassimon et al. 2004, Damnjanovic et al. 2008, Pendharkar 2010, Hauschild and Reimsbach 2015. In theory, McDonald and Siegel's (1986) work is a classical application of the American option, and Geske's (1979) seminal work is based on the European option. The former is a continuous-time model; the latter is a discrete-time model. In the practice of infrastructure planning, Kr€ uger (2012) suggests that, in road investment projects, the investor must wait until a current lane expansion project is completed before they can decide whether to conduct any further expansion at that time or to defer the decisions regarding further expansions. Yoon and Kim (2016) suggest that the decision to either execute or defer a two-stage coal bed methane development project should be deferred for at least after a one-year period. More generally, as Torres-Rinc� on et al. (2021) observed, the frequency of decision-making may not coincide with the frequency of the random process, and an increase in the decision-making frequency increased the size of the valuation problem.
The choice seems to depend on the planning approach and the needs of planning practitioners (e.g. Machiels et al. 2021). A staged investment plan can be formulated as a multi-exercise American option, as in J. Li et al. (2022), who assumed that the investor aims to optimize the timing of investment to maximize project value and treated the problem as being multistaged with optimal stopping time. In this case, optimal timing depends on demand, cost, and other preformulated functions of key project variables. In addition, since there is no analytical solution for American options, the valuation problem needs to be solved by Monte Carlo simulation, binomial tree, or the least-square method.
Alternatively, one can take a staged investment plan as given and formulate it as compound options. This approach can be used to provide preliminary feasibility analysis for pre-planned investment-timing. The problem is not optimal multiple stopping time, but rather the question of how to provide a workable model to explore various investment-timing scenarios for staged investments. An analytical solution is available for the compound option valuation problem. One can also use a binomial tree, decision tree, scenario planning, or numerical methods to solve the problem. However, the problem becomes very complicated for the valuation of large-scale projects when the valuation period increases to an n period. Actually, a European-type option may be used to fit the process of an American-type option if the decision-making intervals are small enough (Geske and Johnson 1984). In this aspect, the SCDCO proposed in this paper can bridge the gap between the European-type and American-type options. The SCDCO has n folds and generalizes the exercise of the deferment option at the beginning of each fold into m n periods. In practice, this model allows the planning practitioners to assess the various scenarios produced by a combination of execution, abandonment, or deferment at each stage of a staged investment plan.

Classic call option model
The classic call option model developed by Black and Scholes (1973) assumes the underlying asset value follows a stochastic process with a geometric Brownian motion to satisfy the following stochastic differential equation, where t is time, l is a drift parameter, r is the volatility of the underlying asset, and W t is the standard Brownian motion. The Black-Scholes model assumes the investor is risk-neutral, and the formula used in the pricing of options is a function of time and the underlying asset value. In the framework of the classic call option model, the option value is zero if the underlying asset value S t is lower than the exercise price K on the expiration date T; otherwise, the option value is the difference between the underlying asset value and the exercise price. In the case of European real-call options, the project value is the "underlying asset value", the investment cost is the "exercise price", and the commencement date is the "expiration date"; Therefore, the expected value of the call option at T is given by With (1), Black-Scholes developed the following call option pricing formula: where N(x) is the standard normal distribution, r is the risk-free rate, r is the volatility of the underlying asset, The proposed SCDCO model was derived on the basis of the Black-Scholes model and follows the related assumptions.

Investment options
A real-option model is developed to evaluate the value of deferment in multi-stage investment infrastructure projects. The proposed deferment investment model assumes the investor has the right to execute or to defer the investment at pre-scheduled commencement date. The discrete-time decision-making process is used in the proposed deferment investment model. The investor has the right to execute or defer the investment at pre-scheduled commencement date and has the same right at the new commencement date after deferment. This study assumes: The investor will choose to execute the investment when the expected cash flow outweighs the investment cost, even if the deferment provides a higher expected value than the value provided by expansion. In other words, the deferment of each stage only occurs when the expected cash flow of the execution is lower than the investment cost.
Each stage of the project cannot be deferred infinitely, and the number of deferment time is assumed to be limited in m i times at ith stage.
Further, the following notations are defined: S t The project value at t. H t The critical value for making a decision at t. V t The expected project value at t after deferment. Q t The expected project value at t after execution. K t The investment cost at t.
Since the commencement date of each stage is deferrable, there are three options at each prescheduled commencement date: 1. To execute the investment when Q t > K t ; 2. To defer the commencement date when Q t < K t and V t > 0; and 3. To abandon the investment when Q t < K t and V t < 0: If the commencement date is not deferrable, there are only two options at the commencement date: 1. To execute the investment when Q t À K t > 0, namely S t > H t , and 2. To abandon the investment when Q t À K t < 0, namely S t < H t : Here, the critical value H t is the value of the underlying asset satisfying the boundary condition Q t À K t ¼ 0:

A one-stage-deferrable model
A one-stage investment model is presented here to observe model behavior. For a one-stage investment project, assume that the investor has the flexibility to defer the investment before investing. Let S t and K t define the project value and the investment cost, respectively, at time t: If the investment is not deferred, the project value at the commencement date T 1 is S T 1 À K T 1 : If the investor defers the project, the commencement date T 1 shifts to T 2 , and the project value at T 2 is S T 2 À K T 2 . The expected project value at T 1 to the investor is where C T 2 ðS T 1 , T 1 Þ is the expected project value at T 2 discounted to T 1 , and (4) can be rewritten as where r is the risk-free rate, H T 1 is the critical value of S T 1 when the investor decides to exercise the option at T 1 , and H T 1 ¼ K T 1 : The expected value C T 2 ðS T 1 , T 1 Þ is a classic call option and can be obtained by where H T 2 is the critical value of S T 2 when the investor decides to exercise at T 2 , and H T 2 ¼ K T 2 : Substituting Equation (6) into (5), the expected project value at T 0 is Let the project value follow a stochastic process with a geometric Brownian motion to satisfy the following stochastic differential equation, where t is time, l is a drift parameter, r is the volatility of the underlying asset, and W t is the standard Brownian motion. The closed-form solution to Equation (7) is where and N i ð½x 1 , x 2 �, ½y 1 , y 2 �; qðtÞÞ is the standard i-variate cumulative normal distribution which has a correlation coefficient matrix qðtÞ: The cumulated normal distribution can be written as a probability: where Z 1 and Z 2 are standard normal deviates. The correlation coefficient matrix qðtÞ is a symmetric matrix, and the value of the correlation coefficient is the correlation of the underlying asset value at different times (Lee et al. 2008). It can be obtained by This model can evaluate projects that have a single investment stage and the opportunity to defer the commencement date. To evaluate a project with multiple stages and multiple deferment times, the valuation model must be extended to a multi-stage model, assuming that the commencement dates of the following stages would be different if the commencement dates of the previous stages were deferred. To observe the behavior of the multi-stage valuation model, a twostage model is first demonstrated below.

A two-stage-deferrable model
When a project has two stages, the assessment of the investment of the second stage will be made at the end of the first stage. The investor has the right to expand the project, defer the commencement date, or abandon the project at the commencement date of the second stage (see Figure 1). The expected project value is composed of the expected values when the second stage is deferred as well as not deferred. The expected value of the first stage depends on the expected value of the second stage. Importantly, the deferment of the first stage changes the commencement date of the second. To observe the model behavior, a two-stage investment model with one deferment time is presented as follows.
Let T f1g, 1 and T f1g, 2 define the commencement date and deferred commencement date of the first stage, T f2g, 1 and T f2g, 2 define the commencement date and deferred commencement date of the second stage when the first stage is not deferred; T 0 f2g, 1 and T 0 f2g, 2 define the commencement date and deferred commencement date of the second stage when the first stage is deferred. The valuation of the option model requires a backward calculation. The expected value of the second stage at t can be considered as the one-stage model, written as where H T f2g, 1 ¼ K T f2g, 1 , and H T f2g, 2 ¼ K T f2g, 2 : However, if the first stage was deferred, the commencement date of the second stage changed. The expected value becomes where H T 0 f2g, 1 ¼ K T 0 f2g, 1 , and H T 0 f2g, 2 ¼ K T 0 f2g, 2 : The expected project value is composed of two expected values: where EV 1 is the expected value of the first stage discounted by the factor e À rðT f1g, 1 À tÞ when the value of exercising the second stage is in the money, and EV 2 is the expected value of the first stage when the value of exercising the second stage is out of the money (the first stage would be deferred). First, the expected value EV 1 can be expressed as Substituting (13) into (16), where H T f1g, 1 is the critical value of S T f1g, 1 when C T f2g, 1 ðS T f1g, 1 , T f1g, 1 Þ À K T f1g, 1 ¼ 0: Second, the expected value EV 2 discounted by the factor e À rðT f1g, 1 À tÞ is Since C T f1g, 2 ðS T f1g, 1 , T f1g, 1 Þ is always positive, EV 2 can be rewritten as EV 2 ðtÞ ¼E½C T f1g, 2 ðS T f1g, 1 , T f1g, 1 Þe À rðT f1g, 1 À tÞ j where C T f1g, 2 ðS T f1g, 1 , T f1g, 1 Þ ¼ E½maxðC T 0 f2g, 1 ðS T f1g, 2 , T f1g, 2 ÞÀ K T f1g, 2 , 0Þe À rðT f1g, 2 À T f1g, 1 Þ �: Equation (20) can be rewritten as where H T f1g, 2 is the critical value of S T f1g, 2 when C T 0 f2g, 1 ðS T f1g, 2 , T f1g, 2 Þ À K T f1g, 2 ¼ 0: Thus, Equation (18) can be rewritten as Therefore, the expected project value is where and The correlation coefficient matrix, ½q h, j � p�p , is a symmetric matrix. It can be obtained by where b In the twostage model, the deferral of the commencement date of the previous stage postpones the commencement date of the successive stage. Therefore, the value of the two-stage model is composed of more expected values than the one-stage model.

A generalized n-stage-deferrable model
Consider further that a project has n stages and is allowed to defer m fig times at the i th investment stage, 1 � i � n. The pre-scheduled commencement date of the i th stage is T i f g, 0 the termination date is T i f g, 1 , and the pre-scheduled investment duration is s i f g, 0 : To simplify the formulation, further denote the index of the i th stage as fig. Figure 1 shows that when the prescheduled commencement date is deferred from T i f g, 0 to T i f g, a fig after a fig deferments, the deferments would create several deferred periods in the i th investment stage, denoted as "period ( fig, j þ 1 Similarly, the deferral of the previous stages will change the commencement date of the i th stage. Therefore, the valuation of the n-stage model is more difficult than the valuation of the two-stage model. Figure 2 shows the decision tree at stage fig, assuming each stage was allowed to be deferred twice (1 � i � n-2). This decision tree indicates that the commencement date of investment stage fi þ 2g depends on the duration of the deferments in investment stage fig and investment stage fi þ 1g. Likewise, the commencement date of investment stage fng depends on the duration of deferments in investment stages fn-1g, fn-2g, … , and f1g. To simplify the expression of different commencement dates, the commencement dates of the investment stages can be expressed as the summation of the deferred periods: where a * ¼ a 1 , a 2 , a 3 , :::, a i , ½ � and a i is the number of times that the commencement date of the i th stage was deferred. To further simplify the formulation, denote the commencement date T i f g, j ða * Þ as T i f g, j : The options differ according to commencement dates. The options and the boundary conditions can be concluded as follows. When the investment stage is deferrable at commencement date T n f g, j ¼ T n f g, m fig , there are three options at each pre-scheduled commencement date: 1. To execute the investment when S T fng, j À K T fng, j > 0; 2. To defer the commencement date when S T fng, j À K T fng, j < 0 and C T fng, jþ1 ðS T fng, j , T fng, j Þ > 0; and 3. To abandon the investment when C T fng, jþ1 ðS T fng, j , T fng, j Þ ¼ 0 and S T fng, j À K T fng, j < 0: When the investment stage is not deferrable at commencement date T n f g, j ¼ T n f g, m fig , there are only two options at the commencement date: 1. To execute the investment when S T fng, j À K T fng, j > 0, and 2. To abandon the investment when S T fng, j À K T fng, j < 0: When the investment stage is deferrable at

A solution to the SCDCO valuation problem
First, in stage fng, according to the aforementioned payoff function, the expected project value of period (fng, m fng þ1) at t can be expressed as Here, the critical value H T fng, m fng þ1 ¼ K T fng, m fng þ1 : For 1 � j � m fng , the expected project value of period (fng, j) at t can be expressed as Since the expected value C T fng, jþ1 ðS T fng, j , T fng, j Þ is always positive, the Equation (6) can be rewritten as where H T fng, j ¼ K T fng, j : Second, in stage fig, for i < n, the expected project value of period (fig, m fig þ1) at t can be expressed as , m figþ1 ðS t , tÞ ¼E½maxðC T fiþ1g, 1 ðS T fig, m fig þ1 , T fig, m fig þ1 ÞÀ K T fig, m fig þ1 , , j) at t can be expressed as In sum, the expected project values of the periods are as follows.
The terminal payoffs in stage fng can be summarized as S T fng, j À K T fng, j when S T fng, j > H T fng, j 0 when S T fng, j < H T fng, j when j ¼ m fng þ 1 ( S T fng, j À K T fng, j when S T fng, j > H T fng, j C T fng, jþ1 ðS T fng, j , T fng, j Þ when S T fng, j < H T fng, j when 1 � j � m fng : The terminal payoffs in stage fig can be summarized as Here, C T f1g, 1 ðS t , tÞ is the value of the n-fold SCDCO, which can be obtained by the following theorem. where Z 1 , Z 2 , :::, Z i are standard normal deviates.

Deferment valuation covering dedicated-asset investments
For multi-stage investment infrastructure projects, part of the costs of the current investment stage could be attributable to the expansions of the subsequent investment stages (Huang and Pi 2009) Since dedicated assets cannot be ignored, for an n-stage project, the project value is the summation of the option value of the different folds, which can be obtained by Here, SCDCO g ðP t g , t g Þ is the value of the g-fold SCDCO, P t g is the project value generated by investment at stage fgg at t g , and t g ¼ T fgg, 0 : The investment cost of the g-fold SCDCO at T fqg, j is I g, T fqg, j , i � q � n:

Numerical application: a large-scale sanitary sewerage project
The preliminary investment plan for a multi-stage BOT sanitary sewerage project in Taiwan is used to demonstrate how multi-stage investment infrastructure projects can be evaluated by the SCDCO model. The computed results are used to discuss the impact of the dedicated asset investments, deferment times, asset value, and investment cost on the value of multi-stage projects.

Project background
This application project is a part of an ambitious sewerage privatization program designated under the BOT model. In this project, the investor had an exclusive a fig þ1 , T fig, a fig þ1 , tÞ, 1 d 1 ðH T fig, 1 , T fig, 1 , tÞ fig, a fig þ1 , T fig, a fig þ1 , tÞ, 1 right to build and operate sanitary sewerage systems in designated service areas for 35 years. The total construction period runs from 2010 to 2028, and the total estimated cost is approximately NT $15.8 billion. There were four stages in the project; the construction work in each stage would include common sewer systems, house connections, and water treatment and disposal plants.
The investment in one stage gives the concessionaire the option to invest in the next stage at a predetermined time. The concession to invest in a four-stage BOT project can thus be viewed as a sequential option. The common sewer systems are composed of a trunk sewer, submain sewer, branch sewer, and lateral sewer. In order to allow for possible future expansion, the trunk requires an upfront investment to reserve capacity and sewer transition. When a section of the public sewer system is completed, the house connection can begin. Finally, to meet the scheduled operational deadlines, when the construction work for water treatment and disposal commences, the common sewerage system also commences. In order to retain the flexibility to cover uncertain demand, expansion of treatment capacity is planned when the volume of wastewater reaches a predetermined limit in a given area. The treatment and disposal work requires upfront investment in a reserve capacity, spares, and common facilities to prepare for possible future expansion. The expansion plan covered the estimated quantity of future sewage produced, calculated on the basis of historical data. At the end of each stage, the host government must approve future demand estimates before the subsequent stage can be started. The construction period was estimated to be five years for the first stage, four years each for the second and third stages, and six years for the fourth stage, with an estimated cost of NT $4.60 billion, NT $3.13 billion, NT $3.19 billion, and NT $4.16 billion, respectively. When the concession period expires, the investor transfers the facilities to the host government. The total estimated operating income during the concession period was NT $18.96 billion.

Underlying asset values (S 0 ), investment costs (K)
The underlying asset values of the staged investment were calculated by discounting the net cash flows of each stage by the CAPM-based discount rate. The cash flows were calculated by the earnings before interest, tax, and depreciation (EBITDA) of the investment. The 10.57 billion initial asset value served as the base-case initial asset value. The investment cost increased annually if the commencement dates were postponed. All costs were subject to an annual increase due to factors such as inflation. The annual rate of increase is based on Taiwan's Consumer Price Index (CPI) from 2005 to 2009 (see Table 1). The 1.39% mean value served as the base-case annual increasing rate of the investment cost.

Dedicated assets
In this project, part of the costs of the common sewer and the treatment and disposal works are attributable to future expansions of the subsequent investment stages. The SCDCO model can be used for the valuation. The dedicated assets of the sanitary sewage project were estimated by the reserve capacities, preparation work, and common facilities for future expansions. This paper assumed the estimated costs of dedicated assets accounted for 12% of the first stage work, 4% of the second stage, and 3% of the third stage. These costs were allocated evenly for the subsequent stages of the investment. Table 2 summarizes the results of the estimation.

Deferrable durations (s)
Assuming the host government assesses the demand and decides the commencement dates annually, the final decision regarding whether to abandon future expansion would be made after the commencement date was deferred twice. That is, the commencement date of each stage was allowed to be deferred twice, and the deferral durations (s) were one year. Table 3 presents a summary of the parameters.

Risk-free rate (r), asset volatility (r), and discount rate
To obtain the CAPM-based discount rate, a portfolio of 11 utility companies was used as a proxy to estimate the underlying asset volatility. These companies, which included gas, water, and energy companies, have an important common feature: They are all operated in specific concession zones or market areas. The companies are chosen as sensible industrial proxies for the sanitary sewage project primarily because of their concessional characteristics. In the CAPM, asset volatility is a measure of business risk and is influenced by capital market performance or systematic risk in some significant ways. Weekly volatilities of the portfolio and market are estimated from the weekly Taiwan Economic Journal Database utility company and market data for the 5year observation periods between January 2005 and December 2009. The 0.128 mean value is served as the base-case asset volatility. The volatility is lower than the market volatility, which is likely a result of the monopolies or the exclusive concession rights those public utilities enjoy.
Market returns and industry betas for 5-year observation periods are also estimated. The discount rate is calculated by CAPM based on these estimates. The 7.78% CAPM-based discount rate served as the base case discount rate. The risk-free rate was estimated by a set of 5-year data from Taiwan's 10-year treasury bond monthly spot rate. The 2.30% mean value served as the base-case risk-free rate. The results are shown in Table 4.

Impact of deferment option
When the commencement dates are not deferrable, the project value can be obtained using the SCCO model or by assuming that the number of deferments (m) is zero in the SCDCO model. As summarized in Table 5, the introduction of a deferment option into each stage of the project increased the project value from NT$0.7861 billion to NT$0.8772 billion. It is noted that without the deferment option, the project value increased by NT$0.4025 billion. With the deferment option, the project value increases by NT$0.3928 billion. Although the difference is not substantial, this result suggests that the existence of the deferment options reduces the risk of losing the sunk cost of dedicated assets, and thus the value of the dedicated assets is reduced accordingly.

Impact of dedicated assets
As shown in Table 5, if one treats dedicated assets as pure sunk costs and ignores their usage for future expansions, the project value is NT$0.7861 billion. However, if one treats them as prior investments for future usage, the project value is increased to NT$1.1886 billion. The value further increases to NT$1.27 billion when deferment options are introduced. Note that the project value that includes a consideration of dedicated assets investment was calculated by Equation (48) and that the project value ignoring the dedicated assets was obtained from Equation (38). The project values without deferment   (38) and Equation (48), assuming there is no deferment time. Figure 3 shows the sensitivity of project value with respect to the underlying asset value. Increasing the number of deferment times will increase the project value in cases of different underlying asset values. However, the marginal benefit of increasing deferment time are diminishing. In Figure 4, the initial project value is further increased from three to nine times, and the project value increases are quite limited. Figure 5 shows the impact of increasing investment costs on project value. When the investment cost is increased, the increase in the number of deferment time at each stage of investment reduces the project value on a diminishing scale. When the growth rate reaches a relatively higher level, for example, 6%, and the deferment time is increased four times, the increase in project value is more limited.

Impact of asset volatility
As shown in Figure 6, increasing asset volatility increases project value. An increase in the number of deferment options increases project value, and the increase of project value follows the increase of volatility in a monotonously increasing way. Of note, asset volatility is more influential than deferment time on project value, and the marginal benefit of increasing the deferment time is decreasing. Additionally, the    SCDCO model always produces higher project value than does the SCCO model. Figure 7 shows increasing discount rate decreases project value, and increasing deferment time increases project value in a diminishing way. Note that discount rate is more influential than deferment time on project value. A comparison of the impacts of the underlying asset value, the growth of investment cost, the asset volatility, and the discount rate reveals three important observations. First, as shown in Figure 3, the level of influence of the deferment time on project value is diminishing as the underlying asset value increases or decreases. In Figure 4, when the underlying asset value increases from three to nine times, the level of influence diminishes to an insignificant level.

Impact of discount rate
Second, the growth of the staged investment cost has a more significant negative impact on project value. However, this influence is also diminishing as the deferment time increases. In an inflationary environment, additional project value provided by deferment will be substantially reduced, although at a diminishing rate. Conversely, in a deflationary environment, deferment will increase project value substantially at a diminishing rate. Therefore, inflation or deflation is an important factor to consider in the deferment decision.
Finally, both asset volatility and discount rate have a significant impact on project value, and increasing deferment time increases project value at a diminishing rate. As a result, the real value of deferment is not its ability to increase project value but its flexibility for risk, change, or budget management without sacrificing project value. Overall, the extension of the SCCO model into the SCDCO model can provide important new insights deriving from the deferment options that cannot be examined by the SCCO model. The SCDCO also produces higher project value due to the deferment options. This result is consistent with the general real-option theory that flexibility increases value.

Conclusions
To manage risks and changes and to overcome resource constraints, it is important to provide the investor with more flexibility in the execution of staged infrastructure investments. Previous real-option studies recognized this issue and proposed a sequential compound call option (SCCO) that includes an option to abandon a project when subsequent investments are deemed infeasible. The SCCO model can be used for the valuation of multi-stage investment projects that grant the investor abandonment rights. It can also be extended to cover the valuation of demand risk, dedicated assets, and technological obsolescence.
Although the SCCO model is a promising valuation tool, it ignores the possibility that the investor may want to defer and request more information rather than exercising straightforward abandonment. This paper thus extended the SCCO model to cover the deferment option, and the new model is referred to as a sequential compound deferrable call option (SCDCO). The SCDCO generalized the exercise of the n-fold deferment option into m n periods, and a closed-form solution is derived accordingly. In theory, the generalization and analytical solution allow investors to fit the multi-exercise American-type option by smaller deferment intervals.
In practice, a real-world project was used to show how the model can be applied. The application produced several important insights. First, the introduction of a deferment option can create value. The more the number of exercising deferment options is allowed, the higher the project value, but the marginal benefit of increasing the deferment time is diminishing. Second, the introduction of a deferment option can reduce the value of dedicated assets. The reason is deferment options can mitigate the risk of losing the sunk cost of dedicated assets, and thus, the value of dedicated assets is limited. Third, increasing investment costs can quickly reduce project value when the deferment time is increased. Therefore, in an inflationary environment, the expansion should be executed as soon as possible. Conversely, in a deflationary environment, it is more beneficial to defer the expansion. Finally, although the influence of deferment option on project value is relatively small in comparison with other factors, the real value of the option is its flexibility for risk, change, and budget management.
To conclude, the results suggest that the SCDCO model has practical value. First, one can conceive a multi-stage plan for the investment and expansion of a major infrastructure complex or network. Hence, the SCDCO model can be used to assess the value of the staged plan under various scenarios of execution, abandonment, and deferment.
Although the SCDCO model has practical value, the application of this model depends on if we could develop carefully produced investment plans. Future research should explore how to make use of this newly developed analytical tool and other real-option models in the infrastructure planning process, as indicated by Machiels et al. (2021). For example, whether the Program Evaluation and Review Technique or Monte Carlo Simulation can be applied to produce various investment-timing scenarios of staged investment plans seems to be an open issue. If the issue can be resolved, the compound option models may be used in parallel to assess and adjust the plans.

Disclosure statement
The authors declare that there are no conflicts of interest.