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Probabilistic Cash Flow-Based Optimal Investment Timing Using Two-Color Rainbow Options Valuation for Economic Sustainability Appraisement Yonggu Kim 1 1

2

*

ID

, Keeyoung Shin 2 , Joseph Ahn 1 and Eul-Bum Lee 1, *

Graduate School of Engineering Mastership, Pohang University of Science and Technology, 77, Cheongam-ro, Nam-gu, Pohang-si 37673, Korea; [email protected] (Y.K.); [email protected] (J.A.) Technical Research Laboratories, POSCO, 6261, Donghaean-ro, Nam-gu, Pohang-si 37859, Korea; [email protected] Correspondence: [email protected]; Tel.: +82-10-5433-8940

Received: 8 August 2017; Accepted: 29 September 2017; Published: 1 October 2017

Abstract: This research determines the optimal investment timing using real options valuation to support decision-making for economic sustainability assessment. This paper illustrates an option pricing model using the Black-Scholes model applied to a case project to understand the model performance. Applicability of the project to the model requires two Monte Carlo simulations to satisfy a Markov process and a Wiener process. The position of project developers is not only the seller of products, but it is also the buyer of raw materials. Real options valuation can be influenced by the volatility of cash outflow, as well as the volatility of cash inflow. This study suggests two-color rainbow options valuation to overcome this issue, which is demonstrated for a steel plant project. The asymmetric results of the case study show that cash outflow (put option) influences the value of the steel plant project more than cash inflow (call option) does of which the discussion of the results is referred to a sensitivity analysis. The real options valuation method proposed in this study contributes to the literature on applying the new model, taking into consideration that investors maximize project profitability for economic sustainable development. Keywords: economic sustainability appraisement; Monte Carlo simulation; probabilistic cash flow; rainbow real options; optimal investment timing

1. Introduction Since the U.S. was hit by the Global Financial Crisis in 2008, the world economy has declined; a situation that continues to the present time. According to the Steel Statistical Yearbook, the steel industry has not been immune to this decline, making efficient and accurate project profitability of the upmost importance [1]. In this context, Korean steel companies have been seeking to open new markets through overseas business to ensure their economic sustainability. However, a new steel plant project engenders a large, fixed, and irreversible cost [2]. An investment in a steel plant project should be a very cautious decision. Market imperfections provide an important opportunities to create radical technologies and innovative business models based on sustainable entrepreneurship [3]. Especially, in order to succeed in sustainable entrepreneurship in the unstable market, enterprises should continuously develop. In other words, the development of enterprises is a continual success of investment. In order for investors to continue to invest successfully, it is necessary to predict the risks of investment projects and to identify sustainable investment opportunities. Therefore, it is important to consider the risks and opportunities together to find the optimal investment timing in project decisions due to the economic irreversibility [4,5]. Sustainability 2017, 9, 1781; doi:10.3390/su9101781

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The discounted cash flow (DCF) method is a long-used valuation method for steel plant projects. DCF is a method of measuring the future cash flows of a project by converting the gains into the present value (PV), and is currently the most widely used valuation method for decision making by investors [6]. However, DCF does not systematically reflect uncertainties caused by sudden changes in the environment. If any project cannot presently generate cash flow, but is highly likely to generate cash flow in the future, investors may miss a good opportunity due to DCF not reflecting this future potential inflow. In addition, DCF is limited in that it does not adequately evaluate option values that can flexibly determine investments for investors depending on the situation. A real options valuation (ROV), a method to overcome these shortcomings, is suitable for assessing the value of a project because it can estimate the value of business flexibility due to changes in the environment and consider future uncertainties. Project developers are both the sellers of products and the buyers of raw materials. In other words, they are sensitive to the volatility of the price of products in terms of the sellers, and are sensitive to the volatility of the price of raw materials in terms of the buyers. Traditional ROV analyzed only one side of the seller or buyer [4,7–16]. In this paper, a steel plant project, which is a case study, is a large seller of steel products and a large buyer of iron ores and coal. In this case, the cash flow of the project is separated into cash inflow of the sellers and cash outflow of the buyers and applied to the model. Therefore, the major contribution of this study is to present a model that simultaneously considers both the seller of products and buyer of raw materials to solve this issue of the traditional ROV. This study also confirms the suitability of steel plant projects by applying an ROV that suits the characteristics of OPM. Then, the results of the model’s application to a case study of a steel plant project is described to verify that the cash outflow of the project should be considered. A sensitivity analysis is performed to check how differences between the volatilities of cash inflow and cash outflow affect the option value. 2. Literature Review In economics, an asset price moves in a random walk, assuming an efficient market hypothesis [17]. In the efficient market hypothesis, the PV completely reflects the past record of asset prices, and the market reacts directly to the latest information about the asset. If these hypotheses are met, the asset price is said to follow the Markov process [18]. Over time, the variables that change uncertainly in the continuous time process follow the stochastic process. The process of changing these variables is called the Wiener process, which is a probability process with an annual mean of 0 and an annual variance of 1 [19]. In thermal physics, this process behaves similarly to the motion of a molecule due to the many small impacts from other molecules, and is called Brownian motion. An option provides investors with the right to buy or sell something until maturity at a specific contracted price depending on the type of option (call or put), but investors are not obliged to exercise this right [20]. The underlying assets for options include stocks, stock indices, interest rates, currencies, crude oil, insurance contracts, weather, and even real assets, such as projects. ROV dealing with the value of diverse projects through various option types has been actively researched, as are the following papers. The Black-Scholes model is common in the OPM [21]. The Black-Scholes partial differential equations can be derived using Ito’s lemma. Trigeorgis [22] focused on easily introducing an option to expand, contract, abandon, and defer projects through analysis of comprehensive industrial examples, such as raw material development, rural urban development, alternative business models, flexible production systems, and elsewhere. Lee et al. [23] used ROV as a solution to financial barriers to government guarantees in implementing green building projects. They measured ROV by estimating the value of the government guarantees for risks to uncertainties of that asset value. Lee et al. [24] obtained the ROV for a sequential investment in commercial energy retrofit projects on building performance risks as a case study. They also provided investors with advanced management knowledge of investments in performance risks with more accurate project information by assessing potential risks, barriers, and revenues. Zhao and Tseng [2] studied the ROV for an optimal infrastructure investment in preparation for the expansion of a public parking garage project. They identified the value of improving infrastructure flexibility

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for infrastructure expansion by applying ROV. Mayer and Kazakidis [25] acquired values for project capacity options and project shutdown options for a mine production system design and applied ROV to the investment decision process. Each type of ROV was only represented by a call or put option depending on the project situation of the ROV. Studies using ROV to determine the optimal investment timing of projects have steadily developed. Trigeorgis [8] investigated the effects of competition on the optimal timing to start a project through an option pricing model (OPM). The author said that the presence of competitors can speed up a company’s planned investment and the expected preemptive competition is similar to dividends on options. Benaroch and Kauffman [10] analyzed the timing of the deployment of information technology projects and a range of issues associated with the application of ROV to problems in capital budgeting. The authors provided a formal rationale for the validity of the Black-Scholes model in relation to the capital budgeting methods that might be used to evaluate projects, and conducted sensitivity analyses to support verification. Kauffman and Li [12] developed a continuous-time probability model to determine the optimal timing for technologically controlled selection within the ROV framework. They also suggested that technology adopters should postpone their investments until the target technology dominates the market where the probability of achieving a critical mass reaches a threshold. Bøckman et al. [4] presented an ROV for choosing the optimal timing and capacity for small hydropower plants. They presented ROV along with continuous expansion of how to evaluate the small hydropower plant projects affected by uncertain electricity prices and suggested price limits to initiate the project. These studies have been conducted to evaluate the value of technology or business using ROV’s deferring option, and investigate optimal investment timing in various projects. The abovementioned studies focused on inward project cash flow volatility only and did not consider outward cash flow volatility simultaneously. Odeyinka et al. [15] studied 26 identified significant risk factors for baseline forecasting of construction cost flow, and proposed a predictive mode for risk assessment. Maravas and Pantouvakis [16] conducted a study to calculate activity duration and cost through an uncertain project cash outflow based on the payment methods or the fluctuations in capital requirements. In a construction project, Hwee and Tiong [11] confirmed the influence of cash inflow and outflow on the sensitivity analysis, according to five factors: duration, over/under measurement risk, variation risk, material cost variances, predicted cash inflow, and cash outflow in progress. Kenley and Wilson [7] found that external cash flow is more relevant to the model than internal cash flow when forecasting net cash flows in a construction project. These studies emphasized the importance of predicting cash outflows rather than cash inflows in project implementation. However, they have common limitations. That is, these studies focus on the construction phase of the project, missing the valuation of the entire lifecycle project over several decades. In order to overcome these limitations, this paper performs the estimation and forecasting of cash inflow and cash outflow over several decades. The manufacturing enterprises fundamentally generate cash flows by constructing a factory, buying raw materials, operating a factory, and selling products. Factory construction, raw material purchase, and factory operation cause cash outflow, and product sales lead to cash inflow. In particular, prices of raw materials and products fluctuate with time. Both inward cash flow volatility and outward cash flow volatility are significant concerns in investment decisions. Therefore, this paper presents a model to consider both the volatility of cash inflows and cash outflows. The main component of cash inflow is the project’s revenue, which is the unit price multiplied by the sales volume. These studies considered ROV as a major factor in the volatility of unit prices and/or volatility of sales volumes. However, the volatility of cash outflow can also be a major part of the project’s value depending on the nature of the project. The main components of cash outflow are capital expenditure (CAPEX) and operating expenditure (OPEX). CAPEX refers mainly to the initial investment cost of the project in a construction period. The volatility of CAPEX was identified through various cases of final project costs that differed from contracted project costs [9]. OPEX refers to the materials, labor, overhead, and other costs required in a business period. In particular, several studies addressed fluctuations in material costs [13,14,26].

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Chen [13] conducted a significant studybehavior of inter-cluster and intra-cluster responses the volatility responses to the volatility of long-term of raw material prices. Dwyer et al. to [14] researched of long-term behavior of raw material prices. Dwyer et al. [14] researched the volatility of various the volatility of various commodity prices, including steel products, on short-term price dynamics commodity including products, short-term price for financial investors for financial prices, investors based onsteel long-term data.onJacks et al. [26] has dynamics used historical data since 1700 to based on long-term data. Jacks et al. [26] has used historical data since 1700 to analyze the impact analyze the impact of commodity price volatility on poor countries and found that commodity price of commodity price volatility on poor countries and found that commodity price volatility does not volatility does not vary with output. Their papers used long-term price estimation and short-term vary output.through Their papers used long-term price estimation andclaims short-term estimation price with estimation the volatility of raw material prices. Their focus price on raw material through the volatility of raw material prices. Their claims focus on raw material prices themselves prices themselves and need to take advantage of the forecasting part of the project value needed for and company's need to take advantage of the forecasting part of we the use project needed theclaim company’s the ongoing development. In this paper, the value volatility that for they in the ongoing development. In this paper, we use the volatility that they claim in the estimation phase estimation phase and further use the forecasting phase of ROV to evaluate the project value. To and further use the forecasting phase of ROV to evaluate the project value. To overcome the gap in overcome the gap in the current literature [4,7–16] presented within this paper is the optimal the current literature presented this paper is theoption, optimal investment timingvolatilities by means investment timing by[4,7–16] means of an ROV within of two-color rainbow which can consider of an ROV of two-color rainbow option, which can consider volatilities in both the cash inflow and in both the cash inflow and cash outflow. Rainbow option valuation represents options whose payoff cash outflow. Rainbow option valuation represents options whose payoff depends on the multiple depends on the multiple underlying assets [27]. In other words, the value of the options is determined underlying assets and [27].circumstances In other words, of theassets. options is determined by the conditions and by the conditions of the the value underlying circumstances of the underlying assets. 3. Optimal Investment Timing for a Project 3. Optimal Investment Timing for a Project This study analyzes how ROV applies to projects. Applicability of the project to ROV requires This study analyzes how (MCSs) ROV applies to projects. Applicability ofathe project to ROV requires two Monte Carlo simulations to satisfy a Markov process and Wiener process [28]. First, two Monte Carlo simulations (MCSs) to satisfy a Markov process and a Wiener process [28]. First, this study proposes a cash flow model using a MCS to satisfy the Markov process in the project.this A study proposes a cash flow model using a MCS to satisfy the Markov process in the project. A MCS is MCS is a method of iteratively extracting the PV of cash flows at random according to a probability a method of iteratively extracting the PV of cash flows at random according to a probability distribution distribution of input data, such as CAPEX, OPEX, and revenue. In practice, project cash flows change of input data, such CAPEX, OPEX, revenue. In practice, project flowscontinuously. change continuously continuously overastime. Thus, the and overall project value can alsocash change If the over time. Thus, the overall project value can also change continuously. If the distribution of the rate distribution of the rate of return on the PV of the cash flows from the second MCS is a standardized of return on the PV of the cash flows from the second MCS is a standardized normal distribution with normal distribution with a mean of 0 and variance of 1, the Wiener process is also satisfied. Figure 1 a mean of 0a and the conditions Wiener process is also satisfied. Figureto 1 illustrates a flow diagram illustrates flowvariance diagramofof1,the for satisfying applicability a project from the Blackof the conditions for satisfying applicability to a project from the Black-Scholes model to ROV. Scholes model to ROV.

Figure 1. Flow Figure 1. Flow diagram diagram for for the the application application of of real real options options valuation valuation to to aa steel steel plant plant project project (source: (source: authors’ design, 2017). 2017). authors’ design,

Projects occur in large, heavy industry, and thus require caution in making project investments Projects occur in large, heavy industry, and thus require caution in making project investments due to the irreversibility of the investment. This section presents a framework for project investment due to the irreversibility of the investment. This section presents a framework for project investment decisions using ROV of two-color rainbow options, which considers the volatilities of cash inflow decisions using ROV of two-color rainbow options, which considers the volatilities of cash inflow and and outflow. Figure 2 illustrates the framework of ROV and can be divided into two phases: an outflow. Figure 2 illustrates the framework of ROV and can be divided into two phases: an estimation estimation phase and a forecasting phase. The estimation phase determines the distribution of the phase and a forecasting phase. The estimation phase determines the distribution of the underlying underlying asset price through the historical data of CAPEX, OPEX, and revenue, which affect the asset price through the historical data of CAPEX, OPEX, and revenue, which affect the ROV. The ROV. The distribution derived from the estimation phase is used as an important parameter of the distribution derived from the estimation phase is used as an important parameter of the ROV. The ROV. The forecasting phase predicts the project value by deriving the ROV by changing the time to forecasting phase predicts the project value by deriving the ROV by changing the time to maturity maturity from one year to 20 years later. The forecasting value can be used to determine the optimal from one year to 20 years later. The forecasting value can be used to determine the optimal investment investment timing since it represents the project value at that time. timing since it represents the project value at that time.

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Figure 2. Framework of ROV of two-color rainbow options for optimal investment timing (source: Figure 2. Framework of ROV of two-color rainbow options for optimal investment timing (source: authors’ design, 2017). authors’ design, 2017).

The following describes in detail each of the three stages shown in Figure 2. The following describes in detail each of the three stages shown in Figure 2. 3.1. Cash Flow Model 3.1. Cash Flow Model Projects can be used as real assets in an asset price determination through the cash flow model. Projects can be used as real assets in an asset price determination through the cash flow model. Therefore, it is necessary to apply a cash flow model appropriate for the characteristics of the project. Therefore, it is necessary to apply a cash flow model appropriate for the characteristics of the project. To construct a cash flow model, the first step is to define the profitability impact factors affecting the To construct a cash flow model, the first step is to define the profitability impact factors affecting project value. This study presents three major factors from components of a pro forma income the project value. This study presents three major factors from components of a pro forma income statement, which provides the following profitability impact factors: CAPEX, OPEX, and revenue. statement, which provides the following profitability impact factors: CAPEX, OPEX, and revenue. The net present value (NPV) of the project can be calculated as the PV of cash inflow (revenue) The net present value (NPV) of the project can be calculated as the PV of cash inflow (revenue) minus the PV of cash outflow (CAPEX and OPEX) using the DCF method. However, the minus the PV of cash outflow (CAPEX and OPEX) using the DCF method. However, the disadvantage disadvantage of the DCF method is that it does not account for the volatility of profitability impact of the DCF method is that it does not account for the volatility of profitability impact factors and, factors and, therefore, does not adequately reflect the risks in the future. To overcome the weakness therefore, does not adequately reflect the risks in the future. To overcome the weakness of the DCF of the DCF method, some studies proposed evaluating the value of the cash flow model considering method, some studies proposed evaluating the value of the cash flow model considering the variability the variability of profitability impact factors using MCS [29–31]. The probability distributions of of profitability impact factors using MCS [29–31]. The probability distributions of profitability impact profitability impact factors can be drawn by probability distribution fitting, which can help provide factors can be drawn by probability distribution fitting, which can help provide an accurate value an accurate value forecast for the project. Probability distribution fitting is the fitting of a probability forecast for the project. Probability distribution fitting is the fitting of a probability distribution over distribution over collected data with repeated measurements of statistical variables, and a procedure collected data with repeated measurements of statistical variables, and a procedure for selecting for selecting a statistical distribution best suited to a dataset [32]. The authors calculate the probability a statistical distribution best suited to a dataset [32]. The authors calculate the probability distribution distribution fitting using the commercial statistical software, @Risk for Excel Version 6.3.1 (Palisade Corporation, Ithaca, NY, USA).

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fitting using the commercial statistical software, @Risk for Excel Version 6.3.1 (Palisade Corporation, Sustainability 2017, 9, 1781 6 of 16 Ithaca, NY, USA). The MCS The of the stochastic processprocess extractsextracts the random outputs of the process. MCS of the stochastic the random outputs of stochastic the stochastic process.The The probabilityprobability distribution is used as input data to account for the uncertainty of risks. Two MCSs are distribution is used as input data to account for the uncertainty of risks. Two MCSs are used to meet purposes. The first provides thethe distribution theunderlying underlying asset usedseparate to meet separate purposes. The MCS first MCS provides distribution of of the asset and the second yields the volatility of rate the rate of return underlying asset a variable and the second MCS MCS yields the volatility of the of return for for thethe underlying asset asasa variable required forThe theinput ROV.data The for input for the firstprobability MCS is the probabilityofdistribution of the required for the ROV. the data first MCS is the distribution the profitability profitability impact factors, and the input data for the second MCS is the probability distribution impact factors, and the input data for the second MCS is the probability distribution of cash inflowof cash inflowalso andresult outflow, which the1 first MCS. Table thetwo process of the and outflow, which from the also firstresult MCS.from Table summarizes the1 summarizes process of the MCSs. two MCSs.

Table 1. Summary of Monte Carlo simulations for the ROV (source: authors’ design, 2017).

Table 1. Summary of Monte Carlo simulations for the ROV (source: authors’ design, 2017).

Monte Carlo Monte Carlo Simulation

Input Data Input Data

Output Data Output Data

Object

Object

Simulation

Probability distribution of the model Probability distribution of the presentCash valuesflow of Cash flow model (to present values of cash inflow (to satisfy Markov process) and cash outflow inflow and outflow satisfy Markov process)

Probability distributions Probability distributionsof of First MCSprofitability impact factors

First MCS

profitability impact factors

Probability distribution of the Probability distribution of the cash inflow and Probability distribution of the Volatility estimation (to Probability distribution of the inflowconverted and outflow estimation present values of cash inflow and cashoutflow into a standardizedVolatility normal present values of cash inflow satisfy Wiener process) converted into a standardized (to satisfy Wiener process) and outflow outflow distribution normal distribution

Second MCS

Second MCS

3.2. Volatility Estimation

3.2. Volatility Estimation

The purpose of the second MCS is to simulate the volatility of the rate of return on the underlying

The purpose of the second by MCS is to simulate volatility of return on the underlying asset. This is obtained converting the firstthe MCS outputsoftothe therate PV and uses them as inputs for the asset. Thissecond is obtained by converting the first MCS outputs to the PV and uses them as inputs MCS. Using the probability distribution of the underlying asset from the first MCS, for it is the secondpossible MCS. Using the probability distribution underlying asset from the first MCS, it is to transform the distribution of the rateofofthe return for the underlying asset into a standardized possible tonormal transform the distribution of the for asset into a standardized distribution with a mean of 0rate andofa return variance ofthe 1 tounderlying satisfy the Wiener process. The formula for obtaining thea converted is as follows: of 1 to satisfy the Wiener process. The formula for normal distribution with mean of 0PV and a variance obtaining the converted PV is as follows:𝑃𝑉 = 𝑆𝑡𝑑. × rand(x ) + Mean (1) 𝑛

𝑛

where Std. is a value of the deviation from the probability distribution of the underlying PVstandard (1) n = Std. × rand(xn ) + Mean asset, Mean is the value of the mean from the probability distribution of the underlying asset, rand(xn) generatordeviation of the standardized distribution, and n isofthe number of trials where Std.isisa random a value variable of the standard from thenormal probability distribution the underlying from the second MCS. asset, Mean is the value of the mean from the probability distribution of the underlying asset, rand(xn ) The rate of return can is expressed using Equation (2) [33]:

is a random variable generator of the standardized normal distribution, and n is the number of trials 𝑃𝑉𝑛 + 1 from the second MCS. 𝑘𝑛 = ln ( ) (2) 𝑛 The rate of return can is expressed using Equation (2) 𝑃𝑉 [33]: where kn is the rate of return.   PVreturn n + 1 is used as the volatility in the ROV. The result of the deviation for the rate of

k n = ln

PVn

(2)

3.3 .Two-Color Rainbow Options Valuation

where kn is the rate of return. The fundamental hypothesis of OPM is that the value of underlying assets should be greater The result of the deviation for the rate of return is used as the volatility in the ROV.

than zero. If NPV of a project is taken as the underlying asset in the ROV, the hypothesis could not be satisfied because the underlying assets can have a zero, or negative, value. Therefore, various 3.3. .Two-Color Rainbow Options Valuation studies focused on the ROV with the volatility of cash inflow (revenue) from a project as the The fundamental hypothesis of OPM is thatAdditionally, the value of the underlying should be greater underlying asset [2,10,12,22–24,34–38]. volatility assets of both cash outflow andthan cash inflow a criticalisfactor affecting value of the project. Thus, it isthe reasonable to usecould exoticnot options zero. If NPV of ais project taken as the the underlying asset in the ROV, hypothesis be for a project, such as the assets rainbow option. satisfied because the underlying can have a zero, or negative, value. Therefore, various studies study uses rainbow of option n = 2,(revenue) called two-color [27]. focused on theThis ROV with thethe volatility cashwith inflow from arainbow projectoptions as the valuation underlying Rubinstein [39] stated that two-color rainbow options are options for two volatile assets that cannot

asset [2,10,12,22–24,34–38]. Additionally, the volatility of both cash outflow and cash inflow is a critical

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factor affecting the value of the project. Thus, it is reasonable to use exotic options for a project, such as the rainbow option. This study uses the rainbow option with n = 2, called two-color rainbow options valuation [27]. Rubinstein [39] stated that two-color rainbow options are options for two volatile assets that cannot be interpreted as options for only one underlying asset. They also categorized two-color rainbow options into 10 types. Among the 10 types of options, this study uses a dual-strike call/put type suitable for steel plant projects, so the equation at maturity is: max[(Sin − Kin ), (Kout − Sout ), 0]

(3)

where Sin and Sout are the underlying assets for cash inflow and outflow, respectively, Kin and Kout are the exercise prices for cash inflow and outflow, respectively, Sin 6= Sout , Kin 6= Kout , max[(Sin − Kin ), 0] means call options at maturity, and max[(Kout − Sout ), 0] means put options at maturity. European options are those that can only be stroked at maturity, and American options are those that can be stroked at any time within maturity. To apply OPM in ROV, it is necessary to consider the possibility of exercising the rights to defer before maturity. Therefore, ROV is likely to be an American option. European options need to approximate American options so that investors can exercise the right before maturity. This approximation is called Black’s approximation [40]. This method calculates the prices CT and Ct of European options with maturity T and t (t < T) using OPM, where the price that satisfies the condition of max (CT , Ct ) determines the price of the American options. Applying Black’s approximation, this study obtains an optimal invest timing value Vop with the following equation:   Vop = max

  Sin,t e−qt N(din, 1 ) − Kin,t e−rt N (din,2 ) ,   Kout,t e−rt N (−dout,2 ) − Sout,t e−qt N (−dout,1 ) ,  0

(4)

where:   S +(r −q+ 21 σin 2 )t ln Kin,t +(r −q− 12 σin 2 )t √ in,t √ √ , din,2 = = din,1 − σ t  S σin t S σin t ln Kout,t +(r −q− 12 σout 2 )t ln Kout,t +(r −q+ 12 σout 2 )t √ out,t out,t √ √ , dout,2 = = dout,1 − σ t σout t σout t 

ln

din,1 = dout,1 =

Sin,t Kin,t



(t = 1, 2,·····, T, where t is a maturity of the option). N(x) is a cumulative probability distribution function of x for a standardized normal distribution, Sin,t is PV of cash inflow as the underlying asset, Kin,t is PV of cash outflow as the exercise price, σin is a volatility of the rate of return on cash inflow, Sout,t is PV of cash outflow as the underlying asset, Kout,t is PV of cash inflow as the exercise price, σout is a volatility of the rate of return on cash outflow, r is a risk-free rate, and q is an opportunity cost for project delay (the dividend payout). The value of options has a correlation of 1 with the value of the project. Therefore, investors can choose the time t from the highest option value as the optimal investment timing. 4. Case Study 4.1. Background Steel plants are a component of an especially heavy industry and operate with a collection of high-cost equipment that affects CAPEX. OPEX is also quite high because plant reliability and automation are critical in a continuous process industry that operates 365 days per year. In addition, plant operations usually require revamps that involve replacing major equipment for each process every 15 to 20 years. The quality of steel products is not only heavily influenced by operational experience, but also the quality of materials, especially that of iron ore and coal. The price of steel

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products has a positive correlation with the price of materials (refer to Figure 3). As Figure 3 shows, the price volatility of raw materials and steel products has increased sharply since 2004. Therefore, to overcome these issues and make reliable investment decisions, investors must evaluate options to defer a steel plant and to propose optimal investment timing. Sustainability 2017, 9,project 1781 8 of 16

Figure 3. Price trends in raw materials and steel products [41]. Figure 3. Price trends in raw materials and steel products [41].

4.2. Data Collection and Parameters of a Project 4.2. Data Collection and Parameters of a Project This study examines a case of cash flow for a steel plant project C (Brazil, 2011), which contains This studyimpact examines a case of cash flow steelfor plant C that (Brazil, 2011), which contains profitability factors assumed based onfor thea data the project project at time. In traditional DCF profitability impact factors assumed based on the data for the project at that time. In traditional methods, all profitability impact factor values are fixed and each factor has only one value. However, DCF methods, profitability impact factor values are fixed and eachimpact factor has only value. if these valuesallvary due to uncertain environments, each profitability factor canone have a However, if these values vary dueprovides to uncertain eachthe profitability factor can have probability distribution, which inputenvironments, data to determine probabilityimpact distributions of the a probability which provides to revenue, determine the probability distributions PVs of cashdistribution, inflow and cash outflow usinginput MCS.data First, which is the profitability impact of of cash the cash inflow, obtained by multiplying production the unit price of the product. thefactor PVs of inflow andiscash outflow using MCS.the First, revenue,by which is the profitability impact Production is assumed be a constant value of 3 million tons, and the distribution ofof profitability factor of the cash inflow, to is obtained by multiplying the production by the unit price the product. impact factor is obtained through the annual of the unitand price the product of [41]. Second, Production is assumed to be a constant valuevolatility of 3 million tons, theofdistribution profitability the distributions of CAPEX and OPEX, which are profitability impact factors of cash outflow, as impact factor is obtained through the annual volatility of the unit price of the product [41]. are Second, follows: CAPEX reflects the cost growth in the PERT distribution based on cases of over 160 analyzed the distributions of CAPEX and OPEX, which are profitability impact factors of cash outflow, are as designCAPEX build projects consistsinofthe raw material and otherbased costs on to produce product. The follows: reflects[42]. theOPEX cost growth PERT distribution cases of the over 160 analyzed distribution of profitability impact factorofisraw obtained by reflecting volatility of the raw material design build projects [42]. OPEX consists material and otherthe costs to produce the product. The costs [41]. The source data are used as the input for the probability distribution fitting using the distribution of profitability impact factor is obtained by reflecting the volatility of the raw material statistical program @Risk. The probability distribution fitting makes the probability distributions fit costs [41]. The source data are used as the input for the probability distribution fitting using the a set of data accumulated for 15 years (from 1996 to 2011) for the variable profitability impact factors. statistical program @Risk. The probability distribution fitting makes the probability distributions fit Some of additional factors in Table 2 required to obtain the cash flow model are determined by a set of data accumulated for 15 years (from 1996 to 2011) for the variable profitability impact factors. calculation. The cost of capital according to the weighted average cost of capital (WACC) consists of Some of additional factors in Table 2 required to obtain the cash flow model are determined by the cost of equity (CoE), cost of debt before tax (CoD), debt-to-capital ratio (DCratio), and corporate tax calculation. cost of capital according weighted costasset of capital (WACC) consists rate (tax). The In this case, the cost of equityto is the derived usingaverage the capital pricing model [20]. The of themodel cost of equity (CoE), cost of debt before tax (CoD), debt-to-capital ratio (DC ), and corporate ratio also includes the risk-free rate (r), market risk premium (Rm), and beta (β). The cost of debt taxrefers rate (tax). this case,interest the costrate of equity is derived using the capital asset pricing model [20]. The to theIn borrowing in project financing: model also includes the risk-free rate (r), market risk premium (Rm ), and beta (β). The cost of debt CoE = r + β × (Rm − r) (5) refers to the borrowing interest rate in project financing: WACC = (1 − DCratio) × CoE + DCratio × CoD × (1 − tax) (6) CoE = r + β × (Rm − r) (5) To calculate the project’s profitability, it is essential to obtain a discount rate (μ), which is an index of project risks: WACC = (1 − DCratio ) × CoE + DCratio × CoD × (1 − tax) (6) μ = WACC + Country risk premium (7)

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To calculate the project’s profitability, it is essential to obtain a discount rate (µ), which is an index of project risks: µ = WACC + Country risk premium (7) Sustainability 2017, 9, 1781 Sustainability 2017, 9, 1781 SustainabilitySustainability 2017, 9, 17812017, 9, 1781

Profitability Impact Factors Profitability

Table 2. Case information for the cash flow model.

Table 2. Case information for the 2011) cash flow model. Project C (Brazil, Table 2. Case information for the cash flow model. Table 2. Case information for the cash flow model. Table 2. CaseDeterministic information for Project the cash flow model. C (Brazil, 2011) Project CProbability (Brazil, 2011)Model Unit DeterministicProject C (Brazil, 2011) Model Probability Model

9 of 16 9 of 16 9 of 16 9 of 16

Source of Data

(Using @Risk) Source of Data (Using Deterministic Project Unit Model C (Brazil, 2011)Probability Model Profitability Source of Data (Using Deterministic Impact Factors @Risk) Fixed Value Probability Variation Unit Model Profitability Probability Model Source of Data (Using Deterministic FixedModel Value (for Probability Probability Variation Impact Factors @Risk) Unit Profitability Source of Data (Using Model (for (for DCF) ROV) Fixed Value (for Distribution Probability Variation Impact Factors @Risk) Unit Model Distribution (for ROV) FixedDCF) Value (for Probability Variation Impact Factors @Risk) DCF) Distribution (for ROV) Fixed Value (forDCF) Probability Variation(for ROV) Distribution Min:3800 3800 Min: Distribution (for ROV)Min: 3800 MillionMillion DCF) CAPEX 3800 PERT Mode: 3882 3800 PERT Mode: 3882 CAPEX Million Min: 3800 USD/Project CAPEX USD/Project 3800 PERT Mode: 3882 Million Max:3990 3990 Max: CAPEX Million USD/Project 3800 PERT Min: 3800 Mode: 3882 Max: 3990 USD/Project3800 CAPEX PERT Mode: 3882 Max: 3990 USD/Project Max: 3990Mean: 813 Million OPEX 813 Normal Million Mean:813 813 Mean: Million USD/Year Std.: 130 Normal 813 813 Normal OPEX OPEX Million Mean: 813 USD/Year Std.:130 130 Std.: USD/Year OPEX 813 Normal Million USD/Year Mean: 813 Std.: 130 OPEX 813 Normal USD/Year Std.: 130 Million Mean: 1920 Revenue 1920 Normal Million Mean: 1920 USD/Year Std.:1920 281 Revenue 1920 Normal Million Mean: 1920 Million Mean: USD/Year Std.: 281 Revenue Normal Revenue 19201920 Normal USD/Year Std.:281 281 USD/Year Std.: Million Mean: 1920 Revenue 1920of data: project plan Normal  Additional factors (source documents) USD/Year  Additional factors (source of data: project plan documents) Std.: 281  Construction Period/Business Period: 4 Years/20 Years  Additional factors (source of data: project plan documents)  Construction Period/Business Period: 4 Years/20 Years Corporate tax rate: 34% factors Construction Period/Business Period: 4 Years/20 Years Additional (source of data: project plan documents) • Additional factors (source of data: project plan documents)  Corporate tax rate: 34% Debt-to-Capital ratio: 0.5 (debt: 50%, Equity: 50%)  Corporate tax rate: 34%  Construction Period/Business Period: 4 Years/20 Years 50%)  Debt-to-Capital ratio: 0.5 (debt: 50%, Equity:  Risk free rate: 3.691%  Debt-to-Capital ratio: 0.5 (debt: 50%, Equity: 50%)  Corporate tax rate:  Risk free34% rate: 3.691%  Market risk premium: 7.219% Riskratio: free rate: 3.691%50%, Equity: 50%)  Debt-to-Capital 0.5 (debt:  Market risk premium: 7.219% Beta: 0.91  rate: Market risk premium: 7.219%  Risk free 3.691%  Beta: 0.91 Cost of debt before tax: 4.311%  Beta: 0.91  Market risk Cost premium: of debt7.219% before tax: 4.311% WACC: 4.873%  Cost of debt before tax: 4.311%  WACC: 4.873%  Beta: 0.91 Country4.873% risk premium: 3.400%  WACC: Country risk4.311% premium: 3.400%  Cost of debtDiscount before tax: rate:premium: 8.273% 3.400%  Country risk  4.873% Discount rate: 8.273%  WACC:  Discount rate: 8.273%  Country risk premium: 3.400% 4.3.  Results Discount rate: 8.273% 4.3. Results

4.3. Results The case study makes several assumptions. In Table 2, the order of CAPEX spending goes from The case study makes several assumptions. In Table 2, the order of CAPEX spending goes from 4.3. Resultsequity The several In Table 2, the aorder CAPEX spending goes from to case debt.study Loan makes repayment is aassumptions. method of fully amortizing loan of and starts after five years 4.3. Results equity to debt. Loan repayment is a method of fully amortizing a loan and starts after five years from equity to debt. Loan repayment is a method of fully amortizing a loan and starts after five years from the beginning of a business period. Principal repayment is evenly repaid during the remaining The case makes assumptions. In Tablerepayment 2, the orderis ofevenly CAPEX spending goes the study beginning of several a business period. Principal repaid during thefrom remaining Thethe case study makes several assumptions. Tableis2,athe order of CAPEX spending goes from beginning of This a business period. Principal repayment is evenly repaid during the remaining business period. study assumes that all In CAPEX depreciable asset and is amortized a equity to debt. Loanperiod. repayment is a method of fully amortizing a loan and startsasset afterand five is years from on business This study assumes that all CAPEX is a depreciable amortized on a business period. This study assumes that all CAPEX asset andNet is amortized on afrom straight-line basis. All output is sold every year, soamortizing thereisisanodepreciable inventory of goods. working capital equity to debt. Loan repayment is a method of fully a loan and starts after five years the beginning of a business repayment is evenly repaid during the remaining straight-line basis. Allperiod. output Principal is sold every year, so there is no inventory of goods. Net working capital straight-line basis. Allperiod. output is sold is every year, there is noof inventory of goods. Netremaining working capital is the cost operate the plant and spent oneso year the business period and recovered in the beginning ofThis a to business Principal repayment isahead evenly repaid during the business period. study assumes that is year a depreciable asset and isperiod amortized on a business is the cost to operate the plant andall is CAPEX spent one ahead of the business and recovered in is the cost to operate the plant and is spent one year ahead of the business period and recovered in the last year of the business period. period. This study thatevery all CAPEX a depreciable asset is amortized a straight-line thebasis. last year of the business period. straight-line All assumes output is sold year, soisthere is no inventory ofand goods. Net workingon capital the last year cash of theflow business period.into the construction period and business period, each of which Project is divided Alltooutput is the sold every so there is no inventory ofbusiness goods. working capital cost to Project cash flowand isyear, divided into construction period and Net business period, each is of which isbasis. the cost operate plant is spent onethe year ahead of the period and recovered inthe Project the cashcash flowinflow is divided the construction period business period, impact each of factors which constitutes and into outflow associated with theand three profitability constitutes the cash inflow and outflow associated with the three profitability impact factors operate the and iscash spent one year ahead ofassociated the business period andprofitability recovered in the last year of the last year ofplant the business period. constitutes inflow andstudy outflow with the threewith impact factors presented inthe Tables 2 and 3. This calculates project profitability a deterministic NPV. presented in Tables 2 andinto 3. This study calculates projectand profitability with a deterministic NPV. cash flow is divided the construction period business period, each of which the Project business period. presented in Tables 2 and 𝑁3. This𝑅𝑡study calculates project profitability with a deterministic NPV. NPV = ∑outflow , soconstruction the deterministic NPV = profitability 823.34 Millionimpact USD factors 𝑅𝑡 𝑡−1 constitutes thecash cashflow inflow and associated with the threeand 𝑡=1 𝑁 (1+𝜇) Project is divided into the period business each of(8) which NPV = ∑𝑁 , so the deterministic NPV = 823.34 Millionperiod, USD 𝑅𝑡 𝑡−1 (8) 𝑡=1 (1+𝜇) ∑ NPV = , so the deterministic NPV = 823.34 Million USD (8) 𝑡=1 (1+𝜇) 𝑡−1 presented in Tables 2 and 3. This study calculates project profitability with a deterministic NPV. constitutes thetcash outflow associated with the three profitability impact presented where is theinflow project and period, N is the total project period, Rt is annual cash flow, and μ factors is the discount where t is the project period, N is the total project period, Rt is annual cash flow, and μ is the discount in Tableswhere 2 and 3. This study calculates project profitability with a deterministic NPV. t is the project period, N is the total project period, R t is annual cash flow, and μ is the discount rate. NPV ∑ , so the deterministic NPV = 823.34 Million USD (8) rate. rate. N Table where t is the project period, N is the totalflow project t is annual cash flow, andUSD). μ is the discount 3. data period, sheet forRProject C (2011, Unit: million R Cash Table 3. tCash flow for Project CNPV (2011, Unit: millionMillion USD). USD NPV = , so data the sheet deterministic = 823.34 (8) ∑Table 3. Cash rate. t−1 flow data sheet for Project C (2011, Unit: million USD). Periodt=1 (1 + µ )Construction (4 Years) Business (20 Years) Period Construction (4 Years) Business (20 Years) Period (20 Year 1 Construction 2 3(4 Years)4 5 ··· Business 21 22 Years)23 Year3. Cash flow 1data sheet 2 for Project 3 4 (2011, Unit: 5 ··· 21 22 23 Table C million USD). (1409) outflow (950) (991) (1152) (1369) · · · (1413) Year 1 is (950) 2 total 3 project 4 period, 5 · · · 21 22cash(1406) 23 where t is the Cash project period, N the R is annual flow, t Cash outflow (950) (950) (991) (1152) (1369) ·· · (1413) (1409) (1406) CAPEX (950) (1152) (950) - Business ·· ·· · (1413) - Years) outflow (950) (4 (950) (991) (1369) · (1409) (1406) Years) discount Period rate. Cash CAPEX Construction (950) (950) (950) (950) ··· (20 OPEX (120) (1003) (1003) (1003) (1003) CAPEX (950) (950) (950) (950) · · · Year 2 3 45 ··· 22 23 24 OPEX 1 (120) (1003) 21 (1003) (1003) (1003) Financial (41) (82) (82) ··· (1003) (149) (1003) (143) (1003) (138) OPEXcost (950)- (991)- (1152) (120) (1003) Cash outflow ··· (1406) (138) (1282) Financial(950) cost (41) (1369) (82) (82)(1413) ··· (1409) (149) (143) Tax cost (284) ··· (149) (261) (263) (265) Financial (41) (82) (82) (143) (138) CAPEX (950) ···(284) - ··· (261) Tax (950) (950)- (950)(263) (265) CashTax Inflow 1985 · · · (261) 1985 1985 1985 (284) · · · (263) (265) OPEX Cash Inflow - - (120) (1003) (1003) (983) 1985(1003) ·· · (1003) 1985 1985 1985 Revenue 1920 ·· ·· · 1985 1920 1920 1920 Cash Inflow 1985 · 1985 1985 ··· 1920 1920 Financial costRevenue- - (41)(82) (82) ···1920(149) (143) (138) 1920 (132) Depreciation 65 ··· 1920 65 65 65 Revenuesaving 1920 1920 1920 ··· (263) 65 65 65 TaxDepreciation saving - - -(284) ··· 65 (261) (265) (267) Net cash flow (950) (991) (1,152) 616 · · · 572 575 579 Depreciation saving (950) 65 · · · 65 65 65 Net cash flow (1,152) 572 575 579 Cash Inflow -(950) -(950) (991) 1985 ···616 1985··· 1985 1985 1985 Net cash flow (950) (950) (991) (1,152) 616 ·· · 572 575 579 Revenue 1920 ··· 1920 1920 1920 1920 Depreciation saving 65 ··· 65 65 65 65 Net cash flow (950) (950) (991) (1,152) 616 ··· 572 575 579 703

24 24 (1282) 24 µ and (1282) (1282) (983) (983) (132) (983) (132) (267) (132) (267) 1985 (267) 1985 1920 1985 1920 65 1920 65 703 65 703 703

is the

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Table 3. Cash flow data sheet for Project C (2011, Unit: million USD). Period

Construction (4 Years)

Year 1 Cash outflow (950) CAPEX (950) OPEX Financial cost Tax Cash Inflow Revenue Depreciation saving Net cash flow (950) Sustainability 2017, 9, 1781

2 (950) (950) -

3 (991) (950) (41) -

Business (20 Years) 4 (1152) (950) (120) (82) -

5 (1369) (1003) (82) (284) 1985 1920

··· ··· ··· ··· ··· ··· ···

21 (1413) (1003) (149) (261) 1985 1920

22 (1409) (1003) (143) (263) 1985 1920

23 (1406) (1003) (138) (265) 1985 1920

24 (1282) (983) (132) (267) 1985 1920

-

-

-

65

···

65

65

65

65

(950)

(991)

(1,152)

616

···

572

575

579

703 10 of 16

Apart from NPV model, this study can build modelflow as a probabilistic Apart fromthe thedeterministic deterministic NPV model, this study cana cash buildflow a cash model as a NPV model that of profitability factors.impact The authors the first show MCS probabilistic NPVreflects modelthe thatvolatility reflects the volatility ofimpact profitability factors.show The authors using the probability distributions of the profitability impact factors for the case in Table 2. The number the first MCS using the probability distributions of the profitability impact factors for the case in Table ofThe simulation is setrepetitions to 10,000 to a dependable simulation. The simulation. left side of Figure 4 2. numberrepetitions of simulation is ensure set to 10,000 to ensure a dependable The left shows the probability distribution for the PV of cash inflow and the right side shows that of cash side of Figure 4 shows the probability distribution for the PV of cash inflow and the right side shows outflow. Theoutflow. figures also thealso means and deviations, asdeviations, well as the upper lower 5% that of cash The show figures show thestandard means and standard as welland as the upper values according to the probability distribution fitting. and lower 5% values according to the probability distribution fitting.

Figure Project C C obtained obtained from from the the first firstMCS. MCS. Figure4.4.Distributions Distributionsof ofPV PV of of cash cash inflow inflow and and outflow outflow for for Project

From probabilistic NPV NPV assuming assuming that that the the mean mean From these these distributions, distributions, itit is is possible possible to to obtain obtain aa probabilistic value is a representative value for these distributions. Therefore, the probabilistic NPV of Project C value is a representative value for these distributions. Therefore, the probabilistic NPV of Project C is is$632.96 $632.96 (15,063.46 14,430.50) million USD. If investors have a specific investment strategy, (15,063.46 − −14,430.50) million USD. If investors dodo notnot have a specific investment strategy, the the investment can be approved by a DCF criteria that NPV of the project is greater than investment can be approved by a DCF criteria that NPV of the project is greater than zero. However,zero. the However, theNPV probabilistic NPV can vary depending oninflows the PVsand of cash inflows and outflows. probabilistic can vary depending on the PVs of cash cash outflows. Thecash deterministic The deterministic NPV ($823.34 million USD) is also one of the probabilistic NPVs. However, in the NPV ($823.34 million USD) is also one of the probabilistic NPVs. However, in the extreme cases of extreme cases of probabilistic NPVs in Figure 4, if the cash inflow has a value below 5% ($11,907 probabilistic NPVs in Figure 4, if the cash inflow has a value below 5% ($11,907 million USD) and million andhas the acash outflow value above 5%USD), ($16,182 the NPV project the cashUSD) outflow value abovehas 5%a($16,182 million themillion NPV ofUSD), the project willofbethe negative will be negative ($−4275 million USD) and the investment must be rejected. To prepare for ($−4275 million USD) and the investment must be rejected. To prepare for these uncertainties,these it is uncertainties, it is critical to determine the optimal investment timing that maximizes the project critical to determine the optimal investment timing that maximizes the project value. value.The second Monte Carlo simulation is performed to estimate the volatility of rate of return of the The PVs second Carloand simulation performedthe to estimate volatility of rate values of return of project’s for Monte cash inflow outflow. is Substituting mean andthe standard deviation from the PVs of forcash cashinflow inflowand andoutflow outflow. theEquation mean and deviation the project’s distribution ofSubstituting Project C into (1),standard the authors arrive values at the from the distribution of cash inflow and outflow of Project C into Equation (1), the authors arrive at results in Table 4. the results in Table 4. Table 4. PV of cash flow for a steel plant project. Case

Case Project C Project C

Table 4. PV of cash flow for a steel plant project. PV of Cash Inflow PV of Cash Outflow

PV of Cash Inflow PVin,n = 1918.12 × rand(xin,n) + 15,063.46 PVin,n = 1918.12 × rand(xin,n ) + 15,063.46

PV of Cash Outflow PVout,n = 1077.67 × rand(xout,n) + 14,430.50 PVout,n = 1077.67 × rand(xout,n ) + 14,430.50

Equation (2) is used to perform the second Monte Carlo simulation 10,000 times to determine the volatility of the rate of return on the project’s PVs for cash inflow and outflow. The results are shown in Table 5.

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Equation (2) is used to perform the second Monte Carlo simulation 10,000 times to determine the volatility of the rate of return on the project’s PVs for cash inflow and outflow. The results are shown in Table 5. Table 5. Volatility of the rate of return of a steel plant project’s PV. Case

Volatility Estimation for Cash Inflow

Volatility Estimation for Cash Outflow

Project C

18.61%

10.68%

The results in Table 5 are used as the volatility for the ROV. All elements to apply ROV have been collected, and Table summarizes the elements. Sustainability 2017, 9,61781 11 of 16 Table 6. Elements totocalculate plantproject’s project’sROV. ROV. Table 6. Elements calculatethe the steel steel plant

Notation Sin,t Sin,t Kin,t Kin,t σσinin out,t SS out,t KK out,t out,t σout σout r µr qμ q

Notation

Description PV of cash inflow as the underlying asset PV of cash inflow as the underlying asset PV of cash outflow as the exercise price PV of cash outflow as the exercise price volatility the rate rateofofreturn return cash inflow volatility of of the onon cash inflow PV outflowasasthe theunderlying underlying asset PVof ofcash cash outflow asset PVof of cash cash inflow price PV inflowasasthe theexercise exercise price Volatility of the rate of return on cash outflow Volatility of the rate of return on cash outflow Risk-free rate Risk-freerate rate Discount Discount Notion of opportunity costrate for project delay Notion of opportunity cost for project delay Description

Project C (Brazil, 2011) Project C (Brazil, 2011) $15,063.46 million USD $15,063.46 million USD $14,430.50 million USD $14,430.50 million USD 18.61% 18.61% $14,430.50 million USD $14,430.50 million USD $15,063.46 million USD $15,063.46 million USD 10.68% 10.68% 3.691% 3.691% 8.273% 8.273% 4.582% 4.582%

The elements Sin,t ,SK S,out,t , and were derived from the first MCS, and the elements σin and The elements in,tin,t , K,in,t Sout,t , andKKout,t out,t were derived from the first MCS, and the elements σin and σout were derived fromfrom the second MCS. Otherwise, r isrthe risk-free rate as as thethe mean London σout were derived the second MCS. Otherwise, is the risk-free rate mean LondonInter-Bank InterOffered Rate for the past years, µ is a discount rate based the on capital asset pricing model; and q is Bank Offered Rate for15 the past 15 years, μ is a discount rateon based the capital asset pricing model; the opportunity of the project from the market price of risk calculated as µ − r [19]. and q is the cost opportunity cost ofderived the project derived from the market price of risk calculated as μ − r [19]. Substituting the elements in Table 6 for each project and the maturity of the option, t, from one the elements Table the 6 forresults each project and the of To the illustrate option, t, from one year to 20Substituting years into Equation (4)inyields illustrated inmaturity Figure 5. the optimal year to timing, 20 years the into time Equation (4) is yields the horizontal results illustrated in Figure 5. To illustrate the optimal investment (year) on the axis and the option price (million USD) is investment timing, (year) on the 5, horizontal andisthe option price (million USD) on on the vertical axis. Inthe thetime legend ofisFigure the call axis option the option price when cashis inflow the vertical axis. In the legend of Figure 5, the call option is the option price when cash inflow is the is the underlying asset, and the put option is the option price when cash outflow is the underlying underlying asset, and the put option is the option price when cash outflow is the underlying asset. asset. Finally, the rainbow option indicates the price of the two-color rainbow options when it is the Finally, the rainbow option indicates the price of the two-color rainbow options when it is the underlying asset, considering both cash outflow.In InFigure Figure5,5, the orange shows underlying asset, considering both cashinflow inflowand and cash cash outflow. the orange starstar shows the optimal investment timing and option value derived from the rainbow option, while the the optimal investment timing and option value derived from the rainbow option, while the blue starblue star shows timingand andoption optionvalue value derived from option, which is the shows the the optimal optimal investment investment timing derived from thethe callcall option, which is the conventional method. conventional method.

Figure 5. Options values determinethe the optimal optimal project timing. Figure 5. Options values toto determine projectinvestment investment timing.

5. Discussion In order to maintain sustainable entrepreneurship in an unstable market, enterprises should develop steadily [43]. For the development of the enterprise, investors need to focus on long-term growth rather than short-term vision [44–46]. This study suggests the two-color rainbow option, which is a more robust model of real options valuation, to respond to the long-term unstable market for investors. The conventional method, real options valuation, focused on cash inflow volatility

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5. Discussion In order to maintain sustainable entrepreneurship in an unstable market, enterprises should develop steadily [43]. For the development of the enterprise, investors need to focus on long-term growth rather than short-term vision [44–46]. This study suggests the two-color rainbow option, which is a more robust model of real options valuation, to respond to the long-term unstable market for investors. The conventional method, real options valuation, focused on cash inflow volatility when evaluating2017, the project Sustainability 9, 1781 value as call options. However, the proposed method, the two-color rainbow 12 of 16 option, considers cash inflow and cash outflow simultaneously when evaluating project value as call is associated withoptions. CAPEX The and OPEX is a major sub-factor CAPEX, and options and put inwardvolatility, cash flowconstruction volatility iscost associated with revenueofvolatility, raw materialselling price isprice a major of OPEX. Based onOutward the case study results, this paper suggests the product is a sub-factor major sub-factor of revenue. cash flow volatility is associated that itCAPEX is moreand reasonable to consider both cash inflow cashsub-factor outflow when the correlation with OPEX volatility, construction cost is and a major of CAPEX, and raw between material raw material average price and product unit average price is 0.9144 (refer to Figure 3), as in the price is a major sub-factor of OPEX. Based on the case study results, this paper suggests thatsteel it is plant Steeltoplant companies typically link steel product prices to raw material prices to reduce more project. reasonable consider both cash inflow and cash outflow when the correlation between raw risk. Therefore, correlation between raw materials steel products is close This result is material averagethe price and product unit average price isand 0.9144 (refer to Figure 3), astoin1.the steel plant similar in steel industries as well as oil link and steel gas industries, power generation and sorisk. on. project. Steel plant companies typically product prices to raw materialindustries, prices to reduce Thus, thesethe industries need to consider the volatility cash outflowismore cash Therefore, correlation between raw materials and of steel products close than to 1. the Thisvolatility result is of similar inflow. in steel industries as well as oil and gas industries, power generation industries, and so on. Thus, these The above study of steel plant project effectively demonstrates the of difference between industries need case to consider thethe volatility of cash outflow more than the volatility cash inflow. the conventional method (real options valuation) and the demonstrates proposed method (two-color rainbow The above case study of the steel plant project effectively the difference between the options). The rainbow option result (15 years, $2199.79 million USD) differs from those for the call conventional method (real options valuation) and the proposed method (two-color rainbow options). option (eightoption years,result $2051.56 million USD) that earlier gauge optimal investment The rainbow (15 years, $2199.79 million USD)studies differsused from to those for the call option (eight timing.$2051.56 The steel plant project caseearlier studystudies is dominated the put option by meanstiming. of cash The outflow, years, million USD) that used toby gauge optimal investment steel as well as thecase call study optionisby means of by cash whenby determining optimal timing. Therefore, the plant project dominated theinflow put option means of cash outflow, as well as the call authors suggest investors should the volatility both cash inflowsthe and cash outflows option by meansthat of cash inflow whenconsider determining optimal in timing. Therefore, authors suggest in aninvestors investment decision based the implied option and NPV. if investors make an that should consider theon volatility in both cash value inflows and cashEven outflows in an investment investment decision onlyoption the DCF method, the investment can bemake approved without deferring decision based on theusing implied value and NPV. Even if investors an investment decision the project the NPV is greater than zero. However, investors who want to maximize profitssince can using only since the DCF method, the investment can be approved without deferring the project benefit more from making decisions based on the method proposed in this study. In economics, the NPV is greater than zero. However, investors who want to maximize profits can benefit more rational investors demand compensation forproposed risk, so in thethis price of an asset necessarily reflects the from making decisions based on the method study. In economics, rational investors magnitude of the risk that the asset [20]. rational investors in this will choose the demand compensation for risk, so thehas price ofThus, an asset necessarily reflects the paper magnitude of the risk option make the[20]. bestThus, return, if the project is fully reflected in risks. Thethe results aretosummarized in that thetoasset has rational investors in this paper will choose option make the best Figure return, 6ifbelow. the project is fully reflected in risks. The results are summarized in Figure 6 below.

Figure 6. Summary of the project values by evaluation methods. Figure 6. Summary of the project values by evaluation methods.

This paper additionally confirms the conditions under which the proposed two-color rainbow options and existing real options valuations yield the same results. The parameters σin and σout affect the option value, where σin represents the volatility of the rate of return on cash inflow, and σout represents that for cash outflow. In this study, the sensitivity analysis of σin and σout is conducted to check the change of the options value as the volatility of the underlying asset changes. Table 7 shows the sensitivity analysis for σin and σout.

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This paper additionally confirms the conditions under which the proposed two-color rainbow options and existing real options valuations yield the same results. The parameters σin and σout affect the option value, where σin represents the volatility of the rate of return on cash inflow, and σout represents that for cash outflow. In this study, the sensitivity analysis of σin and σout is conducted to check the change of the options value as the volatility of the underlying asset changes. Table 7 shows the sensitivity analysis for σin and σout . Table 7. Sensitivity analysis for the optimal investment condition. Volatility of Cash Inflow 15% 5% 10% Volatility of cash outflow

15% 20% 25%

8th year Call 15th year Put 14th year Put 13th year Put 13th year Put A

20% 1633 2046 2605 3173 3732 B

C

9th year Call 9th year Call 14th year Put 13th year Put 13th year Put

25% 2212 2212 2605 3173 3732

9th year Call 9th year Call 9th year Call 13th year Put 13th year Put

30% 2785 2785 2785 3173 3732

9th year Call 9th year Call 9th year Call 9th year Call 13th year Put

35% 3343 3343 3343 3343 3732

9th year Call 9th year Call 9th year Call 9th year Call 9th year Call

3882 3882 3882 3882 3882

A is optimal investment timing (Year), B is option value (Million USD), and C is option type dominated by the Two-color rainbow options

Note: An asymmetric relationship with cash inflow and cash outflow is expressed in color. Blue region is dominated by the cash inflow and is affected by the call option. Red region is dominated by the cash outflow and is affected by the put option.

Table 7 and Figure 7 show that option value has an asymmetric relationship with both cash outflow and cash inflow. The two-color rainbow options consist of two option values: the call option (volatility of cash inflow) and put option (volatility of cash outflow). Basically, the greater the volatilities of cash inflow and outflow, the greater the option value is. Since volatility is in proportion to the risk, it means that the values of risks increase. As the volatilities increase, the optimal investment timing increases, and the put option has a greater optimal investment timing than the call option. In Table 7, the call option has a condition governing the value of the rainbow option: the volatility of cash inflow should be about 10% greater than that of cash outflow. In finance, call options generally have favorable trading conditions compared to put options. In this case, the Black-Scholes model without dividends is applied. The Black-Scholes model applied in this paper is based on opportunity cost instead of dividends. Therefore, ROV is represented by the asymmetry of the volatility effect. The intuitive analysis, not technical analysis, is as follows: a problem that needs to consider the order of cause and effect; Cause: prices of iron ore and coal rose or fell. Effect: steel prices go up or down. This logic, created by the practices of the steel plant project, explains that the cash outflow affects the cash inflow through the phenomenon that the led cause affects the lagged result. Therefore, this can be seen that put options occurs as a more important factor than the call options do because put options occur as a led cause in the steel plant project. In the case study of steel plant project, the two-color rainbow option evaluation model proposed in this paper can provide more effective results to investors when the difference between inward cash flow volatility and outward cash flow volatility is not large. This means that the two-color rainbow option evaluation model can help investors achieve sustainable entrepreneurship with rational decision-making. Therefore, this study confirms the necessity to consider the volatility of cash outflow during the project planning stage through the sensitivity analysis.

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Figure7.7.Options Optionsvalue valueofofthe thevolatility volatilityofofcash cashinflow inflowand andoutflow. outflow. Figure

Conclusionand andFuture FutureWork Work 6.6.Conclusion Foreconomic economicsustainability, sustainability,investment investmentdecisions decisionsrequire requiredue dueconsideration considerationofofthe thefeatures featuresofof For steelplant plantprojects, projects,which whichare areirreversible. irreversible.Although Althoughtraditional traditionaldiscounted discountedcash cashflow flowmethods methodsare are steel conveniently implemented using datasheets, they cannot account for the volatility of the risk factors conveniently implemented using datasheets, they cannot account for the volatility of the risk factors thataffect affectthe thevalue valueofofprojects. projects.Therefore, Therefore,an anapproach approachtotosupport supportdecision-making decision-makingthat thatcan canbe beeasily easily that integrated into the discounted cash flow method, and that reflect these risks of projects, is needed. integrated into the discounted cash flow method, and that reflect these risks of projects, is needed. Thisstudy studyanalyzed analyzedaasteel steelplant plantproject projectand andshowed showedthat thatthe thevolatilities volatilitiesininthe thepresent presentvalue valueofofthe the This projectcash cashoutflow outflowand andinflow infloware aredecisive decisivefactors factorsininthe theproject projectvalue, value,unlike unlikemodels modelspresented presentedinin project previous studies. To apply these factors, unlike regular real options valuation, this study appliedreal real previous studies. To apply these factors, unlike regular real options valuation, this study applied optionsvaluation valuationtotothe theexotic exotictwo-color two-colorrainbow rainbowoptions. options.Effectively, Effectively,this thisstudy studyconfirmed confirmedthat thatthe the options valueof of options options was was dominated value of of cash outflows rather thanthan that value dominated by bythe thevolatility volatilityininthe thepresent present value cash outflows rather for cash inflows through a casea study. By altering the two of uncertainty, the sensitivity analysis that for cash inflows through case study. By altering thefactors two factors of uncertainty, the sensitivity providesprovides significant insights insights for optimal timing and clarified dominant factor forfactor a type analysis significant for investment optimal investment timing and the clarified the dominant of option. This study contributes a finding that suggests that investors prioritize cash outflow for a type of option. This study contributes a finding that suggests that investors prioritize when cash investing in projects for in sustainable development. projects are to on ROV, outflow when investing projects for sustainable If development. If support projects decision-making are to support decisionsuch ason a refinery plant, plant, aviation automobile industry, and semiconductor making ROV, such as apower refinery plant, power industry, plant, aviation industry, automobile industry, and industry, where the project is considered both seller and buyer positions, the authors suggest the semiconductor industry, where the project is considered both seller and buyer positions, theusing authors two-color rainbow options model. suggest using the two-color rainbow options model. However, this this study study has has two two limitations. limitations. First, First,the theBlack-Scholes Black-Scholesmodel modelconsiders considersonly onlythe the However, volatility of the underlying asset and does not consider the volatility of the exercise price. To overcome volatility of the underlying asset and does not consider the volatility of the exercise price. To this matter, studythis proposed incorporating the volatilities of cash inflow andinflow cash outflow as overcome thisthis matter, study proposed incorporating the volatilities of cash and cash different types of options based on an exotic option valuation. However, the proposed option model outflow as different types of options based on an exotic option valuation. However, the proposed does not account correlation the two factors, so itfactors, is possible the results be less option model doesfor notthe account for theofcorrelation of the two so itthat is possible that may the results realistic. it is necessary to presenttoa present model besides Black-Scholes model inmodel a future may be lessTherefore, realistic. Therefore, it is necessary a modelthe besides the Black-Scholes in study. Second, this study considered multiple profitability impact factors in the cash flow model, but a future study. Second, this study considered multiple profitability impact factors in the cash flow assumed the cash flow is constructed in each profitability factor model, butthat assumed that themodel cash flow model is independently constructed independently in each impact profitability becausefactor the study performed the first Monte Carlo based on each profitability impact impact because the study performed thesimulation first Monte Carlo simulation based on factor each without a relationship among profitability impact factors. Future proposals can offer more accurate and profitability impact factor without a relationship among profitability impact factors. Future proposals reasonable estimations cashreasonable flow usingestimations statistical theories datausing mining methodstheories to determine the can offer more accurateofand of cashorflow statistical or data correlation among each profitability impact factor. Recent advances in artificial intelligence technology mining methods to determine the correlation among each profitability impact factor. Recent advances help investors make decisions through sustainability more accurate incanartificial intelligence technology can economic help investors make assessment decisions using through economic forecasting techniques to using replace traditional such as Monte to Carlo simulation. Fuzzy theory, sustainability assessment more accuratemethods, forecasting techniques replace traditional methods,

such as Monte Carlo simulation. Fuzzy theory, artificial neural networks, and machine learning will help researchers develop methodologies to generate more accurate predictions and reasonable investment decision methodologies.

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artificial neural networks, and machine learning will help researchers develop methodologies to generate more accurate predictions and reasonable investment decision methodologies. Author Contributions: Yonggu Kim developed the concept and drafted the manuscript. Keeyoung Shin collected and analyzed the data. Joseph Ahn provided extensive advice throughout the study regarding the research design, methodology, findings, and revised the manuscript. Eul-Bum Lee revised the manuscript and supervised the overall work. All authors read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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