The Suez Canal Area Development Project's Investment Opportunity: A

The Suez Canal Area Development Project’s Investment Opportunity: A Real Options Approach ∗

Boping Tiana, Mahmoud A. Eissaa,b,† Guangqiang Tenga, Qi Liangc aDepartment of Mathematics, Harbin Institute of Technology, Harbin 150001, China bDepartment of Mathematics, Faculty of Science, Menouﬁa University, Menouﬁa 32511, Egypt cDepartment of Business Administration, Sun Yat-sen University, Guangzhou, China

Abstract. The importance of the Suez Canal in world trade shows through its strategic role such that providing a gateway between Eastern and Western world. Nearly 10% of all world trade pass through the Canal. But Suez Canal revenues are still small relative to its importance and unique position. So, the Egyptian government launched two inter-related major projects to transform the Suez Canal region into an international logistics, maritime and industrial hub which will offer enhanced opportunities for investment across all economic sectors including: ports and logistics, maritime services, industry, ICT, renewable energy (RE) and other areas of opportunity. Egypt has a gap on the power supply in the last three years. The government is putting a strategy plane to reach 20% of the total electricity generated from RE by 2020 Vs 9.1% in 2013 to address this energy deficit. But, one of the challenges facing the Egyptian strategy plane is how to attract RE investments?, though the RE investment are irreversible and uncertain such that the RE investment are long-term, costly and depend on a feed-in tariff system. In this paper, we discuss a real option framework for use in RE investment and apply this framework on RE investment opportunity under the Suez canal area development Project (SCADP) in the light of steps taken by the Egyptian Government to support investment in this sector. A real option framework is modeled to optimizing the time of launching the project to maximizing the utility to assess the value, and assess the value of deferred option and abandonment option. At any stage in the project, the model can inform a strategic options to defer or abandon project. Numerical methods form one of the important tools of options valuation and especially in cases where there is no closed form analytic formula. Here, we construct new two numerical methods, the first one is drifting split- step Theta Milstein (DSSθM) methods and the second one is modified split-step Theta Milstein (MSSθM) methods for solving Itôstochastic differential equations (SDEs). We examine commonly used methods DSSθM and MSSθM with another two methods, finite difference methods (FDM) and Monte Carlo (MC) method in options valuation for investments with uncertainty. A new split- step methods are integrated with concepts of Black-Scholes option pricing theory and economic principles of cost, value, risk and flexibility. Finally, we discuss the deferred and abandonment options for the project, 140 MW integrated solar combined cycle power plant in Kuraymat.

Keywords. Stochastic diﬀerential equation, Real option, Renewables energy, Egypt, Suez Canal .

1 Introduction

In recent years many efficient numerical methods are constructed for solving different types of stochastic differential equations (SDEs) with different properties (see [12, 15, 19, 41]). Split- step Theta (SSθ) methods have attracted a lot of attention due to its advantage in flexibility

∗This research was partly ﬁnanced by NSFC grant 71350005 and State Natural Sciences Foundation Monumental Projects 13&ZD166. †E-mail:[email protected] (Corresponding Author).

1 and stability. The split-step backward Euler (SSBE) method has been given [13, 26]. In order to improve the numerical properties based on the work of Higham et al. [13], many numerical methods depend on split-step methods and Milstein method have been introduced in [9, 11, 40, 43, 54]. In this paper, we derive the drifting split-step Theta Milstein (DSSθM) methods and modified split- step Theta Milstein (MSSθM) methods for use on SDEs. Numerical methods are needed for real option valuation in cases where analytic solutions are either unavailable or not easily compatible. In the area of options valuation the subject of numerical methods is very broad. There are many different types of numerical methods can be applied to real option valuation, approximation of partial differential equations (PDE) and approximation of stochastic process for underlying asset are available. There are several candidates models for the stochastic evolution of the underlying asset [25]. An overview of two of numerical methods available in the context of Black-Scholes-Merton [18, 45]. Brennan et al. [31] considered finite difference methods (FDM) which are governed by solving the underlying PDE. Monte Carlo (MC) approach is introduced by Boyle [38] give simulation of stochastic process. The comparative study of FDM and MC method for pricing European option was considered by [10]. In addition to basic usage of the methods, the methods are typically tailored to fit into a specific problem at hand (see [1, 34, 44]). In this study, we present the applicability of the two families of methods DSSθM and MSSθM that can be used to approximate a stochastic process arising from real options analysis for underlying asset and comparing with another two numerical methods FDM and MC method in assessment the uncertainty investment. In international trade, the Suez Canal has a strategic role such that providing a gateway between Asia and Europe. Nearly 10% of all world trade and 20% of world containers pass through the Canal. Its annual revenues do not exceed $ US 5 billion dollars [17]. So, The Egyptian Government launched two inter-related major projects to transform the Suez Canal region into an international logistics, maritime and industrial hub by exploiting the economic potential of the unique position of the Suez Canal. The government expects a world-class value-added services hub which will offer enhanced opportunities for investment across all economic sectors including ports and logistics, maritime services, industry, ICT, renewable energy (RE) and other areas of opportunity. There are many research studies demonstrate the economic value of the Suez Canal and discussed future vision and current projects for the Suez Canal area (see [5, 14, 20, 30, 32, 49]). Recently, The Egyptian government has faced increasing difficulties in satisfying growing electricity demand. Final consumption is dominated by the residential and industrial sectors, with respective shares of 42.3 , and 31.4 percent [22]. The installed capacity 1 has been slow to grow and has increasing peak load (i.e., peak demand) 2 Iman and Nadine discussed the energy security in Egypt [21]. Egypt plans to expand its capacity of electricity generation from RE. Renewables in Egypt are mainly hydro power, wind and solar energy. Renewables share of electricity generation is 9 % only [16]. Recently, Egypt has adopted an ambitious plan to reach 20% of the total electricity generated from RE by 2020. The Egyptian government develops policies to encourage local and international investors to invest in projects to produce electricity from RE sources. In September 2014, Egypt issued the feed-in tariffs for electricity projects produced from solar and wind sources (see [35, 46, 51]). So, in this work we introduce investment opportunities in the RE sector in the light of the Suez Canal projects and policies are taken by the Egyptian Government to encourage investment in this sector. The renewable energy is uncertain investment such that it is long-term, costly and depend on a feed-in tariff system. A. Dixit et al. [2] addressed the subject of investment under uncertainty. Here, we follow the real options approach (ROA) to address the real option valuation (ROV) of an investment in a RE and the optimal time to invest under a number of different payments settings

1 Installed capacity is the maximum electric output a generator can produce under specific conditions. 2Peak electricity load is defined as the maximum electrical power that a system can supply for a sustained period above its average supply level, so as to meet peak demand. Such peak demand is dictated by the highest point of customer consumption of electricity at a given point in time (e.g., one hour), and is affected by seasonal factors (e.g., excessively hot or cold weather).

2 [23, 24]. Fernandes et al. [7] present a review of the current state of the art in the application of ROA to investments in non-renewable and renewable energy sources. Valuation of wind energy projects are introduced by L. M. Abadie et al. [28]. The approach of Solar and wind energy production as a ROA are discussed in [3, 27]. Our work introduce, a real option framework for use in RE investment. The real option framework consider the volatility in RE price during the project lifetime and the development lag between launch the project and start the production. Also, The real option framework diﬀers of a previous work since, the new numerical methods DSSθM and MSSθM methods are integrated with option theory and the four economic elements cost, value, risk and ﬂexibility to value a real option. The paper is organized as follows. In section 2, we derive the DSSθM and MSSθM methods. The projects to development the Suez Canal area are presented. The investment opportunity in sector RE in Egypt under the Suez canal projects and Egyptian government policies is discussed in section 3. In Section 4, we present a real option frame work. Numerical applications for the solar thermal in Egypt is discussed in section 5.

2 The Split-step Theta Milstein methods

We consider the ItˆoSDEs of the form

dy(t) = f(y(t))dt + g(y(t))dW (t), t ∈ [t0,T ],

y(t0) = y0, (2.1) where f(y(t)) is the drift coefficient, g(y(t)) is the diffusion coefficients. Wiener process W (t) is defined on a given probability space (Ω, F,P ) with a filtration {Ft}t≥0 which satisfies the usual conditions, whose increment ∆W (t) = W (t + ∆t) − W (t) is a Gaussian random variable N(0, ∆t). In this section, we derive new two families of Split-step Theta Milstein methods, the DSSθM and MSSθM methods of SDEs (2.1). The numerical solution of SDEs (2.1) using drifting split-step methods have appeared over the past several years in the literature. Higham et al. [13] discussed the SSBE method possessing strong order 0.5

∗ ∗ yn = yn + hf(yn), ∗ ∗ yn+1 = yn + g(yn)∆Wn. (2.2) Wang et al. [39] presented the drifting split-step backward Milstein (DSSBM) method of strong order 1.0.

∗ ∗ yn = yn + hf(yn), 1 y = y∗ + g(y∗ )∆W + g(y∗ )´g(y∗ ) (∆W )2 − h , (2.3) n+1 n n n 2 n n n and modiﬁed split-step backward Milstein (MSSBM) method 1 y∗ = y + h[f(y∗ ) − g(y∗ )´g(y∗ )], n n n 2 n n 1 y = y∗ + g(y∗ )∆W + g(y∗ )´g(y∗ )(∆W )2. (2.4) n+1 n n n 2 n n n Let us rewrite the SDEs (2.1) in the following form:

dy(t) = f(t, y(t))dt + g(t, y(t))dW (t), y(t0) = y0, t ∈ [t0,T ], (2.5) where f(t, y(t)) is the drift coefficient and g(t, y(t)) is the diffusion coefficients. For SDEs (2.5), Similar to DSSBM and MSSBM methods, Wang et al. [40] introduced a drifting split-step backward balanced Milstein (DSSBBM) method

∗ ∗ yn = yn + hf(tn, yn), 1 ∂g y = y∗ + g(t , y∗ )∆W + g(t , y∗ ) (t , y∗ ) (∆W )2 − h + C (y∗ − y ), (2.6) n+1 n n n n 2 n n ∂y n n n n n n+1

3 and modiﬁed split-step backward balanced Milstein (MSSBBM) method 1 ∂g y∗ = y + h[f(t , y∗ ) − g(t , y∗ ) (t , y∗ )], n n n n 2 n n ∂y n n 1 ∂g y = y∗ + g(t , y∗ )∆W + g(t , y∗ ) (t , y∗ )(∆W )2 + C (y∗ − y ), (2.7) n+1 n n n n 2 n n ∂y n n n n n n+1

2 where Cn = c0(tn, yn)h + c2(tn, yn) (∆Wn) − h . Ding et al. [54] considered the SSθ methods for SDEs (2.5)

∗ ∗ yn = yn + (1 − θ)hf(tn, yn) + θhf(tn, yn), θ ∈ [0, 1], ∗ ∗ yn+1 = yn + g(tn, yn)∆Wn, (2.8) which is generalization of SSBE method (θ = 1). Guo et al. [43] presented modiﬁed split-step composite θ-Milstein (MSSCTM) methods of strong order 1.0 for SDEs (2.5). 1 ∂g y∗ = y + h[(1 − θ)f(t , y ) + θf(t , y∗ ) − g(t , y∗ ) (t , y∗ )], θ ∈ [0, 1], n n n n n n 2 n n ∂y n n 1 ∂g y = y∗ + [λg(t , y∗ ) + (1 − λ)g(t , y )]∆W + g(t , y∗ ) (t , y∗ )(∆W )2, (2.9) n+1 n n n n n n 2 n n ∂y n n n where the values of parameter θ and λ are in [0, 1]. Huang et al. [9] introduced SSθ methods with strong order 0.5 as follows for SDEs (2.5).

∗ ∗ yn = yn + θhf(tn + θh, yn), θ ∈ [0, 1], ∗ ∗ ∗ yn+1 = yn + (1 − θ)hf(tn + θh, yn) + g(tn + θh, yn)∆Wn, (2.10) which is also generalization of SSBE method. In this paper, for SDEs (2.5), similar to previous methods, we use SSθ methods (2.10) on the drift function instead of using the backward Euler method on the drift function in DSSBM method to present the drifting split-step Theta Milstein (DSSθM) methods

∗ ∗ yn = yn + θhf(tn + θh, yn), (2.11) ∗ ∗ ∗ yn+1 = yn + (1 − θ)hf(tn + θh, yn) + g(tn + θh, yn)∆Wn 1 ∂g + g(t + θh, y∗ ) (t + θh, y∗ )[(∆W )2 − h], (2.12) 2 n n ∂y n n n and modiﬁed split-step Theta Milstein MSSθM methods 1 ∂g y∗ = y + h θf(t + θh, y∗ ) − g(t + θh, y∗ ) (t + θh, y∗ ) , (2.13) n n n n 2 n n ∂y n n ∗ ∗ ∗ yn+1 = yn + (1 − θ)hf(tn + θh, yn) + g(tn + θh, yn)∆Wn 1 ∂g + g(t + θh, y∗ ) (t + θh, y∗ )(∆W )2, (2.14) 2 n n ∂y n n n In order to facilitate the description, we study the DSSθM and MSSθM methods (2.11-2.14) of SDEs (2.1) as the following form. The DSSθM methods

∗ ∗ yn = yn + θhf(yn), (2.15) 1 y = y∗ + (1 − θ)hf(y∗ ) + g(y∗ )∆W + g(y∗ )´g(y∗ )[(∆W )2 − h]. (2.16) n+1 n n n n 2 n n n and the MSSθM methods 1 y∗ = y + h θf(y∗ ) − g(y∗ )´g(y∗ ) , (2.17) n n n 2 n n 1 y = y∗ + (1 − θ)hf(y∗ ) + g(y∗ )∆W + g(y∗ )´g(y∗ )(∆W )2, (2.18) n+1 n n n n 2 n n n

4 where yn is approximation to y(t), θ is a free parameter, with increments ∆Wn := W (tn+1) − W (tn) are independent N(0, h)-distributed Gaussian random variables and y(0) = y0. Moreover, yn is {Ftn }-measurable at the mesh-point tn. In our previous work [8, 29], we proved the convergence of order 1.0 and mean-square stability for the two families of methods DSSθM and MSSθM for SDEs (2.1, 2.5), respectively. In this paper, the new two DSSθM and MSSθM numerical methods (2.15-2.18) are examine of the geometric Brownian motion (GBM) is a especial case of 2.1 which is used to model of real option valuation (ROV).

3 Suez Canal projects and investment opportunities

In international trade, the Suez Canal has a strategic role such that providing a gateway between Asia and Europe. The current length of the Suez Canal is 193.3 km since its establishment in 1869. The Suez Canal was considered the most important artery and waterway for the world trade between the production sources and the consumption markets see Figure 1. Nearly 10% of all world trade and 20% of world containers pass through the Canal [17]. Thus, it has a great impact in aﬀecting the states economies [30]. So far, Egypt has only beneﬁted from the Suez Canal through collecting fees from ships and vessels crossing it. Its annual revenues do not exceed $ US 5 billion dollars.

Figure 1: Importance of Suez Canal

Suez Canal faces threat through several projects that compete against the Suez Canal from some states and the world competition. But, the growth in world trade coupled with the need for market proximity and responsive supply chains present several opportunities for Egypt to maximize the returns on this unique asset. Especially in relation to Far Eastern manufacturers and commodity producers who are looking for locations that are close to their customers and able to provide them with less expensive options to store and further process their products for the European, Middle Eastern and African markets. Further opportunities come from the expansion of maritime and port services oﬀered to the shipping industry, midway between East Asia and Western Europe. The Egyptian government aims to build on these opportunities to transform the Suez Canal region into a global hub and to ensure the long-term growth of the Egyptian economy by exploiting the economic potential of the unique position of the Suez Canal. The Egyptian Government launched two inter-related major projects that are taking place in the Suez Canal region. A Report by British Expertise ([17] Spring 2015) discussed transition in Egypt between infrastructure and development. This report aims to advise on business opportunities in Egypt, and equip its reader with practical guidance on how to capitalise on these opportunities and do business in Egypt. In this context the report sets out to answer the question, what sectors and projects oﬀer the best business opportunities? and explain Suez Canal development projects as follows.

5 3.1 New Suez Canal The project was launched on 5 April 2014 to add an extra lane to the Suez Canal. The total project cost valued about $ 8.2 billion dollars. The project already was accomplished and operated in 6 August 2015. The New Suez Canal project is 72 km long and includes 35 km of the dry excavation works, with an estimated volume of 258,000,000 m3, and 37 km of dredging and deepening works, with an estimated volume of 242,000,000 m3 see Figure 2.

Figure 2: New Suez Canal project

The project serves the master project Suez Canal area development project (SCADP) and aims to: increase attractiveness and competitiveness of the Suez Canal, provide two-way navigation for half of the length of the Canal, minimize transit time through the Canal from 20 hours to 11 hours, minimize waiting time at the northern entrance from 11 hours to 3 hours, accommodate ships that draw 66 ft draft throughout the Canal in both directions, increase nominal capacity of the Canal from 49 ships/day to 97 ships/day in 2023.

3.2 Suez Canal Area Development Project (SCADP) The SCADP represents the ﬁrst integrated and structured approach to transform the Suez Canal region into an international logistics, maritime and industrial hub see Figure 3.

Figure 3: Suez Canal Area Development Project (SCADP)

The project has identiﬁed the objectives as Export and international trade development, based on the unique geographic position of the Suez Canal. Creating new centres of growth in the region through diversiﬁcation and expansion of existing activities, such as supply chain management and logistics, processing, port and maritime services, and tourism. Ensuring long term economic growth

6 by integrating the above activities and attracting foreign investment. Leveraging this development opportunity to adopt best practices in sustainable development, including environmental, ﬁnancial and socio-economic sustainability, as well as technological excellence. Job creation and investment, with the project expected to create millions of new jobs, shifting the demographic burden from the over-populated capital, Cairo, and other highly populated areas, to the Suez Canal governorates and Sinai. Providing additional revenues to the government of Egypt. Increasing the volumes of cargo transport due to increase in export and import of goods after implementation of the proposed projects in the Suez Canal region. Increasing the share of logistics value-added services in GDP after implementation of the proposed projects in the Suez Canal region. Increasing the share of multimodal operations in national and international supply chains. Developing Sinai peninsula. The master plan of the project was accomplished in March 2015 by Dar Al Handasah Egypt company (Presentation of the master plan can be seen in [57]). The master plan include three key plans for logistics, industrial and maritime transport services, as well as an inward investment strategy and business plan, an investment promotion strategy, and a strategic impact assessment. The master plan shown the industrial activities of SCADP in Figure 5.

Figure 4: The industrial activities of SCADP

The total project cost is likely to reach $ 15 billion, The new development would boost annual revenues from the Suez Canal, which is operated by the state-owned Suez Canal Authority, to $ 13.5 billion by 2023 from $5 billion currently. The government expects a world-class value-added services hub which will oﬀer enhanced opportunities for investment across all economic sectors including: • Ports and logistics

• Maritime services • Industry • ICT • Renewable energy (RE) The renewables sector is highly promising with strong potential for solar and wind farm development, and for the establishment of clean energy industries. • Other Areas of Opportunity The integrated nature of the SCZone is such that there will be numerous opportunities for residential, commercial, mixed use developments, as well as key social infrastructure.

To ﬁnd out more details about the project and investment opportunities, you can visit The Suez Canal zone web site http://www.sczone.com.eg/English/. In the following, under the SCADP, we discuss the investment opportunities in sector of RE.

7 3.3 Renewable energy sector in Egypt In recent years, the Egyptian Electricity Holding Company (EEHC) has faced increasing diﬃculties in satisfying growing electricity demand. Final consumption is dominated by the residential and industrial sectors, with respective shares of 42.3, and 31.4 percent [22]. The installed capacity has been slow to grow and has increasing peak load. Figure 5a and 5b present the evolution of demand and installed Capacities over the Next 20 years.

(a) (b)

Figure 5: Evolution of demand and installed capacities over the next 20 years, source: [47]

Egypt has a gap on the power supply side, which caused recurring power cuts in 2012, 2013 and 2014. This gap will increase if the lack of investment in energy generation, both conventional and renewable continued. The government is putting a strategy plans in place to address this energy deﬁcit. The RE is poised to play a leading role. Egypt has great natural potential to generate energy from renewables, especially wind, solar and biomass. The country enjoys a total annual global solar irradiance of up to 2.6 T W h/m2 and a total annual sunshine duration of up to 4,000 hours. Two third of the country area has a solar energy intensity more than 6.4 KW h/m2 day (an annual global solar insolation of 2300 KW h/m2 year)[17]. Egypt enjoys excellent wind regimes, the wind energy resources are available in large regions of the eastern and western Deserts of the River Nile and parts of Sinai with average annual wind speed from 7 to 8 m/s. In some areas especially on the Red sea cost the wind speed approaches 10 m/sec or even higher [53]. Figure 6a and 6b show the potential of solar and wind energy in Egypt.

(a) Source: GeoSun Africa (b) Source: NREA, Annual report 2012/2013

Figure 6: Potential of solar and wind energy in Egypt

The World Bank expected the ’shimmering Egyptian deserts to ’host more and more large-scale solar projects that harness its natural advantage. The World Bank also acknowledged Egypts wind power advantage, describing the Gulf of Suez as ’one of the most promising zones for wind power use on Earth.

8 Renewables in Egypt are mainly hydro-power, wind and solar energy. Fossil fuels have the lions share of electricity generation 91 %, 75 % is for natural gas, 16 % for petroleum products, and only 9% is for renewables (equivalent to 14.6 TWh in FY 2012/2013). There is 7.7 % are hydro-power generated, 1.2 % wind, and 0.1 % solar of the 9 % renewables . The above distribution is in line with the installed capacity of renewables in Egypt [16]. Egypt still has the potential to expand its capacity of electricity generation from RE. Recently, Egypt has adopted an ambitious plan to reach 20% of the total electricity generated from RE by 2020 including 12% wind, 6% hydro and 2% solar. The target is expected to be met by scaling up wind energy capacities to reach 7200 MW in year 2020, as well as reaching the solar energy target of 3500 MW installed capacities up to 2027 [16]. For example, in 2015, the total capacity of wind energy was 750 MW [42] as a compared to 550 MW in 2013 and 140 MW of solar energy in 2014 [37] as compared to zero in 2010 [36]. The Egyptian Government has introduced policies to implement RE strategy. Five policies has been approved to foster the increasing of RE energy contribution as following: 1. Public Competitive Bidding Issuing tenders internationally requesting private sector to supply power from RE energy projects. 2. Third party access (TPA) Investors are allowed to build and operate RE power plants to satisfy their electricity needs or to sell electricity to other consumers though the national grid. 3. Feed in Tariff (FIT) In September 2014, the government passed the key Feed-In Tariff Law, triggering wide interest from international developers and investors. The main parameters of the Feed-In Tariffs are: • Solar power stations: The value of the tariff is divided into five scales according to the production capacity of the station and the value of the tariff will be fixed during the contract period which reaches 25 years. • Wind power stations: The value of the tariff is calculated based on two contractual periods the first is 5 years and the second is 15 years, to reach a total period of 20 years. • Land allocation: Through the use of craft scheme for a period of time equal to the contract period. Also, the land will be giving with just 2% of the total power generated revenue from the plant. As well as, the customs will be 2% of the total items cost. • Electricity: Produced through renewable energy stations has priority access to the electricity grid. • Government support and guarantee: For power stations that exceed 500 KW include low-interest credit facilities. 4. Net Metering In January 2013, EgyptERA adopted a net-metering policy that allows small-scale renewable energy projects to feed in electricity to the grid. Generated surplus electricity will be discounted from the balance through the net-metering process. 5. Quota system heavy industries consumption will be obliged to use a percentage of its electricity consumption from RE sources.

One of the challenges facing the Egyptian government to implement RE strategy is how to attract investments in RE, though the RE investment are irreversible and uncertain. The RE investment are long-term, costly, depend on a feed-in tariﬀ system. These problems are addressed as real option framework in [2, 3, 27], which takes into account investment irreversibility, uncertainty and the ﬂexibility to RE investment. In the next section, we introduce a real option framework for use in RE investment and apply this framework on RE projects in Egypt in the light of steps taken by the Government to support investment in this sector.

9 4 Real Option framework

The valuation of real option plays an important role in the real option planning. The framework for real options provides an especial viewpoint in valuating investments with uncertainty. There are many diﬀerent methods can be applied to real option valuation (ROV). ROV methods can be categorized into two sections analytical and numerical methods. They can be further divided into subsections as represented in Figure 7. M. Schulmerich [33] gave an overview in-depth discussion and mathematical descriptions of some speciﬁc methods.

Figure 7: Classiﬁcation of ROV methods

The ROV process can be divided to five steps as follows [50]: • Step 1: Finding uncertainty investment opportunity. • Step 2: The probability distribution of the uncertainties is approximated. • Step 3: Know and analyze available real options. • Step 4: Real option valuation. • Step 5: Develop real options mind-set: by comparing the value of the options and cost to obtain options, a set of strategies and decisions can be reached. Meanwhile, the mind-set regarding flexibility available and different is established.

We consider the RE investment opportunity under the SCADP are uncertainty investment opportunity. RE investors often choose to adopt a deferred option. In this section, we develop a model to assess the value of deferred option and abandonment option. At any stage in the project, the model can inform a strategic options to defer or abandon project. Based on the particular characteristics of real options in RE investment projects, two applications will be presented, the first application considers a deferred option to invest into a RE project where the cash flows are uncertain. We assume that the payoff from the project will be given at a specific time of the lifetime of the project, this specific time is the time of the energy plant will be operated. In the second case we consider an abandonment option. We assume that project has some salvage value. In this real option framework, we distinguish the numerical methods of ROV. We examine the applicability of the DSSθM and MSSθM methods of ROV and comparing with FDM and MC method. The DSSθM and MSSθM methods are integrated with option theory and the four economic elements cost, value, risk and flexibility to value a real option. The real option framework consider the volatility in RE price during the project lifetime, the development lag between launch the project and start the production, and neglect the problem of raising funds for the project. The decision maker is facing an uncertain utility stream for investment. The valuation of real options help the decision maker to evaluate the investment opportunity.

4.1 Frame work application In the RE investment, the renewables is the underlying asset. The value of the asset is based upon two variables, the estimated of the installed Capacity (MW) of the energy plant and the pricing

10 system. To value a RE investment as an option, we need to make assumptions about a number of variables:

1. Available of the RE resource: At the outset since this is not known with certainty, The availability of renewables has to be estimated. The investor can estimate the installed Capacity (MW) of the energy plant and produced energy (KWh) by environmental assessment studies. 2. Estimated cost of establish the energy plant: The estimated development cost is the exercise price of the option. The cost of establish the energy plant can be estimated by feasibility studies for the projects. 3. Time to expiration of the option: The life of a RE option can be defined as a contract period that period will be the lifetime of the option. For example, the contract in the sector of RE is long-time contract approximately from 20 to 25 years. 4. Variance in value of the cash flows: The variance in the value of the cash flows is determined by two factors, variability in the pricing system of the RE, and variability in the estimate of available of the RE. In the more realistic case where the average of the RE resources and the RE price can change over time, the option becomes more difficult to value. 5. Cost of Delay: Since, the net production revenue cannot be started instantaneously, a time lag has to be allowed between the decision to establish the RE plant and the actual production is the cost of delay 3.

6. Salvage value of the project: The abandonment into RE investment is estimated at the same point in project lifetime. In this model, the new two DSSθM and MSSθM numerical methods are examine. Geometric Brownian Motion (GBM) is used to model of ROV. Suppose that we are seeking a valuation to a project with a finite lifetime t ∈ [0,T ]. The cash flows y from the investment are stochastic with a standard deviation σ. Hence, the evolution of cash flows over time are described as

dy(t) = ry(t)dt + σy(t)dW (t), t ∈ [0,T ], (4.1) where r is the risk-free interest rate. In the following, we derive a valuation for the investment case study problem (4.1) which is a especial case of (2.1) by using the DSSθM and MSSθM numerical methods. The SDEs (4.1) describe the pathes of cash ﬂows for the lifetime of RE investment t ∈ [0,T ]. The path values of y(t) can be calculated iteratively by DSSθM and MSSθM methods which are introduced in previous section. The future steps depend on the type of real options.

• The deferred option

If we assume that, a project requires an initial up-front investment of I (Initial cost), and that the present value of expected cash inﬂows computed right at time T is y(T ). The value of the defer option at time T is denote by V (y, T ) gives as

V (y, T ) = e−rT E[max(y(T ) − I, 0)]. (4.2)

3If the cash ﬂows are evenly distributed over time, and the exclusive rights last n years (20 years), the annual 1 1 1 cost of delay can be written as: n = 20 = 5% a year. Note, though, that this cost of delay rises each year , to 19 in 1 year 2, 18 in year 3 and so on, making the cost of delaying exercise larger over time

11 • The abandonment option

To illustrate the value of the abandonment option, we assume that y(T ) is the value of the investment project at time T , and S is the fixed salvage value from abandoning the project at any time. The abandonment values of the project can be compared to the value of the cash flows at time T . If the value of abandonment is greater than the value from continuing the project, then the decision maker consider abandoning the project. The payoff from owning the abandonment option can be given as follows

A(y, T ) = e−rT E[max(S − y(T ), 0)]. (4.3)

The value of the real option can be determined by calculating the expected value in (4.2, 4.3 ) for a given n paths as an approximation to the expected value. The value of y(t) can be determined using DSSθM and MSSθM methods to the SDEs (4.1) for each path. Finally, we compare the value of real options (4.2, 4.3) with the value of real options which are computed by FDM and MC method which are discussed in [4] to show the eﬃciency of our two methods DSSθM and MSSθM.

5 Case Study: 140 MW Integrated Solar Combined Cycle Power Plant

In this section, we present numerical solutions for an actual case study of RE investment and analyze the results. We test the evaluation model for deferred and abandonment options using DSSθM and MSSθM numerical methods. We demonstrate the efficiency of two new methods on real option framework by comparing with FDM and MC method. To illustrate the use of real option framework to assess the RE investment in Egypt under the SCADP in the light of steps taken by the Egyptian Government to support investment in this sector. considering an actual project of solar combined cycle power plant in Kuraymat, Egypt with installed capacity 140 MW. The project is one of 3 similar projects that are being implemented in Africa. The total area of the integrated solar field is about 644,000 m2 and the total solar collector is about 1920 solar collector containing 53760 mirrors. The Total cost is about $340 Million and the development lag is 4 years [37]. The risk interest rate is 8.75 % at July 30th, 2015 in Egypt[52]. To estimate the cash flow (CF) from investment, we use the next information of the project. The capacity of the project is 140 MW producing solar share of 20 MW. The annual total produced energy (GWh) is shown in Table 1, [35]. The value of the tariff of this Plant capacity project is fixed and equal $0.1434 per KWh during the contract period which reaches 25 years [6]

Table 1: The total Produced Energy (GWh) per year 2010/2011 2011/2012 2012/2013 Average 206 479 230 305

Given this information, the inputs cash flows to the real option model can be estimated as follows: Current Value of the CF = y = Value of the developed reserve discounted back the length of the development lag at the dividend yield = 0.1434 ∗ 305 ∗ 25/(1.9)2 = $302.8878 Million. Under the Government’s plan to reduce financial support for energy, the government issued its decision No. 1257 of 2014 to increase electricity prices gradually for about five years. Furthermore, under a five-year tariff reform program the government announced that, the price of the electricity generated from RE will be increased annually at the same rate as wholesale electricity through 2019 and every five years to evaluate the price of the electricity generated from RE. Using a five-year tariff reform program, we estimate the variance in solar energy prices 10 %.

12 • Valuation The deferred option

The inputs to the deferred option model can be estimated from given information as follows:

Table 2: Parameters used in the investment option case. Parameter Symbol Value Unit

Current CF from investment y(0) 302.8878 Dollar m Fixed investment cost I 340 Dollar m Time to invest T 25 Years S.d. of cash ﬂows σ 0.1 Risk-free discount rate r 0.0875

We will use the closed-form solution to benchmark the numerical results. A close resemblance to the pricing of an European call option 4 with the Black-Scholes equation [18]. Plugging the given parameters into the closed-form Black-Scholes equation yields Vexact = $264.7410 Million . Clear interpretation of this result is that a company should not pursue the project if there is an opportunity available with a greater value.

• Two split-step theta methods We derive a numerical solution with the DSSθM and MSSθM methods for the investment option. In addition to parameters listed in Table 2, we have additional parameter θ = 0.8, h = 0.145 such that the sample size is N = T/h and we compute 5000 diﬀerent discredited Bronian paths over lifetime (M = 5000). We get from (4.2)

˜ inv VDSSθM = 264.7531 ˜ inv VMSSθM = 264.7442

If we compare the value of both methods with the exact solution, we ﬁnd that the values of both methods are very close to the exact solution. Moreover, note that the investment is valuated naturally in the whole domain with both methods. Comparing the option values, we note that the error in both methods is approximately the same and decreases rapidly as the length of the time step decreases. Figure 8 shows the mean-square error at time T versus the step-size h is analyzed in log-log diagram.

Figure 8: The MS error for the split-step Theta Milstein methods.

4 In finance, a European option can be exercised only at the expiration time of the option, while an American option can be exercised at any point of time during the option lifetime. Given the price of underlying security P and the strike price S, the payoff for a call option is defined as max P − S, 0 and for a put option as max S − P, 0.

13 Figure 9: The value of the investment option (blue) and the 95% conﬁdence level with a MC simulation in comparison to the analytical solution(red).

• Monte Carlo simulation We begin the analysis by running the MC simulation with the parameters given in Table 2. 6 Using a sample size of nmax = 1.5 · 10 and 95% conﬁdence level, the simulation yields:

˜ inv VMC = 264.8050 ± 0.3161 as the value of investment option. We note that the value is reasonably close to the exact value. To investigate the convergence properties we run the simulation with smaller sample 4 sizes, descending evenly to nmin = 5 · 10 . The results of the simulation are presented in Figure 9 along with the 95% conﬁdence level.

Figure 10: Absolute error for the case study in log-log scale for the explicit and implicit FDM.

• Finite diﬀerence method Finally, we solve the investment option case with FDM. We derive a numerical solution both with the explicit and implicit interpolation scheme. In addition to parameters listed in Table 2, we have to set additional parameters for the grid. Limiting the domain to X = 900 with N = 250 nodes and using M = 105 time steps, we obtain: ˜ inv VF DM,exp = 264.7458 ˜ inv VF DM,imp = 264.7362

Comparing the values to the exact solution, we note that the values are very close to the exact solution with both methods. Moreover, note that the investment is valuated naturally

14 in the whole domain with FDM, which is not possible for example with the MC method due to path independency. Corresponding error plot of the values in log-log scale is given in Figure 10.

Generally, it is rather difficult to rigorously compare different types of numerical methods as they are constructed in a different way. Using the accuracy of the numerical solution as only metric is problematic since by increasing the number of iterations by one step does not equal increasing the grid size by one node. To investigate these two dimensions, we run additional simulations with the case study parameters given in Table 2 with varying requirements for the simulation. First we seek solutions with a condition that the absolute error is smaller than 0.1 percent. Then we examine what is the accuracy of the numerical methods when the wall-clock time on simulation is fixed approximately to one second. Comparison of the numerical methods for the investment case with respect to fixed absolute error and wall-clock time in seconds is presented in Table 3. The inputs correspond the random sample size for the MC method, the grid size with notation (N,M) for fixed X = 900 for both FDM and the number sample size and paths (M,N) for the DSSθM and MSSθM methods. The upper value from 95% confidence level was used in the MC comparison.

Table 3: Comparison of the numerical methods MC FDM-Exp FDM-Imp DSSθM MSSθM

Inputs 100.106 (80, 9000) (80, 9000) (5000, 172) (5000, 172) Value (V˜ ) 264.8050 264.7458 264.7362 264.7531 264.7442 Clock time 48.2573 0.7747 0.6002 0.0875 0.0586 ˜ Error ( V −Vexact ) 0.00024165 0.00001804 0.00001804 0.0000457 0.0000121 Vexact

We ﬁnd that, the MSSθM methods outperforms all the other methods in eﬃciency in the case study with given parameters. The performance for the MC method is clearly the worse as the clock time is close to a minute while other are able to perform calculation under half a second.

• Abandonment option

Next we turn to study and present the results for the abandonment option. We assume that option to abandon the project is valid with a ﬁxed salvage value $ 200 Million (S = 200). Summary of the parameters of abandonment option is given in Table 4.

Table 4: Parameters used in the investment option case. Parameter Symbol Value Unit

Current CF from investment y(0) 302.8878 Dollar m Salvage value S 200 Dollar m Time to invest T 25 Years S.d. of cash ﬂows σ 0.1 Risk-free discount rate r 0.0875

We note that, a closed-form solution for the abandonment problem does not exist and thus numerical methods are required to derive a solution. Following the discussion and results in the deferred option where numerical methods were compared, we derive a solution with the MSSθM methods and implicit FDM for abandonment option. The reasoning behind choosing two methods

15 instead of one is that we are able to compare and verify the numerical results as analytical solution is not available. Guided by the previous discussion and results, we use the MSSθM method for the abandonment option. Using parameters listed in Table 4, we have additional parameter θ = 0.8, h = 0.145 such that the sample size is N = T/h and we compute 5000 diﬀerent discredited Bronian paths over lifetime (M = 5000). We get ˜ inv VMSSθM = 0.0026 Note that the derived value represents the value of the abandonment option at the beginning of the project and thus does not provide relevant information during the project. That is, it can be used only to verify is it reasonable to sign a contract with the abandonment option. Clearly, if the cost of option exceeds the value of the option one should not sign the contract. To further verify the result and obtain additional information on the abandonment option, we solve the corresponding problem with the implicit FDM, which was chosen over the explicit method due to superior convergence properties. Limiting the domain to X = 900 with N = 200 nodes and using M = 300 time steps yields

˜ inv VF DM,imp = 0.0025. We note the value is relatively close to the value derived with the MSSθM methods verifying the both results to some extent.

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