PRINCETON UNIVERSITY Wind Farm Valuation Kimlee Wong 13th April 2009 Professor Warren B. Powell & Professor Birgit Rudloff Submitted in partial fulfillment of the requirements for the degree of Bachelor of Science in Engineering Department of Operations Research and Financial Engineering Princeton University I hereby declare that I am the sole author of this thesis. I authorize Princeton University to lend this thesis to other institutions or individuals for the purpose of scholarly research. Kimlee Wong I further authorize Princeton University to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. Kimlee Wong i Acknowledgements I would like to thank my advisers Professor Warren B. Powell and Professor Birgit Rudloff for all their support and input into this thesis. Professor Powell formed an energy group during the month of September, 2008 at Princeton University which consisted of graduate students as well as fellow operation research and financial engineering undergraduates. He was generous and encouraged me to participate in the group to perform research pertaining to wind farm valuation. Professor Powell generously guided me through the areas of difficulty and has set the standards for my research. As a student in her class, Financial Risk Management, taught at Princeton University, Professor Rudloff has been instrumental to my understanding of risk measures and risk analysis included in this thesis. She has been encouraging and enthusiastic about my senior thesis involving renewable energy, and has helped me think of hedging strategies for wind farm operations. I have learnt a lot from my interaction with both Professor Powell and Professor Rudloff and I am very grateful for the experience. I would also like to thank the Operation Research and Financial Engineering department and to a greater extent, Princeton University, for the abundant resources made available to all undergraduates. The guidance and generous support offered throughout my academic career will not be forgotten. Finally, I would like to thank my parents and family as well as my girlfriend for their endless support and encouragement. ii Content Acknowledgements ......................................................................................................... ii Content ........................................................................................................................... iii 1 Wind Farm Project Valuation ................................................................................. 1 1.1 Introduction ..................................................................................................... 1 1.2 Wind Farm Economics ................................................................................... 3 1.2.1 Generation plant cost .................................................................................. 3 1.2.2 Transmission cost........................................................................................ 5 1.2.3 Cost of Capital ............................................................................................ 6 1.2.4 Tax and Subsidies ....................................................................................... 7 1.3 Wind Farm Cost Structures ............................................................................. 9 1.3.1 Total Investment Cost Structure ................................................................. 9 1.3.2 Operation and Maintenance Cost Structure .............................................. 11 1.4 Economies of Scale ....................................................................................... 11 2 Managing Financial Risk in Wind Farm Investments .......................................... 14 2.1 Financial Risk ............................................................................................... 14 2.1.1 Value-at-Risk ............................................................................................ 15 2.1.2 Expected Shortfall ..................................................................................... 16 2.1.3 Average Value-at-Risk.............................................................................. 17 2.2 Hedging Risk ................................................................................................ 17 2.2.1 Electricity Futures Contracts..................................................................... 18 2.2.2 Direct Hedging and Cross Hedging .......................................................... 19 2.2.3 Electricity Storage ..................................................................................... 21 2.2.4 Hedging congestion risk of bilateral transactions ..................................... 22 3 Electricity Spot Prices ........................................................................................... 24 3.1 Looking At the Market Data ......................................................................... 24 3.2 Models for Electricity Spot Prices ................................................................ 26 3.2.1 Mean Reversion in Log of Price ............................................................... 26 3.2.2 Lognormal Spot Price Model .................................................................... 27 iii 3.2.3 Seasonality ................................................................................................ 28 3.2.4 Volatility ................................................................................................... 29 3.3 Mean Reverting Log Price and Lognormal model Parameters ..................... 31 3.3.1 Mean Reverting Log Price ........................................................................ 31 3.3.2 Lognormal Price Model ............................................................................ 33 4 Real Options.......................................................................................................... 36 4.1 Traditional Project Valuation Tools.............................................................. 36 4.2 Real Options Analysis................................................................................... 37 4.2.1 Real Options Calculations......................................................................... 37 4.2.2 Real Options versus Traditional methods ................................................. 39 4.3 Project Valuation Using Real Options .......................................................... 40 4.4 Conclusion .................................................................................................... 40 5 Projected Cash Flow Simulation and Profit Distribution Forecast ....................... 42 5.1 PJM Electricity Spot Price simulation .......................................................... 42 5.2 Bear Creek Wind Power Project ................................................................... 43 5.3 Cost and Revenue Projections ...................................................................... 45 5.3.1 Investment Cost ........................................................................................ 46 5.4 Bear Creek Expected Yearly Cash Flow ...................................................... 47 5.4.1 EBIT .......................................................................................................... 47 5.4.2 Net Income ................................................................................................ 48 5.4.3 Cash Flow Statement ................................................................................ 52 5.5 20 and 30 Year Investment Outlook ............................................................. 55 5.6 Sensitivity Analysis ...................................................................................... 57 5.7 Conditions for Profitability ........................................................................... 60 5.7.1 30 Year Investment ................................................................................... 60 5.7.2 20 year ....................................................................................................... 62 5.7.3 10 Year ...................................................................................................... 64 5.8 Risk Analysis ................................................................................................ 66 5.8.1 NPV Value at risk ..................................................................................... 68 5.8.2 NPV Expected Shortfall ............................................................................ 69 5.8.3 NPV Average Value at Risk ..................................................................... 69 6 Conclusion ............................................................................................................ 71 Appendix 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A1 References ..................................................................................................................... R1 iv 1 Wind Farm Project Valuation 1.1 Introduction The attention given to renewable energy resources has attracted large amounts of investment, and as a result, money from both the private and public sector is being poured into this area of development. Although it is clear that our future must depend on renewable energy resources, it is important to determine whether these renewable energy projects are economically sustainable. Figure 1-1Wind farm in Sydney, Australia The employment and development of renewable energy projects will depend on how they are valued by the capital markets. It will be the individual investors or corporations who are willing to expose themselves to the associated risks that decide 1 whether a project will be developed. The purpose of this thesis is to provide a valuation framework for these investors to incorporate into their decision making when investing in energy projects, in this case wind farms. It is important to note that wind farm development follows a similar process to that of other power generation plants. The two main differences are that wind farms must be located in high wind speed areas and that their size does not make a serious environmental impact. Figure 1-2 shows the typical agreements that should be secured before starting construction [1]. Shareholder agreement Insurance Site agreement Connection agreement Project Company Construction agreement Power purchase agreement Loan agreement O & M agreement Figure 1-2 Prior to forming the agreements in Figure 1-2, the investors and developers will want to hire a financial analyst to value the worth of their wind farm project. Depending on the expected investment time horizon, this analyst will model the potential revenue generation and cost of operating the wind farm. Risk measures should also be employed to help the investors minimize and manage their risk during the life time of the wind farm. Through this process, information will be made available to help formulate some of the agreements by setting benchmarks necessary for profit. 2 To start the framework, let us first begin with understanding the economics behind developing, installing and operating a wind farm. It is important for the financial analyst to understand the distribution of installation and operating costs, as well as the government and state support available to renewable projects through tax breaks and subsidies. 1.2 Wind Farm Economics Wind is an intermittent source of power, and some of the best locations for turbines are remote from main load centers. This gives rise to extra costs in generation, distribution and transmission, as well as the cost associated with the intermittency of wind. A simplified approach to calculating the cost of electricity is described below. We divide the overall cost of a wind farm into four areas: generation costs, transmission costs, cost of capital, and tax and subsidies. 1.2.1 Generation plant cost Generation cost incorporates all the cost required to bring the wind farm into working condition. This includes the installation cost, the operation and maintenance cost, and the cost of balancing day ahead trading contracts of electricity. 3 1.2.1.1 Installation cost The installation of a wind farm consists of various costs, ranging from the planning process to the actual equipment and construction. Table 1-1 summarizes each major installation cost component and lists their main attributes. Table 1-1 Wind energy conversion (WEC) Civil construction Grid connection Planning Licensing • Turbines, blades (everything on top of the tower). Mostly foreign turbine manufacturers. Price varies with demand, quantity, currency and more. • Towers, variety of styles and heights, available in the states, or imported. • Control Systems, computer equipment, cabling, interconnection apparatus, generators • The base of each turbine requires a lot of concrete, and depends on factors such as soil conditions, terrain, tower height, turbine size, wind conditions, transportation to site, etc • Cost of control room for housing computers, regularly required inventory items, and space for human occupancy • Interconnection costs (substations) to deliver electricity from a wind farm to transmission lines • Access to roads and clearing of land for wind farm which are affected by terrain and local conditions • Site selection involves research and consultation cost • Cost of development, gaining land use and approval from local authorities that regulate such projects. This process will require lawyers, lobbyists, and engineering fees in an effort to gain approval. 1.2.1.2 Operation & Maintenance Operation and maintenance costs are ongoing and are necessary to keep the wind farm in standard condition to continue electricity generation. These costs include the repair of the wind energy conversion system, the monitoring of electricity production, the overall human and capital management of the wind farm, the insurance coverage, and in the case of leased land, rent. All these costs will typically be paid on a monthly basis, and depend on the useful life the wind turbines. 1.2.1.3 Balancing cost The balancing cost is the cost a wind farm must pay for backup generation. Due to the typical day-ahead commitments to utility companies, the wind farm owner will have to make up the difference when the generated electricity falls short of the contracted amount. This could either be a daily cost or a monthly cost depending on the method of balancing the deficiency. The three options open to the wind farmer are purchasing 4 electricity on the spot market, investing in storage facilities or other generators to meet the demand [2]. 1.2.2 Transmission cost The transmission cost is the cost of transmitting electricity from the wind farm to the utility grid. It is heavily dependent on the location of the wind farm site as well as its power rating. Figure 1-3 Electrical transmission lines The transmission lines chosen must be able to hold enough capacity to handle the potential peak output of the wind farm. However, because of intermittency, the peak output of the wind power is unlikely to occur more than 30% of the time in transmission lines from the substations to the utility grid [2]. It is this inefficiency in transmission line use and their physical distance away from where electricity is needed, that tend to make both wind transmission costs and line losses high. 5 A simplified approach of the capital cost for connecting and transmitting electricity can be calculated using the equation [3]: Capital Cost = L × D + S, R (1-1) where: L = Transmission line cost ($/mile), R = Thermal line rating 1 (1/kW), D = Line length (mile), S = substation cost of transmitting ($/kW). 1.2.3 Cost of Capital For large projects, financing with a loan from a bank or financial institution is preferable as it greatly reduces the capital requirements of the developer. The main obligation is that the loan repayment has priority over the income before being distributed to the shareholders. The main factors that influence the cost of financing are the credit rating of the owner, the proposed debt to equity ratio, and whether a contract is in place for the sale of wind generated electricity. The loan can be secured through limited recourse financing 2 or alternatively, using the assets of a willing large parent to secure the loan, which will appear as a liability on its balance sheet. The lenders of debt will prefer to be reassured by a stable large company, and therefore the method a developer chooses to secure his loan will greatly influence the cost of capital available to him. In the wind energy industry, it is not unusual for the debt to equity ratio to be as high as 80%. If a limited recourse loan is made before the construction phase, then a 1020% interest rate are typical [2]. This will be significantly reduced through refinancing once the project is commissioned because many of the construction risks are no longer significant. 1 The current carried by a given transmission line conductor which results in the maximum allowable conductor temperature for a particular set of weather parameters 2 When a loan is secured based on the projected cash flow of the underlying project 6 1.2.4 Tax and Subsidies Renewable energy projects enjoy sharp reductions or exemptions from local government property taxes. In addition, income tax burdens on wind farms are low due to generous federal and state corporate income tax breaks. The combination of tax breaks and subsidies for a wind farm owner dwarf many of the electricity costs described in the above sections. Here are some important ones that developers should consider [4]. 1.2.4.1 Tax breaks The Federal Production Tax Credit (PTC) Under present law, an income tax credit of 2.1 cents/kilowatt-hour is allowed for the production of electricity from utility-scale wind turbines. It was created under the Energy Policy Act of 1992 and recently under the economic recovery legislation passed by Congress and signed into law by President Obama on February 17, 2009, has been extended to 2012. It is available for the first 10 years of operation. 30% Investment Tax Credit (ITC) This is an alternative option to claiming the Production Tax Credit and is temporarily available until 2012. Opting to convert the PTC into an ITC will also allow wind facilities to be leased, or subject to a sale and leaseback, without a loss of the credit. This option is available for small wind turbines of up to 100 kW. Modified Accelerated Cost Recovery System (MACRS) For wind projects, this entitles the wind farm owner to a six-year accelerated depreciation schedule. This benefit allows a higher depreciation deduction before income tax. The depreciation schedule for 10 to 20 year investment projects are reproduced below. 7 Table 1-2 1st year 2nd Year 3rd Year 4th Year 5th Year 6th Year 20% 32% 19.2% 11.52% 11.52% 5.76% This federal accelerated depreciation benefit can be carried through to a state level. For states with tax codes that fully conforms to the federal system, it allows a reduction in state corporate income tax as well. Reduction in state and local property, sales and other taxes 3 State tax reductions are specific to each individual state. Here are some examples where states have reduced or eliminated other taxes to encourage and aid wind farmers [5]: • Iowa reduced its property tax and made exemption for Sales & Use taxes. • West Virginia allowed a 90% reduction in its Business & Occupation taxes, and Property taxes. • Wisconsin, Minnesota and Kansas have exempted Value Added Property tax, while North Dakota enjoys 70% reduction in Property tax as well as equipment exemption from Sales tax. 1.2.4.2 Subsidies Many subsidies are given in addition to tax breaks. Besides typical federal and state subsidies, the government has included subsidies that directly affect utilities and consumers. Renewable portfolio standards (RES) This has been adopted by 28 states and uses market mechanisms to ensure that a growing percentage of electricity is produced from renewable sources. The RES provides a predictable, competitive market, within which renewable generators compete with each other to lower prices. 3 Refer to the Database for State Incentives for Renewables and Efficiency for a full reference of state and federal incentives. URL: http://www.dsireusa.org/index.cfm?EE=1&RE=1 8 Green energy programs In many states, utility companies are mandated to pay premium prices for renewable energy sources. Apart from the tax breaks and subsidies mentioned above, Congress has passed many other benefit packages to renewable energy developers. This includes federal agency support for transmission lines and sites, generous grant schemes, as well as providing $217 million of funding in research and development. Additional measures have been to apply pressure on non-renewable plants through higher taxes, green credit and emission level targets. 1.3 Wind Farm Cost Structures From the preceding sections of the chapter, it is clear that the total cost (from its planning phase to installation, and then to the generation of electricity) of a wind farm is extremely reliant on a number of variables. In order to follow a consistent framework for the valuation of wind farms, we use investment and running costs structures obtained from public wind energy industry research. This will give a general cost model for the average wind farm. We will incorporate the two cost structures introduced below into our valuation framework: the total investment cost structure and the operation and maintenance cost structure. 1.3.1 Total Investment Cost Structure The investment cost of a wind farm will greatly depend on its size and total power output. Below is a summary table from Hau [6] of various investment cost components, as illustrated earlier in the chapter, as a proportion of the wind turbine cost. The table involves two types of large inland wind farm projects financed by limited partnership companies: One is connected to a medium-voltage system and the other to a high-voltage system. 9 Wind farms with total power output of up to about 15 MW can still be connected to a medium-voltage grid. Under favorable conditions such as firm ground for standard foundations, a nearby grid connecting point and an existing connection to the mediumvoltage system or transformer substation, the site-related plant cost can be limited to about 20% of the wind turbines cost. Table 1-3 Wind Park Medium-voltage system Proportion % 13 wind turbines 1 MW High-voltage system Proportion % 32 wind turbines 1.5 MW Wind turbines Ex-works price incl. 20 kV transformers, transportation, erection and commissioning Site-related costs Foundations Civil work, access roads, fences gats 100 100 3.80 1.08 5.91 2.55 Electrical infrastructure: Internal cabling Switchgear 2.23 5.75 Grid connection: Modification to existing transformer substation, utilities substation, feeder Remote monitoring Land lease during construction Environmental compensation fee Expert reports 4.21 0.10 0.22 1.07 0.33 5.99 0.43 0.72 1.68 1.85 7.43 5.22 5.36 4.78 2.40 1.90 2.16 3.13 0.38 29.99 129.99 40.69 140.69 General contractor: Planning, management and guarantees Purchase of lease contracts, preliminary planning with building permit Bridging loan Financing and legal costs, bank fees Aircraft warning lights Total site-related costs Total investment costs The table shows the distribution of costs as a proportion of the wind turbine cost. E.g. Total investment costs for a wind farm is 129.99% of the cost of wind turbines for the medium-voltage system and 140.69% for the high-voltage system. 10 On the other hand, large wind farms with higher power outputs that are greater than 15 MW have to be connected to a high-voltage grid. The larger distances to be covered by the feeder result in greater cost for grid connection. In many cases, a completely new transformer substation needs to be built and financed. Larger wind turbines will often have other cost-increasing factors such as foundation work. The total site related cost can rise up to 40% of the cost of wind turbines for large wind projects. 1.3.2 Operation and Maintenance Cost Structure Total annual running cost of a wind farm can be split into the following table. The percentages are taken from Hau’s study [6], and represent the operation and maintenance cost as a proportion of the cost of wind turbines. We see that the typical total running cost of a wind farm ranges from 3.5% to 5.3% a year. The biggest cost is associated with repairs, followed by monitoring, land lease and maintenance. Table 1-4 general operation and maintenance cost structure of wind farms. Routine maintenance (Service and maintenance contracts) 0.8-1.0 % Repair reserves 1.0-1.5 % Insurances 0.5-0.8 % Land lease 0.5-1.0 % Monitoring and administration 0.7-1.0 % 1.4 Economies of Scale The size of a wind farm project and the size of the wind turbine itself will vary depending on the amount of electricity the developer intends to generate. Costs of components per unit size tend to decrease as size increases, and through economies of scale, the construction costs per unit manufactured decreases as more wind turbines are manufactured (at least to the point where equipment and personnel are adequate). However, because the mass of the wind turbines’ materials increases at a cubic rate to its 11 rotor diameter, and the power rating increases with the square of its rotor diameter, there will be a critical size that increases the cost per kW of maximum power [7]. To make estimates of the cost saved through economies of scale, we can study the learning rates of wind turbine manufacturers. This will define a relationship between cost and cumulative capacity. Let K be the cumulative wind farm capacity installed, c(.) the cost of one standardized wind turbine unit, c0 the intercept for the first wind turbine unit cost and b represent the slope of the learning curve [ 8]: C ( K ) = c0 K b . (1-2) Coulomb and Neuhoff [8] expanded (1-2) to include an engineering-based scale, describing the cost of a single wind turbine unit as a sum of the different changes experienced by the components that it is comprised of. D µx3 Dref c ( D, H ) = c 0 3 + µx 2 D D ref D D ref 2 2 + µx1 D D ref H H ref 3α *C + ( µ − 1) Kb, (1-3) where xi = the fraction of turbine mass which scales with the rotor diameter with exponent i, Dref = reference rotor diameter, D = profile rotor diameter, Href = reference hub height, H = profile hub height, C = capacity factor, α = wind shear exponent. 12 With this model, Coulomb and Neuhoff estimated that with every doubling of global installed capacity, costs of wind turbines per installed capacity fall by 12.7%. Normally we think in terms of cumulative production capacity rather than number of volume doublings n, so we use the equation: n= ln( x 2 / x1 ) . ln 2 (1-4) x 2 is the unit of capacity you are seeking, and x1 is the capacity of the first unit. Given Coulomb and Neuhoff’s learning rate estimate, it is possible to quantifiably measure the cumulative cost of purchasing wind turbines. Having described the component cost structures of wind farms, the next chapter introduces the risks associated with wind energy generation. It illustrates popular risk measures that are used in the industry and the hedging strategies that could raise wind farm revenue. Figure 1-4 Acciona´s wind turbine manufacturing plant in Navarra, Spain 13 2 Managing Financial Risk in Wind Farm Investments When dealing with energy projects, there are often more things to consider than other industries. The source of wind farm revenue, electricity (like other energy markets) is very different from money markets. Energy markets are hard to model due to their dynamic interplay between demand drivers like seasonality and convenience yields, and supply drivers that include producing, transferring, storing and selling electricity. These various uncertainties cause complex energy price behavior, and gives rise to the strong need for risk management when making financial investments in the industry. Although managing and operating wind farms have many different types of risk, we limit ourselves to the discussion of financial risk management. When evaluating long term projects such as wind farms, it will be important to realize that minimizing the long term financial risk requires managing the daily, monthly or yearly uncertainty of the business. This will require understanding the available financial instruments to incorporate into hedging strategies that reduce future uncertainty in wind farm revenue. 2.1 Financial Risk In finance, the best known risk is market risk and credit risk. The former is the risk associated with changes in financial positions due to changes in their underlying components. In the case of wind farms, this would be the change in projected worth due to changes in electricity spot prices. Credit risk, on the other hand, is the risk of not 14 making loan repayments to creditors due to defaults. This is an important type of risk for investors to consider before investing in any project because it quantifies the likelihood of bankruptcy. The following subsections introduce three typical risk measures used for investment projects. They are value at risk, expected shortfall (Conditional Value at Risk) and average value at risk. 2.1.1 Value-at-Risk Value at risk (VaR) is a risk measure that finds the potential loss over a certain confidence interval. It is commonly used by investment banks to capture the expected loss in their portfolio values from potential adverse market movements. This then can be used as benchmarks for money that should be kept in reserve to weather potential crisis. There are three key elements in VaR. They are: a specified loss value, a fixed time period over which risk is assessed, and a confidence interval. When N days is the time horizon and α is the confidence level, VaR is the loss corresponding to the (100- α )th percentile of the distribution of the portfolio over the next N days [9]. When using the VaR measure, we want to make statements of the form: “We are α percent confident that we will not lose more than V dollars in the next N days”. Represented mathematically, we let X = random variable describing a future net worth distribution of profit and loss. α ∈ (0,1). If P[ X < q ] ≤ α ≤ P[ X ≤ q ], qα+ = sup{q | P[ X < q ] ≤ α }, (2-1) and VaRα ( x) = −qα+ ( x) = inf{m | P[ X + m < 0] ≤ α }, (2-2) then q is the α − quantile of X. 15 Figure 2-1 The plot shows the value distribution of a project. The 5% VaR shows 95% confidence that the project will not earn less than $4,500,000 over its expected lifetime. The popularity of VaR stems from its simplicity. It answers an important question in finance, “Given a tolerance level, when do things go bad?” Still, VaR is not without shortcomings. First of all, VaR does not look at the distribution of the tails. It fails to consider the question, “if things go bad, how bad can it get?” Another problem with VaR is that it is not sub-additive. This will be an obstacle for portfolio managers as following the VaR measure could discourage diversification. Regardless of these weaknesses, VaR is the most popular risk measure among both regulators and risk managers [10]. 2.1.2 Expected Shortfall Expected shortfall (or conditional VaR) is a way of improving the VaR measure. This does not mean expected shortfall (ES) should replace VaR, but rather supplement it. 16 ESα ( x) = − E[ X | X ≤ qα+ ( x)], (2-3) qα+ = sup{q | P[ X < q ] ≤ α }. (2-4) where ES is basically taking the expectation in the tail. Its sensitivity to the shape of the loss distribution in the tail makes it an extremely useful measure. It answers the question VaR can’t, “if things go bad, how bad can we expect it to be?” 2.1.3 Average Value-at-Risk Average value at risk (AVaR) is a coherent risk measures. This requires AVaR to satisfy four properties: monotonicity, translation invariance, sub-additivity and homogeneity. AVaR at a level α ∈ (0,1) is given by AVaRα ( x) = 1 α α ∫ VaRβ ( x)dβ . (2-5) 0 For continuous distributions AVaR coincides with the expected shortfall. However, for non-continuous distributions, AVaR may differ from expected shortfall exceeding VaR. In this case, it is represented by AVaRα ( x) = − 1 α (E[X 1{ }]− q P[X ≤ q ]− α ). + x ≤ qα+ α + α (2-6) The second term drops out if there are no jumps at the value at risk. 2.2 Hedging Risk Under state-ownership, utility companies were allowed to earn a regulated rate of return above their cost of capital. Once the regulators approved the construction costs of a power generating plant, the costs would be passed onto consumers through regulated 17 electricity prices over the life of the investment, independent of fluctuation in the evolving supply and demand conditions. This meant most of the investment risks in generating electricity were allocated to the consumers instead of the producers. However, once electricity market reforms took place much of the investment risk was shifted from consumers to producers. The competitive but volatile markets has lead generation companies, power marketers and load serving entities (LSE) to seek certainty in their costs and revenues through hedging practices, contracting and active trading [11]. Such activities involve quantifying, monitoring and controlling trading risks in the wholesale and retail power markets. Thus, hedging achieves value enhancement by reducing the likelihood of financial distress and ensuing costs, or by reducing the variance of taxable incomes and its associated present value of future tax liabilities. The following subsections introduce some of the financial instruments and strategies that are used to hedge against uncertainty in the wind energy industry. 2.2.1 Electricity Futures Contracts Electricity forward contracts are the primary instruments used in electricity price risk management [11]. They represent the obligation to buy or sell a fixed amount of electricity at a pre-specified contract price, known as the forward price, at a certain time in the future. Electricity forwards are essentially custom-tailored supply contracts between a buyer and a seller, where the buyer is obligated to take power and the seller is obligated to supply. The payoff of a forward contract promising to deliver one unit of electricity at price F at a future time T is Payoff of a forward contract=(ST - F), where ST is the electricity spot price at time T. Electricity future contracts have the same payoff structure as electricity forwards. However electricity futures, like other futures, are highly standardized in contract specifications, trading locations, transaction requirements, and settlement procedures. Generators such as independent power producers (IPP) are the natural sellers (short side) 18 of the electricity futures contracts while LSE such as utility companies often appear as the buyers (long side). When hedging against electricity spot price movements, we will consider using futures contracts as oppose to forward contracts because they are more reflective of higher market consensus and transparency than the forward price. In addition, futures contracts are more relevant to the issue of hedging because the majority of electricity futures are settled by financial payments rather than physical delivery. 2.2.2 Direct Hedging and Cross Hedging Tanlapco, Lawarree and Liu [12] performed a statistical study of direct and cross hedging strategies using futures contracts in an electricity market. They concluded that direct hedging strategies are promising and that cross hedging performs better than zero hedging. Direct hedging is when the futures contracts used to hedge are based on the spot market being evaluated. It can be done either through a fully hedged position or by a risk-minimizing framework. Cross hedging uses futures contracts from different markets. Other available futures contracts can be used as long as there is significant correlation between the spot prices and the other future prices. The correlation from these other futures can be due to supply as well as demand factors. Tanlapco, Lawarree and Liu used future contracts such as natural gas and crude oil to capture the supply factors, and index futures such as the DOW Jones Industrial Average and the Standard and Poor 500 to capture the demand factors. Cross hedging can offer a better risk reduction than direct hedging when the link between spot and futures market is weak. It is also more effective when there is not enough trading in the futures market corresponding to the market being hedged (thin market). Tanlapco, Lawarree and Liu describe statistical techniques to identify the effective futures markets for hedging. They also calculate the amount of future contracts that minimize risk by using data from actual markets. Using their results, we take the position of an electricity supplier that commits to selling power in the spot market at some time in 19 the future. The supplier (in our case wind farm owner) takes a hedged position by trading futures contracts in the market. The intent is not to make a profit but rather to protect the owner from price risks. For every mega watt hour (MWh) that the wind farm anticipates to seek at time t, b represents the number of MWh in future contracts that is being used for hedging. If b >0, the wind farm sells futures at time t (short hedge). If b <0, the wind farm buys futures contracts at time t (long hedge). If b=0, it is not using futures contracts for hedging. If b =1, then it sells the same amount of electricity in futures contracts as it does with electricity in the market (fully hedged). Let St denote the random spot price at time t for which the company will sell 1 MWh of power, Ft-1 and Ft stand for the price of nearby futures contracts at time (t-1) and t, respectively. Nearby futures are futures in a market, that are closest to the maturity or delivery date, denoted by (t+k). For each MWh that the company sells at time t, the value (V) of its hedged position is given by: V = St+b(Ft-1-Ft). The two-period hedging strategy is to sell b MWh of the nearby futures contract at time (t-1). At time t when the anticipated spot market transaction is to take place, the company closes out its future position by buying the same nearby futures contract. Tanlapco, Lawarree and Liu illustrate how a fully hedged position (b=1) is implemented and shows that the perfect hedge only works when the difference between the spot and futures prices stay constant over time. In reality, changes in the spot and future prices are not going to be identical. Therefore, the company needs to find a value b not equal to 1 that will minimize risk. This equates to finding the risk minimizing hedge ratio (optimal hedge ratio). To simplify the analysis, the number of MWh to buy or sell is normalized. Let V denote the value of the hedged position. The mean and variance of V are then E[V]=E[ St]+bE[(Ft-1-Ft)], 20 Var[V]= , where =Variance of futures contract price at time t, =Covariance between contemporaneous spot and futures prices. The variance is then minimized to derive the optimal hedge ratio b. Following the first-order condition, we arrive at the solution: . The higher the covariance between spot and futures prices, the higher the futures market position for every MWh to be sold in the spot market. 2.2.3 Electricity Storage The main setback for wind energy is the uncertainty and uncontrollability of its energy source, wind. The popular method to deal with this problem has been to come up with ways to convert electrical energy from the wind farm into a different form energy that can be stored, such as kinetic and potential energy. These range from pumped storage facilities and flywheels to underground compressed air facilities. Some even consider building an additional generating plant to meet the unsatisfied demand. Unfortunately, at present time, there is no efficient and cheap method of storing electricity. For intermittent energy sources like wind and solar energy, a breakthrough in battery technology is needed to enhance their competitiveness with conventional energy sources. The back-up storage readily available from batteries will make wind farm a reliable source for utility grids during sudden low wind speeds. Furthermore, because high wind speeds mostly occur during the night when electricity demands are low, storage will enable the wind farm to store electricity and release it throughout the following day. Other benefits include storing electricity when wholesale prices are low, and selling it when prices are high. 21 A study by Hedman and Sheble [13] investigated whether it is necessary to make large investments into a pumped storage facility just to hedge when there are other methods like options purchasing, direct and cross hedging strategies. The study concluded that the purchasing of financial options were financially competitive with pumped storage, however further research would be needed to determine if there is a preferred method. For a financial analyst, predicting the value added through financial options would be a daunting task. This is because not only would a forecast of electricity prices be necessary, but forecasts of derivative securities like call and put options and forward contracts will be required. For the purpose of valuating wind farms, the financial analyst should consider the cost and value of installing storage facilities rather than incorporating financial options into their valuation method. 2.2.4 Hedging congestion risk of bilateral transactions Rights are required for using transmission networks and rules are needed for rationing transmission usage when networks become congested. There is a major proposal for using financial instruments as transmission rights in the U.S.: the point to point financial transmission rights (FTR). 22 An FTR entitles its holder to receive compensation (or pay) for transmission congestion charges that arise when the grid is congested. The congestion charge (or payment) associated with one unit of FTR is equal to the difference between the two location prices of one unit of electricity resulting from the re-dispatch of generators out of merit order to relieve transmission congestion FTRs can be viewed as an instrument for hedging their exposure to congestion cost risk. The hedging properties of FTR make them ideal instruments for converting historical entitlements of firm transmission capacity into tradable entitlements. It holds the owner of such entitlements harmless, while enabling them to cash out when someone else can make more efficient use of the transmission capacity covered by these entitlements. In other words, FTR makes it relatively easy to preserve status quo while opening up the transmission system to new and more efficient use 4. Having introduced some options available to the analyst to quantify the risks associated with wind energy generation, we proceed to model future electricity spot prices. This will be instrumental for forecasting wind farm revenue and performing risk measures and strategies to combat uncertainty. 4 The hedging function of FTR may not be perfect due to changing network operating conditions and the potential inherent trading inefficiency. 23 3 Electricity Spot Prices The purpose of this chapter is to simulate potential electricity rate movements over long time horizons. The motivation behind this is intuitive since the price at which a wind farmer can sell his electricity will determine the revenue of the business. This chapter is important when performing profit analysis of wind farms because it allows the financial analyst to generate profit distributions. We start this chapter with an examination of electricity spot prices, and use wholesale day-ahead historical price data from the PJM West trading hub 5. PJM is the regional transmission organization that manages high voltage electric grids and the wholesale electricity market that serves 13 states and the District of Columbia 6. The PJM data starts on the 1st of January, 2001 and ends on the 25th of November, 2008. We will model the spot prices using a mean reverting Brownian motion model. This is done by first calibrating its parameters, and then extracting the seasonal components and growth trends that are apparent in the day-ahead PJM electricity spot market. 3.1 Looking At the Market Data The PJM day-ahead spot prices we obtained capture a good number of years. As we can see from the graph of PJM electricity spot prices below, the market appears to be 5 We used 2001-2008 Wholesale Day Ahead Price data from the EIA website http://www.eia.doe.gov/cneaf/electricity/wholesale/wholesale.html 6 For the list of states, please see appendix. 24 both entering a steady rise in price post 2002, as well as a more volatile stage. The plot 50 100 150 200 250 also shows a strong yearly seasonal component. Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 2001 2002 2003 2004 2005 2006 2007 2008 Figure 3-1 Time Series of PJM Wholesale Day Ahead Electricity Spot Prices, 2001-2008. (Incomplete day to day prices) The graph does not include weekends, and for the years 2006-2008, the quoted spot prices are not regularly spaced. To overcome this, we use an S-plus function called “align” on the price of the data 7. This function interpolates missing points to create a data set of regularly spaced points. Figure 3-2 shows the resulting plot. In addition, we transform the data into log prices for reasons that will be presented shortly. As with Figure 3-1, the log prices reveal an upward trend and cyclical behavior 7 See appendix for S-plus code to perform function “align”, seasonal and remainder component extraction. 25 5.5 5.0 4.5 4.0 3.5 3.0 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 2001 2002 2003 2004 2005 2006 2007 2008 Figure 3-2 Time Series of PJM Wholesale Day Ahead Electricity Log Spot Prices (2001-2008) . 3.2 Models for Electricity Spot Prices The two main mean reverting models for energy markets are mean-reverting price, and mean-reverting in the log price. The former is known best for capturing the distribution of energy prices while the latter is commonly used in electricity markets. Because of our intent and purpose of electricity spot price analysis, we will only examine the log price model. It is used in energy markets, and has been stated to be particular applicable to electricity [14]. For the purpose of comparison, we will also model the wholesale electricity spot market as a lognormal model. We will then decide on the best fit for the PJM data by incorporating seasonality components, as well as a volatility measure. 3.2.1 Mean Reversion in Log of Price Mean reversion in the log of the spot price is particularly useful because it maintains the non-negativity of spot prices. We will only be using a single factor mean reverting model, a drawback of which is that it forces the implied black-equivalent average volatility of the price distribution to go to zero over a long period of time. 26 Because of this, caution must be used when using single factor mean reverting models for valuing long term options. The mean reverting log price model is given by 8: xt ≡ ln(S t ) , dxt = α (b − xt )dt + σdz t , where: (3-1) (3-2) S = spot price, t = time of observation, α = rate of mean reversion, σ = volatility, b = long-term equilibrium of spot price, z t = Brownian motion, dz t ~ N (0, dt ) . 3.2.2 Lognormal Spot Price Model Lognormal models are the most famous and are typically used for non-energy markets. It is simple and flexible in its implementation and is represented by equation 33 (for a change in price from time t to time t + dt ): dS t = µS t dt + σS t dz t . where: (3-3) S = spot price, t = time of observation, µ = the drift rate, σ = volatility, z t = Brownian motion, dz t ~ N (0, dt ) . 8 Mean Reversion Log Price equation taken from Chapter 5 of Pilipovic’s “Energy Risk” [14]. 27 This change in price over time dt has two components. One is the deterministic function, µSt dt while the other is the random component, σS t dz , which we take to be a Brownian motion. 3.2.3 Seasonality Energy prices are assumed to have a seasonality component since people tend to use more power during the summer and the winter. For our electricity spot price modeling, we take this component to be a function of an underlying spot price, plus a seasonal factor 9: S t = S tUnd + Seasonality effects, where: S t = S tUnd + β A cos(2π (t − t A )) + β SA cos(4π (t − t SA )) , (3-4) dS t = dS tund − {2πβ A sin( 2π (t − t A )) + 4πβ SA sin( 4π (t − t SA ))}dt , (3-5) St = spot price at time t, S tUnd = underlying spot price value, β A = annual seasonality parameter, t A = annual seasonality centering parameter (time of annual peak), β SA = semi-annual seasonality parameter, t SA = semi-annual seasonality centering parameter. When we start to model the mean reverting log price and lognormal price models, we will substituted the underlying spot price in equation 3-5 by the respective change in prices given by equation 3-2 and equation 3-3. 9 Seasonal equation is taken from Chapter 5 of Pilipovic’s “Energy Risk” [14]. 28 3.2.4 Volatility To estimate the volatility of electricity spot prices empirically, we observe it at fixed time intervals (e.g. days, week, or month). Define: N + 1 = Number of observations, S i = Stock price at end of ith interval, with i = 0, 1, …, n, τ = Length of time interval in years, and let S u i = ln i , S i −1 (3-6) for i = 1, 2, …, n. The usual estimate, s, of the standard deviation of u is given by 10 2 s= 1 N 2 1 N − u ui . ∑ i n(n − 1) ∑ N − 1 i =1 i =1 (3-7) ^ It follows that σ itself can be estimated in annual terms as σ by ^ σ= s τ . (3-8) Historical volatilities are calculated based purely on the underlying historical spot prices, and therefore are a volatility measure of past price behavior. Historical events typically are not good indications of the future, especially in financial markets, so we should also consider implied volatilities of electricity option. Because market implied volatility represents what the market expects the uncertainty in the underlying option to be up until its maturity, it is a great indicator of future volatility in electricity spot prices. 10 Historical volatility as defined by Hull, J.C., Options, Futures, and Other Derivatives (Sixth Edition), Prentice Hall, 2006 (reference [9]). 29 The New York Mercantile Exchange (NYMEX) provides financially settled monthly future contracts (JM) based on the daily floating price for each peak day of the month at the PJM western hub. It also trades put and call options (JO) on these futures as far out as three years 11. The unit price of a JM contract is quoted in USD/ MWh, and is used throughout the data in this chapter. To find the implied volatility of the JO call options, we can use the inverse BlackScholes function available in Matlab 12, and plot the call options’ volatility smiles for each month by varying the strike price. Please refer to the appendix for the volatility smiles plots of JO. A volatility smile is a phenomenon that shows the lognormal equivalent volatilities for the options of the same time to expiration and on the same settlement price to be different across different strikes. When the actual price distribution has fatter tails than the lognormal distribution, the out-of-the-money calls and puts tend to show volatilities that increase as option strikes make the options go further and further out-ofthe-money. Unfortunately for the call options expiring in 2011, there are not enough points to make the result significant. We find that there are only 2 points we could consider. For the years 2010 and 2009, there are distinctive smirks in the plots. We also notice that as the time to expiration increases, the implied volatility decreases. This implies that the market believes electricity spot prices are more volatile in the short run than in the long run. Since the volatility smiles are not flat, it is hard to choose a volatility value and take it to be the best indicator of the future volatility of stock prices. In addition, we only have reasonable future option volatility expectations up to 2010, which may not be sufficient for the purpose of our analysis of long time horizons. Because of these issues, we refrain from using the implied volatility in our models. Instead, the historical volatilities will be implemented. Because the historical data used to calculate the historical volatilities are relatively current (2001-2008), it should be a reasonable estimate of the volatility of future spot prices, at least for the purpose of our analysis. 11 12 All option and futures data were taken from the NYMEX website http://www.nymex.com/JM_desc.aspx See appendix for volatility smiles Matlab code and plots. 30 3.3 Mean Reverting Log Price and Lognormal model Parameters We used the method of least squares to calibrate the mean reverting log price model and lognormal price model parameters. An optimization program was run to minimize the residuals between the historical prices and our models for electricity spot prices. Using our historical volatility equation 3-7, we found the log returns spot price of PJM to be 0.119 and the historical volatility of spot price returns to be 0.1262. For the mean reverting log price model, we assumed the long run equilibrium rate, b, to be around $90 per MWh. Although this figure is highly subjective, it is a reasonable estimate of what we imagine electricity spot prices to be 20-30 years from now. Over the last couple of years, electricity prices have rarely fallen below $40 per MWh, and taking into account the upward trend of our data, electricity prices should continue to increase in the future. $90 per MWh might not be an accurate estimation, but it does seem plausible. Even though it might not be the actual future long term expectation of the log spot price, it will allow the model to stay realistic within long time frames. The calibration process used the expected value of the stochastic variable dzt , which is N ~ (0, dt ) . This respectively reduced the mean reverting log price and lognormal price from dxt = α (b − xt )dt + σE[dz t ] , dS t = µS t dt + σS t E[dz t ] , to dxt = α (b − xt )dt , (3-9) dS t = µS t dt . (3-10) 3.3.1 Mean Reverting Log Price Results from Excel Solver: σ 0.1190 t1c t 2c 0.0833 0.5833 α β1 β2 b dt 0.0736 0.0003 0.0003 4.5 0.0027 31 Using the calibrated parameters, the plot of the seasonality component is produced below in Figure 3-3: Figure 3-3 Derivative of Seasonality with respect to time obtained from optimization The seasonality component shows two distinct peaks. This is consistent with electricity prices. Typical prices will increase over the summer, reaching a peak around July, and then a smaller increase during the winter. Adding the seasaonality component to the log price mean reverting model, we can compare it to the historical spot prices to see how it fares, it is illustrated below in Figure 3-4. 32 Figure 3-4 Log Price Mean Reverting Model and Historical spot prices The model does a good job at following our electricity spot prices. There is one specific period where the model does not capture the movements well, October 2001February 2002. However, although there is a lot of fluactuation in the daily prices, it generally does a good job in capturing the upward and downward movements (even though by different magnitudes). Next, this model will be used to simulate future spot prices. We continue to use the parameters from the linear program but with the addition of the stochastic term. Before finishing with this result though, let us look at the lognormal price model. 3.3.2 Lognormal Price Model Results from Excel Solver: µ 0.0367 σ 0.1262 t1c 0.0833 t 2c 0.5833 β1 0.0193 β2 0.0192 The plot of the resulting seasonality is produced below in Figure 3-5: 33 Figure 3-5 Derivative of Seasonality with respect to time obtained from optimization As expected with the previous log mean reverting price model, there are two peaks during each year. Again, we find that a second peak is much smaller compared to the first. We proceed to compare the historical spot prices of our lognormal model with the added seasonality component, as we did with the log mean reverting price model. Figure 3-6 shows that the lognormal model captures the seasonal trends and the upward trend we saw in the historical spot prices. 34 Figure 3-6 Lognormal Model and Historical spot prices Both models, the log mean-reverting model and the lognormal model seem to capture the basic charasterictics of the spot price well; seasonality and the upward trend. The log mean reverting model performs slightly better in terms of its deviation from the historical spot prices. This and the fact that it is recommended for electricity prices [14], have become the deciding factors for choosing the mean reverting log price model to simulate our spot price forecasts. The next chapter introduces the idea of real option modelling and discusses the different methods that can be used for a financial analyst to value projects. The result will be a valuation tool that incorporates what has been discussed so far: wind farm cost structures and revenue forecast through spot price movements. 35 4 Real Options This chapter provides an overview of real options and addresses the need to supplement traditional valuation tools with real option analysis. Much of the following is a direct interpretation of “Project Valuation Using Real Options” by Kodukula and Papudesu [15]. Before introducing the main real options valuation methods, we look at the traditional tools commonly used. This will then lead to the introduction of real options analysis, where it is shown to be more appropriate for real projects. 4.1 Traditional Project Valuation Tools The quality of project valuation is related to its validity in addressing three important factors: 1. Cash flow streams through the entire project life cycle. 2. Discount rate used to discount the future cash flows to account for their uncertainty 3. Availability of management’s contingent decisions to change the course of the project Project cash flows consist of investment costs and net revenue in the production phase. Investment cost cash flow is typically comprised of development phase costs and production phase capital costs. These costs, in the case of a wind farm project, for example, would include building the wind farm and connecting it to the utility grid, or the installation costs discussed in Chapter 1. 36 The net revenue cash flow is the difference between the revenues and the costs in the production phase of the project. The crux of the project valuation lies in estimating these two cash flow streams over the entire project life cycle and discounting them back to today’s value using the appropriate discount rate. The three main traditional tools that take into account some or all of the three factors into their project calculations are: 1. Discounted cash flow analysis 2. Monte Carlo Simulation 3. Decision Tree Even though these three tools look very different, the building blocks for the calculations are the same for each case. They are provided by the present value of the cash flow streams. 4.2 Real Options Analysis Real options analysis (ROA) is far more complex compared to traditional tools and requires a higher degree of mathematical understanding. The theoretical framework of real options solutions is complex, whereas the calculations involved are relatively simple. The real options solutions are based on models developed for pricing financial options, and are listed below in three different categories: partial differential equations, simulations and lattice models. 4.2.1 Real Options Calculations 1) Partial differential equations The partial differential equation method involves solving a partial differential equation with specified boundary conditions that describe the change in option value with respect to measurable changes to certain variables in the market. Three methods to solve the partial differential equation are: • Closed form solutions using Black-Scholes and other similar equations 37 • Analytical approximations • Numerical methods A challenge with using the Black-Scholes solution is its lack of transparency where the underlying logic behind the calculation is unclear. It also makes certain assumptions on the underlying asset that may not be true. For instance, it assumes a lognormal distribution of the underlying asset value, which may not be true with the cash flows related to real assets. It also assumes an increase in the underlying asset value is continuous as dictated by volatility and does not account for jumps, which are not uncommon in real projects. Furthermore, it only allows one strike price for the option, which can change for a real option. 2) Simulations Simulation is similar to the Monte Carlo technique for DCF analysis. It involves simulation of thousands of paths the underlying asset value may take during the option life given the boundaries of the cone of uncertainty as define by the volatility of the asset value. Monte Carlo simulations are very efficient for European style options, however when wanting to value American style options, the computation can be a very daunting task. 3) Lattices Lattices look like decision trees and lay out the evolution of possible values of the underlying asset during the life time of the option. An optimal solution to the entire problem is obtained by optimizing the future decisions at various decision points and folding them back in a backward recursive fashion into the current decision. The binomial method offers transparency by showing the project values in the future for a given expected payoff and the rational decisions one would make. The idea is that as the uncertainty clears in the future, management can make appropriate decisions at that time by comparing expected payoff with the investment costs. It is recommended to employ both the binomial and the Black-Scholes methods when valuating real options. This is because the latter can verify the results and give more insight. 38 4.2.2 Real Options versus Traditional methods This section briefly outlines the advantages of using real option valuation methodology as oppose to the traditional methods mentioned in section 3.1. Here it is clearly pointed out that when it comes to flexibility in management decisions and other uncertainties, ROA is preferred. 1) Discounted Cash Flow versus Real Options DCF takes into account the downside potential by using a risk-adjusted discount rate whereas ROA captures the value of the project for its upside potential by accounting for the proper managerial decisions that would presumably be taken to limit the downside risk. 2) Decision Tree Analysis versus Real Options Analysis • DTA can account for both private and market risks, but ROA addresses only the market risk. The solutions to real options problems will not be valid for private risk, because the theoretical framework behind the solution development does not apply to it. • DTA accounts for the risks through the different probabilities of a project’s various outcomes. While it basically considers only two, three, or a few mutually exclusive possible outcomes, ROA accounts for a wide range of outcomes. This makes a difference in the discount rate used to discount the cash flows. There is no general consensus in the finance community on what the most appropriate discount rate is for decision trees, whereas a risk free rate is established to be appropriate for ROA. In the absence of market risks, DTA is more appropriate for project valuation, but ROA is a better tool when such risks exist. ROA is most valuable when there is high uncertainty with the underlying asset value and when management has significant flexibility to change the course of the project in favorable direction. When both market risks and private risks exist as well as opportunities for contingent decisions to change the 39 future course of the project, ROA in combination with DTA often provides better valuation than either individual method. 4.3 Project Valuation Using Real Options Critics argue that “no arbitrage” condition is impossible with real assets because they are not as liquid as financial assets, and therefore option pricing models are inappropriate for real options valuation. We believe that “no arbitrage” condition is only a limitation of the model and can be overcome easily by proper adjustments. Practitioners have used three different types of adjustment: 1. Use an interest rate that is slightly higher than the riskless rate in the option pricing model. 2. Use a higher discount rate in calculating the discounted cash flow value of the underlying asset. 3. Apply an “illiquidity” discount factor to the final option value Since most real options are not traded assets, application of the financial option models to real options is questioned by some critics. The real options proponents, however, argue that you can create a replicating portfolio for a real option on paper, which should suffice (Damodaran, 2002). Some (Amram and Kulatilaka, 2000) suggest that instead of simply assuming private risk, the practitioners should look hard to correlate the risk – even partially- with a portfolio of traded securities. 4.4 Conclusion This chapter has shed light into the types of real options that are used in practice. Conventional option pricing is useful, but when it comes to real option pricing, is insufficient to be the sole evaluation method of the project. It was discussed how traditional tools should be supplemented with ROA calculations to provide better 40 intuition and flexibility in areas of high uncertainties. After learning about the value added through ROA, the next step is to formulate a clear outline to follow when performing our valuation on wind farm projects. For our purpose, we will use DCF together with Monte Carlo simulations. This method seems most appropriate for our valuation method of a wind farm because the main uncertainty lies in electricity spot prices. The most straight forward approach for a financial analyst will be to simulate the expected revenue and apply the wind farm cost structures to obtain a yearly cash flow. Then discounting back to present value, the analyst will be able to perform risk analysis on the profit and loss distribution of the project. To illustrate this valuation method, the next chapter puts the methods that have thus far been described into practice. 41 5 Projected Cash Flow Simulation and Profit Distribution Forecast Once the financial analyst understands the cost structures of a wind farm and is able to simulate future spot prices, the next step will be to employ a real option valuation method to value the project. This chapter will illustrate the results an analyst could obtain and the analysis performed to understand the specific areas raising or lowering the worth of such an investment. We will perform our valuation analysis on a Pennsylvania wind farm that is already in operation, the Bear Creek Wind Farm. The objective is to determine the long run profitability of investing in the project, and the conditions necessary for its success. Real option valuation involving simulations together with the discounted cash flow method are used. The process incorporates the cost structures described in chapter 1 and the PJM electricity spot price simulations generated in chapter 3. The future cash flows are to be determined and the net present value calculated. Through numerous simulations, we will obtain cash flow and net present value distributions and perform risk analysis. 5.1 PJM Electricity Spot Price simulation Below are the simulation paths of electricity spot prices for the years 2009 to 2018. We have transformed the log spot prices of our mean reverting model into spot 42 prices by exponentiation. We run 25 simulations to obtain a sample collection of spot price movements. PJM Electricity Spot Price Simulations (2009-2018) 140 120 $USD/MWh 100 80 60 40 20 02-Apr-2008 28-Dec-2010 23-Sep-2013 Date 19-Jun-2016 16-Mar-2019 Figure 5-1 PJM Electricity Spot Price Simulation The erratic movements observed in electricity spot markets are captured in our simulations, and it shows that most spot prices are within the price range of $40-$100 per MWh. When examining the Bear Creek project in Pennsylvania, we will simulate 2000 spot price scenarios over a 10 year horizon 13. With a distribution of electricity price movement scenarios, the expected annual revenue streams can be estimated. 5.2 Bear Creek Wind Power Project The Bear Creek Wind Power Project is Pennsylvania’s newest wind farm. It is a 24 MW wind energy facility located in the Pocono Mountain region, and is expected to produce over 75 million kilowatt-hours of wind energy annually. The project was made possible by commitments from PPL Energy Plus to purchase the output of the project and leading wind energy customers such as the University of Pennsylvania and PEPCO Energy Services. The farm consists of 12 Gamesa 2.0 MW wind turbines, and has been 13 Matlab code for simulations are given in appendix. 43 operating since February 2006 14. For the purpose of our analysis, we imagine Bear Creek to be scheduled to open in 2009, and that we have been assigned to value this investment. Figure 5-2 Bear Creek Wind Power Project, PA To proceed, assumptions are made about the project. Because generating wind forecasts are not within the scope of this thesis, we will assume that the wind blows at a constant speed throughout the day at Bear Creek, and that the wind farm operates 24 hours. We will also not draw distinctions between off-peak or on-peak electricity spot prices. From our understanding of federal and state tax incentives and subsidies, we imagine that the investors are eligible for PTC or ITC, as well as the MACRS tax incentives. For the state of Pennsylvania, there is a Wind and Geothermal Incentives Program and a $1 million grant from the Pennsylvania Energy Development Authority (PEDA). In addition, 100% of the project system value is exempted from Property tax 15. We will assume the Bear Creek wind farm developers are eligible for all of the above. We will also assume the first 2.0 Gamesa wind turbine unit costs $3 million16, and that every doubling of installed capacity, costs of wind turbines per installed capacity fall by 12.7% 17. We expect the cost inflation to be 1% per year. 14 Information of Pennsylvania Wind Farms such as bear Creek can be found on Pennsylvania Wind Working Group website, http://www.pawindenergynow.org/pa/farms.html. 15 Pennsylvania Incentives for Renewables and Efficiency, http://www.dsireusa.org/library/includes/map2.cfm?CurrentPageID=1&State=PA&RE=1&EE=1 16 Wind Turbines have been quoted online to be between $3-6 million. 44 From the amount of wind power the wind farm has been projected to generate, we could solve for the capacity factor of the wind farm. However, capacity factors are not totally meaningful because the kWh may be generated at a time when the electricity is not really needed or when electricity is available at less cost from other generating sources. PJM is the largest grid operator in the country and has determined that the wind class average capacity factor is around 13% 18. We take this to mean that every day, the wind turbines are able to transmit 13% of their nameplate capacity (on average) to the purchasing utility. 5.3 Cost and Revenue Projections To project the revenue, a program is run that multiplies total daily MWh by the PJM capacity factor, and then multiplies it by the daily price of electricity over a specified time horizon. Having previously illustrated the general cost structure for wind farms, we can easily simulate the cash flow projections. Finally we sum all the net present value cash flows (not forgetting to discount appropriate back to today dates 19) within each simulation. This is represented mathematically by: T Hs = ∑ t =1 S t CV (1 + i )t , (5-1) where S t = spot price at time t, C = capacity factor, V = full capacity of wind farm, i = discount rate, T = time horizon being evaluated, H s = total present value of revenue cash flows in simulation s. 17 Recall this figure was taken from Coulomb and Neuhoff [8] This figure represents that amount of generating capacity that can reliably contribute during summer peak hours and which can be offered as unforced capacity into the PJM capacity markets. Taken from PJM website http://www.pjm.com/documents/manuals/~/media/documents/manuals/m21.ashx 19 2/1/2009 US Treasury bond rates are used from yahoo finance, http://finance.yahoo.com/bonds. 18 45 To create a cash flow projection, we will need to define the expenses and depreciation schedule. Using economies of scale, the cost structure and the accelerated depreciation tax incentive illustrated in chapter 1, we can calculate the expected expense and depreciation cash flow. We begin by examining a 10 year investment horizon. 5.3.1 Investment Cost We can use our economies of scale assumption to produce the total cost of the 12, 2.0 MW Gamesa wind turbines. Unit 1 2 3 4 5 6 7 8 9 10 11 12 Capacity Unit Cost 2 $3,000,000.00 4 $2,619,000.00 6 $2,447,048.74 8 $2,286,387.00 10 $2,210,056.13 12 $2,136,273.55 14 $2,064,954.21 16 $1,996,015.85 18 $1,962,414.60 20 $1,929,379.00 22 $1,896,899.52 24 $1,864,966.81 Total $26,413,395.41 Cost For a wind farm of this size, we can assume it will need to be connected to a highvoltage system, requiring extra costs of around 40% of the wind turbines installation cost 20. Investment costs Wind turbines Planning, tech. infrastructure and financing 20 100% $26,413,395.41 40% $10,565,358.16 Following the cost structure introduced in chapter 1. 46 I. Expense Cost ($ 000) Year 1 2 3 4 5 6 7 8 9 10 Maintenance contract 264 269 275 280 286 292 297 303 309 316 Insurance 211 216 220 224 229 233 238 243 248 253 Land rent 264 269 275 280 286 292 297 303 309 316 Repair reserves 396 404 412 420 429 437 446 455 464 473 Administration 264 269 275 280 286 292 297 303 309 316 Total operating cost 1400 1428 1456 1486 1515 1546 1577 1608 1640 1673 II. Depreciation Schedule ($ 000) Year Depreciations ($) 1 5283 2 8452 3 5071 4 3043 5 3043 6 1521 7 0 8 0 9 0 10 0 With both the simulated electricity prices and the expected initial investment and annual costs for 10 years, the next step is to determine the wind projects expected net cash flow. 5.4 Bear Creek Expected Yearly Cash Flow When calculating the projects cash flow, we will split it into three sections. First we start by calculating its earnings before interest and tax (EBIT), then the net income, and finally the net project cash flow over 10 years. 5.4.1 EBIT To find the project’s EBIT, we use our electricity spot price simulation to generate possible revenue cash flow streams 21. EBIT is defined as: EBIT = Revenue - Operating Expenses + Operating Income. We present the yearly EBIT distribution below in figure 5-3. It is the 10 year expected EBIT of the Bear Creek Wind Farm. The first few years are very negative because the depreciation has been deducted from our revenue. This will be added back later when 21 EBIT Matlab code given in appendix for calculation and plots. 47 calculating net cash flow. EBIT starts to break even in year 7, but this is because the depreciation schedule has ended. Year 1 1000 500 0 -5.8 -5.6 -5.4 Year 2 -5.2 -5 1000 500 0 -9 -8.5 -8 -7.5 6 6 1000 500 0 -6 F r e q u e n c y x 10 Year 3 -5.5 -5 -4.5 -4 x 10 Year 4 1000 500 0 -4 -3 -2 -1 6 6 x 10 Year 5 1000 500 0 -4 -3 -2 -1 1000 500 0 -3 x 10 Year 6 -2 -1 0 6 1000 500 0 -1 0 6 x 10 Year 7 1 2 1000 500 0 -1 x 10 Year 8 0 1 2 6 1000 500 0 -1 0 1 2 3 3 6 x 10 Year 9 1 1000 500 0 -2 x 10 Year 10 0 6 x 10 2 4 6 x 10 $USD Figure 5-3 The Bear Creek simulated yearly EBIT cash flow distributions (2009-2018). 5.4.2 Net Income To determine the net income, we must deduct other costs like interest and tax payments. By including these two extra costs, we will see how they affect the projects 48 yearly net revenue. Bear Creek was funded by developers Global Wind Harvest and Community Energy Inc., which partnered Babcock and Brown and Central Hudson Energy Group, Inc. to make the project possible. Since these investors are publically traded companies, we imagine that they would be able to secure a loan with a good rate, around 8%. We previously assumed that the Bear Creek project is eligible for the Pennsylvania Wind and Geothermal Incentives Program, as well as the Pennsylvania Energy Development Authority (PEDA) Grant. This means the investors will be able to apply for a loan at a fixed interest rate (generally the prime rate minus 1%) for terms of up to 10 years (equipment) or 15 years (real estate), and be awarded $1 million from the state. Under the Wind and Geothermal program, the loans for energy production projects are limited to $5 million. In the case where the project receives the $5 million, the investors will have to borrow the rest to start the project. The size of the loan should be the capital cost required to install and construct the wind farm minus the state loan and grant. This will amount to a $ $35,978,753 loan from a financial institution, and another $ 5,000,000 from Pennsylvania State. The developers and investors, some of whom are big corporations, should be able to secure the loans by using their assets as collateral. They could be required to put down capital too, but we will assume that this is not the case. Earnings before Tax (EBT) Let us first find the yearly interest payments to the state and the financial institution. The state loan is worth $5 million today with a capital cost interest rate of 1.46% 22, while the other loan is $36.0 million with an interest rate of 8%. Equation 5-5 mathematically represents the annual annuity payment formula. 22 This is the 1/2/2009 10 year treasury rate 2.46% - 1% (Wind and Geothermal Incentive Program) 49 An = Annuity:Error! Bookmark not defined. PV 1 1 1 − i (1 + i ) T , (5-5) where PV = present value of loan, i = interest rate, T = payment time periods. Solving for A, we obtain an annual interest payment of $4,616,748 to the financial institution and $541,022 to the state. This is a total annuity payment of $5,157,770 every year. Earnings after Tax Let us assume that the wind farm is subject to a 35% corporate tax which includes all state and federal taxes. Because depreciation is deducted before tax, the wind farm depreciation schedule saves the owners money by decreasing the taxable income. The amount saved is called a depreciation tax shield, and is the depreciation amount multiplied by the tax rate. The tax shield schedule from the accelerated depreciation is given below: Depreciation Tax Shield Year Schedule Depreciation Potential Tax Shield ($ 000) 1 2 3 20% 32% 19% 5283 8452 5071 1849 2958 1775 4 11.5% 3043 5 11.5% 3043 6 5.8% 1521 7 0% 0 8 0% 0 9 0% 0 10 0% 0 1065 1065 532 0 0 0 0 The PTC is another tax incentive we must include. As mentioned in chapter 1, PTC offers 2.1 cents/kilowatt-hour (adjusted for inflation) for the first 10 years of a wind farms operation. To calculate the amount of money paid by the government, we first calculate the annual kW at full capacity. Bear Creek generates 576 MW a day and is operating 365 days a year. 50 576000 kW day kW × 365 = 210,240,000 day year year We calculated previously that the PJM capacity factor was 13%. Therefore, we expect PTC to allow the wind farm owner to claim: 210,240,000 kWh 1 1 × 13% × $0.021 = $573,955 year kWh year Expanding from our EBIT calculation, we now find the yearly net income of the project 23. Net Income = EBIT(1-Tax) - interest payment or preferred stock dividends The net incomes plots in figure 5-4 look a lot less profitable than the EBIT plots. The expected net income is never in the money, instead the highest expected values appears to be -$5 million. The distributions are uni-modal and slightly skewed to the right, which suggests it could be a lognormal distribution. 23 The code for calculations and plots for net income and cash flows are in the appendix. 51 Year 1 1000 500 0 -10.2 -10 Year 2 -9.8 F r e q u e n c y -10 -9.5 -1.35 6 x 10 Year 3 1000 500 0 -10.5 1000 500 0 -9.6 -1.4 -9 -1.3 -1.25 7 x 10 Year 4 1000 500 0 -8.5 -9 -8 -7 -6 6 1000 500 0 -9 -8 -7 6 x 10 Year 5 -6 -5 x 10 Year 6 1000 500 0 -7 -6 -5 -4 6 6 x 10 Year 7 1000 500 0 -6 -5 -4 -3 -2 x 10 Year 8 1000 500 0 -6 -5 -4 -3 6 1000 500 0 -6 -5 -4 -3 -2 -2 6 x 10 Year 9 -1.2 x 10 Year 10 1000 500 0 -6 -5 -4 -3 6 x 10 -2 6 x 10 $USD Figure 5-4 The Bear Creek simulated yearly net income cash flow distributions (2009-2018). 5.4.3 Cash Flow Statement Cash flow is similar to net income, however it does not include the deduction of depreciation from the income statement. The cash flow statement accounts for all the movement of money within a company, therefore the depreciated amount needs to be added back in. Deprecation is relevant for accounting and taxation. 52 Cash Flow Statement = Net Income + Depreciation. The following plots are the results of yearly net income plus the depreciation we took away from the EBIT simulation. Since the costs are not random variables (in our case), the plots are basically shifts. In addition, it is important to note that because the EBIT was never in the money, the project was unable to benefit from the accelerated depreciation schedule. It had no income to be taxed. 1000 500 0 -5 Year 1 -4.8 -4.6 1000 500 0 -4.2 -5.5 -4.4 Year 2 -5 -4.5 -4 6 1000 500 0 -5.5 F r e q u e n c y -5 -4.5 6 x 10 Year 3 x 10 Year 4 1000 500 0 -3.5 -6 -4 -5 -4 -3 6 1000 500 0 -6 -5 -4 6 x 10 Year 5 -3 -2 1000 500 0 -6 x 10 Year 6 -5 -4 -3 6 1000 500 0 -6 -5 -4 -3 -2 1000 500 0 -6 x 10 Year 8 -5 -4 -3 6 1000 500 0 -6 -5 -4 -3 -2 -2 6 x 10 Year 9 -2 6 x 10 Year 7 -3.5 1000 500 0 -6 6 x 10 x 10 Year 10 -5 -4 -3 -2 6 x 10 $USD Figure 5-5 Bear Creek yearly cash flow distributions 53 It is clear from our yearly cash flow histograms that the Bear Creek wind farm is not going to be profitable in 10 years. This strongly suggests a longer term investment commitment. To emphasis this result, let us calculate the net present value of the project. Net Present Value 90 80 70 Frequency 60 50 40 30 20 10 0 -5 -4.5 -4 -3.5 $USD -3 -2.5 7 x 10 Figure 5-6 Bear Creek net present value distributions Looking at the histogram above, the bear creek wind farm is not a good 10 year investment. The distribution is nowhere in the money and reaches losses of up to -$47 million. The mean is around -$40.7 million and it is skewed towards the right. After reaching this conclusion for the 10 year investment horizon, the financial analyst might want to project out further. The next two sections will look at a 20 year and 30 year time horizon respectively. 54 5.5 20 and 30 Year Investment Outlook We will skip the EBIT and net income yearly distribution plots and instead just plot the net present value of the project over 20 and 30 years. Projecting further into the future increases the uncertainty of our electricity spot price model, so caution should be taken when interpreting the final result. In the following distributions, two changes are made to our calculations. After 10 years of operation, the wind farm will no longer be able to claim PTC, so this will be removed. Also, the MACRS schedule will be different according to the duration of operation. 20 Year Outlook The projection out to 20 years does not change the profitability of the project by much. The mean is around -$37.6 million. Net Present Value 80 70 60 Frequency 50 40 30 20 10 0 -5.5 -5 -4.5 -4 -3.5 $USD -3 -2.5 -2 -1.5 7 x 10 Figure 5-7 Bear Creek 20 year net present value distributions 55 30 Year Outlook Unfortunately, even the 30 year projection is extremely poor. The mean profit has shifted only slightly to -$36.0 million. This result raises some important issues. It is clear that even after paying all the debt required to install the wind farm, the revenue generated barely covers the annual operation costs of running the business. Net Present Value 80 70 60 Frequency 50 40 30 20 10 0 -5.5 -5 -4.5 -4 -3.5 -3 $USD -2.5 -2 -1.5 -1 -0.5 7 x 10 Figure 5-8 Bear Creek 20 year net present value distributions We will perform sensitivity analysis on certain variables to identify which factors are stunting the projects profitability. The variables considered are level of taxation, operation and maintenance cost, capital cost, debt to equity ratio and the capacity factor 24. To make the information comparable, we take the average NPV across the different variables. 24 Matlab code for sensitivity analysis is shown in the appendix. 56 5.6 Sensitivity Analysis The sensitivity analysis is performed on the 30 year investment horizon. First we consider the capital cost, capacity factor, tax and operation and maintenance costs. We vary these parameters from -20% to 20% of assigned base levels. They are $36,978,754, 13%, 35% and $1,399,910 (5.3% of the wind turbine cost) respectively. 7 -1 x 10 Operation & Maintenance Capacity Factor Capital Cost Tax -1.5 -2 -2.5 NPV($) -3 -3.5 -4 -4.5 -5 -5.5 -6 -20 -15 -10 -5 0 Percent Variation 5 10 15 20 Figure 5-9 NPV Sensitivity Analysis: Operation & maintenance cost, capacity factor, capital cost and taxation. From our sensitivity analysis plot, the wind farm is most sensitive to the capital cost. The capital cost is by far the largest cost for the wind farm. It takes the wind farm very long to pay back the debt from the financial institution. In addition, the wind farm barely generates enough revenue to pay the annual operating costs (with 13% capacity factor). Next is the capacity factor of the wind farm, followed by the operation and maintenance costs and tax. Reductions in the capacity factor could be caused either by low availability of the turbines or by low wind speeds. At PJM, wind projects can only request Capacity Interconnection Rights up to the capacity value determined by the 57 Reliability Pricing Model (RPM 25), which is currently 13%. Tax does not affect the NPV much because the earnings before taxation are negative almost every year. Even when income is above zero, it is usually dwarfed by the PTC. The following sensitivity analysis involves the interest rate and debt to equity ratio. Depending on the interest rate, there will be an optimal debt and equity ratio which maximizes the expected profit of the wind farm. Depending on its source of funding or credit rating, the interest rate could be a serious factor in profitability. Below is the sensitivity of the project to interest rates, varied from 5% to 25%. 7 -2 x 10 -3 NPV($) -4 -5 -6 -7 -8 5 10 15 Interest rate 20 25 Figure 5-10 NPV sensitivity analysis of interest rate As we would expect, the wind farm owners would prefer to have as low an interest rate on their financial loan as possible. The NPV of the project declines rapidly with every percentage increase in the rate. It is important to note that the impact of the interest rate depends on the loan size being borrowed. This is because the interest rate is 25 The Reliability Pricing Model (RPM) is PJM’s capacity-market model. 58 embedded in the wind farm annuity payment. This means that if the government can subsidize a larger portion of the wind farm capital cost, the interest rate will be less influential. For the debt to equity plot, we will let the interest rate be 8%. 7 0 x 10 -0.5 -1 NPV($) -1.5 -2 -2.5 -3 -3.5 -4 0 10 20 30 40 50 60 Debt to Equity Ratio 70 80 90 100 Figure 5-11 NPV sensitivity analysis of wind farm debt to equity ratio The net present value is very sensitive to the debt to equity ratio when the ratio is less than 10% and rather insensitive for ratios larger than 10%. This is due to the state loan that is offered at a much lower rate than the bank loan. However, because the state loan has a limit of $5 million, once the debt exceeds this value, the investors must turn to financial institution that lend at a much higher rate. It is clear from the plot that the cost of capital is a big burden on the wind farm. Another thing to point out is that if the investors cannot pull together 90% of the capital (which will be almost always be the case), then they can borrow up to 100% of debt without much difference in NPV. 59 Our overall sensitivity analysis has identified both the capital cost and the capacity factor as the two factors that most influence wind farm profitability. They are independent of one another, and so the alteration of one will have no impact on the other. 5.7 Conditions for Profitability This section describes the conditions that could make a wind farm profitable in 30, 20 and 10 years. Following from the previous section, capital cost and capacity factor were identified as the main factors that determine the financial worth of a wind farm. 5.7.1 30 Year Investment For the 30 year investment horizon, we will first look at the capital cost that will make an investor indifferent about investing in the wind farm, that is, when average NPV equals 0. 7 4 Capital Cost Vs 30 Year Average NPV x 10 3 Average NPV ($) 2 1 0 -1 -2 -3 -4 0 10 20 30 40 50 60 Percent of Capital Cost 70 80 90 100 Figure 5-12 The change in Average NPV with respect to changes in the percent of wind farm capital cost From Figure 5-11, an investor would be indifferent about the project when the capital cost is reduced by about 55% of its original value. Next, we look at the capacity factor. 60 7 6 Capacity Factor Vs 30 Year Average NPV x 10 4 Average NPV ($) 2 0 -2 -4 -6 -8 0 5 10 15 20 25 Capacity Factor 30 35 40 45 Figure 5-13 The change in Average NPV with respect to changes in the capacity factor From the above plot of the change in capacity factors, the wind farm has an average NPV of 0 when the capacity factor reaches about 25%. Having realized the individual benchmarks that the capital cost and capacity factor need to reach in order to make the wind farm profitable, the next step is to determine the combination of these two factors that will make investors indifferent about the Bear Creek wind farm as an investment venture. The results are approximations of when NPV is $0 because it is difficult to exactly pinpoint a specific level of NPV when our model consists of numerous simulations over a long time horizon of 30 years. The table below represents a summary of the variables required to reach a NPV of around $0 26. To include the affect of interest rates, we will consider three different interest rate values, 1%, 6% and 11%. 26 A Matlab script was run to find the indifference points. 61 Capital Cost 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Interest Rate 1% 6% 10.60% 10.65% 11.80% 11.95% 13.00% 13.45% 14.20% 15.05% 15.35% 16.65% 16.55% 18.20% 17.80% 19.75% 19.00% 21.30% 20.20% 22.95% 21.45% 24.50% 11% 10.65% 12.05% 14.00% 16.00% 17.95% 19.95% 22.00% 24.00% 26.00% 27.95% Table 5-1 Capacity factors with respect to percent of capital cost and interest rate (30 year). Column one shows the capital cost as a percentage of its original value. Column 2 to 4 shows the capacity factor required to achieve a NPV of around $0 with the corresponding capital cost and interest rates. The capacity factors are not affected much by interest rates when the capital costs are small. However, their impact (interest rates) is more notable for capital costs larger than 0.5. With a PJM capacity factor of around 13%, the minimum conditions required for investor indifference are 1) at least 70% subsidy on wind farm capital costs and 2) a loan with an interest rate of no more than 11% (payback in 10 years). 5.7.2 20 year To look at the wind farm as a 20 year horizon investment, the same plots are created. The percentage of capital cost that provides $0 NPV in 20 years is around 50%. 62 7 3 Capital Cost Vs 20 Year Average NPV ($) x 10 2 Average NPV ($) 1 0 -1 -2 -3 -4 0 10 20 30 40 50 60 Percent of Capital Cost 70 80 90 100 Figure 5-14 The change in Average NPV with respect to changes in the percent of wind farm capital cost The capacity factor needs to reach around 30% to make an investor indifferent about investing in the wind farm. 7 3 Capacity Factor Vs 20 Year Average NPV ($) x 10 2 1 Average NPV ($) 0 -1 -2 -3 -4 -5 -6 -7 0 5 10 15 25 20 Capacity Factor 30 35 40 45 Figure 5-15 The change in Average NPV with respect to changes in the capacity factor Considering 10% intervals of capital costs, the corresponding capacity factor that produces $0 NPV is found. 1%, 6% and 11% interest rates are used as was done with the 30 year horizon. 63 Capital Cost 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Interest Rate 1% 6% 10.70% 10.70% 12.20% 12.45% 13.75% 14.95% 15.25% 16.45% 16.75% 18.45% 18.30% 20.45% 19.85% 22.45% 21.30% 24.45% 22.95% 26.35% 24.45% 28.35% 11% 10.70% 12.55% 15.55% 17.55% 20.10% 22.65% 25.20% 27.70% 30.35% 32.90% Table 5-2 Capacity factors with respect to percent of capital cost and interest rate (20 year). If we assume that the PJM capacity factor must stay around 13%, then the conditions required for investor indifference are 1) Around 75% or more in subsidies for capital cost and 2) Obtain a loan for remaining capital cost that has no more than 11% interest. 5.7.3 10 Year Finally we examine the capacity factor and capital cost that produces an NPV of $0 over a 10 year time horizon. As shown in the below plot, about 40% of the capital cost will make an investor indifferent about investing in the project. 7 2 Capital Cost Vs 10 Year Average NPV x 10 1 Average NPV ($) 0 -1 -2 -3 -4 -5 0 10 20 30 40 50 60 Percnt of Capital Cost 70 80 90 100 Figure 5-16 The change in Average NPV with respect to changes in the percent of wind farm capital cost 64 Holding everything else constant, the capacity factor needs to reach around 40% for an investor to be indifferent about undertaking the project for a 10 year investment. 7 1 Capacity Factor Vs 10 Year Average NPV x 10 0 Average NPV ($) -1 -2 -3 -4 -5 -6 0 5 10 15 25 20 Capacity Factor 30 35 40 45 Figure 5-17 The change in Average NPV with changes in the capacity factor The following table illustrates the necessary capacity factor for changing levels of capital costs. Capital Cost 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Interest Rate 1% 6% 10.75% 10.75% 13.15% 13.40% 15.50% 16.40% 17.85% 19.50% 20.25% 22.55% 22.65% 25.65% 25.00% 28.65% 27.40% 31.75% 29.75% 34.75% 32.15% 37.80% 11% 10.75% 13.65% 17.50% 21.30% 25.15% 28.95% 32.80% 36.65% 40.35% 44.25% Table 5-3 Capacity factors with respect to percent of capital cost and interest rate (10 year). 65 If we assume that the PJM capacity factor must stay around 13%, then the conditions required for investor indifference are 1) Around 80% or more in subsidies for capital cost and 2) Obtain a loan for remaining capital cost that has no more than 11% interest. For present time feasibility, it is unlikely that the PJM rate of 13% will change in the near future. In order for investors to be interested in wind farm projects, heavy subsidies are required to lower the capital costs of wind turbine installation and wind farm construction. For long time horizons of 10, 20 and 30 years, the subsidies should be of magnitudes 80%, 75% and 70% respectively. The fact that a difference of a 20 year investment horizon only reduces the required subsidy by 10% emphasizes the burden that initial capital cost places on operating a wind farm for profit. Concluding this section, we have also seen that investor indifference is more sensitive to loan interest rates when capital costs are high. 5.8 Risk Analysis After a financial analyst has determined the potential profitability of the project, a risk analysis should be performed to measure the probability of the amount of capital that should be left un-invested to weather potential side tracks from the expected outcome. The results obtained for 10, 20 and 30 year investment horizons showed that the project was not worth investing in. Therefore, for the purpose of illustrating risk measures, let us assume that the Bear Creek wind farm was awarded a 50% reduction in initial capital costs from the government, and that the state of Pennsylvania removed the $5 million cap on their state loan for renewable projects. In addition, PJM has afforded 18% of capacity to Bear Creek due to its up to date technology that ensures more efficient transmission to the utility grid. Given these assumptions, the NPV distribution of Bear Creek is found. 66 Net Present Value 150 Frequency 100 50 0 -2 -1 0 1 2 $USD 3 4 5 7 x 10 Figure 5-18 Distribution of wind farm NPV Having the NPV distribution for the Bear Creek wind farm project, we are now in a position to quantify certain investment risks to better prepare investors for the type of profitable outcome they can expect. In chapter 1 we introduced value at risk (VaR), expected shortfall (ES) and average value at risk (AVaR). Depending on the risk tolerance of the investors, a confidence level will be chosen. Typical values are 95% or 97.5% or 99% of confidence. These confidence levels translate to finding the 5%, 2.5% and 1% percentile of the distribution. Before we proceed, Microsoft’s @Risk program is used to fit our histogram to an appropriate distribution by performing chi-square tests 27. @Risk has fitted the NPV distribution to a lognormal distribution. The 1%, 2.5% and 5% percentiles are -$8,799,761, -$6,809,412 and -$4,709,812 respectively. 27 @Risk summary table is given in appendix. 67 Figure 5-19 @Risk lognormal distribution of NPV 5.8.1 NPV Value at risk The NPV data fits well to the lognormal distribution, and since @Risk has provided the 95%, 97.5% and 99% confidence values, we have already determined the values of the VAR. 5.0% 2.5% 1.0% VaR $4,709,812 $6,809,412 $8,799,761 68 5.8.2 NPV Expected Shortfall The expected shortfall can be found both empirically or with the formula [16]: z2 exp - 2 σ , ES = α 2Π ( where ) (4-7) σ = standard deviation, z = cumulative normal at σ (e.g. 1.645 for a normal distribution). Since we have the distribution and the VaR at the 5%, 2.5% and 1% percentile, all we have to do is sum up all the values below their corresponding values at risk, and divide by the number of simulations that fell below that level. This leads us to the following answers: 5% 2.50% 1% $4,709,812 $6,809,412 $8,799,761 VaR $7,256,816 $8,733,605.71 $10,532,735.11 Expected Shortfall 5.8.3 NPV Average Value at Risk To find the average value at risk using our empirical results, equation 2-3 is used. AVaRα ( x) = − where 1 α (E[X 1{ }]− q P[X ≤ q ]− α ), + x ≤ qα+ α + α X = random variable describing NPV, α = 1-confidence level, qα+ = NPV corresponding to α level. 5.0% 2.5% 1.0% Expected Shortfall $7,256,816 $8,733,605.71 $10,532,735.11 AVaR $7,256,816 $8,733,605.71 $10,532,735.11 69 The results show that the expected shortfall and AVaR are identical, suggesting that the VaR does not fall between nodes and that no jumps occur at that point. The risk measures offer the investors more information about the project and allow them to plan for potential crisis. The VaR measure has showed that for a 50% subsidy, state funded loan and 18% PJM capacity factor, the investors can be 95% confident that they will not lose more than $4.7 million. In the off chance that they do, they can expect to lose on average of about $7.3 million. Depending on investor risk tolerance, the investors will decide whether the project is worth the risk. They will weigh the expected profit of $8.5 million against the 5% probability of losing an average of $7.3 million. 70 6 Conclusion The success of developing a wind farm project will be unique to the environment and policies it is subject to. These will include state subsidies that help the installation and running of the wind farm as well as the site location and terrain that influence the construction costs. However different each project might be, it is important to have a general method for valuating them. This will not only help when performing comparisons between different sites, but more importantly, also help understand the conditions that increase the chances of wind farm profitability. Before the actual construction and installation of a wind farm begins, agreements need to be formulated to remove some uncertainties. These agreements include a connection and power purchasing agreement with a utility company, a loan agreement from a financial institution, an operation and maintenance agreement, site and construction agreements, as well as shareholder and insurance agreements. Investors will hire financial analysts to value the projects feasibility and worth, and to help set the benchmarks when contracting agreements. Apart from the general revenue and cost projections, the financial analyst will want to perform standard risk measures. This will be subject to the investors risk tolerance, and should be thought about carefully when making the decision to invest. Unfortunately for the assumptions we took when valuating Bear Creek, the NPV distribution showed very unprofitable results. There would have been no motivation for a financial analyst to continue with the VaR and ES risk measures in this case. Simulating the electricity spot prices was the most challenging when forecasting the wind farm revenue. It is possible to leave out the simulation effort and assume a set 71 price of electricity over the horizon. However, this is a highly simplified approach and does not take into consideration the volatility of energy markets. In this thesis we used a one factor mean reverting model to simulate future spot prices. A better model would be a double factor mean reverting model that incorporates forward or future prices into the simulation. This requires first generating a futures contract model and then using it as an input for the spot price simulation. With a futures contract model at hand, the financial analyst would also be able to check the direct and cross hedging strategies that contribute to the value of a wind farm. Other improvements to the simulation model include the volatility selection. Historical volatilities do not take into account future expectations of spot price movements, and generally should not be used when extrapolating into the future. We attempted to use implied volatilities but unfortunately the 2 and 3 year puts and calls were not heavily traded. This provided too few points to use for analysis, and so was abandoned. When finding the value of a project, the analyst will want to incorporate real option valuation. This thesis used simulations of potential electricity price paths to account for the uncertainty in future price movements. To enhance this, it will be worthwhile to include wind farm managerial decisions. Such decisions could involve 1) Purchasing more wind turbines to increase revenue if the capital costs decrease enough during the 30 year horizon. 2) Selling of the wind farm if its ongoing value exceeds a predetermined specified amount (like a strike price). 3) Adding Renewable Portfolio Standards (RPS) into the valuation process to account for the value added through energy credits received from conventional energy sources to offset their carbon footprints. From our analysis of the Bear Creek wind farm, important considerations have been realized. The first issue to clarify is the investment horizon of the investor. Our analysis identifies that wind farms are not short term investments. For investment horizons of less than 20 years, it is unlikely to achieve a positive NPV. The main reason for this is the large liability from the construction and installation phase. The ratio of the initial capital costs to the annual operating profit was around 35:1. The time required to 72 repay this cost spans over 2 to 3 decades. Investors interested in wind farms must realized that profits cannot be made until this liability has been paid off. Although our analysis only considered the Bear Creek wind farm, the cost structure would have been similar for other wind farm projects. The difference between the Bear Creek project and other projects would be the state support and subsidies as well as the electricity spot prices (outside the PJM area). The other important factor identified as influencing the profitability of the Bear Creek project was its capacity factor. PJM has allocated 13% of capacity to wind power generators. This is because of the intermittency of wind and the obligation PJM has to serving the high demand of electricity every day. Unfortunately for Bear Creek, selling 13% of its nameplate capacity severely limits operating profit, not to mention its ability to repay the large loan to install the wind farm. Tax incentives and government subsidies are very important for current wind farm economics. The Production Tax Credit (PTC) contributed around $573,955 to the Bear Creek wind farm annually, which was 30%-50% of its annual operating profits. The Investment Tax Credit (ITC) for Bear Creek was inferior to PTC and thus was not taken. ITC offers 30% tax credit for the first 8 years of wind farm life. Unfortunately, because income only became positive after year 10, there were no realized tax deduction incentives. ITC would be preferable only if profits began to be positive no later than year 4. The results from Bear Creek would have given the financial analyst sufficient information about the contracts and agreements necessary to make the project worthwhile. Given the actual risk tolerance of the investors, the analyst can find the conditions that will achieve investor confidence to invest. For Bear Creek, we found that the shareholders must be committed to a 20 to 30 year investment. 10 year investments would require a higher level of reliability and efficiency in capacity transmission which is not available (at a feasible price) with current technology. Even with a long term investment horizon of 20 to 30 years, the connection and power purchasing agreement with PJM must also be negotiated to allow for higher capacity to be transmitted. This negotiation might be successful if PJM can trust the wind farm to be reliable and be able to consistently provide a higher proportion of its nameplate capacity. 73 The strategies to achieve a certain level of capacity were introduced in section 2.2 and involved the purchasing of electricity futures or installation of electrical storage units. Hedging the uncertainty of wind capacity with electricity futures will allow the wind farm to achieve the level of capacity but at an extra fixed cost (depends on the price of futures contracts and financial options). The extra cost of purchasing financial options to hedge could add value to the project if the extra capacity allocated to the wind farm from PJM generates additional revenue to offset the cost. To test this, general wind forecasts are required to determine the likelihood of Bear Creek electricity production to fall below its promised capacity. We did not attempt to examine this method because wind forecasting is not within the scope of this thesis. This is however recommended to be added if the financial analyst is able and wishes to improve the method of valuation. Installing an electrical storage facility is the other option. With backup storage, PJM can feel more confident about allocating more capacity to wind farms. Once again, the wind farm will have to weigh the installation cost of the facility and the added profit obtained through supplying backup electricity. We have seen through the analysis of Bear Creek that the main hindrance of wind farm profitability is the high initial capital cost. To mitigate the affect this has, the government should heavily subsidize wind farm construction and installation cost. Our example of Bear Creek suggests that it is unlikely for a wind farm to be solely owned by a private entity. Due to the necessary government subsidies on the capital cost, it should be considered to approach the government (or state) to enter into renewable projects as a joint partner. This is not to suggest regressing back to state ownership but rather to give investors more confidence to enter into these projects and to accelerate the return investors seek when investing in risky ventures. Although secondary to capital cost, the analyst should strongly suggest negotiating loan agreement. Investors should seek to obtain state loan agreements due to their lower interest rates (under 2%) as well as low rates of interest from financial institutions. For this purpose, larger state loans should be made available. States should work towards removing standard caps on the size of state loans, and instead make it specific to the size and requirements of each project. However as the bear Creek results have shown, even if wind farms can obtain a state loan to cover the entire cost, we found 74 that a 30 year wind farm investment would still need a 30% government subsidy on the overall wind farm cost to be profitable. The future for wind farms seems to lie within the development of two main industries. They are the wind power manufacturing industry and the electrical storage technology industry. Breakthroughs in storage technology will greatly help reduce the intermittency problem utility companies have with wind power and lower capital costs will make wind farm investments more attractive. 75 Appendix Footnote 6 List of states served by PJM Delaware, Illinois, Indiana, Kentucky, Maryland, Michigan, New Jersey, North Carolina, Ohio, Pennsylvania, Tennessee, Virginia, West Virginia and the District of Columbia. S-Plus Commands Footnote 7 Function align Import data Dates01.08 and Rawdates excel files. Rawdates are the dates with prices without alignment and Dates01.08 are the everyday dates from year 2001 to 2008. > d=as(Dates01.08[,1],"timeDate") Create a time series vector: > SP.ts=timeSeries(positions=as(Rawdates$Trade,"timeDate"),data=Rawdates$Wtd) Use the align function to interpolate missing values: > a=align(SP.ts,d,how = "interp",error.how = "nearest"); Matlab code Footnote 12 Implied Volatility Smiles plots % % Sort JOdata by year of expiration % Year=9:11; % JO09=zeros(500,8); % JO10=zeros(500,8); % JO11=zeros(500,8); % i=1; % j=1; % k=1; % for row=1:length(JOdata) % if JOdata(row,2)==9 % JO09(i,1:8)=JOdata(row,1:8); A1 % i=i+1; % end % if JOdata(row,2)==10 % JO10(j,1:8)=JOdata(row,1:8); % j=j+1; % end % if JOdata(row,2)==11 % JO11(k,1:8)=JOdata(row,1:8); % k=k+1; % end % end % % % Match Option Maturity to Maturity and appropriate risk free % MaturityRiskfree09=zeros(length(JO09),2); % MaturityRiskfree10=zeros(length(JO11),2); % MaturityRiskfree11=zeros(length(JO10),2); % % for row=1:length(JO09) % for m=1:12 % if JO09(row,1)==m % MaturityRiskfree09(row,1)=Maturitiesdata(m,2); % if m<=3 % MaturityRiskfree09(row,2)=USTBdata(1,1); % end % if m>3 && m<=6 % MaturityRiskfree09(row,2)=USTBdata(2,1); % end % if m>6 % MaturityRiskfree09(row,2)=USTBdata(3,1); % end % end % end % end % % for row=1:length(JO10) % for m=1:12 % if JO10(row,1)==m % MaturityRiskfree10(row,1)=Maturitiesdata(12+m,2); % MaturityRiskfree10(row,2)=USTBdata(3,1); % end % end % end % % for row=1:length(JO11) % for m=1:12 % if JO11(row,1)==m % MaturityRiskfree11(row,1)=Maturitiesdata(24+m,2); % MaturityRiskfree11(row,2)=USTBdata(4,1); % end % end % end % % % Sort JO09Month by Month of expiration % ImpliedVolatility09=zeros(12,65); % Implied volatility for the options of each month in 2009 % ImpliedVolatility10=zeros(12,10); % Implied volatility for the options of each month in 2010 % ImpliedVolatility11=zeros(12,2); % Implied volatility for the options of each month in 2011 A2 % JO09OpPrice=zeros(12,65); % OpPrice=0; % K=0; % JO09Strike=zeros(12,65); % for m=1:12 % % n=1; % for l=1:length(JO09) % if JO09(l,1)==m % Price0=FutPrice09(l,m); % JO09OpPrice(m,n)=JO09(l,4); % OpPrice=JO09OpPrice(m,n); % JO09Strike(m,n)=JO09(l,3); % K=JO09Strike(m,n); % r=MaturityRiskfree09(l,2)/100; % T=MaturityRiskfree09(l,1)/365; % ImpliedVolatility09(m,n)=blsimpv(Price0,K,r,T,OpPrice, true); % n=n+1; % end % end % end % % figure(1); % for o=1:12 % subplot(4,3,o); % plot(ImpliedVolatility09(o,1:65),JO09Strike(o,1:65),'x'); % title(o); % xlabel('Implied Volatility'); % ylabel('Strike'); % end % % JO10OpPrice=zeros(12,10); % JO10Strike=zeros(12,10); % OpPrice=0; % K=0; % for m=1:12 % %m=6; % n=1; % for l=1:length(JO10) % if JO10(l,1)==m % Price0=FutPrice10(l,m); % JO10OpPrice(m,n)=JO10(l,4); % OpPrice=JO10OpPrice(m,n); % JO10Strike(m,n)=JO10(l,3); % K=JO10Strike(m,n); % r=MaturityRiskfree10(l,2)/100; % T=MaturityRiskfree10(l,1)/365; % ImpliedVolatility10(m,n)=blsimpv(Price0, K, r, T, OpPrice, true); % n=n+1; % end % end % end % % figure(2); % for o=1:12 % subplot(4,3,o); A3 % plot(ImpliedVolatility10(o,1:10),JO10Strike(o,1:10),'x'); % title(o); % xlabel('Implied Volatility'); % ylabel('Strike'); % end % % JO11OpPrice=zeros(12,2); % OpPrice=0; % K=0; % JO11Strike=zeros(12,2); % for m=1:12 % %m=6; % n=1; % for l=1:length(JO11) % if JO11(l,1)==m % Price0=FutPrice11(l,m); % JO11OpPrice(m,n)=JO11(l,4); % OpPrice=JO11OpPrice(m,n); % JO11Strike(m,n)=JO11(l,3); % K=JO11Strike(m,n); % r=MaturityRiskfree11(l,2)/100; % T=MaturityRiskfree11(l,1)/365; % ImpliedVolatility11(m,n)=blsimpv(Price0, K, r, T, OpPrice, true); % n=n+1; % end % end % end % % figure(3); % title('Year 11'); % for o=1:12 % subplot(4,3,o); % plot(ImpliedVolatility11(o,1:2),JO11Strike(o,1:2),'x'); % title(o) % xlabel('Implied Volatility'); % ylabel('Strike'); % end A4 Implied Volatility 2009 200 100 0 0 0.5 1 Implied Volatility 10 0 0.2 0.4 Implied Volatility 200 100 0 200 100 0 Strike 0 0.5 1 Implied Volatility 5 Strike 200 100 0 3 0 0.5 Implied Volatility 8 Strike Strike Strike 0 0.5 Implied Volatility 7 200 100 0 0 0.5 1 Implied Volatility 11 Strike 200 100 0 0 0.5 1 Implied Volatility 4 Strike 200 100 0 2 Strike Strike Strike Strike Strike 1 200 100 0 0 0.2 0.4 Implied Volatility 200 100 0 200 100 0 200 100 0 200 100 0 0 0.5 1 Implied Volatility 6 0 0.2 0.4 Implied Volatility 9 0 0.5 Implied Volatility 12 0 0.2 0.4 Implied Volatility Figure A-1 Volatility smiles for JO call options expiring in the year 2009. Implied Volatility 2010 100 50 0 0 0.2 0.4 Implied Volatility 10 0 0.2 0.4 Implied Volatility 200 100 0 100 50 0 Strike Strike 100 50 0 0 0.2 0.4 Implied Volatility 5 0 0.2 0.4 Implied Volatility 8 Strike Strike Strike 0 0.2 0.4 Implied Volatility 7 100 50 0 0 0.2 0.4 Implied Volatility 11 Strike 200 100 0 0 0.2 0.4 Implied Volatility 4 Strike 100 50 0 3 2 Strike Strike Strike Strike Strike 1 100 50 0 0.2 0 0.4 Implied Volatility 100 50 0 100 50 0 100 50 0 100 50 0 0 0.2 0.4 Implied Volatility 6 0 0.2 0.4 Implied Volatility 9 0 0.2 0.4 Implied Volatility 12 0 0.2 0.4 Implied Volatility Figure A-2 Volatility smiles for JO call options expiring in the year 2010. A5 Strike Strike Strike 2 90 80 70 0.28 0.26 0.24 Implied Volatility 5 90 80 70 0.28 0.26 0.24 Implied Volatility 8 90 80 70 0.28 0.26 0.24 Implied Volatility 11 90 80 70 0.3 0.25 0.2 Implied Volatility Strike Strike Strike Strike 1 90 80 70 0.28 0.26 0.24 Implied Volatility 4 90 80 70 0.28 0.26 0.24 Implied Volatility 7 90 80 70 0.28 0.26 0.24 Implied Volatility 10 90 80 70 0.28 0.26 0.24 Implied Volatility Strike Strike Strike Strike Strike Implied Volatility 2011 3 90 80 70 0.28 0.26 0.24 Implied Volatility 6 90 80 70 0.28 0.26 0.24 Implied Volatility 9 90 80 70 0.28 0.26 0.24 Implied Volatility 12 90 80 70 0.3 0.25 0.2 Implied Volatility Figure A-3 Volatility smiles for JO call options expiring in the year 2010. Footnote 13 Spot Price Simulation % Inputs for electricity path simulation %sigma=.11904; %Volatility %tc1=1/12; %tc2=7/12; %alpha=0.07385; %Beta1=.00026; %Beta2=.00026; %b=4.5; %dt=1/365; %LnSt=zeros(365*30,2000); %days=1:1:365; %t=zeros(1,365*30); % make a 10 year seasonality vector which repeats each year %for i=1:30 % if i>1 % t((i-1)*365+1:i*365)=days; % else % t(i:365)=days; % end %end % generate 2000 log price simulations A6 %for j=1:2000 %dLnSt=zeros(365*30,1); %dSea=zeros(365*30,1); %LnSt(1,j)=log(52.43); %dz=sqrt(dt)*randn(365*30,1); % for i=2:365*30 % dSea(i)=-(2*pi*Beta1*sin(2*pi*(t(i)/365-tc1))+4*pi()*Beta2*sin(4*pi()*(t(i)/365-tc2))); % dLnSt(i)=alpha*(b-LnSt(i-1,j))*dt+sigma*dz(i); % LnSt(i,j)=dSea(i)+dLnSt(i)+LnSt(i-1,j); % end %end % %St4=exp(LnSt); Footnote 21 EBIT, Net Income and Cash Flow % %Calculating EBIT % %Annual Cost streams % %DepreciationSchedule=[0,.03750,.07219,.06677,.06177,.05713,.05285,0.04888,0.04522,0.04462,.04461,. %04461,.04461,.04461,.04461,.04461,.04461,.04461,.04461,.04461,.0295,0,0,0,0,0,0,0,0,0,0]; % Depreciation=zeros(1,31); % Expenses=zeros(1,31); % costinflation=0.01; % Windturbinecost=26413395.41; % OtherCapitalCost=Windturbinecost*.4; % Expenses(2)=0.053*26413395.41; % for i=2:31 % if i==2 % Depreciation(i)=DepreciationSchedule(i)*Windturbinecost; % Expenses(i)=Expenses(2); % else % Depreciation(i)=DepreciationSchedule(i)*Windturbinecost; % Expenses(i)=Expenses(i-1)*(1+costinflation); % end % end % % % Annual Revenue streams % % daily average generation for a typical Penn wind farm (Bear Creek) % % http://www.pawindenergynow.org/pa/farms.html % % 2.0 MW rated capacity with a 35% capacity factor. % % Assume wind farm works all day. % CapacityFactor=.357; % PJMCapacityFactor=.13; % OperatingHoursPerDay=24; % TotalMaxCapacity=24; % generation=OperatingHoursPerDay*TotalMaxCapacity*PJMCapacityFactor; % % daily revenue for a penn wind farm % drev=St4*generation; % % Annual revenue % AR=zeros(31,2000); % for j=1:2000 % for i=2:31 % if i==2 % AR(i,j)=sum(drev(1:365,j)); % %leap year A7 %% elseif i==4||i==8 %% AR(i,j)=sum(drev(i*365:(i+1)*365+1,j)); % else % AR(i,j)=sum(drev((i-2)*365:(i-1)*365,j)); % end % end % end % % %EBIT % EBIT=zeros(31,2000); % for i=1:31 % for j=1:2000 % EBIT(i,j)=AR(i,j)-Expenses(i)-Depreciation(i); % end % end % % % EBT % % Annuity Payments % StateGrant=1000000; % if (Windturbinecost+OtherCapitalCost-StateGrant)<=5000000 % StateLoan=Windturbinecost+OtherCapitalCost-StateGrant; % else % StateLoan=5000000; % end % tenyear=0.0246; % 10 year 1/2/2009 treasury rate % StateRate=tenyear-0.01; % BankLoan=Windturbinecost+OtherCapitalCost-StateLoan-StateGrant; % r=0.08; % T=10; % StateAnnuity=StateLoan*StateRate/(1-(1/(1+StateRate)^T)); % BankAnnuity=BankLoan*r/(1-(1/(1+r)^T)); % Annuity=StateAnnuity+BankAnnuity; % EBT=zeros(31,2000); % EBT(2:11,:)=(EBIT(2:11,:)-Annuity); % EBT(12:31,:)=(EBIT(12:31,:)); % % % Net Income % Tax=0.35; % PTC=.021*1000*generation*365; % NetIncome=zeros(31,2000); % for j=1:2000 % for i=2:31 % if EBT(i,j)<0 && i<=11 % NetIncome(i,j)=EBT(i,j)+PTC; % elseif i<=11 % NetIncome(i,j)=EBT(i,j)*(1-Tax)+PTC; % elseif i>11 && EBT(i,j)<0 % NetIncome(i,j)=EBT(i,j); % elseif i>11 % NetIncome(i,j)=EBT(i,j)*(1-Tax); % end % end % end % % % Cash Flow % % Add back in depreciation since this nether really left A8 % CashFlow=zeros(31,2000); % for j=1:2000 % CashFlow(:,j)=NetIncome(:,j)+Depreciation'; % end % % % discount cash flows with appropriate treasury rate % DiscountedCashFlow=zeros(31,2000); % for j=1:2000 % for i=2:31 % if i==2 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.004*1); % elseif i>2 && i<=3 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0088*2); % elseif i>3 && i<=4 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0114*3); % elseif i>4 && i<=6 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0172*((i-1)/5)*5); % elseif i>6 && i<=8 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0207*((i-1)/7)*7); % elseif i>8 && i<=11 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0246*((i-1)/10)*10); % elseif i>11 && i<=21 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0322*((i-1)/20)*20); % elseif i>21 && i<=31 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0283*((i-1)/30)*30); % end % end % end % % sum discounted cash flows cash flows % NPV=zeros(1,2000); % for i=1:2000 % NPV(i)=sum(DiscountedCashFlow(:,i)); % end % % hist(NPV,100); % xlabel('$USD'); % Title('Net Present Value'); % ylabel('Frequency'); % mean(NPV) % % % for i=2:31 % % subplot(6,5,i-1); % % hist(CashFlow(i,:)) % % end Footnote 24 Sensitivity Analysis Code for capacity factor sensitivity analysis (Capital cost, taxation, interest rate, debt to equity ratio all done similarly) % % Capacity Factor, -20% to +20% % % %Calculating EBIT % %Annual Cost streams % DepreciationSchedule=[0,.20,.32,.192,.1152,.1152,.0576,0,0,0,0]; % Depreciation=zeros(1,11); A9 % Expenses=zeros(1,11); % costinflation=0.02; % Windturbinecost=35217860; % OtherCapitalCost=Windturbinecost*.4; % Expenses(2)=0.053*Windturbinecost; % for i=2:11 % if i==2 % Depreciation(i)=DepreciationSchedule(i)*Windturbinecost; % Expenses(i)=Expenses(2); % else % Depreciation(i)=DepreciationSchedule(i)*Windturbinecost; % Expenses(i)=Expenses(i-1)*(1+costinflation); % end % end % % % Varying the capacity factor % AvNPVCF=zeros(1,41); % l=1; % for k=-20:1:20 % %Annual Revenue streams % CapacityFactor=.35+(k/100); % PJMCapacityFactor=CapacityFactor; % OperatingHoursPerDay=24; % TotalMaxCapacity=24; % generation=OperatingHoursPerDay*PJMCapacityFactor*TotalMaxCapacity; % % daily revenue for a penn wind farm % drev=St*generation; % % Annual revenue % AR=zeros(11,2000); % for j=1:2000 % for i=2:11 % if i==2 % AR(i,j)=sum(drev(1:365,j)); % %leap year % % elseif i==4||i==8 % % AR(i,j)=sum(drev(i*365:(i+1)*365+1,j)); % else % AR(i,j)=sum(drev((i-2)*365:(i-1)*365,j)); % end % end % end % %EBIT % EBIT=zeros(11,2000); % for j=1:2000 % for i=1:11 % EBIT(i,j)=AR(i,j)-Expenses(i)-Depreciation(i); % end % end % % % EBT % % Annuity Payments % StateLoan=5000000; % StateGrant=1000000; % tenyear=0.0246; % 10 year 1/2/2009 treasury rate % StateRate=tenyear-0.01; % BankLoan=Windturbinecost+OtherCapitalCost-StateLoan-StateGrant; A10 % r=0.08; % T=10; % StateAnnuity=StateLoan*StateRate/(1-(1/(1+StateRate)^T)); % BankAnnuity=BankLoan*r/(1-(1/(1+r)^T)); % Annuity=StateAnnuity+BankAnnuity; % EBT=zeros(11,2000); % EBT(2:11,:)=EBIT(2:11,:)-Annuity; % % Net Income % Tax=0.35; % NIT=0.05; % % Convert MW to KW % AnnualPTC=generation*1000*365*.021; % NetIncome=zeros(11,2000); % for j=1:2000 % if EBT(:,j)<0 % NetIncome(2:11,j)=EBT(2:11,j)-BankLoan*(NIT)+AnnualPTC; % else % NetIncome(2:11,j)=EBT(2:11,j)*(1-Tax)+AnnualPTC; % end % end % % % Cash Flow % % Add back in depreciation since this nether really left % CashFlow=zeros(11,2000); % for j=1:2000 % CashFlow(:,j)=NetIncome(:,j)+Depreciation'; % end % % % discount cash flows with appropriate treasury rate % DiscountedCashFlow=zeros(11,2000); % for j=1:2000 % for i=2:11 % if i==2 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.004*1); % elseif i>2 && i<=3 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0088*2); % elseif i>3 && i<=4 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0114*3); % elseif i>4 && i<=6 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0172*((i-1)/5)*5); % elseif i>6 && i<=8 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0207*((i-1)/7)*7); % elseif i>8 && i<=11 % DiscountedCashFlow(i,j)=CashFlow(i,j)*exp(-0.0246*((i-1)/10)*10); % end % end % end % % sum discounted cash flows cash flows % NPV=zeros(1,2000); % for i=1:2000 % NPV(i)=sum(DiscountedCashFlow(:,i)); % end % AvNPVCF(l)=mean(NPV); % l=l+1; % end % A11 % plot(AvNPVCF) Sensitivity plots %PercentVariation=[-20:1:20]; %AXIS([-20 20 -5000000 20000000]) %axis manual %hold on %plot(PercentVariation,AvNPVOM,'.',xlabel('Percent %Variation'),ylabel('NPV($)')) %plot(PercentVariation,AvNPVT,'o',xlabel('Percent %Variation'),ylabel('NPV($)')) %plot(PercentVariation,AvNPVCF,'.-',xlabel('Percent %Variation'),ylabel('NPV($)')) %plot(PercentVariation,AvNPVCC,'x',xlabel('Percent %Variation'),ylabel('NPV($)')) %legend('Operation & Maintenance','','Taxes','','Capacity %Factor','','Capital Cost') %hold off; Footnote 26 Investor Indifference Points % NPV1=zeros(10,400); % Indifferent=zeros(10000,2); % l=1; % for q=10:10 % %Calculating EBIT % %Annual Cost streams % DepreciationSchedule=[0,.03750,.07219,.06677,.06177,.05713,.05285,0.04888,0.04522,0.04462,.04461,.0 4461,.04461,.04461,.04461,.04461,.04461,.04461,.04461,.04461,.0295,0,0,0,0,0,0,0,0,0,0]; % Depreciation=zeros(1,31); % Expenses=zeros(1,31); % costinflation=0.01; % Windturbinecost=26413395.41*q/10; % OtherCapitalCost=Windturbinecost*.4; % Expenses(2)=0.053*26413395.41; % for i=2:31 % if i==2 % Depreciation(i)=DepreciationSchedule(i)*Windturbinecost; % Expenses(i)=Expenses(2); % else % Depreciation(i)=DepreciationSchedule(i)*Windturbinecost; % Expenses(i)=Expenses(i-1)*(1+costinflation); % end % end % % % Annual Revenue streams % % daily average generation for a typical Penn wind farm (Bear Creek) % % http://www.pawindenergynow.org/pa/farms.html % % 2.0 MW rated capacity with a 35% capacity factor. % % Assume wind farm works all day. % for w=98:98 % CapacityFactor=.357; % PJMCapacityFactor=(w)/1000; % OperatingHoursPerDay=24; % TotalMaxCapacity=24; % generation=OperatingHoursPerDay*TotalMaxCapacity*PJMCapacityFactor; % % daily revenue for a penn wind farm % drev=St4*generation; % % Annual revenue % AR=zeros(31,1000); A12 % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % for i=2:31 for j=1:1000 if i==2 AR(i,j)=sum(drev(1:365,j)); %leap year % elseif i==4||i==8 % AR(i,j)=sum(drev(i*365:(i+1)*365+1,j)); else AR(i,j)=sum(drev((i-2)*365:(i-1)*365,j)); end end end %EBIT EBIT=zeros(31,1000); for i=1:31 for j=1:1000 EBIT(i,j)=AR(i,j)-Expenses(i)-Depreciation(i); end end % EBT % Annuity Payments StateGrant=1000000; if (Windturbinecost+OtherCapitalCost-StateGrant)<=5000000 StateLoan=Windturbinecost+OtherCapitalCost-StateGrant; else StateLoan=5000000; end tenyear=0.0246; % 10 year 1/2/2009 treasury rate StateRate=tenyear-0.01; BankLoan=Windturbinecost+OtherCapitalCost-StateLoan-StateGrant; T=10; r=0.06; StateAnnuity=StateLoan*StateRate/(1-(1/(1+StateRate)^T)); BankAnnuity=BankLoan*r/(1-(1/(1+r)^T)); Annuity=StateAnnuity+BankAnnuity; EBT=zeros(31,1000); EBT(2:11,:)=(EBIT(2:11,:)-Annuity); EBT(12:31,:)=(EBIT(12:31,:)); % Net Income Tax=0.35; PTC=.021*1000*generation*365; NetIncome=zeros(31,1000); for j=1:1000 for i=2:31 if EBT(i,j)<0 && i<=11 NetIncome(i,j)=EBT(i,j)+PTC; elseif i<=11 NetIncome(i,j)=EBT(i,j)*(1-Tax)+PTC; elseif i>11 && EBT(i,j)<0 NetIncome(i,j)=EBT(i,j); elseif i>11 NetIncome(i,j)=EBT(i,j)*(1-Tax); A13 % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %% %% %% %% % end % end end end end % Cash Flow % Add back in depreciation since this nether really left CashFlow=zeros(31,1000); for j=1:1000 CashFlow(:,j)=NetIncome(:,j)+Depreciation'; end CashFlow1=zeros(31,1); for i=2:31 CashFlow1(i)=mean(CashFlow(i,:)); end % discount cash flows with appropriate treasury rate DiscountedCashFlow=zeros(31,1); for i=2:31 if i==2 DiscountedCashFlow(i)=CashFlow1(i)*exp(-0.004*1); elseif i>2 && i<=3 DiscountedCashFlow(i)=CashFlow1(i)*exp(-0.0088*2); elseif i>3 && i<=4 DiscountedCashFlow(i)=CashFlow1(i)*exp(-0.0114*3); elseif i>4 && i<=6 DiscountedCashFlow(i)=CashFlow1(i)*exp(-0.0172*((i-1)/5)*5); elseif i>6 && i<=8 DiscountedCashFlow(i)=CashFlow1(i)*exp(-0.0207*((i-1)/7)*7); elseif i>8 && i<=11 DiscountedCashFlow(i)=CashFlow1(i)*exp(-0.0246*((i-1)/10)*10); elseif i>11 && i<=21 DiscountedCashFlow(i)=CashFlow1(i)*exp(-0.0322*((i-1)/20)*20); elseif i>21 && i<=31 DiscountedCashFlow(i)=CashFlow1(i)*exp(-0.0283*((i-1)/30)*30); end end % sum discounted cash flows cash flows NPV1(q,w)=sum(DiscountedCashFlow); if NPV1(q,w)>0 && NPV1(q,w)<=100000 Indifferent(l,:)=[q w]; l=l+1; end if (NPV(q,w,e)>=-50000 && NPV(q,w,e)< 0) Indifferent(l,:)=[q w e]; l=l+1; end A14 Footnote 27 @Risk Summary Table @RISK Fit Statistics Performed By: Princeton Affiliate Date: Thursday, April 09, 2009 7:56:59 PM Input Lognorm Distribution Statistics Minimum (12,855,832.84) Maximum 40,510,793.62 Mean Mode 6,800,380.13 4332839.7148 [est] (42,771,456.03) +Infinity 6,800,023.91 5,079,305.17 Median 6,014,865.96 6,219,683.38 Std. Deviation 7,665,551.91 7,652,651.12 Skewness 0.50 0.47 Kurtosis 3.55 3.39 Percentiles 5% (4,757,573.25) (4,709,812.10) 10% (2,308,969.92) (2,527,400.85) 15% (822,826.66) (984,673.08) 20% 445,888.52 283,507.64 25% 1,606,664.97 1,402,123.37 30% 2,571,841.18 2,431,421.84 35% 3,576,910.44 3,406,617.72 40% 4,381,330.34 4,351,429.70 45% 5,163,038.89 5,283,941.35 50% 6,011,844.75 6,219,683.38 55% 7,081,792.23 7,173,646.32 60% 7,996,754.55 8,162,006.62 65% 9,029,730.38 9,204,114.74 70% 10,099,305.41 10,325,423.05 75% 11,474,295.50 11,562,645.33 80% 12,833,971.36 12,974,305.57 85% 14,664,605.57 14,666,125.22 90% 16,640,135.96 16,867,954.81 95% 20,447,519.24 20,287,610.60 Chi-Squared Test Chi-Sq Test (Binning Information) Anderson-Darling Test Kolmogorov-Smirnov Test A15 References [1] T. Burton, D. Sharpe, N. Jenkins and E. Bossanyi , Wind energy handbook. , J. Wiley & Sons, Chichester, UK (2001). [2] Dale, Lewis & Milborrow, David & Slark, Richard & Strbac, Goran, 2004. 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