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 ...................................................................................................................... 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. "Total cost
estimates for large-scale wind scenarios in UK," Energy Policy, Elsevier, vol. 32(17), pages
1949-1956, November.
[3] Joseph F. DeCarolis, David W. Keith, The economics of large-scale wind power in a
carbon constrained world, Energy Policy, Volume 34, Issue 4, March 2006, Pages 395-410,
ISSN 0301-4215, DOI: 10.1016/j.enpol.2004.06.007.
URL: http://www.sciencedirect.com/science/article/B6V2W-4D09G8J2/2/2bea57f41c3e062518fa69e201d3ad0b
[4] American Wind Energy Association (AWEA), legislative page
URL: http://www.awea.org/legislative/american_recovery_reinvestment_act.html
[5] Schleede, Glenn R., “The True Cost of Electricity from Wind Power And Windmill
Availability Factors”, April 2003
URL: http://www.mnforsustain.org/windpower_schleede_costs_of_electricity.html
[6] Hau, Erich, “Wind Turbines: Fundamental, Technologies, Application, Economics”, 2nd
edition. Springer, 2006
[7] Johnson, G.L., “Wind Energy Systems”, 1985, Prentice-Hall Englewood Cliffs, NJ
[8] Coulomb, L. & Neuhoff, K., "Learning curves and changing product attributes: the case of
wind turbines," Cambridge Working Papers in Economics 0618, Faculty of Economics,
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