The Pennsylvania State University

The Graduate School College of Agricultural Sciences

ASSESSING THE PROFITABILITY OF ANAEROBIC DIGESTERS ON DAIRY

FARMS IN PENNSYLVANIA:

REAL OPTIONS ANALYSIS WITH MULTIPLE JUMP PROCESSES

A Thesis in

Agricultural, Environmental, and Regional Economics

by

Elizabeth R. Leuer

© 2008 Elizabeth R. Leuer

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

May 2008

ii

The thesis of Elizabeth R. Leuer was reviewed and approved* by the following: Jeffrey A. Hyde Associate Professor of Agricultural Economics Thesis Adviser Jeffrey R. Stokes Associate Professor of Agricultural Economics Tom L. Richard Associate Professor of Agricultural and Biological Engineering Stephen M. Smith Professor of Agricultural, Environmental, and Regional Economics Head of the Agricultural Economics and Rural Sociology Department *Signatures are on file in the Graduate School.

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Abstract

Many factors have led to an increased focus on the development of renewable

energy sources. Anaerobic digesters, which convert methane into energy in the form of

electricity, natural gas, or heat, are one such technology. Previous economic research has

shown that subsidies are required for most farms to adopt this technology profitably.

However, that research has not fully addressed issues related to carbon credit trading, net

metering laws, and possible jumps, positive or negative, in the value of digesters.

Carbon credits and net metering are two benefits a farmer may receive for

implementing a digester. In some cases, farmers are eligible to receive carbon credits,

which have a monetary benefit, when operating an anaerobic digester. Net metering laws

effectively modify the costs and returns associated with electricity generation. Before

these laws, farmers often received a bill from the electric company each month, even

though they were generating their own electricity.

This research employs a real options approach with multiple jump diffusion

processes to address the issue in light of these mitigating factors. Real options analysis is

superior to net present value (NPV) calculations because it takes into account the

uncertainty associated with the investment as well as the fact that most investments are at

least partially irreversible. That is they exhibit sunk costs, which cannot be recovered

once the investment has been made.

A jump is defined as any sudden shock to an asset’s value that causes the value to

change by more than the market’s usual variation. It is important to include jumps in this

model because asset values follow a jump diffusion process if market shocks occur. I

define changes in policy or improvements to technology as the jumps that could affect a

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digester’s value. A jump may or may not occur in a given time period. The size of the

jump also varies and small jumps are more likely than large jumps.

In this setting, the value of the underlying asset follows Geometric Brownian

Motion (GBM) in the absence of a jump. When jumps are present, the value follows both

GBM and the jump. My research represents a modification of a model developed by

Martzoukos and Trigeorgis (2002), which allowed them to implement jumps of multiple

types. Additionally, I employ Monte Carlo simulation techniques to simulate paths

throughout the state space, resulting in a distribution of option values.

I use the modified methodology to value European and American call options.

These represent the value of the option to purchase a digester and solids separator at the

end of a five-year period, in the European option case, or during this time period, in the

American option case. I analyzed two different dairy herd sizes, 1,000 and 2,000 cows.

For those farms that have a digester without a solids separator, the American

option values for a 1,000- and 2,000-cow herd are $35,000 and $108,400, respectively,

when carbon credits and net metering are included. These American option values

increase to $109,300 and $318,700, respectively, for herd sizes of 1,000 and 2,000 cows

when the farm sells solids as compost. If the same farms use the separated solids as

bedding material, the option values increase to $257,900 and $715,300. These results are

sensitive to the price the farm receives for electricity sales and the initial capital cost of

the digester. Carbon credit prices and composting costs have little effect on the option

values.

The inclusion of jumps as well as parameters of the jumps also have an effect on

option values. Without jumps, European option values for 1,000-cow farms that use the

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solids for bedding are $76,000, compared to $257,900 when jumps are included in the

model. This indicates that these jumps positively affect option values. The results were

also sensitive to the time to maturity, standard deviation of a jump, mean jump size, and

jump frequency. An increase in the time to maturity, standard deviation of a jump, and

jump frequency increased the option value, while increasing the mean jump size

decreased the option value.

These results suggest that the option to invest increases in value with herd size

and the addition of a solids separator, particularly when the farm uses the solids as

bedding. The price the farm receives for selling excess electricity also affects the option

value. Option values increase with the inclusion of net metering and carbon credit

trading. This suggests that some policies and programs increase the expected

profitability of anaerobic digesters and, consequently, may increase farm-level adoption.

Adding jumps to the model provides a more accurate picture of option value because they

incorporate the uncertainty of market shocks. Values for the jump parameters were

difficult to estimate. Because sensitivity analysis indicates that option values are

extremely sensitive to jump parameter values, additional research to parameterize the

jumps is necessary.

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TABLE OF CONTENTS List of Figures .................................................................................................................. viii List of Tables ..................................................................................................................... ix Acknowledgements............................................................................................................. x Chapter 1 Introduction ........................................................................................................ 1

1.1 Background............................................................................................................... 1 1.2 Process of Anaerobic Digestion................................................................................ 7 1.3 Utilization of Digester By-Products.......................................................................... 9

1.3.1 Methane Use ...................................................................................................... 9 1.3.2 Effluent Use ..................................................................................................... 10

1.4 Outline of Thesis..................................................................................................... 11 Chapter 2 Literature Review............................................................................................. 12

2.1 Digester Literature .................................................................................................. 12 2.1.1 Case Studies ..................................................................................................... 12 2.1.2 Separator Technology ...................................................................................... 15 2.1.3 Costs................................................................................................................. 16 2.1.4 Scale Factors and Community Digesters ......................................................... 16 2.1.5 Financial Analyses of Policies ......................................................................... 18

2.1.5.1 Net Metering Laws ................................................................................... 18 2.1.5.2 Carbon Credits .......................................................................................... 20 2.1.5.3 Other Policies............................................................................................ 21

2.2 Investment Analysis................................................................................................ 22 2.2.1 Capital Budgeting ............................................................................................ 23 2.2.2 Real Options..................................................................................................... 25

2.2.2.1 Real Options Application in Agriculture .................................................. 26 2.2.2.2 Real Options with Jump Processes ........................................................... 27

2.3 Chapter Summary ................................................................................................... 29 Chapter 3 Methodology .................................................................................................... 31

3.1 Stochastic Capital Budgeting Model ...................................................................... 31 3.1.1 Economic Benefits ........................................................................................... 31 3.1.2 Economic Costs ............................................................................................... 34 3.1.3 Taxes and Depreciation.................................................................................... 35 3.1.4 Net Present Value Calculations ....................................................................... 36 3.1.5 Variables and Parameters................................................................................. 36

3.1.5.1 Model Parameters ..................................................................................... 39 3.1.5.2 Model variables......................................................................................... 41

3.1.6 Stochastic Simulations ..................................................................................... 45 3.2 Real Options Model ................................................................................................ 45

3.2.1 Real Options Framework ................................................................................. 46 3.2.2 Option Values in the Presence of Jumps.......................................................... 49 3.2.3 A Modified Approach to Modeling Options in the Presence of Jumps........... 54

3.2.3.1 Finding the Asset Value at Time t ............................................................ 55 3.2.3.2 Defining Movements in Asset Value........................................................ 56 3.2.3.3 Defining the probability of a jump............................................................ 58

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3.2.3.4 Defining the Magnitude of a Jump ........................................................... 58 3.2.3.5 Defining Geometric Brownian Motion..................................................... 58 3.2.3.6 Option Valuation....................................................................................... 59

3.3 Stochastic Simulations ............................................................................................ 60 3.4 Chapter Summary ................................................................................................... 60

Chapter 4 Results .............................................................................................................. 61 4.1 Base Case Results ................................................................................................... 61

4.1.1 Results without a solids separator.................................................................... 62 4.1.2 Results including a separator with the end product sold as compost............... 62 4.1.3 Results including a separator with the solids used on-farm as bedding .......... 65

4.2 Sensitivity Analysis ................................................................................................ 65 4.2.1 Electricity Price................................................................................................ 67 4.2.2 Carbon Credit Price.......................................................................................... 69 4.2.3 Capital Cost...................................................................................................... 71 4.2.4 Composting Cost.............................................................................................. 73

4.3 Sensitivity of Option Values to Jumps and Their Parameters ................................ 73 4.3.1 Without Jumps ................................................................................................. 75 4.3.2 Time ................................................................................................................. 75 4.3.3 Standard Deviation of Jumps ........................................................................... 78 4.3.4 Mean Jump Size............................................................................................... 80 4.3.5 Jump Frequency ............................................................................................... 82

4.4 Chapter Summary ................................................................................................... 82 Chapter 5 Conclusions ...................................................................................................... 84

5.1 Discussion of results ............................................................................................... 85 5.2 Research Suggestions.............................................................................................. 88

References......................................................................................................................... 91 Appendix......................................................................................................................... 106

Section A.1 Avoided Electricity Purchase................................................................. 106 Section A.2 Electricity Sold........................................................................................ 107 Section A.3 Bedding Savings ..................................................................................... 108 Section A.4 Compost Sold.......................................................................................... 108

Benefits Calculation............................................................................................ 108 Opportunity Cost Calculation ............................................................................. 109 Yearly Cost to Turn Compost Calculation.......................................................... 109 Profits from Composting..................................................................................... 109

Section A.5 Carbon Credits ........................................................................................ 109 Section A.6 Renewable Energy Credits...................................................................... 110 Section A.7 Operating and Maintenance Costs .......................................................... 112 Section A.8 Financing Costs....................................................................................... 112

Yearly Payment................................................................................................... 113 Interest Expense and Principal Payment............................................................. 113

Section A.9 Depreciation ............................................................................................ 113 Depreciation of Engine-Generator...................................................................... 113 Depreciation of Digester ..................................................................................... 113

Section A.10 Taxes ..................................................................................................... 113

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List of Figures

Figure 1.1 Potential Methane Uses…………………………………………………….. 10 Figure 2.1 Asset Value over Time with a Jump……………………………………….. 28

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List of Tables Table 3.1 Parameters for Stochastic Capital Budgeting Model........................................ 37 Table 3.2 Variables Specified in the Model...................................................................... 42 Table 3.3 Parameters used in Real Options Model........................................................... 57 Table 4.1 Simulation Results and Option Values of Investment in Digester with No

Solids Separator .................................................................................................... 63 Table 4.2 Simulation Results and Option Values of Investment in Digester with a Solids

Separator and Solids sold as Compost .................................................................. 64 Table 4.3 Simulation Results and Option Values of Investment in Digester with a Solids

Separator and Solids used as Bedding .................................................................. 66 Table 4.4 Sensitivity of Results for Varied Electricity Prices for Herd Sizes of 1,000 and

2,000 cows ............................................................................................................ 68 Table 4.5 Sensitivity of Results to the Price of Carbon Credits for Herd Sizes of 1,000

and 2,000 cows...................................................................................................... 70 Table 4.6 Sensitivity of Results to Capital Cost for Herd Sizes of 1,000 and 2,000 cows72 Table 4.7 Sensitivity of Results to Composting Cost for Herd Sizes of 1,000 and 2,000

cows ...................................................................................................................... 74 Table 4.8 Sensitivity of Results to the Inclusion of Jumps for Varied Electricity Prices for

Herd Sizes of 1,000 and 2,000 cows..................................................................... 76 Table 4.9 Sensitivity of Results to Changes in Time, T, for Herd Sizes of 1,000 and 2,000

cows ...................................................................................................................... 77 Table 4.10 Sensitivity of Results to Changes in Standard Deviation, σ1, for Herd Sizes of

1,000 and 2,000 cows............................................................................................ 79 Table 4.11 Sensitivity of Results to Changes in Mean Jump Size, k1, for Herd Sizes of

1,000 and 2,000 cows............................................................................................ 81 Table 4.12 Sensitivity of Results to Changes in Jump Frequency, λ1, for Herd Sizes of

1,000 and 2,000 cows............................................................................................ 83

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Acknowledgements I would like to thank USDA/CSREES for their financial support of this work. Thank you to Dr. Jeffrey Hyde for his helpful suggestions and guidance throughout the research and writing process. I wish to express my appreciation to Dr. Tom Richard and Dr. Jeffrey Stokes for serving on my thesis committee. Sincerest gratitude to my family and friends for their support during my schooling. I could not have finished my degree without the phone calls and e-mails from my four sisters, Mary, Cathy, Patty and Judy, and two brothers, Tom and Dan. I have valued my mother’s wisdom, patience, and listening skills during my joys and struggles these past two years. I am grateful to my husband Bob for all the love that he has given me.

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Chapter 1 Introduction

1.1 Background

For a host of reasons, U.S. government leaders and citizens are increasingly

seeking alternative sources of energy. In 2006, President Bush wrote, “For the sake of

our economic and national security, we must reduce our dependence on foreign sources

of energy – including on the natural gas that is a source of electricity for many American

homes and the crude oil that supplies gasoline for our cars,” (National Economic Council,

2006). Prior to this, states were identifying ways to increase alternative energy usage. In

2004, for example, the Commonwealth of Pennsylvania enacted the Alternative Energy

Portfolio Standards Act, which requires electric utilities to purchase energy derived from

renewable and environmentally beneficial sources (Pennsylvania Act 213, 2004).

Furthermore, many scientists believe that greenhouse gases, such as carbon

dioxide, nitrous oxide, and methane, trap heat inside the atmosphere, thereby effectively

raising the earth’s temperature. Their fear is that a prolonged increase in temperature

could have serious repercussions such as rising ocean levels and droughts. This is

related to production agriculture because ruminants, such as dairy cattle, produce and

release methane during the digestive process. Additionally, anaerobic storage of manure

(e.g., an anaerobic lagoon) also produces methane (US EPA, 2006b). The United States

Environmental Protection Agency (US EPA) estimates that agriculture was responsible

for 7.4 percent of all greenhouse gas emissions in 2005 (US EPA, 2007a).

The Clean Water Act (CWA) and Clean Air Act (CAA), as well as state level

legislation, put additional pressure on farms to reduce pollution. For example, as a result

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of the CWA, it is federally mandated that all concentrated animal feeding operations

(CAFOs) must apply for a National Pollutant Discharge Elimination System (NPDES)

permit and develop a nutrient management plan (NMP)1. The goal of these rules is to

prevent manure run-off from entering rivers, streams, and other bodies of freshwater (US

GPO, 2003). While the CAA currently does not require farms to monitor emissions, the

U.S. EPA is conducting a study to determine farm emission standards (US EPA, 2006c).

Some states, like Pennsylvania, have enacted even tougher regulations. CAFOs and

concentrated animal operations (CAOs)2 in Pennsylvania must write a NMP and some

must create an odor management plan.

There are renewable sources of energy that a farm may implement to solve some

of these problems. Wind turbines are one example of such a technology. While the

turbines create renewable energy, they do not reduce odor or methane emissions from

manure. They also are sensitive to location, needing to be in a windy place for optimal

energy production (US DOE, 2007a). Solar energy may also be created on a farm and

used in a variety of ways (e.g., to dry corn, heat water). Solar energy is also considered a

renewable source of energy, but, like wind energy, solar energy does not reduce odor or

methane emissions from manure (US DOE, 2007b)3.

Anaerobic digesters represent a potential solution to a portion of our energy,

methane, and pollution problems. They are an alternative source of energy found on

1 The EPA defines a CAFO in two different ways. In the first definition, a dairy is considered a CAFO if it houses 700 or more mature dairy cows that are confined for more than 45 days per year. The second definition identifies CAFOs as dairies that have 200 or more cattle confined for 45 days/year and the farm discharges pollutants into navigable water or waters of the United States (US EPA, 2008). 2A CAO is defined as, “agricultural operations where the animal density exceeds two AEU's per acre on an annualized basis,” (Pennsylvania Act 38, 2005, p. 10). An animal equivalent unit (AEU) is defined as 1,000 lbs of live animal(s). Interested readers should view Pennsylvania Act 38, 2005. 3 Readers interested in more information about renewable energy technologies should visit http://www.eere.energy.gov.

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some dairy, hog, and poultry farms across the United States. They utilize manure to

produce methane, which is burned in an engine or boiler to create electricity or heat. This

process also decreases methane and odor emissions, and may reduce manure run-off.

To encourage the adoption of digesters, the federal government has created

AgSTAR, which describes itself as,

“...a voluntary effort jointly sponsored by the U.S. Environmental Protection Agency (EPA), the U.S. Department of Agriculture, and the U.S. Department of Energy. The program encourages the use of methane recovery (biogas) technologies at the confined animal feeding operations that manage manure as liquids or slurries. These technologies reduce methane emissions while achieving other environmental benefits (US EPA, 2007b).”

In 2006, AgSTAR reported that there were 100 operational digesters on farms in the

United States and estimated that 6,900 farms in the U.S. could potentially utilize digesters.

If all of these farms adopted digesters, the approximate energy generated would be over

6.3 billion-kilowatt hours (kWh)4. In addition, 1.3 million metric tons of methane would

be eliminated, which is the equivalent of reducing automobile usage by 4.7 million cars

(US EPA, 2006a). These numbers indicate expanding the number of digesters in the US

is plausible and if it occurred, there would be clear environmental benefits.

The first anaerobic digesters appeared on farms in the United States during the

1970s Energy Crisis. It was hoped that these would provide a renewable source of

energy, as well as profits, for farms. Unfortunately, many digesters in the 1970s were not

successful for a variety of reasons. AgSTAR’s Handbook lists the following reasons for

digester failure:

4 The average US home uses 10,656 kWh per year or about 29.2 kWh per day (US DOE, 2008).

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• farmers lacking the skills and time necessary to keep the digester

functioning,

• incorrect equipment for the digester,

• digester designs that were not customized to fit the farm’s manure

handling practices,

• high maintenance and repair costs due to design,

• limited educational opportunities or technical support for the digester

manager,

• diminished monetary returns or no returns, and

• attrition of farms due to non-digester factors (US EPA, undated).

A digester’s large capital cost is a significant barrier to adoption. For example,

AgSTAR estimates that installing a covered lagoon and heated digester would cost

between $200 and $450 per 1,000 pound animal unit (AU), while RCM Digesters, a

leading company in the industry, estimates the cost of a plug-flow digester as $887–

$1757 per cow depending on herd size and digester options (US EPA, 2002b; McEliece,

2007). Despite this, there is a renewed interest in digesters driven by the need to find

alternative sources of energy that would reduce odors, improve water quality, and lower

greenhouse gas emissions (Martin, 2004).

Since the 1970s, engineers have devoted considerable time to improving digester

efficiency. For example, Converse, Graves, and Evans (1977) published one of the

earliest papers on anaerobic digestion of dairy manure. Ten years later, Durand et al.

(1987) examined the optimization of digester temperature, hydraulic retention time, and

manure consistency. Similarly, Massé, Masse, and Croteau (2003) explored digester

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temperature fluctuations and their relationship to biogas production. Malaspina et al.

(1996), Ndegwa et al. (2005), and Sung and Santha (2003) studied digester design.

Callaghan et al. (2002), El-Mashad and Zhang (2006), and Zhang et al. (2007)

investigated the use of food waste in a digester by itself or combined with manure. The

bacteria that live inside the digester, methanogens, were examined by McHugh et al.

(2003) and LeClerc, Delgènes, and Godon (2004).

In addition to digester improvements since the 1970s, there are opportunities and

policies available today that were not available then. These opportunities and policies

affect digester profitability and include items such as net metering, carbon credits, and

grants.

Net metering refers to a governmental policy that requires utility companies to

pay digester operators for generated electricity. A meter is installed on the farm that

keeps track of the farm’s electricity use and generation. If the reading at the end of the

month indicates the farm used more electricity than it generated, it must pay for the

deficit. On the other hand, if the reading indicates the farm generated more electricity

than it used, the utility company cannot send the farm a bill. When net metering laws are

not in place, utilities can charge for every instance in which a farm uses more electricity

than it generates, which may occur several times throughout the course of the month.

Furthermore, the utility could include standby or demand charges on a farm’s utility bill.

These are fees the utility assesses for being a back-up source of electricity (Birge, 2007).

Carbon credits are traded on the Chicago Climate Exchange (CCX) and a farm

with a digester can receive them in one of two ways. First, credits may be received if the

farm reduces its greenhouse gas emissions. For instance, if the farm’s previous manure

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handling system was an open cover lagoon, the farm may be eligible for credits. A farm

may also receive carbon credits for renewable energy generated. 0.4 carbon credits are

realized for every 1,000 kW, or one mega-watt (mWh), of electricity produced. A farm

typically works with an aggregator, which is an organization that pools carbon offsets

from several farms (or other eligible agents) and sells the offsets as credits on the

exchange (Subler, 2006). Typically, a farm gets about one-half of the credits’ value due

to costs of verifying the on-farm gas production and other miscellaneous aggregator costs

(Six, 2007).

The members of the Climate Exchange are required to reduce their carbon

emissions by a specified percentage each year. To do this, they must either implement

practices that reduce emissions or purchase carbon credits for the year in which the

reduction must occur. For example, a firm that needs to reduce emissions in 2007 would

buy Vintage 2007 carbon credits to meet its goal (Chicago Climate Exchange, 2007a).

Vintage 2007 carbon credits have traded anywhere from approximately $3.25-$5.00

(Chicago Climate Exchange, 2007b).

The federal government and some states offer grants and loans to farms that are

interested in building digesters. The 2002 Farm Bill makes guaranteed loans and grants

available to farms interested in implementing renewable energy technology and energy

efficiencies. The minimum grant request is $2,500 and the maximum request is $500,000

(USDA- Rural Development, 2007). On a state level, Pennsylvania’s Energy Harvest

program offers grants to projects like an on-farm anaerobic digester. This program is

competitive and there is no minimum, maximum, or required match (Campbell, 2007).

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Despite these improvements and policies, it is unclear if this new generation of

digesters is profitable. Because of this uncertainty, the objective of this research is to

explore the profitability of digesters on dairy farms in Pennsylvania. Two specific

sources of large uncertainty, technology and policy changes, could drastically change the

value of a digester to a farmer. These changes are modeled as jumps in this research. A

change in technology could be an improvement to digester efficiency or a new alternative

energy that is more economical and environmentally friendly than a digester. Policy

changes could come in the form of direct assistance for the farmers, such as grants or

low-interest loans. Laws that require others to purchase or subsidize energy from a

digester represent another type of policy.

This study will illustrate what conditions and policies are necessary to make a

digester profitable. Furthermore, this research will also evaluate the effects of changing

the parameters of the jumps included in the model. The remainder of this chapter

provides information necessary to understand the benefits and costs of anaerobic

digesters.

1.2 Process of Anaerobic Digestion

Anaerobic digestion is the process by which manure is turned into methane,

without the presence of oxygen. Manure is made up of water and solids. The manure’s

total solids (TS) are comprised of volatile solids (VS) and ash. When manure enters the

digester, the VS are digested by several types of bacteria. In the early stages of digestion,

the bacteria break the manure into simple fatty acids, carbon dioxide, and hydrogen. The

simple fatty acids are later digested by methanogens, thereby creating biogas, which is

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approximately sixty percent methane. The manure remains in the digester anaerobically

for a number of days (Converse, 2001; Lusk, 1998).

There are three general types of digesters: covered lagoon, complete mix, and

plug-flow (US EPA, 2002b). The type of digester a farm selects depends upon the TS of

the manure and other materials entering the digester (Wilkie, 2005). For example, a farm

using a flushed-water system to clean manure out of its barn would select a covered

lagoon digester, while a farm with a scraped manure system would utilize a plug-flow

digester (Lusk, 1998). A dairy farm in Pennsylvania would be most likely to install a

plug-flow digester.

Several factors influence the amount of biogas produced by the digester and the

percent methane in the biogas. The digester must be the right temperature and at a pH

(approximately 6.4-7.4) that is conducive to bacteria growth in order to best operate

(Converse, 2001). In colder climates, the digester must be heated so that digestion may

take place. In warmer climates, however, ambient temperature digesters may be used.

These rely upon the outside temperature to facilitate the digestion process (US EPA,

undated). This could mean that a smaller net energy gain exists for digesters in colder

climates because some of the energy produced by the digester will be used to heat the

manure. On the other hand, it could also mean that digesters in colder climates produce

more biogas because their internal temperature is constant, while a digester in a warmer

climate is dependent upon warm air temperatures.

A digester manager needs to spend time with the digester on a regular basis.

Lazarus and Rudstrom’s (2007) case study digester required about one hour of daily

monitoring and maintenance between the engine and digester. This consisted of

9

checking the pH level and temperature of the digester. From time to time additional

maintenance, such as an oil change, is necessary. AgSTAR’s FarmWare estimates

operating and maintenance costs are 5% of the total capital cost (US EPA, undated).

1.3 Utilization of Digester By-Products

The digestion process yields two valuable products, methane and effluent. The

methane is only valuable if it is used to generate energy. Likewise, the value of the

effluent depends upon how it is used. This section describes the potential uses and

related values of both methane and the effluent.

1.3.1 Methane Use

The created methane can be flared (burned) or used as electricity, heat, or natural

gas (Figure 1.1). If the methane is used for electricity, the electricity is generated in an

engine or turbine, depending on the size of the farm. The electricity can be used on the

farm, sold to a utility, or both, depending on the farm’s cost of electricity and the price it

receives for excess electricity, as well as its electricity usage. In a situation where the

methane is used for heat, it is used like natural gas or propane. It is burned in a boiler,

thereby creating heat in the form of hot water and steam. The energy from the boiler can

be used to heat the water for the milking parlor, holding areas for the cattle, milk houses,

equipment rooms, and even a house on the farm (Lusk, 1998). Additionally, a farm

could choose to sell its biogas to a natural gas company, but it would need to find a way

to connect to the natural gas pipeline (Krich et al., 2005). Because of the expense of

10

purchasing an engine-generator or boiler, a farm typically selects one or the other, while

few sell biogas directly.

Figure 1.1 Potential Methane Uses

1.3.2 Effluent Use

The farm has only two options for effluent use. It can be used as fertilizer or

separated into liquid and solids portions. Using the effluent as fertilizer poses potential

pollution problems. If a farm applies too much manure to its land, it can lead to water or

air pollution problems (US EPA, 2007a; US GPO, 2003). The second option requires

investment in a solids separator, which separates the effluent into liquid and solid

Natural Gas Pipeline Flare

Natural Gas Waste

Heat

Engine Generator

Electricity

Farm Energy

Power Grid

Boiler

Heat

Methane

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portions. The liquid portion goes to long-term storage, but the solid portion has value. It

can be used as bedding material for the livestock or it can be sold as soil amendments

(Goodrich, 2005). If used as bedding, the solids may be used immediately, resulting in a

savings for the farm (Martin et al., 2003). If a farm decides to sell the solids as soil

amendments, then the solids must compost for a period. A farm must consider how it

will market the solids and who will buy them, as well as the space required to dry the

solids (Wright et al., 2004; Rynk et al., 1992).

1.4 Outline of Thesis

The remainder of this thesis provides details of this research. In the following

chapter, a review of relevant production and financial literature pertaining to digesters is

covered. This is followed by a development of the methodology employed to evaluate

digester profitability, results from this model, and a discussion of how the results can be

applied to improve digester success on farms.

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Chapter 2 Literature Review

A large amount of research has focused on the technical aspects of methane

digesters. In addition, investment analysis tools have been broadly applied in the

economics literature. Thus, it is important to thoroughly review both sets of literature.

The first section of this chapter covers digester literature and the financial models applied

to digesters. This discussion leads into the second section of this chapter, which

examines investment analysis literature.

2.1 Digester Literature

Both engineers and economists have investigated the profitability of anaerobic

digesters, mostly in the form of case studies. While case studies are useful, they do not

provide a generalized perspective of digester profitability. This section begins by

summarizing these case studies and listing financial tools available to producers

considering this technology.

2.1.1 Case Studies

Within academia, digester economics have been examined on a case-by-case basis

by various researchers. For example, Martin et al. (2003) studied waste stabilization,

pathogen reduction, and biogas production on a 550-cow dairy that operated a digester in

New York. The digester created approximately 43,000 cubic feet of biogas per day, of

which 59.1 percent was methane. It had the potential to generate 495,000 kWh of

electricity annually, which translated to $45,000 in electricity savings and sales.

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Martin (2005) described a farm in Wisconsin with a different type of arrangement

with the electric company. Alliant Energy owned the engine-generator and paid the farm

$0.015 per kWh, which equated to about $18,400 per year. Because Alliant owned the

engine-generator, operating and maintenance costs were insignificant to the farm. The

farm saved an estimated $60,000 per year on bedding and generated $8,600 from the sale

of excess solids. The total benefits of the digester totaled approximately $87,000. The

dairy saw a reduction in odor, methane emissions, pathogens, and carbon dioxide

emissions, and an increase in farm income.

Lazarus and Rudstrom (2003; 2007) estimated future cash flows for a digester in

Minnesota. They examined what would happen to their case study farm in the future and

analyzed a hypothetical digester’s profitability by varying grants, loans, and subsidies.

Initially, the case farm received grants, low-interest loans, and subsidies to offset project

costs. Furthermore, during the first six years of digester operation, the farm was paid

$0.0725 per kWh of electricity generated. At the time Lazarus and Rudstrom’s (2007)

work was published, the farm was negotiating a new contract with the power company.

It appeared that the farm would receive significantly less than before; probably between

$0.03-$0.04 cents/kWh. If the farm did not receive financial incentives and simply

financed the digester from its own funds, it required an electricity price of $0.08 per kWh

to break-even.

Lusk (1998) documented 23 digesters and briefly discusses the costs and benefits

associated with each. The farms included cattle, hog, and poultry operations and the

digesters ranged in age from pre-implementation to 26 years. Each farm’s profile

consisted of:

14

• farm type and size,

• digester type and size,

• digester and maintenance history,

• lessons learned, and

• cost and savings information.

The responses varied widely, which may have been due to differences in implementation

date, type of digester, and amount of construction completed by the farmer, rather than

the contractor. While the obvious benefits of the digester, electricity and heat, were

mentioned in the savings information section, many of the farms also listed odor

reduction and bedding savings in this portion. The dairy farms cited that anywhere from

five to fifteen minutes were required each day for maintenance. The necessary daily

maintenance was more varied on swine and poultry farms. The lessons learned portion of

each case study provided interesting insight into digester operation. Most of the

responses in this section commented on the costs, digester design, or equipment.

In 2004, Kramer published a casebook about 11 digesters mostly located on dairy

farms throughout the upper Midwest. Dairy herd sizes ranged from 675 to 3,750 head.

Like those in Lusk’s study, most farms generated electricity with the methane from the

digester. The summary for each case included information about the digester

specifications, biogas use, revenues/savings, cost estimates, required maintenance, and

lessons learned. Similar to other work, many of the surveyed farms added decreased odor,

bedding savings, and weed seed reduction as benefits. Because of discrepancies among

the maintenance costs responses (e.g., no response, different units), it is difficult to

determine if this cost was similar between the case study farms. As case studies, the

15

lessons learned from Kramer and Lusk may not be relevant to a generalized group of

representative farms.

2.1.2 Separator Technology

Martin (2004 and 2005); Wright and Inglis (2003); Wright et al. (2004); Gooch et

al. (2006); Kramer (2004); Lusk (1998); and Goodrich (2005) analyzed individual farms

that implemented separator technology. Each estimated the value of the solids, and

whether they were used for bedding or sold as soil amendments. Their estimates of

benefits were very different. For example, Kramer (2004) estimated a farm would realize

a savings of $32 per cow per year if the solids were used for bedding, while Gooch et al.

(2006) found that the savings were between $60-100 per cow per year.

There is little research that analyzes the profitability of selling solids as compost.

One case study described a 550-head dairy farm that experienced sales of 1,825 cubic

yards of digested solids per year and received $16 per cubic yard of digested solids,

which amounted to $29,200 in revenue (Martin 2004). These numbers indicated that

each cow produces approximately 3.32 cubic yards of compost per year. Wichert (2004)

estimated a cow generates 3.00 cubic yards of compost per year.

There are concerns that using digested manure solids may increase pathogens,

thereby increasing mastitis in the herd (Cornell Waste Management Institute, 2006). If

solids used as bedding cause an increase in mastitis cases, this would decrease the

benefits of utilizing solids as bedding. Because researchers are still in the early stages of

understanding this problem, it is difficult to quantify these costs and they are not included

in the model.

16

2.1.3 Costs

A digester requires a large capital investment. RCM Digesters estimated these

costs varying from $887-$1,757 per cow depending on herd size (McEliece, 2007). In

focus group meetings in Iowa, a low economic return was the leading concern of farmers

considering adoption (Garrison and Richard, 2005). Similarly, Scruton, Weeks, and

Achilles (2004a and 2004b) also identified finances as the most significant hurdle to

widespread adoption of anaerobic digesters on farms.

Other important barriers include digester design, excessive maintenance

requirements, and little specialized support for installation or servicing (Scruton, Weeks,

and Achilles, 2004a; Scruton, Weeks and Achilles, 2004b). Faulty equipment and

management were also reasons for non-operational digesters (Kramer, 2004; Lorimor and

Sawyer, 2004). Any of these management or equipment issues could cause financial

difficulties, thus leading to digester shutdown.

2.1.4 Scale Factors and Community Digesters

Limited analysis of potential scale economies associated with digesters has been

completed. RCM Digester’s cost estimates showed that the investment cost per cow was

much lower for larger herds (McEliece, 2007). Likewise, AgSTAR recommended that

farms consider if they are “big enough” to have a digester, which also suggests digester

scale factors exist (US EPA, undated).

Researched have suggested the economics of digesters may improve if they were

community-based, which hints economies of scale may be present (Garrison and Richard,

17

2005; Criner et al, 1986). In its simplest form, a community-based digester could consist

of a few farmers investing in a digester together. However, it is possible that a

community-digester could involve farms, restaurants, and any other firm with digestible

waste. Community digesters like these are already operating in Europe and Asia. In the

United States, at least one community digester has operated since 2003 at the Bay of

Tillamook in Oregon (Port of Tillamook Bay, undated).

Ghafoori and Flynn (2006a) analyzed nine community digester scenarios for a

county in Canada. Each scenario used manure from all of the 61 confined feeding

operations in the county, but the number, size, and locations of digesters varied across the

scenarios. The scenario with one centralized digester had the lowest cost per unit of

power produced. The cost to generate power in each of the scenarios, which ranged from

$218-$278 per megawatt hour (mWh), was greater than the monthly average power price

of $30-$100 per mWh. The authors concluded that only the capital costs of a community

digester were scalable, while the transportation costs associated with hauling the manure

to the centralized digester were not.

In a different study, Ghafoori and Flynn (2006b) performed an economic analysis

of a pipeline for a community digester. The pipeline would run from each participating

farm to the digester. In this way, the farm would not need to transport manure to a

centrally located digester. The objective was to determine if a pipeline was less

expensive than trucking manure to a centrally located digester. They found that pipeline

transport exhibits economies of size and costs were minimized when manure was

between 12-14% TS. Furthermore, a two-way pipeline, which can either pump manure

18

into the digester or send effluent back to the farm, was more cost-effective than two

separate pipelines.

2.1.5 Financial Analyses of Policies

Policy makers continue to assess options to make digester implementation more

appealing. Some of these opportunities, such as carbon credits and net metering, may

increase the income realized from a digester. Other policies, such as nutrient and odor

management plans, which require farmers to change current manure handling practices,

encourage digester adoption as an alternative to current practices. With these types of

policies, digester implementation allows farms to continue doing business, rather than

exiting the industry because they could not adhere to government guidelines.

2.1.5.1 Net Metering Laws

In 2006, Pennsylvania enacted net metering laws, which have a direct benefit to

farms with digesters (Pennsylvania Public Utility Commission, 2006). Prior to the

passage of these laws, a farm may have been generating more electricity than it used each

month, but if it periodically produced less electricity than it was consuming, the utility

could charge the farm for each instance of excess use (Birge, 2007). In addition, utility

companies could also charge farms with a “standby” or “demand” fee to remain an

alternate power source.

Under net metering, the utility must consider the total electricity generated by the

farm compared to the farm’s consumption throughout the month. If the monthly

generation is greater than the monthly consumption, the farm does not receive a bill. A

19

farm may be compensated for the net excess electricity at a rate negotiated with the

power company (Pennsylvania Public Utility Commission, 2006). While the language in

the regulation suggests that net metering should be implemented immediately, it will take

longer for these rules to be put into practice due to the deregulation of the electric

industry and different types of electric companies (e.g., co-ops, utilities) (Birge, 2007).

Limited analysis of net-metering laws has been reported. Scruton, Weeks, and

Achilles, (2004a) examined Vermont’s net-metering laws and their effects on digester

profitability. At the time, the authors described Vermont’s net metering laws as some of

the most progressive in the United States because a farm could use energy from the

digester to run several electric meters. Referred to as group net metering, these laws

allowed a farm to use the digester energy in any farm related business or residence, even

if it was on a different meter than the main farm. Most states allow a farm to use only a

single meter to utilize energy from a digester, which may not allow a farm to take full

advantage of all the power it generates. It should be noted that Pennsylvania allows an

operation to aggregate several meters, with a few limitations (Pennsylvania Public Utility

Commission, 2006).

In the presence of group net metering, revenues increased because a farm realized

the retail rate of electricity for more of the electricity it used than when only single net

metering was available (Scruton, Weeks, and Achilles, 2004a). In other words, in single

net metering, a farm may receive the retail rate for the electricity used by one meter, but

if the farm has other electricity meters, it cannot receive the retail rate for operating these

meters. Instead, the farm can sell the excess electricity for the wholesale rate, which is

typically less than the retail rate. While improved revenues due to group net metering

20

laws helped digester profitability, the authors indicated these benefits were not enough to

overcome the high capital costs of a digester.

From an engineering standpoint, when net metering laws are in place, the

digester and engine generator can be designed to be smaller because these components

need not be large enough to produce at peak-levels (Scruton, Weeks, and Achilles,

2004a). Rather, the digester can be engineered to produce electricity for average power

use, resulting in a smaller design and decreased capital costs. Like group net metering

laws, the authors suggested the benefits of a smaller digester do not make up for the

digester’s capital cost.

2.1.5.2 Carbon Credits

In addition to net metering laws, carbon credits may improve the economic

returns of digesters. Farms that implement a digester are eligible for carbon credits if

they are reducing their emissions of greenhouse gases. These credits have a monetary

value (Subler, 2006). A farm with an open cover lagoon that builds a digester is eligible

for carbon credits because it is reducing its methane emissions. Carbon credits can also

be received for generation of renewable energy (Six, 2007)5.

The impact of carbon credits on the digester purchase decision has not been well

assessed. Ghafoori, Flynn, and Checkel (2007) analyzed community digester scenarios in

Red Deer County, Alberta, Canada and calculated the carbon credit value needed to cover

the costs of the digesters. In that study, a carbon credit was received when carbon

emissions are displaced. Energy generation from an anaerobic digester created fewer 5 For every 1mWh a farm generates, it receives an alternative energy credit. The farm can sell this credit to a utility or on the carbon credit market.

21

emissions than other power sources, thereby displacing carbon emissions from other

sources, such as coal. In the best scenario, a centrally based digester for the county, $125

per tonne ($113 per ton) of carbon dioxide per year was necessary for a digester to break-

even.

2.1.5.3 Other Policies

As previously discussed, the U. S. and state governments are making a

concentrated effort to reduce air and water pollution from animal feeding operations. The

CWA mandates that CAFOs must apply for a NPDES permit, which allows the

government to regulate pollution from a farm. Additionally, permitted CAFOs must

develop a NMP, which is a plan outlining how a CAFO will utilize plant nutrients

through best management practices (US GPO, 2003). The Commonwealth of

Pennsylvania enacted its own nutrient management legislation in 2005 (Pennsylvania Act

38, 2005). This requires that all concentrated CAOs develop a NMP. Additionally, some

farms may need to establish an odor management plan.

An anaerobic digester may impact a farm’s NMP. Depending upon how it is

managed, digested manure may have fewer nutrients available. This is accomplished in

at least two ways. First, the manure entering the digester, also called influent, has a

greater volume than the manure exiting the digester. Manure entering the digester is

about 87 percent water and 13 percent TS. Ash makes up about 15 percent of the TS and

VS are 85 percent of the TS. The total mass of the manure is lowered in the digestion

process; this reduction is about 5 percent (Aldrich, 2005). In addition to a mass reduction,

if farms choose to use the solids from the digester as bedding or sell the solids as compost,

22

there are (potentially) fewer nutrients for a farmer to manage. Essentially this means that

producers only need to determine how to manage the liquid portion of the manure.

At this time, the CAA does not require farms to monitor emissions, but this is

likely to change. In 2006, air emissions were being monitored on 2,568 volunteer farms

across the United States by the Environmental Protection Agency. The data gathered

from that study will be used to form future federal air emission regulations (US EPA,

2006c).

Moreover, digesters are an effective way to decrease odor from manure.

Numerous authors list odor control as one of the benefits of a digester (Aldrich, 2005;

Wilkie, 2005; US EPA, 2002a). Ndegwa et al. (2005) measured volatile fatty acids

(VFA), which cause odor in manure, in swine slurry pre- and post-digestion. Before

digestion the manure had an average VFA concentration of approximately 640

milligrams (mg) per liter (L). After digestion the VFA had dropped to between about 75

and 85 mg per L. Manure with a VFA concentration of greater than 520 mg per L is

considered to have an unacceptable odor, while manure with a VFA concentration of less

than 230 mg per L is not considered an odor problem. Because of its ability to decrease

odor, digester implementation could help a farm comply with odor regulations issued by

state or federal government.

2.2 Investment Analysis

When a firm makes a large investment, it is usually as the result of intensive

financial analysis. There are numerous tools available to firms to assist in the decision-

making process. Capital budgeting is often employed because of its simplicity. The

23

decision rule in capital budgeting indicates that one should invest if a project’s

discounted benefits are larger than its discounted costs. In other words, investment

should occur if the project’s net present value (NPV) is positive. While this is

straightforward, often future benefits and costs are difficult to forecast. In addition,

capital budgeting does not account for the value of waiting to invest. Real options

analysis is a type of model that can account for these factors. This section provides an

overview of capital budgeting, as well as its strengths and weaknesses. I then discuss real

options analysis and examples of its applications in agriculture.

2.2.1 Capital Budgeting

Capital budgeting is one approach to evaluating investment decisions. It requires

the benefits and costs of the investment, as well as the discount rate and project life to

calculate the project’s NPV.

( )[ ]∑=

+−=T

0t

ttt )r1/(CostsBenefitsNPV (2.1)

where t = 0 to T indexes time, r is the discount rate, and Benefits, Costs, and Taxes

represent marginal cash inflows and outflows, respectively (Brealey and Myers, 2000)6.

Once the marginal cash flows of a given project are estimated, capital budgeting is

straightforward, making it a widely used method of financial analysis.

The discount rate represents the opportunity cost of capital. It can be selected in a

variety of ways. For example, it might be based on current interest rates or a firm may

6 There are various capital budget models available to calculate the NPV of digesters. The University of Florida as well as University of California-Davis offer on-line resources to calculate the NPV of digesters (deVries et al., 2007; California Biomass Collaborative, 2007). AgSTAR offers free software, FarmWare, to analyze the costs and benefits of digesters (US EPA, undated).

24

define a specified rate of return for all projects. Conversely, a firm might choose

different discount rates depending on the risk associated with a project. A discount rate

may vary between individuals and across firms.

Internal rate of return (IRR) and payback period are decision rules based on

break-even NPVs. An IRR is the discount rate necessary to yield a NPV equal to zero. It

assumes that the project’s lifespan and marginal benefits and costs are as estimated.

Payback period calculates the time period in which a project’s NPV will equal zero. The

marginal benefits and costs, as well as the discount rate are considered known in this

form of analysis (Ross, Westerfield, and Jordan, 2000).

While capital budgeting has its merits, it does not directly account for a project’s

uncertainty. It has been criticized because of its assumptions that investment decisions

are fully reversible and that an investment is now or never. Implicitly, capital budgeting

methods employ these assumptions that are unlikely to hold for most real world projects.

Investment decisions are often uncertain. For example, it is difficult to predict a

project’s future benefits and costs (Mun, 2002). Additionally, assigning a discount rate

is somewhat arbitrary. Investments often contain an element of irreversibility. If a firm

chooses to disinvest in a project, it does not completely recover its initial cash outlay.

That is, there are sunk costs, which a firm cannot recoup (Dixit and Pindyck, 1994).

Moreover, capital budgeting assumes that a decision must be made now or never.

In fact, Lander and Pinches (1998) fault capital budgeting models for failing to account

for the value of the decision to wait. They add that this methodology does not consider a

firm’s ability to make changes if future events are different than predicted.

25

2.2.2 Real Options

The indicated shortcomings of capital budgeting can be addressed by applying

real options techniques when making financial decisions. Real options are similar to

financial options, in that a decision maker holds the right, but not the obligation, to buy or

sell an item. In fact, the modeling framework for real options is largely based on Black

and Scholes’ (1973) financial option pricing formula, which can be used to find European

option values in a continuous time setting7.

The investment rule in real options analysis is different from that of capital

budgeting. In capital budgeting, an investment decision is made when the NPV is greater

than zero. Real options analysis, on the other hand, requires the NPV to be greater than a

trigger value, which is greater than zero when returns are uncertain. If a firm opts to

invest, it exercises its option (Dixit and Pindyck, 1994).

Real options are used in a range of fields and there are several types and ways to

model them. Telecommunications, utilities, oil and gas, airlines, computers,

manufacturing, and automobiles are examples of industries that employ real options to

aid in decision-making (Mun, 2002). Different types of real options include the firm’s

decision to wait, change the scale of the project, abandon, switch inputs or outputs, grow,

or a combination of these things (Trigeorgis, 1996). There are four primary models used

when assessing real options: continuous time, finite-difference, binomial, and trinomial

or other lattice models (Lander and Pinches, 1998).

7 A European option may only be exercised at its maturity. An American option, on the other hand, may be exercised at any time between its purchase and its maturity date. To value a European option, one can use continuous time modeling techniques, while American options generally use discrete time modeling.

26

2.2.2.1 Real Options Application in Agriculture

One of the earliest examples of real options analysis in agriculture examined the

adoption of freestall housing on dairy farms in Texas (Purvis et al., 1995). Because the

building of a barn is partly irreversible and the decision can be delayed, it was suitable to

use a real options approach to determine the optimal investment. When uncertainty and

irreversibility were not accounted for, the decision to build a freestall barn was positive.

However, when uncertainty and irreversibility were included in the calculation, the

optimal investment trigger was greater than the expected returns from this technology. In

this case, the best decision was to postpone investment.

Engel and Hyde (2003) applied Purvis et al.’s model to the decision to switch

from a traditional milking system to a robotic milking system. They found the expected

life of the new technology was the most influential factor in the decision. If the robotic

system would last longer than the current traditional milking system, then it was a wise

investment decision.

Real options analysis was used to determine what milk prices would cause firms

to enter or exit dairy farming in the state of New York (Tauer, 2006). The entry and exit

decisions are like options in that the entry decision can be viewed as a call (option to buy)

and the exit decision as a put (option to sell). Their research indicated that smaller farms

entered and exited at higher prices than larger farms. A 50-cow dairy’s entry price was

$23.71 and its exit was $13.48, while the entry price for a 500-cow dairy was $17.52 and

the exit price was $10.84. Similarly, the entry and exit decision of coffee farmers was

27

assessed and exit prices of $0.47 per pound and $0.14 per pound were found (Luong and

Tauer, 2006).

Real options were used to explore the profitability of a digester on a dairy farm in

Pennsylvania (Stokes, Rajagopalan, and Stefanou, 2006). Their results indicated that as

price received for excess electricity increased or as the percent electricity sold increased,

the value of the option decreased. They also varied volatility levels and the difference

between the risk free and dividend rates. As each of these parameters increased, the

value of the option also increased. These results suggest that grants are necessary to

entice producers to implement this technology.

2.2.2.2 Real Options with Jump Processes

Recent work valuing real options has focused on methods that incorporate market

shocks, or jumps, into their valuation. Jumps occur when the price of the option changes

by more than normal market fluctuation. One might think of what happens to the price of

a stock when a company announces a merger. The news represents a shock and the stock

price changes significantly (Figure 2.1).

28

Figure 2.1 Asset Value over Time with a Jump

One of the earliest option valuation works with jumps was developed because,

“…. the Black-Scholes solution is not valid, even in the continuous limit, when the stock price dynamics cannot be represented by a stochastic process with a continuous sample path. (p. 126, Merton, 1976).”

This model resembled the Black-Scholes model, with jumps added to relax the

continuous time component. Like the Black-Scholes model, this model could be used to

value European options. Later work developed a discrete time model that allowed

American options to be valued in the presence of one jump (Amin, 1993).

Additional research developed a model to value options in the presence of

multiple jumps (Martzoukos and Trigeorgis, 2002). This method allowed European or

American options to be valued. European option values may be calculated using a

t T 0

Asset Value

Time

29

modified Black-Scholes model or estimated using the same discrete time framework

developed to value American options.

2.3 Chapter Summary

Digester research thus far has consisted mainly of case studies. These case

studies serve well to provide a stepping-stone and resource to producers considering this

technology. Furthermore, the suggestions for improvement and problems encountered

are thoroughly discussed in these studies; however, the cases do not explain how these

items may affect other digesters that may be similar or dissimilar. These case studies

provide valuable insights to the model developed in this research.

Furthermore, many of the case studies do not mention the effect of new policies

and market opportunities, such as carbon credits and net metering. While other research

outside of the case studies does address these issues, it is a small body of literature. Akin

to the case studies, much of this additional research is not easily applied to other farms

adopting this technology.

To assess the profitability of digesters, financial analysis that incorporates new

technologies, opportunities, and policies must be conducted. Capital budgeting is one

approach to analyze the profitability of a project, but it does not account for the

uncertainty or irreversibility of this decision. A digester has a high degree of

irreversibility and, because of new and changing policies and other factors, often the

future cash flows are very uncertain. For a project like this, a capital budget model is not

the best approach. Real options analysis is a method by which one can account for this

uncertainty and irreversibility. Incorporating jumps into the real options model allows

30

the digester value to be modeled using the same assumptions and movements that a stock

would follow.

This purpose of this research is to fill the gaps that have been left by the case

studies and limited literature pertaining to the analysis of new policies. Moreover, it will

assess the profitability of the new policies, opportunities, and technologies intended to

improve digester profitability to determine if, in fact, they are doing what they seek to

accomplish. Finally, it will examine the effect of incorporating jumps into a real options

model by testing the model with jumps, without jumps, and with changes to the jump

parameter values. The next chapter will develop the real options model used to appraise

these policies and the profitability of a digester.

31

Chapter 3 Methodology

This chapter explains the method used to value the real option of investing in an

anaerobic digester project when the digester’s value follows a jump diffusion process.

There are two model components used to estimate the option values. First, a stochastic

capital budgeting component generates a distribution of asset values. The second

component estimates the option value. As described below, information from the

stochastic capital budgeting model is integrated into the real options framework to

calculate option values.

3.1 Stochastic Capital Budgeting Model

A stochastic capital budgeting model simulated the NPV of the digester and the

digester’s value. Hyde and Engel (2002) applied a similar method to compare traditional

milking parlors to robotic milking systems (RMS). Using Monte Carlo simulations, they

determined the break-even RMS purchase price required for a farmer to be more

profitable using robotic milkers than traditional means of milking. Preliminary results

from this research showed the expected profitability of anaerobic digesters (Leuer, Hyde,

and Richard, forthcoming). In this case, the investment was considered profitable

anytime the expected NPV is greater than zero.

3.1.1 Economic Benefits

The economic benefits of a digester consist of several components. As is standard

with capital budgeting, only the marginal cash flows associated with the investment are

32

assessed. For example, while milk production is a part of farm income, it is not impacted

by the digester, so it is excluded as a benefit (or cost) to the digester. The sale of

electricity, on the other hand, only exists if a digester is operating.

There are several factors that may be included as benefits, depending upon

decisions made by the farmer. This section briefly describes each source of benefits.

Later, the valuation of each benefit within the context of this research is discussed. The

appendix provides the formulas used in the computation of the benefits and costs.

Avoided electricity purchase – Once the farm begins to generate electricity, it is

able to avoid purchasing at least a portion of its electricity from the utility. The value of

the benefit depends upon the retail price of electricity.

Electricity sold - The digester may produce more electricity than the farm can

use. When this occurs, the excess electricity may be sold to a utility. Often this is at a

price well below the retail rate.

Bedding savings - If a farm opts to utilize a solids separator, the solids may be

sold for bedding or used to replace current bedding materials on the farm. In this model,

I assume that bedding materials are used on-farm.

Compost sales - A farm may also choose to sell composted separated solids as a

soil amendment. The value of the compost depends upon many factors, including how

long it has been composted. I calculate the cost of preparing the solids for sale by

accounting for the opportunity cost of land used to dry the solids and the cost of operating

a tractor to turn the solids as part of the composting process (Rynk, et al., 1992).

Carbon credits - Carbon credits have been traded on the CCX since 2003. Each

credit traded at the CCX represents a reduction of one metric ton of carbon dioxide (CO2)

33

equivalent. Although firms (including farms) are not required to participate, they do

have the option of buying or selling credits at the current market price (Chicago Climate

Exchange, 2007a). A farm can only receive credits if its digester reduces methane

emissions. For example, if prior to digester implementation, the farm’s manure handling

system was emitting methane, the implementation of a digester will reduce the farm’s

methane emissions (Subler, 2006).

Carbon credits are calculated by using the minimum of either the methane

captured/combusted by the digester or the per animal head default emissions factor issued

by the CCX. Methane captured/combusted can fluctuate between farms and is also

dependent upon the digester type, so I apply the CCX’s baseline figure for Pennsylvania

of 4.41 carbon credits per lactating cow per year when a plug-flow digester is present

(McComb, 2007). The baseline figures vary by state, animal species, and animal size.

As I discussed earlier, a farm works with an aggregator to sell these credits and the

aggregator receives about fifty percent of the revenue generated from the sale of carbon

credits.

Renewable energy certificates (RECs) - If a farm with a digester generates

alternative energy, it can receive a REC for every mWh of energy it produces. Farms in

Pennsylvania may sell these to utility companies, if allowable under contract, or sell them

as carbon credits. I assume that farmers receive additional carbon credits for renewable

energy. For each mWh produced, a farm receives 0.4 carbon credits (Subler, 2006).

Again, the aggregator receives about fifty percent of the revenues.

Some research suggests that the fertilizer value of the effluent is different from

the influent. I follow Mehta (2002) and do not assign a value to this. Most sources agree

34

that the amount of phosphorous and potassium remains the same pre- and post-digestion

(Ndegwa, 2005; Converse, 2001; Lusk, 1998). Additionally, the total nitrogen in the

manure remains the same after digestion, but the distribution of nitrogen containing

compounds changes (Ndegwa, 2005; Lusk, 1998). Digestion breaks down organic

nitrogen into a form that is more immediately available to plants. Martin et al. (2003)

and Martin (2005) found that the increase in ammonia nitrogen (NH4-N) after digestion

was statistically significant; however, the values of these changes are difficult to quantify.

Consequently, the benefits associated with the digester may be slightly underestimated in

this work.

There may be additional benefits such as odor reduction, waste heat, or weed-seed

reduction, but the exact nature of the benefits is neither well understood nor easily

quantifiable. Additionally, while an item such as odor reduction may be part of a farm’s

decision to adopt this technology and is certainly a social benefit, this analysis focuses

strictly on the farm’s monetary benefits from implementation of a digester.

3.1.2 Economic Costs

There are a few costs associated with an anaerobic digester. These relate to the

purchase and financing of the system as well as its operation and maintenance.

Capital cost- The cost of the digester is quite large and I assume that a farm

makes a down payment of twenty percent of the digester’s cost at time zero. There is a

cost associated with the farm using its own equity to finance a portion of the digester,

which is included in the NPV calculation. The remaining eighty percent of the digester’s

cost is financed and there is a principal and interest expense associated with this.

35

Depending upon the scenario, the capital cost may include the cost of the digester

as well as a solids separator. The estimates include costs for the mix tank, raw manure

pumping and mixing, total site piping, digester, engine building, engine generator,

hydrogen sulfide filter, installation labor, utility charge, startup, contingencies,

engineering/site assistance/grant work required, construction observation and assistance,

travel, and insurance and performance bonds. The cost of a separator is included in

scenarios where applicable (McEliece, 2007).

Operating and maintenance - Labor and parts are needed to maintain and

operate the digester over time. In this model, this cost occurs in each of t time periods

and is assumed constant over time.

3.1.3 Taxes and Depreciation

The benefits from a digester are subject to taxes and the digester equipment is a

depreciable asset. The taxes and depreciation associated with a digester can affect a

digester positively or negatively. The assumptions of these parameters are as follows.

Depreciation- The digester is depreciated via two calculations. The engine-

generator is depreciated separate from the rest of the digester because it can be classified

as a piece of farm machinery. I depreciate the remaining components of the digester

together as a single-purpose livestock facility (Lazarus and Rudstrom, 2007).

Taxes- I used a 33% marginal tax rate. The total benefits of the digester less the

operating and maintenance costs, interest expense, and depreciation, also called taxable

income (TI), are subject to this tax rate. The principal portion of the financing expense is

36

not part of the TI. If the TI is positive, the taxes will be a cost incurred by the farm, but if

the TI is negative, the farm will experience a decrease in its tax bill.

3.1.4 Net Present Value Calculations

NPV was defined in Equation 2.1 in the previous chapter. The standard decision

rule is that investment should occur when a project’s NPV exceeds zero. Because I

assume a mixture of debt and equity capital, I use the weighted average cost of capital

(WACC) to discount the net cash flows in each time period, t. WACC is a type of

discount rate that combines the cost of debt and equity capital. WACC is defined as

follows

[ ] tedt (E/A)I (D/A)I )-(1WACC +τ= (3.1)

where τ signifies tax rate, Id is the cost of debt capital, D represents the farm’s total debt,

A represents the farm’s total assets, Ie is the cost of equity capital, and E denotes the

farm’s total equity.

3.1.5 Variables and Parameters

Each component of the economic benefits and costs is calculated using

parameters and variables. Parameters are constant within a simulated scenario and

variables are randomly generated within a scenario. This section details the parameters

and variables included in the stochastic capital budgeting model.

37

Table 3.1 Parameters for Stochastic Capital Budgeting Model

Parameter Units Value(s) Data Source(s)

Herd size Milk cows 1,000 and 2,000 Assumed Downtime of digester

Percent 10 Kramer, 2004 and Lazarus and Rudstrom, 2004

Digester costa

$ 1,000 cows - 1,149,379

2,000 cows - 1,890,224 McEliece, 2007

Separator cost

$ 1,000 cows - 76,500 2,00 cows - 115,625

McEliece, 2007

Price realized for sale of electricity

$/kWh 0.01, 0.03 (base case), 0.05, 0.10

Lazarus and Rudstrom, 2003 and Garrison and Richard, 2005

Cost to turn compostb

$/cubic yard 1,000 cows - 1.22 2,000 cows - 1.09

Rynk et al., 1992 and USDA/NASS, 2007

Opportunity cost of land given up for compostc

$/acre 46.50 Pennsylvania Agricultural Statistics Service, 2006

Compost generated d Cubic feet/cow/day

0.24 Wichert, 2004; Martin, 2004; and Wright, 2001

Farm electricity use kWh/cow/year 811 Ludington and Johnson, 2003

Engine efficiency Percent 25 Lusk, 1998; Giesy et al.,2005; and US EPA, undated

Carbon credits received for methane reduction

Carbon credits per cow

4.41 McComb, 2007

Carbon credit value $/ metric ton of CO2 equivalent

3.70 Chicago Climate Exchange, 2007a

RECs Carbon credits per 1 mWh

0.4 Subler, 2006

Carbon credit aggregator costs

Percent of carbon credit sales

50 Six, 2007

Operation and maintenance

$/kWh generated 0.015 Lazarus and Rudstrom, 2003 and Krich et al., 2005

Cost of debt capital Percent 8 Assumed Cost of equity capital

Percent 10 Hyde, Stokes, and Engel, 2003

Tax rate Percent 33 Assumed Down payment Percent of cost 20 Assumed

38

Total assets pre- digester

$ Without separator 1,000 cows- $2,576,910 2,000 cows- $4,259,040 With separator 1,000 cows- $2,758,510 2,000 cows- $4,536,540

Assumed

Total equity pre-digester

Percent of assets 80 Assumed

Total debt-pre-digester

Percent of assets 20 Assumed

Total assets post-digester

$ Without separator 1,000 cows- $3,422,210 2,000 cows- $5,678,720 With separator 1,000 cows-$3,678,010 2,000 cows- $6,048,720

Assumed

Total equity post-digester

Percent of assets 60 Assumed

Total debt post-digester

Percent of assets 40 Assumed

Digester depreciation

Years 10 Lazarus and Rudstrom, 2007

Length of loan Years 10 Assumed Yearly digester depreciatione

Dollars 1,000 cows-$ 596,707 2,000 cows- $1,038,469

McEliece, 2007

Engine-generator depreciation

Years 7 Lazarus and Rudstrom, 2007

Yearly engine generator depreciation

Dollars 1,000 cows- $250,066 2,000 cows- $403,357

McEliece, 2007

a. Digester costs include separate mix tank, raw manure pumping and mixing, total site piping, digester, hydrogen sulfide filter, installation labor estimate, utility charge estimate, startup cost, contingencies, engineering/site assistance/ grant required work, construction observation and assistance, travel, and insurance and performance bond (McEliece, 2007).

b. I indexed suggested figures from Rynk et al. (1992) using data from USDA-NASS (2007) and Pennsylvania Agricultural Statistics Service (2006). This cost assumes that a farm uses an 85 hp tractor with a loader attachment (1-yard bucket) to turn the compost 4 times/year (Rynk et al., 1992).

c. I used Rynk et al. (1992) and Pennsylvania Agricultural Statistics Service (2006) to determine the amount of land necessary to compost digested solids and cost of using this land for composting purposes.

d. Where appropriate, the information from the data sources may be subjected to a 65% solids loss during digestion and/or a 50% mass loss during composting (Tiquia, Richard, and Honeyman, 2002 and Topper, 2007).

e. The digester depreciation includes separate mix tank, raw manure pumping and mixing, total site piping, digester, hydrogen sulfide filter, and installation labor. The cost of the separator is added to this, where applicable. This figure summed with the engine-generator expense does not represent the entire cost of the digester because there are additional costs of a digester (e.g. contingencies or travel) that are not depreciable.

39

3.1.5.1 Model Parameters

Parameters are those items that remain unchanged within a given simulated

scenario, although some, such as herd size, may change across scenarios (Table 3.1). I

assess herd sizes of 1,000 and 2,000 lactating cows. The digester’s costs change with

herd size (McEliece, 2007). By varying herd size, I am able to draw conclusions

regarding economies of scale in digester adoption.

If a farm installs a digester, it must decide if it wants to add a solids separator to

the project. A separator’s cost increases as herd size increases. Of course if a farm selects

the “no separator” option, the cost of the separator is zero.

Several data points are used to determine the compost generated per cow per day.

In Martin’s (2004) case study, a 550-cow farm sells 1,825 cubic yards of compost each

year, or about 0.245 cubic feet of compost per cow per day. The other farm in his study

was a 100-cow herd that generated 38 cubic feet of solids a day, which is about 0.19

cubic feet of compost per cow per day. Three cubic yards of compost per cow per year or,

equivalently, 0.22 cubic feet of compost per cow per day are cited in Wichert (2004).

Some researchers list the amount of separated solids generated per cow per day, from

which one can estimate the amount of compost generated. A farm that separated solids

pre-digestion experienced about one cubic foot of separated solids per cow per day,

which is approximately the same as 0.30 cubic feet of compost from post-digested solids

(Wright, 2001). The average of all these numbers is approximately 0.24 cubic feet of

compost per cow per day and this is what I use as a parameter.

Lazarus and Rudstrom (2004) found that a digester on a Minnesota dairy farm

operated 98% of the time during its first five years of operation. They cautioned that,

40

“Generator running time on other farms has often been less, so other producers considering digester investments might wish to consider a range of scenarios including some with lower generator runtime and/or lower gas output that might result due to management differences or other factors,” (p. 3).

Several cases in Kramer (2004), describe problems with the engine generator that caused

varying amounts of downtime. In a few cases downtime was calculated, but in most it

was only described generally. The digester studied in Lazarus and Rudstrom (2004)

experienced more downtime as it aged. Kramer (2004) showed digesters had downtime

in one year, but not necessarily in the next. Given the broad range, I choose a figure of

10% downtime in each year.

I assume a farm will pay for twenty percent of the digester’s cost from its equity.

Furthermore, since a digester is a large capital expense, I assume that a farm is in a

financially sound position, eighty percent owner equity, pre-digester. Post-digester

implementation, I assume the farm maintains a strong equity position, sixty percent. It is

unlikely that a farm with an equity position of less than sixty percent after digester

implementation would receive financing. The farm makes yearly payments on the

remaining eighty percent of the digester over ten years.

Depreciation is calculated on the engine generator and the digester. The engine

generator is depreciated as a piece of farm machinery. It is depreciated over seven years

using straight-line depreciation. The remaining digester components are depreciated as a

single purpose livestock facility over ten years (Lazarus and Rudstrom, 2007).

41

3.1.5.2 Model variables

Variables are those items that change within a given simulated scenario (Table

3.2). Each variable is assigned a distribution based on information and data found in the

literature. All variables and their distributions are discussed below.

Under the Pennsylvania net metering regulation, the farm avoids payment for all

electricity generated by a methane digester. The value of this benefit is a function of the

retail price of electricity. Based upon United States Department of Energy data on the

residential price of electricity from 1985-2005, I specify a triangular distribution with a

minimum and mode of $0.0739 and maximum of $0.0967 per kWh (US DOE, 2006).

Digester operators may also install a solids separator. If so, they must decide if

they will utilize the solids for bedding on the farm or sell them as compost. Weeks

(2002) estimated that each cow’s manure would provide approximately one cubic foot

per day of separated solids. He indicated that this is more than enough to provide for

bedding for cattle. Furthermore, he commented that the average bedding purchase on a

farm is approximately $50-$100 per cow per year. These numbers are similar to Gooch

et al.’s (2006) who found that farms could save $60-$100 per cow per year on bedding

purchases8. Without data to clarify further the nature of the distribution, I specify a

uniform distribution with a minimum of $50 and maximum of $100.

Likewise, Wright and Ma (2003) indicated that one could charge $10 per cubic

yard of compost generated, while in Martin’s 2004 case study, a farm received an average

of $16 per cubic yard of compost generated. The length of time the solids are able to

8 There are concerns that using digested manure solids may increase pathogens, thereby increasing mastitis cases in the herd (Cornell Waste Management Institute, 2006). Clearly, if herd health declines as a result of using manure solids for bedding, this is a cost a farm would incur. However, because scientists are in the early research stages of this problem and this health cost is difficult to quantify, we do not include this potential cost as part of our analysis.

42

Table 3.2 Variables Specified in the Model

Parameter/Variable Units Specified Distribution Source(s)

Retail electricity price $/kWh Triangular (0.0739, 0.0739, 0.0976)a US DOE, 2006

Bedding savings $ saved/cow /year Uniform (50, 100)b Gooch et al., 2006 and Weeks, 2002

Value of compost created $/cubic yard Uniform (9, 16) Martin, 2004 and Wright and Ma, 2003

Amount of methane in

biogas

Percent Triangular (55, 60, 80) Lusk, 1998; Wilkie, 2005; Shih et al.,

2006; and US EPA, 2002b

Biogas Cubic feet/ lb. VS Triangular (3, 5, 8) Wright, 2001 and Engler et al., 1999

Life expectancy Years Triangular (10, 15, 20) Lusk, 1998 and US EPA, undated

Total Solids Percent Triangular (11, 13.33, 14) ASAE, 2005; US EPA, 2002b; and US

EPA, undated

Pounds of manure Pounds/cow/day Triangular (110, 150, 160) ASAE, 2005; US EPA, undated; and

VanHorn et al., 1994

a. Triangular (minimum, most likely, maximum) b. Uniform (minimum, maximum)

43

compost affects the final product’s quality. Longer drying times result in higher quality

compost, which is more valuable. Since there were limited data available and the quality

is dependent upon a menu of other decisions, I estimate a farm could charge between $9

and $16 per cubic yard (distributed uniformly) of composted materials. This allowed me

to capture uncertainty associated with both quality and market price. As already

mentioned, there are costs associated with drying the solids and these are included in the

calculation. Rynk et al.’s (1992) method is implemented to compute total composting

costs.

The amount of methane in the biogas depends upon a collection of factors

including bedding material, digester management, digester type, and herd diet. Wilkie

(2005) wrote that biogas is approximately 60% methane, while Shih et al. (2006) and the

US EPA (2002b) indicated that methane makes up 60-70% of biogas. Lusk (1998) cited

55-80% as a range of expected methane production. Thus, a triangular distribution is

specified with minimum of 55%, mode of 60% and maximum 80% methane gas.

FarmWare 3.0 used a 15-year lifespan in its estimation of a plug-flow digester’s

profitability (US EPA, undated). Lusk (1998) indicated that well-designed plug-flow

digesters will last 20 years, but also wrote that the failure rate of plug-flow digesters is

63%. Based on this, I specify a triangular distribution for lifespan with a minimum of 10

years, most likely 15 years, and maximum of 20 years.

The biogas production per cow per day and, more importantly, the daily

electricity produced on a per cow basis vary widely depending on the cubic feet of biogas

produced per pound of volatile solids (VS). Because of this, the distribution of biogas

per pound of VS is tested by calculating the values it generates for daily electricity and

44

biogas production per cow to ensure these fall within the ranges defined in the literature.

Lusk (1998) estimated that a 500-cow dairy would create 35,700 cubic feet of biogas per

day or roughly 1,607 kWh. On a per cow basis, these numbers equate to 70 cubic feet of

biogas per day and 3.2 kWh. Similar biogas production figures, 65 and 90 cubic feet of

biogas per cow per day, were found in Mehta (2002) and Scott et al. (2006), respectively.

Three case studies in Kramer (2004) reported actual digester electricity production, which

ranged from 2.9 kWh per cow per day to 3.5 kWh per cow per day.

Few sources listed the biogas produced per pound of VS; this may be because

manure’s composition can vary depending upon diet and differences between animals.

Ranges of 3 to 7 cubic feet of biogas per pound of VS and 3 to 8 cubic feet of biogas per

pound of VS were found in Wright (2001) and Engler et al. (1999). A minimum of 3 and

a maximum of 8 define this distribution; however, these values generate extreme biogas

and electricity production, which will most often not occur. As such, a uniform

distribution does not define this variable well. When 5 cubic feet of biogas per pound of

VS is used as a mode in a triangular distribution, mean electricity and biogas production

are similar to those found in the literature. In this situation, a cow produces 76 cubic feet

of biogas and 3.26 kWh of electricity on a daily basis. I define the distribution as

triangular with the minimum of 3, maximum of 8, and a most likely of 5.

The American Society of Agricultural Engineers (ASAE) estimated that a

lactating dairy cow produces 150 pounds of manure per day, of which 20 pounds (13.3%)

are TS and 17 pounds (11.3%) are VS. This means that VS are approximately 85% of

the TS. Furthermore, a plug flow digester operates best with 11-13% TS (ASAE, 2005;

45

US EPA, 2002b). I assign a triangular distribution with minimum 11%, maximum 14%

and most likely 13.33% to estimate the volume of TS in relation to manure.

The amount of manure produced by each cow is variable. The ASAE (2005)

found 150 pounds per day as the average amount of manure created by a lactating cow.

Conversely, VanHorn et al. (1994) used 125 pounds of manure per day for a cow in the

last 135 days of lactation. The US EPA suggested a lower output; 112 pounds per 1,400-

pound dairy cow (US EPA, 1999). Based on these estimates, I specify a triangular

distribution with minimum 110 pounds, maximum 160 pounds, and most likely 150

pounds of manure per day per lactating cow.

3.1.6 Stochastic Simulations

The basic capital budgeting model was developed in Microsoft Excel. To

implement the Monte Carlo simulations, @Risk, an Excel add-in, is utilized. I use a

constant seed value across simulations, such that the simulated values are constant across

simulations. Thus, it is easy to directly compare scenarios without being concerned that

differences may be due in part to the randomness of the simulation technique. Within

each simulation, 10,000 iterations are performed. The simulated results converge to

stable distributions within this number of iterations.

3.2 Real Options Model

This section begins by describing the basic framework required to model real

options in continuous time. From there, I describe modifications necessary to model

46

jump diffusion processes in discrete time. The variables and techniques used in this

model are defined in the final section.

3.2.1 Real Options Framework

The NPV calculated in the previous section is one tool that may be used to make

an investment decision. If the NPV is greater than zero, a positive investment decision is

made, while if the NPV is less than zero, the investment is forgone. This method does

not account for the value of waiting to make an investment decision; in other words, the

decision is now or never. The value of waiting arises from two factors, uncertainty and

irreversibility. NPV simply values the project’s future cash flows and does not allow for

uncertainty in the future. A stochastic capital budgeting model can implement some of

this uncertainty; however, it does not assume irreversibility. If an investment’s returns

are uncertain and exhibit irreversibility, i.e. sunk costs, a firm may wish to defer

investment until the probability of a profitable project is greater. This delay helps a firm

reduce the risk of uncertainty and sunk costs.

Real options are another tool that may be used to make an investment decision.

Like financial options, real options give the owner of the option the right to buy or sell an

investment. Unlike capital budgeting, real options analysis assumes that the asset’s value

is uncertain over time and that the investment incurs some sunk costs. From these two

factors, the option value is derived.

The NPV is a part of the option valuation. It is summed with the capital cost of

the digester, which gives the value of the digester, or asset, represented by S. The asset

value, in turn, is used to calculate option values. Additionally, the real options model

47

assumes that the asset value is log normally distributed. The standard deviation of the

distribution of the log of the asset value is also a part of the option valuation model.

Trigeorgis (1996) states four basic assumptions of European option valuation:

1. Markets are frictionless.

2. The discount rate or risk-free interest rate is constant throughout

the life of the option.

3. The underlying asset does not pay dividends9.

4. Asset values follow a continuous time stochastic diffusion process,

or Geometric Brownian Motion (GBM).

Because of these assumptions, this relationship can be represented mathematically as:

dZ dt SdS

σ+α= , (3.2)

where S represents the value of the underlying asset,SdS is the change in the asset’s value,

and α represents the expected return on the stock. dt is an increment of time, σ is the

standard deviation of S, and dZ represents an increment of a Wiener process with mean

zero and variance dt (Dixit and Pindyck, 1994).

Alternatively, Equation 3.2 can be expressed in integral form as,

∫∫ σ+σ−α=−

T

0

T

0

2 )t(dZdt)5.0()]0(Sln[)]T(Sln[ , (3.3)

9 Dividends can be a part of option valuation, if the model is altered to account for their effects, but are not part of the most basic European option valuation assumptions. An American option will never be exercised early and its value will be equal to that of the European option, if the asset doesn’t pay dividends (Merton, 1973). If one is valuing American options, dividends are usually a part of the calculation for this reason. The dividends represent the opportunity cost of keeping the option alive (Dixit and Pindyck, 1994).

48

in which S(T) is the asset’s value at time T, S(0) is the initial asset value, and T is the

option’s time to maturity. Using these results, one can derive the Black-Scholes equation,

which gives the closed-form solution of a European call option (Black and Scholes, 1973).

)d(N Xe )d(N )T(SV n2rT

n1−−= , (3.4)

where V represents the option’s value, S(T) is the stock’s value at the option’s maturity,

and N represents a normal cumulative density function of (d1n) and (d2n), which can be

further decomposed into

( )T

T)5.0r(X

S lnd

2

n1σ

⎥⎦

⎤⎢⎣

⎡σ++⎟

⎠⎞

⎜⎝⎛

= (3.5)

and

Tdd n1n2 σ−= . (3.6)

X symbolizes the exercise price of the option and r is the riskless rate of interest.

Dividends may also be a part of the option valuation because they are the

opportunity cost of investing in the asset now, rather than waiting until the option’s

maturity, time T (Dixit and Pindyck, 1994). They are incorporated easily into the above

and are represented by δ. Equations 3.4 - 3.6 are modified as follows,

)* N(d X e*) N(dS(T) eV n2T r

n1T - −−= δ , (3.7)

where

)T(

T)5.0r()X

S ( ln*d

2

n1σ

⎥⎦⎤

⎢⎣⎡ σ+δ−+

= , (3.8)

and

T*d*d n1n2 σ−= . (3.9)

49

3.2.2 Option Values in the Presence of Jumps

The previous section assumes that asset values follow a continuous time process,

but assets can experience large instantaneous changes in their values. In reality, an asset

value may follow a continuous time process and occasionally experience a discrete jump.

For example, the price of a barrel of oil will increase and decrease following a continuous

time process, but an event that affects production, such as war or weather, could cause a

positive or negative price jump. After a jump occurs, the asset’s value again follows a

continuous time process.

My research values the option to build an anaerobic digester when there are

multiple sources of uncertainty. The following section explains a methodology

developed by Martzoukos and Trigeorgis (2002) that can be used to value options in

discrete time when there are several sources of jumps10. Building from their approach, I

describe my own technique to value options in the presence of multiple classes of

uncertainty in the final section.

Martzoukos and Trigeorgis (2002) build upon Amin’s (1993) work by modeling

option values in the presence of multiple jumps. The jumps can represent different

shocks that would affect an asset’s value, e.g. changes in an asset’s technology or supply,

and are assumed independent of each other. They assume N jump classes, with the jumps

indexed by i= 1 to N. They begin with

10 Martzoukos and Trigeorgis (2002) also develop a closed form solution to value European options using a modified Black-Scholes equation. Interested readers should see that paper for additional information.

50

∑=

+σ+δ−=N

1i

ii* dqkdZdt)r(

SdS , (3.10)

where δ* is a term derived from the dividend yield, ki is the average size of jump i, and

dqi is a jump counter. If the ith jump occurs, dqi equals one, where as if the ith jump does

not occur, the jump counter takes a value of zero. The relationship between δ and δ* is

∑=

λ+δ≡δN

1i

ii )k(* , (3.11)

where ]k[Ek ii λ≡λ , with E denoting the expectation operator. λi is the annual frequency

of jump i.

Equation 3.10 may be represented in integral form as,

∑∑∫∫= =

++σ+σ−δ−=−N

1i

n

1q

q,i

T

0

T

0

2*i

)k1ln()t(dZdt)5.0r()]0(Sln[)]T(Sln[ . (3.12)

The double summation in Equation 3.12 captures all occurrences of jumps. Note that n is

a vector with N elements and each element of the vector is equal to the number of jumps

of the ith event. Furthermore, the distribution of the ith jump size is defined as

),5.0(N~)k1ln( 2i

2iii σσ−γ+ , (3.13)

where N represents a normal distribution with mean γi – 0.5σ2i and variance σ2

i, γi is the

mean jump size, and σ2i is the variance of the ith jump. The relationship between ik and

γi can be expressed as

1)exp(k]k[E iii −γ=≡ . (3.14)

51

Martzoukos and Trigeorgis (2002) also specify that the standard deviation of the asset

changes due to the impact of the jump arrivals. They explain this relationship as,

∑=

σλ+σ=σN

1i

2ii

2 . (3.15)

The option is valued over one year, T = 1 year, and discretized into h smaller time

segments, with t = 1,…, h. In each time period, t, a jump may or may not occur. Only

one jump per class may occur, in each period, but multiple jumps may occur in a time

period, if they are from different classes. The probability of jumps occurring and not

occurring, denoted as P(.), is as follows,

Te)e(Te)0n ,1n(P

Te)e(Te)0n ,1n(P

e)e()i all for 0n(P

N

1ii

iN

N

1ii

i1

N

1ii

i

)(T

N

N

Ni,i

TN

TNiNi

)(T

1

N

1i,i

T1

T1i1i

)(TN

1i

Ti

∑ λ−

≠

λ−λ−≠=

∑ λ−

≠

λ−λ−≠=

∑ λ−

=

λ−

=

=

=

λ=λ===

λ=λ===

===

∏

∏∏

Λ

(3.16)

In this model, the asset’s value increases or decreases throughout time by both

GBM and jumps. Its movement can be described in terms of steps. For example, in most

time periods, a jump does not occur and the asset’s value follows GBM. When this

happens, GBM is equivalent to the asset value increasing or decreasing one-step.

Conversely, when a jump occurs, the asset value may greatly increase or decrease, just

like the asset value in Figure 2.1, and it moves several steps. The size of the jump is

defined by the number of steps the asset moves, such that different sized jumps may

52

occur. Jumps with a smaller magnitude are more likely than jumps with a larger

magnitude.

Because jumps are, by definition, rare events, most of the time the asset will go up

or down only one-step, following GBM. The probability of the asset’s value increasing

by one-step is notated as pup and is defined as 5.0pup ≅ . This probability is used to

define the probability of the asset’s value going down one-step, denoted pdown, and is

expressed pdown ≅ (1- pup).

Martzoukos and Trigeorgis’ model begins with the asset value at time 0. From

there, in each time period, t, the asset may go up or down l steps, in their case

l = -42,…,42, where m = 42, for a total of 2 m +1 = 85 possible asset values in the next

time period, t=1. The asset’s value in the next time period is calculated as,

tt) 5.0r()t(Sln)1t(S ln 2* Δσ+Δσ−δ−+=+ λ , (3.17)

with Δt being equal to T/h. The asset value space grows quickly through time and as a

result, Martzoukos and Trigeorgis (2002) establish a floor and a ceiling creating a state

space of 650 asset values. This number was not picked arbitrarily; rather, the authors

used the European option values from their continuous time model to test their European

option values in an asset space of 650. At an appropriate truncation point, the difference

between continuous time and discrete time option values should be nonexistent

(Martzoukos, 2007). That is, the truncation is chosen such that it does not affect option

valuation. Each asset value is spaced σ√Δt apart in the state space, and j is used to

represent the different states within this space; more succinctly it is defined,

j = -325,…,325. Asset values are calculated for each t time period.

53

To value the option, Martzoukos and Trigeorgis assign probabilities to each of the

2 m +1, or 85, possible steps that may occur. These are calculated from the following

expression, when l ≠ ±1

]}t)5.01[(Np

]t)5.01[(Np

]t)5.01[Np

]t)5.01[(Np{ x

)0n ,1n(P

}1tt*) 5.0r()]t(Sln[)]1t(S{ln[P

id

id

iu

iu

N

1i

1kik

2*

Δσ−+−

Δσ+++

Δσ−−−

Δσ+−

==

=±≠Δσ+Δσ−δ−=−+

∑=

≠=

λ

λ

λ

λ

λλ

(3.18)

Ni representing a cumulative normal distribution with the mean and variance defined in

Equation 3.14. This expression is slightly modified for the cases when l = +1 or -1:

]}t)5.01[(Np

]t)5.01[(Np

]t)5.01[Np

]t)5.01[(Np{ x

)0n ,1n(P

i) allfor 0,n(Pp }1tt*) 5.0r()]t(Sln[)]1t(S{ln[P

id

id

iu

iu

N

1i

1kik

iup

2*

Δσ−+−

Δσ+++

Δσ−−−

Δσ+−

==+

=

=+=Δσ+Δσ−δ−=−+

∑=

≠=

λ

λ

λ

λ

λλ

(3.19)

and

54

]}t)5.01[(Np

]t)5.01[(Np

]t)5.01[Np

]t)5.01[(Np{ x

)0n ,1n(P

i) allfor 0,n(Pp }1tt*) 5.0r()]t(Sln[)]1t(S{ln[P

id

id

iu

iu

N

1i

1kik

idown

2*

Δσ−+−

Δσ+++

Δσ−−−

Δσ+−

==+

=

=−=Δσ+Δσ−δ−=−+

∑=

≠=

λ

λ

λ

λ

λλ

(3.20)

These probabilities, in turn, are used to value the option and are inserted into this

expression,

{ }∑=

−=

Δ−

++∗Δσ+Δσ−δ−=−+

=42

42

2*tr

)j 1,V(t tt*) 5.0r()]t(Sln[)]1t(Sln[ P e)j,t(V

λ

λλ

λ (3.21)

with V(t, j) representing the value of an option at a particular time period, t, and state

space, j. Option values are found via backward recursion, which means the option values

at time t=T are first calculated, and those values, in turn, are used to solve for the option

values in the previous time periods. This method is continued back to time t=0, which

yields the expected European option value. The model can readily be altered to compute

the value of an American option, as well.

3.2.3 A Modified Approach to Modeling Options in the Presence of Jumps

The methodology developed by Martzoukos and Trigeorgis (2002) provides the

framework for my option calculations. There are several items that are part of their

option value equation: the stock’s value at each time period, t; the probability of a jump

at each t; and the magnitude of a jump, if one occurs, including GBM. I include these

55

probabilities and values in my option model; however, I use simulations to value options,

rather than an asset space. Each simulation contains 1,000 iterations, which can be

thought of as 1,000 different paths through the asset value space described above.

As such, my calculations of probabilities and stock values are different from

Martzoukos and Trigeorgis’ approach. For example, the probability of a jump occurring

is different between our models. They use Equation 3.16, and I take a more

straightforward approach. Another difference is that they calculate the probability of a

jump being of magnitude l by integrating both jump movement and GBM into this

calculation (Equations 3.17-3.19), but I calculate the probability of jump magnitude l

based on jumps only. GBM is calculated separately in my model. Finally, because

Martzoukos and Trigeorgis use an asset space to calculate the option value, their option

value solution, in Equation 3.20, is a bit different than the standard V(t) = e-rT{max (S(T)

– X, 0)}. Due to the changes I have made to their model, I am able to use an option value

model that is much closer to this standard.

3.2.3.1 Finding the Asset Value at Time t

I begin with time T = 5 years and h =1,000. The asset’s value at time t = 0 is

equal to the mean of the ln(E(NPV) + capital cost), which was calculated by the

stochastic capital budgeting model in a previous step. The capital cost is added back to

the mean NPV because in the real option framework, the capital cost represents the

exercise price while the net cash flows represent the value of the asset. The resulting asset

value is inserted into the following equation to calculate the asset value at the other

values of t.

56

[ ] t*) 5.0r(t*GBM*dq)1t(Sln)t(Sln 2* Δσ−δ−+Δσ++−= λ (3.22)

The following sections provide details about the components of Equation 3.22. Table 3.3

also defines the values given to the parameters in this model.

3.2.3.2 Defining Movements in Asset Value

In my model, there are two forms of movement in asset value, jumps and GBM.

Technology and policy are the two types of jumps in my model. Jump 1 represents a

change in policy and jump 2 is a change in technology. Information about their means

and distribution is included in Table 3.3. Most of the time, a jump will not occur; this

probability is discussed in the next section. In most iterations, asset values follow GBM,

as explained above in Section 3.2.2.

Because data were not available to understand the distribution of these jumps, a

standard normal distribution is used to define the mean and variance of a jump. This

assumption seems logical based on the uncertainty surrounding policies and technology.

For example, historically, policies encouraging farmers to invest in digesters have been

enacted and technology changes have increased productivity, which would indicate a

positive mean jump size. On the other hand, it is possible that a much better form of

alternative energy could be discovered, in which case the new technologies and policies

would be pursued and the mean of a jump distribution would be negative. There is no

prior method of assessing if one is more likely than the other, so a mean jump size of zero

is appropriate.

57

Table 3.3 Parameters used in Real Options Model Variable Definition Value Source

Ln(S(t=0)) Asset value at time t Ln (E(NPV)+ capital cost) Stochastic capital budgeting modelσ Standard deviation of asset Standard deviation of ln(S(t=0)) Stochastic capital budgeting modelT Time to maturity 5 years Assumed h Number of time increments 1000 Assumed t Increments of time {0,…h} Martzoukos and Trigeorgis (2002) Δt Change in time, T/h 1/1000 Assumed r Riskless rate of interest 8% Stochastic capital budgeting modelδ Dividend yield 5% Assumed δ * Adjusted dividend yield 5% Martzoukos and Trigeorgis (2002) α r-δ 3% Assumed l Steps 0}42,...42{ ⊄− Martzoukos and Trigeorgis (2002) GBM Geometric Brownian Motion 1 or -1 Martzoukos and Trigeorgis (2002) dq Jump counter 0 or 1 Martzoukos and Trigeorgis (2002) σ Standard deviation of asset,

adjusted for jump arrivals ( ) 5.02

22211

2 σλ+σλ+σ Martzoukos and Trigeorgis (2002)

N Number of jump classes 2 Assumed Ni Cumulative normal distribution of the ith jump ~(γi -0.5 2

iσ , 2iσ ) Martzoukos and Trigeorgis (2002)

σJ1 Standard deviation of Jump 1 1 Assumed σJ2 Standard deviation of Jump 2 1 Assumed E[k1] Average size of jump 1 0 Assumed E[k2] Average size of jump 2 0 Assumed γ1 Mean of jump 1 0 Martzoukos and Trigeorgis (2002) γ1 Mean of jump 2 0 Martzoukos and Trigeorgis (2002) λ1 Probability of jump 1 occurring in time T 1% Assumed λ2 Probability of jump 2 occurring in time T 1% Assumed

58

3.2.3.3 Defining the probability of a jump

The value of dq, the jump counter, is 0 if a jump does not occur and 1 if a jump

does occur. These values are drawn from a Poisson distribution, where

)].0dqPr(1[)1dqPr(]1[)0dqPr(

i

tii

=−==λ−== Δ

(3.23)

λi is the probability of the ith jump throughout time T. Because a jump is a rare event, λi is

a small number, near zero.

3.2.3.4 Defining the Magnitude of a Jump

l is defined as before, varying between -42 and 42, not including zero. If dq=1, l-

values are selected from a discrete probability distribution that is defined as

)1dq( Pr])t( )5.0[(N)]t( )5.0[( N

)Pr( ii

=Δσ−−Δσ+

=λλ

λ (3.24)

Again, Ni is the cumulative normal distribution, with the mean and variance as defined in

Equation 3.13.

3.2.3.5 Defining Geometric Brownian Motion

GBM represents Geometric Brownian Motion, and it may take on a value of +1 or

-1. The GBM value is also drawn from a discrete probability distribution where both

GBM outcomes have a probability equal to 0.5.

59

3.2.3.6 Option Valuation

After the asset values are calculated, the option values can be found. The

European option value, V, is straightforward because the option can only be exercised at

maturity, t = T.

{ })0 ,X)T(Smax(eV rT −= − , (3.25)

in which the value of X, the exercise price, is the cost of the digester. e-rT is a discount

rate, which allows the value of the option at time t=T to be discounted back to time t=0.

American option pricing is a bit more complex because American options may be

exercised at any point in time. As a result, the value of the option at each increment, t,

must be calculated. At S(T), the call value is simply, X]max[S(T)VA −= , much like

the European option, where VA is the value of the American option. Because the option

value in each time period is computed, the option value at time T need not be discounted.

The option values at other times are found via backward recursion. If the asset

value at time t is less than the sum of the exercise price and the discounted option value

from the time period t+1, then the option value at time t is the option value from time t+1

multiplied by a discount factor. Conversely, if the asset value is greater than the sum of

the exercise price, and the discounted option value at time t+1, the option value at time t

is equal to the asset value at time t less the exercise price. Mathematically, these

relationships are

⎪⎩

⎪⎨⎧

>+

≤++=

Δ

ΔΔ

0)1t(VA*e-X-S(t) if X-S(t)0)1t(VA*e-X-S(t) if 1)V(t*e

VA(t)tr-

t-rt-r

(3.26)

60

3.3 Stochastic Simulations

The real options model is developed in Microsoft Excel, incorporating @Risk for

the Monte Carlo simulations. A random seed value is used across the different

simulations to simulate different paths through the asset space. Each simulation consists

of 1,000 iterations. The option values reported later reflect the mean option values found

through the simulation procedure described here.

3.4 Chapter Summary

This chapter has developed a two-part method to finding option values. In the

first part, the asset is valued using a stochastic capital budgeting model. The average

value of the asset is used to price European and American real options. European options

can be valued using either the closed-form Black-Scholes’ equation or a numerical

method. American options may be valued using the numerical method, as well, but

additional work must be completed to account for their early exercise feature.

Martzoukos and Trigeorgis’ (2002) provide the framework for my option calculations. In

the next chapter, I will apply my method to determine the effects of digester policy and

technology changes on the value of options.

61

Chapter 4 Results

This chapter presents the option values derived from the models described in the

previous chapter. From the stochastic capital budgeting model, I provide the expected

NPV in dollars and the probability that the simulated NPV is greater than zero.

Furthermore, I provide the mean of the simulated distributions for both the European and

American option values. The option values give insight into the project’s expected

profitability under various conditions. The first section of this chapter analyzes option

values across two herd sizes and assesses the effects of net metering and carbon credits.

The second section evaluates the option values’ sensitivity to changes in electricity price,

carbon credits, capital cost, and composting cost. The final section considers the effect of

the jumps by altering their parameters, as well as removing them from the model.

4.1 Base Case Results

I simulate four basic scenarios. I begin by excluding net metering or carbon

credits. In subsequent scenarios, I include carbon credits and net metering individually,

and then together. When operating a digester without a net metering policy in place, I

estimate that a farm pays for 30% of the electricity it uses (Topper, 2008). Excess

electricity is sold to the utility at a rate of $0.03 per kWh based on data described in the

previous chapter.

Each of the four scenarios described above is analyzed with different herd sizes

and separator options. I use farms with 1,000 and 2,000 cows to determine the effect of

62

herd size. Furthermore, I consider the option of adding a solids separator to the project

and the final use of the solids, which may be sold as compost or used as bedding.

4.1.1 Results without a solids separator

Without a separator, the base case scenarios for herd sizes of 1,000 cows do not

have an American option value greater than $35,000 (Table 4.1), indicating that this

project is very unlikely to be profitable. Three of the four base case scenarios for a 2,000

cow herd have an American option value greater than $35,000, but the largest option

value is only $108,400, which is small in comparison to the $1.7 million capital cost of

the project. These results suggest that without a separator the value of the option is small

relative to the project’s cost and that investment would not be profitable.

4.1.2 Results including a separator with the end product sold as compost

A separator increases the cost of the project, but the solids provide a stream of

income to the digester operator. The option value increases for herds of 1,000 and 2,000

cows when a separator is added (Table 4.2). This suggests that although the separator

increases the initial cost of the project, it adds to the potential profitability of the digester.

The option values for a 1,000-cow herd are also small relative to the capital cost. For

example, a European option is $29,500 for a 1000-cow herd that sells the solids in the

base case scenario, while the capital cost is over $1.1 million. When carbon credits and

net metering are added to the simulation, this value increases to $60,000, indicating that

these policies may increase expected profitability. One should note, that the E(NPV)

remains

63

Table 4.1 Simulation Results and Option Values of Investment in Digester with No Solids Separator

Herd Size

Scenario P(NPV>0)a E(NPV) b European Option b, c American Option b,c

Base 0.0% -$525 $7.0 $12.5

Base with Carbon Credits 0.0% -$481 $7.7 $15.8

1,000 cows Base with Net Metering 0.0% -$465 $13.1 $22.1

Base with Carbon Credits and Net Metering

0.0% -$420 $19.9 $35.0

Base 0.0% -$969 $17.0 $28.2

Base with Carbon Credits 0.0% -$815 $26.4 $51.8

Base with Net Metering 0.0% -$761 $38.6 $71.3

2,000 cows

Base with Carbon Credits and Net Metering

0.0% -$608 $61.2 $108.4

Notes: a. All probabilities relate to the percentage of simulated results that exceed zero b. All values are in $1,000 units c. Option values are rounded to nearest $100

64

Table 4.2 Simulation Results and Option Values of Investment in Digester with a Solids Separator and Solids sold as Compost

Herd Size

Scenario P(NPV>0) a E(NPV) b European Option b, c American Option b, c

Base 0.0% -$395 $29.5 $52.4

Base with Carbon Credits 0.0% -$350 $40.4 $74.1

1,000 cows Base with Net Metering 0.0% -$334 $44.7 $82.2

Base with Carbon Credits and Net Metering

0.0% -$290 $60.0 $109.3

Base 0.0% -$489 $96.2 $183.4

Base with Carbon Credits 0.0% -$400 $116.0 $223.9

2,000 cows Base with Net Metering 0.0% -$368 $135.3 $242.7

Base with Carbon Credits and Net Metering

0.1% -$279 $172.6 $318.7

Notes: a. All probabilities relate to the percentage of simulated results that exceed zero b. All values are in $1,000 units c. Option values are rounded to nearest $100

65

largely negative and the P(NPV>0) is 0.0%, suggesting that these policies are not enough

to make a digester a profitable decision.

For a herd with 2,000 cows, the option values increase in all four scenarios, which

suggest there may be economies of scale present. It appears that carbon credits and net

metering effectively increase the expected NPV and option value. The greatest option

value occurs when a farm takes advantage of both net metering and carbon credit sales.

The corresponding option values for this situation are $172,600 (European option) and

$318,700 (American option).

4.1.3 Results including a separator with the solids used on-farm as bedding

If a farm opts to invest in a separator, it also has the option of using the solids as

bedding on the farm, rather than trying to sell them as compost. The results indicate that

bedding is the most profitable way to use the solids (Table 4.3). The expected NPVs are

greater in the bedding scenarios than in the compost scenarios. More importantly, one

can see that the option values and expected NPVs increase as herd size increases and

additional policies are added to each scenario, as shown earlier. At both herd sizes, the

option value associated with this particular analysis exceeds those found earlier. For

example, a farm with 1,000 cows realizes an American option value of $258,000 and a

farm with 2,000 cows realizes an American option value of approximately $715,000.

4.2 Sensitivity Analysis

In the previous section, several base case scenarios were analyzed. The price of

carbon credits, electricity, capital costs, and composting costs were fixed throughout the

66

Table 4.3 Simulation Results and Option Values of Investment in Digester with a Solids Separator and Solids used as Bedding

Herd Size

Scenario P(NPV>0) a E(NPV) b European Option b, c American Option b, c

Base 0.0% -$211 $84.4 $167.3

Base with Carbon Credits 0.3% -$166 $107.7 $208.4

1,000 cows Base with Net Metering 1.4% -$151 $111.6 $216.8

Base with Carbon Credits and Net Metering

9.0% -$106 $133.1 $257.9

Base 24.6% -$126 $238.7 $496.7

Base with Carbon Credits 42.2% -$36 $306.3 $589.2

Base with Net Metering 48.7% -$5 $317.2 $616.1

2,000 cows

Base with Carbon Credits and Net Metering

67.3% $85 $361.4 $715.3

Notes: a. All probabilities relate to the percentage of simulated results that exceed zero b. All values are in $1,000 units c. Option values are rounded to nearest $100

67

valuation. This section shows the impact of altering these items across simulations. This

information may be useful in determining what policies may be effective in the future.

Because the base case scenarios were largely unprofitable for farms that did not utilize a

solids separator, these situations are excluded from additional analyses.

4.2.1 Electricity Price

To this point, I have assumed that a farmer receives $0.03 per kWh for excess

electricity generated, but this may not always be the case. The farm must negotiate a rate

with the power company, which means that a farm may receive nothing for the excess

electricity. On the other hand, a farm may receive significantly more than $0.03 per kWh

if the local utility is interested in purchasing green energy or laws require that the utility

pay the full retail rate for excess electricity. Therefore, I analyze all of these situations to

determine electricity price’s effect on the expected NPV and option value.

Table 4.4 contains information about varying electricity price received for farms

with 1,000 and 2,000 lactating cows. Both the option to use the separated solids as

bedding or to sell them is included. The electricity price takes values ranging from $0.00

to $0.10 for herds of 1,000 cows and $0.00 to $0.05 for herds of 2,000 cows11. A 1,000-

cow herd that utilizes the solids as bedding, experiences a positive expected NPV at a

price of $0.10 and the European option value is $232,000. The American option value in

this scenario is $475,000.

For herds of this size that choose to sell composted solids, the option values are

less than option values for farms that choose the bedding option. At a price of $0.00, the

11 Because the option values and probability of a positive NPV are large at $0.05 for this herd size, $0.10 is not included in the analysis.

68

Table 4.4 Sensitivity of Results for Varied Electricity Prices for Herd Sizes of 1,000 and 2,000 cows

Herd Size Electricity Price

Solids used as Bedding Solids sold as Compost

P(NPV>0) a E(NPV) b European Option b

American Option b

P(NPV>0) a E(NPV) b European Option b

American Option b

$0.00 0.1% -$173 $106 $206 0.0% -$357 $36 $68

$0.03 9.0% -$106 $133 $258 0.0% -$290 $60 $109

$0.05 26.0% -$61 $163 $318 0.0% -$245 $76 $146

$0.10 62.0% $51 $232 $475 15.3% -$133 $136 $280

$0.00 39.5% -$50 $282 $559 0.0% -$413 $119 $222

$0.03 67.3% $85 $361 $715 0.1% -$279 $173 $319

1,000

2,000

$0.05 81.9% $174 $418 $839 7.8% -$189 $220 $423

Notes: a. All probabilities relate to the percentage of simulated results that exceed zero. b. All values are in $1,000 units

69

European option value is $36,000 and the American option value is $68,000 for farms

selling compost. Again, the option values increase as the price of electricity increases.

At $0.03, the European and American option values are $60,000 and $109,000. At $0.10,

the European option is $136,000 and the American option is $280,000.

The results are similar when the herd size is equal to 2,000 cows. The European

and American option values at $0.00 received per kWh when this farm uses the solids for

bedding are $282,000 and $559,000. These values are greater than any of the respective

option values for a 1,000-cow farm. In the case of farms selling compost, however, the

European and American option values are small relative to capital cost at $0.00 received

for electricity, only $119,000 and $222,000, respectively. As the price of electricity

increases, so does the option value. At $0.05, the value of the American option increases

to $423,000. The European option value is $220,000. This indicates that the results are

sensitive to electricity price.

4.2.2 Carbon Credit Price

The price of carbon credits could also vary. In the base cases, a price of $3.70 is

used to value a carbon credit, but market prices are frequently higher or lower than this.

Accordingly, I analyze a carbon credit price of $2.40 and $5.00 for the same types of

herds described in the previous scenario (Table 4.5).

70

Table 4.5 Sensitivity of Results to the Price of Carbon Credits for Herd Sizes of 1,000 and 2,000 cows

Notes: a. All probabilities relate to the percentage of simulated results that exceed zero. b. All values are in $1,000 units

Herd Size

Carbon Credit Price

Solids used as Bedding Solids sold as Compost

P(NPV>0) a E(NPV) b European Option b

American Option b

P(NPV>0) a E(NPV) b European Option b

American Option b

$2.40 5.1% -$122 $122 $243 0.0% -$305 $48 $93

$3.70 9.0% -$106 $133 $258 0.0% -$290 $60 $109

1,000 cows

$5.00 14.1% -$90 $141 $275 0.0% -$274 $64 $114

$2.40 60.7% $53 $355 $680 0.0% -$310 $149 $287

$3.70 67.3% $85 $361 $715 0.1% -$279 $173 $319

2,000 cows

$5.00 73.6% $116 $388 $754 0.3% -$247 $178 $335

71

Table 4.5 illustrates that option values move in the same direction as carbon credit

prices. For example, the European option value declines by $6,000 for a 2,000-cow herd

using the solids as bedding when carbon credits decrease, while on the same farm that

sells the effluent as compost, the American option value decreases by $32,000. This

relationship holds for the European option values as well. A 1,000-cow farm selecting

the bedding option experiences an $8,000 change in the European option value when the

price of carbon credits increases. If the farm were to choose to sell the solids, instead, the

European option value changes by $4,000 as carbon credit price increase.

4.2.3 Capital Cost

The capital cost of a digester with a separator ranges from close to $900,000 to

almost $1,900,000. In recent years, the price of construction materials has increased and

the price of digesters has increased with it (McEliece, 2007). Both the Commonwealth of

Pennsylvania and federal governments have worked to make digesters more affordable

for farms by offering incentives to those that invest in a digester. As such, I investigate

the option value’s sensitivity to a 10% increase or decrease in capital cost.

Across all scenarios, the option values and expected NPVs vary greatly,

depending on the cost of digester (Table 4.6). For example, a 10% reduction in cost

increases the American option value for a farm with 2,000 cows that sells compost from

$319,000 to $391,000, while a 10% increase in cost causes a $50,000 decrease. Similar

results are found for herd sizes of 1,000 cows. For a farm like this that uses the solids as

bedding, the European option increases from $133,000 to $160,000, a $27,000 increase,

when costs decrease.

72

Table 4.6 Sensitivity of Results to Capital Cost for Herd Sizes of 1,000 and 2,000 cows

Herd Size

% Cost Solids used as Bedding Solids sold as Compost

P(NPV>0) a E(NPV) b European Option b

American Option b

P(NPV>0) a E(NPV) b European Option b

American Option b

90% 39.5% -$25 $160 $311 0.0% -$208 $70 $131

100% 9.0% -$106 $133 $258 0.0% -$290 $60 $109

1,000

110% 0.1% -$187 $116 $221 0.0% -$371 $45 $83

90% 91.5% $217 $423 $818 5.7% -$146 $207 $391

100% 67.3% $85 $361 $715 0.1% -$279 $173 $319

2,000

110% 40.1% -$48 $307 $607 0.0% -$412 $143 $269

Notes: a. All probabilities relate to the percentage of simulated results that exceed zero. b. All values are in $1,000 units

73

When capital costs increase 10%, the European option value declines by $19,000, from

$133,000 to $116,000.

4.2.4 Composting Cost

The above scenarios have indicated that a farm is more profitable if it utilizes a

solids separator and uses the solids on farm as bedding. This may be due to the costs of

preparing solids to be sold as compost. I conduct sensitivity analyses to determine the

effect of decreasing or increasing composting costs on option values.

The results in Table 4.7 reveal that expected NPVs and option values increase if

composting costs increase and decrease if composting costs decline. The effect, though,

is small. On a farm with 1,000 cows, both the European and American option remained

approximately the same if the cost of composting increased 10%. A larger farm, i.e.

2,000 cows, experiences a $1,000 change in the European option value when composting

costs decrease 10%. An increase in composting costs did not cause a scenario with a

positive expected NPV to experience a negative expected NPV. Conversely, no scenario

with a negative expected NPV experiences a positive NPV when composting costs

decreased.

4.3 Sensitivity of Option Values to Jumps and Their Parameters

One of the unique features of my model is the addition of the jump processes to

the asset and option valuation. To test the effect of the jump, I compare models with no

jumps to those with jumps. For further analysis, I alter the parameters of the first jump,

policy, to determine the effects of mean jump size, jump standard deviation, and jump

74

Table 4.7 Sensitivity of Results to Composting Cost for Herd Sizes of 1,000 and 2,000 cows

Notes: a. All probabilities relate to the percentage of simulated results that exceed zero. b. All values are in $1,000 units

1,000 cows 2,000 cows % Cost

P(NPV>0) a E(NPV) b European Option b

American Option b

P(NPV>0) a E(NPV) b European Option b

American Option b

90% 0.0% -$287 $59 $108 0.1% -$275 $172 $330

100% 0.0% -$290 $60 $109 0.1% -$279 $173 $319

110% 0.0% -$292 $60 $109 0.1% -$282 $165 $312

75

frequency. In all cases, I evaluate the effects on option values for 1,000- and 2,000-cow

herds that used solids for bedding. I used the results from Table 4.4 as a control in each

simulation and altered the parameters for each of those scenarios to determine the effect

of jump changes. Carbon credits and net metering are included in each situation.

4.3.1 without Jumps

I began by examining the effect of including jumps in the model (Table 4.8). A

1,000-cow herd that receives $0.03 per kWh of electricity sold experiences American

option values of $117,000 and $258,000 without and with jumps, respectively. Similarly,

a 2,000-cow herd selling electricity for the same price, also experiences an increase in

option values when jumps are included in the model. With jumps the American option

value is $715,000; however, when jumps are excluded the option value declines to

$430,000. When a farm with 1,000 cows receives $0.05 per kWh of electricity, the

European option value declines from $163,000 with jumps to $104,000 without jumps.

4.3.2 Time

In my base case scenarios, I assumed that time, T, was equal to five years;

however, it is possible that a farmer could hold the right to exercise this option for a

much longer time. Conversely, a policy, tax credit, or governmental program may only

be available for a shorter amount of time, so I also compared the effect of changing T to

one year and twenty years (Table 4.9).

76

Table 4.8 Sensitivity of Results to the Inclusion of Jumps for Varied Electricity Prices for Herd Sizes of 1,000 and 2,000 cows

Notes: a. All scenarios assume bedding, carbon credits, and net metering b. All values are in $1,000 units

Herd Size Electricity Pricea European Optionb American Optionb

Without Jumps With Jumps Without Jumps With Jumps

$0.00 $47 $106 $76 $206

$0.03 $76 $133 $117 $258

1,000 cows $0.05 $104 $163 $160 $318

$0.10 $190 $232 $336 $475

$0.00 $200 $282 $314 $559

$0.03 $285 $361 $430 $715

2,000 cows

$0.05 $348 $418 $548 $839

77

Table 4.9 Sensitivity of Results to Changes in Time, T, for Herd Sizes of 1,000 and 2,000 cows

Herd Size Electricity

Pricea European Optionb American Optionb

T =1 T =5 T =20 T =1 T =5 T =20

$0.00 $44 $106 $158 $85 $206 $454

$0.03 $63 $133 $187 $126 $258 $499

$0.05 $85 $163 $193 $168 $318 $554

$0.10 $155 $232 $276 $307 $475 $1,256

$0.00 $155 $282 $324 $319 $559 $967

$0.03 $245 $361 $397 $486 $715 $1,113

1,000

2,000

$0.05 $306 $418 $403 $593 $839 $1,219

Notes: a. All scenarios assume bedding, carbon credits, and net metering b. All values are in $1,000 units

78

The results indicate that as T increases the option values increase. The European

option value for a 1,000-cow herd receiving $0.10 per kWh of electricity sold is $232,000,

in the initial scenario, but that value increases to$276,000 when T is twenty years and

decreases to $155,000 when T is 1 year. For a 2,000-cow herd that does not receive

compensation for its sale of electricity, the American option value is $559,000 in the

initial scenario; however when T goes to twenty years, the option increases to over

$967,000. Alternatively, when T decreases to one year, the American option value

decreases to $319,000.

4.3.3 Standard Deviation of Jumps

Each jump is assigned a standard deviation of one in my initial scenarios;

however, this is somewhat arbitrary because data for this parameter are not available.

Because standard deviation is chosen randomly, I analyze the effect of decreasing and

increasing the standard deviation of jump one, σ1 (Table 4.10).

The results indicate that increasing the standard deviation of the jump increases

the option values and decreasing this standard deviation decreases the option value. For

example, the European option value declines from $106,000 to $88,000 for a 1,000-cow

herd receiving no value from selling its electricity when the standard deviation of the

policy jump decreases. On the other hand, the value of this option increases to $167,000

when the standard deviation of the jump doubles. This effect is present in the remaining

results, as well. For a 2,000-cow herd, the American option value when a farm receives

$0.05 per kWh is $839,000 in the initial simulation, but the value decreases to $750,000

when σ1 decreases and this value increases to $1,170,000 when σ1 increases.

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Table 4.10 Sensitivity of Results to Changes in Standard Deviation, σ1, for Herd Sizes of 1,000 and 2,000 cows

Notes: a. All scenarios assume bedding, carbon credits, and net metering b. All values are in $1,000 units

Herd Size Electricity Pricea

European Optionb American Optionb

σ1= 0.5 σ1=1 σ1=2 σ1=0.5 σ1=1 σ1=2

$0.00 $88 $106 $167 $162 $206 $351

$0.03 $118 $133 $194 $215 $258 $410

$0.05 $147 $163 $206 $270 $318 $459

$0.10 $231 $232 $303 $443 $475 $663

$0.00 $272 $282 $390 $495 $559 $847

$0.03 $326 $361 $447 $619 $715 $1,003

1,000

2,000

$0.05 $408 $418 $542 $750 $839 $1,170

80

4.3.4 Mean Jump Size

The mean jump size in the above scenarios was zero, but like the standard

deviation of a jump, it was chosen because it is the mean of a standard normal

distribution. In reality, the mean of the distribution of jumps could be a positive or

negative number, as mentioned above. I examine altering the mean jump size of jump

one, k1, to a positive number. While negative policies certainly may occur, it seems more

likely that a policy would result in a positive change in the value of the underlying asset.

That is, legislators would probably enact laws that encourage digester adoption if it

looked to be a promising source of energy, but would probably not pass laws that directly

discourage digester adoption.

I explore altering the mean jump size to one and five (Table 4.11). Increasing the

jump size decreases option values. European option values on a 1,000-cow farm in a

case where $0.03 per kWh is given for the sale of electricity start at $133,000 when the

mean jump size is zero, but decline to $118,000 and $49,000 when the mean jump size

increases to one and five, respectively. This same relationship is experienced with

American options for this herd. These values begin at $258,000 when the mean jump

size is equal to zero and decrease to $241,000 at mean jump of one and $144,000 at a

mean jump of five. Similarly, a 2,000-cow herd that receives $0.05 per kWh has a

European option value of $839,000 when the mean jump size equals zero, but this value

declines to $781,000 and $604,000 as the mean jump increases to one and then five. This

may occur because there is less variability due to the increased upside potential.

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Table 4.11 Sensitivity of Results to Changes in Mean Jump Size, k1, for Herd Sizes of 1,000 and 2,000 cows

Notes: a. All scenarios assume bedding, carbon credits, and net metering b. All values are in $1,000 units

Herd Size Electricity Pricea

European Optionb American Optionb

E(k1) =0 E(k1)=1 E(k1) =5 E(k1) =0 E(k1)=1 E(k1) =5

$0.00 $106 $91 $36 $206 $180 $103

$0.03 $133 $118 $49 $258 $241 $144

$0.05 $163 $145 $63 $318 $293 $187

$0.10 $232 $205 $106 $475 $445 $331

$0.00 $282 $255 $122 $559 $532 $357

$0.03 $361 $304 $154 $715 $670 $483

1,000

2,000

$0.05 $418 $374 $193 $839 $781 $604

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4.3.5 Jump Frequency

In my final jump parameter sensitivity analysis, I alter the frequency of jump one,

λ1, from 0.01 to 0.05. When this change occurs, option values increase (Table 4.12). A

1,000-cow herd that receives $0.05 per kWh experiences an American option value of

$318,000 in the original scenario, but when the frequency of jumps increases, the option

value is $556,000. For a 2,000-cow herd selling electricity at the same rate, European

option values begin at $361,000 when the jump frequency is 0.01 and increase to

$476,000 when the jump frequency increases.

4.4 Chapter Summary

Results indicate that larger farms are more likely to be profitable investing in an

anaerobic digester than smaller farms. Moreover, the addition of a solids separator to the

digester investment further increases profitability. Option values and expected NPVs

increase as herd size increases and farms enhance the digester project with a solids

separator. This was particularly true if the farm chooses to use the separated solids as

bedding for its herd. Policies and opportunities such as net metering and carbon credits

also increase option values and expected NPVs and are one-step toward increasing

digester adoption on farms. Additional analyses showed that results are sensitive to the

price of electricity and capital cost changes. Changes in composting costs and carbon

credits had minimal impact on option values. The inclusion of jumps in the model

increases option values, as does increasing the time to maturity, standard deviation of a

jump, and jump frequency. Increasing the mean jump size, on the other hand, decreases

option values.

83

Table 4.12 Sensitivity of Results to Changes in Jump Frequency, λ1, for Herd Sizes of 1,000 and 2,000 cows

Notes: a. All scenarios assume bedding, carbon credits, and net metering b. All values are in $1,000 units

Herd Size Electricity Pricea

European Optionb American Optionb

λ1=0.01 λ1=0.05 λ1=0.01 λ1=0.05

$0.00 $106 $193 $206 $394

$0.03 $133 $235 $258 $502

$0.05 $163 $262 $318 $556

$0.10 $232 $310 $475 $726

$0.00 $282 $457 $559 $988

$0.03 $361 $476 $715 $1,101

1,000

2,000

$0.05 $418 $510 $839 $1,238

84

Chapter 5 Conclusions

Renewable energy is being increasingly sought and demanded in this country, and

anaerobic digesters are one source of alternative energy. The initial capital cost of a

digester is quite high, which is a barrier to widespread adoption of this technology. Laws

like net metering and market opportunities such as carbon credits are new revenue

streams that might make digester implementation a profitable decision for farms.

Prior digester research has mostly focused on improving digester efficiency.

Financial analysis of digesters has been based mainly on case studies and the results have

not been generalizable. Additionally, while real options analysis has been used to

address investment decisions in agriculture, real options that incorporate jumps, or

market shocks, have not been evaluated.

This research uses real options analysis to assess digester profitability on dairy

farms in Pennsylvania, including jumps in the option valuation. The jumps represent

shocks that may occur due to changes in policy or technology. While the mean of the

jumps is zero, they can range in value from a large negative number to a large positive

number.

I explored the effect of policies and opportunities to determine if they are

increasing digester profitability. Furthermore, I analyzed the addition of a solids

separator to the digester project’s profitability. These results were further studied

through sensitivity analysis to determine the nature of the relationship between these

opportunities and profitability. Electricity prices, carbon credit prices, capital costs, and

composting costs were tested. Additionally, the jumps were removed from the model, so

85

that jump and non-jump option values could be compared. Finally, the parameters that

defined my policy jump were altered to determine the option values’ sensitivity to these

changes.

5.1 Discussion of results

This work clearly shows that larger dairy farms are more likely to profit from

methane digester technology. AgSTAR suggests that dairies with 500 cows or more will

be profitable with a digester, but my findings suggest that dairies must be larger, in the

range of 1,000 or 2,000 cows or more, to have the potential to be profitable under the

conditions modeled here (US EPA, 2002a). 2,000-cow herds experienced greater

expected NPVs and option values than 1,000-cow herds. More of the expected NPVs

were greater than zero when herd size was 2,000 cows than when herd size was 1,000

cows; this indicates expected positive profitability. With the addition of a solids

separator and new policies and regulations, the project becomes significantly more

profitable for larger herd sizes.

The option of a separator increased the probability of the investment being

profitable. When a separator was not included in the digester investment, all of the

simulations had a negative expected NPV. The option values in this scenario were also

small, relative to the other results. When a separator was a part of the digester project,

the expected NPVs and option values increased. This is due to the increased revenue that

the solids separator generated.

Using the solids as bedding for the herd further increased the profitability of the

project. Farms experienced greater expected NPVs and option values when they opted to

86

use solids as bedding, rather than sell them as compost. This may be because there are no

costs associated with using the solids as bedding, whereas there are costs to compost the

solids. Additionally, by the time the solids are of high enough quality to be sold as

compost, they have lost about 50% of their volume, while solids used for bedding are

used immediately and do not lose volume (Tiquia, Richard, and Honeyman, 2002). This

results in less material for the farm to sell as compost.

Net metering regulations and carbon credit sales also affect the project’s

profitability. While these items increased the expected value of the project’s NPV,

neither, by itself, turned an unprofitable scenario (i.e., one that had a negative mean NPV)

into a profitable scenario. The option values increased as carbon credits and net metering

were included in scenarios. Expected NPVs and option values were greater in the base

case with net metering than in the base case with carbon credits, which may mean that net

metering has a greater effect on digester profitability than carbon credits.

Sensitivity analysis suggests that results are highly sensitive to the price received

for excess electricity sold to the power company. In all the scenarios analyzed, as

electricity price increased, the expected NPV and option values increased. Increased

electricity prices turned some scenarios’ negative expected NPVs into positive expected

NPVs.

Digester profitability is also sensitive to initial capital cost. As capital costs

increased, the expected NPVs and option values declined. Conversely, as capital costs

decreased, mean NPVs and option values increased. The option values varied greatly

between scenarios, which indicate that grants or other financial assistance to farmers

would improve the success of their digester project.

87

Moreover, the results are less sensitive to some other factors. For instance, my

results fluctuated only slightly when I altered the market price of carbon credits. This is

most likely because revenue from carbon credits is a small part of the total benefits

realized from a digester.

Furthermore, the costs of producing and selling compost from the separated

solids have little effect on the distribution of NPVs and option values for the relevant

scenarios. The option values changed very little between the simulated scenarios.

Perhaps the results would be more sensitive if the price realized for the sale of solids

changed. This may be an area for further research.

When jumps were removed from the model, option values declined. This may

occur because removing the jumps from the model removes some of the uncertainty. The

results suggest that changes in policies or improvements in technology could increase

digester adoption.

Sensitivity analysis of the jumps indicated that additional work to determine the

exact nature of the jumps is necessary. For example, increasing the standard deviation of

a jump or the options’ time to maturity increased option values, which suggests that a

change in these parameters increases the volatility, or uncertainty, associated with the

option value. A decrease in option values when the mean jump size increases may be

because the uncertainty of negative cash flows is removed, so there is less uncertainty

about the future. Furthermore, if the frequency of a jump increases the option values

increase, which may occur because of an increased uncertainty. While these results are

useful, parameterizing the model remains difficult. Since results are sensitive to changes

88

in jump parameters, it is difficult to accurately calculate and interpret option values if the

nature of the parameters is unknown.

The analysis of carbon credits and net metering suggest that current policies are

not enough for most farms, even many large ones, to adopt this technology profitably

without a significant reduction in the investment cost. If renewable energy, in the form

of electricity produced by methane digesters on dairy farms, is important to public

policymakers, then large grants or subsidies are needed to induce investment on nearly all

farms of the size commonly found in Pennsylvania and the northeastern United States.

5.2 Research Suggestions

Although little attention has been given to “community digesters,” which two or

more farmers might use, these may provide a way to achieve scale economies across

farms. It is also possible for farmers to combine efforts with other types of businesses

(e.g. restaurants or cheese plants) to mix manure with other waste to be digested. Thus,

investment could occur on a large scale, taking advantage of scale economies.

Some farms incorporate food waste into their digester, which increases biogas

production. The increase varies depending on the type of food, of course, but analyzing a

digester that uses both food waste and manure could pave the way for additional

community digester research. Perhaps shared digesters between the food service industry

and agriculture would be most profitable. There are additional opportunities to explore

economies of scale associated with digesters.

It would be interesting to research the parameter values associated with jumps.

My research indicates that the results are sensitive to changes in the jump parameter

89

values. Additional research could seek to determine the parameter values, in order to

provide more precise option values. This could be accomplished by reviewing historical

data related to energy policy to identify the frequency of policy changes and the impact

(small or large) of the policies on energy markets. Agricultural policy (e.g. Farm Bills

throughout US history) could also be used as a data source to parameterize jumps. Each

policy change might be classified as positive or negative and a magnitude assigned to it

based on its impact. Technology jumps could be assessed by identifying changes, i.e.

jumps, in engine-generator efficiency and assigning a positive or negative numeric value

to each change. Analyzing and determining the nature of the jump parameter values

would provide a better idea of the strength of these results.

Applying these results to other species, types of digesters, biogas uses, or other

locations would also be appropriate. A digester on a dairy farm in Florida is very

different than a digester in Pennsylvania. Additionally, a digester that sells natural gas

avoids one of the large portions of capital cost (the engine-generator) and its profitability

may be much greater than that of a farm selling electricity. Although digesters are found

mostly on dairy farms, there are other types of farms that utilize digesters: ducks,

chickens, turkeys, beef, and hogs. Altering the model to include these species might

show that another type of animal manure would be more profitable to produce methane.

The results found here indicate that grants and low-interest loans are necessary

conditions for most farms to have a profitable digester. More policies like net metering

would also help increase farm-level adoption. Opportunities, such as utilizing the solids

or selling carbon credits, effectively increase profitability, as well, and seeking these out

or creating new ones would be another way to encourage implementation. Additionally,

90

legislation mandating higher prices for excess electricity would help increase digester

profitability.

91

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Appendix This section provides the formulas I used to calculate the benefits and costs associated with an anaerobic digester. I used the parameters and variables defined in Tables 3.1 and 3.2, where appropriate.

Section A.1 Avoided Electricity Purchase

Inputs

• Herd size: 500; 1,000; or 2,000 cows • Pounds of manure per cow per day: Triangular distribution (110, 150, 160) • Percentage of TS in 1 lb of manure: Triangular distribution (11, 13.33, 14) • Percentage of VS in 1 lb of manure: Percentage of TS in 1 lb of manure * 85% of

TS are VS12 • Cubic feet of biogas produced per 1 lb of VS: Triangular distribution (3, 5, 8) • Percentage of methane in biogas: Triangular distribution (55, 60, 80) • Percentage of methane going to engine generator: 100 • Percent efficiency of engine generator: 25 • Percentage of time that digester is down: 10 • kWhs of electricity used per cow per year: 811 • Retail rate of electricity (in dollars): Triangular distribution (0.0739, 0.0739,

0.0976) Formula

1. Pounds of manure per cow per day * herd size * 365 days per year = pounds of manure produced by a herd in a year

2. Pounds of manure produced by a herd in a year * percentage of VS in 1 lb of manure = pounds of VS produced by a herd in a year

3. Pounds of VS produced by a herd in a year * cubic feet of biogas produced per 1 lb of VS = cubic feet of biogas produced by a herd in a year

4. Cubic feet of biogas produced by a herd in a year* percentage of methane in biogas * 1,000 British Thermal Units (BTUs) per cubic foot of methane * percent of methane going to engine generator * engine generator efficiency in percent * (1- percent of time that digester is down) =BTUs produced per year

5. BTUs produced per year * 1 kWh per 3,413 BTUs = kWhs produced by engine generator in a year

6. Electricity used per cow per year * herd Size = kWhs of electricity used by a farm each year

12 A cow produced 20 lbs of TS per day, of which 17 lbs, or 85%, are VS (ASAE, 2005).

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7a. If kWhs of electricity used by a farm each year > kWhs produced by engine-generator in a year, then kWhs produced by engine generator in a year * Retail rate of electricity = Avoided electricity purchase.

7b. If kWhs of electricity used by a farm each year < kWhs produced by engine- generator in a year, then kWhs of electricity used by a farm each year * Retail rate

of electricity = Avoided electricity purchase

Section A.2 Electricity Sold

Inputs

• Herd size: 500; 1,000; or 2,000 cows • Pounds of manure per cow per day: Triangular distribution (110, 150, 160) • Percentage of TS in 1 lb of manure: Triangular distribution (11, 13.33, 14) • Percentage of VS in 1 lb of manure: Percentage of TS in 1 lb of manure * 85% of

TS are VS • Cubic feet of biogas produced per 1 lb of VS: Triangular distribution (3, 5, 8) • Percentage of methane in biogas: Triangular distribution (55, 60, 80) • Percentage of methane going to engine generator: 100 • Percent efficiency of engine generator: 25 • Percentage of time that digester is down: 10 • kWhs of electricity used per cow per year: 811 • Price (in dollars) realized for sale of 1 kWh: 0.01, 0.03, 0.05, or 0.10

Formula

1. Pounds of manure per cow per day * herd size * 365 days per year = pounds of manure produced by a herd in a year

2. Pounds of manure produced by a herd in a year * percentage of VS in 1 lb of manure = pounds of VS produced by a herd in a year

3. Pounds of VS produced by a herd in a year * cubic feet of biogas produced per 1 lb of VS = cubic feet of biogas produced by a herd in a year

4. Cubic feet of biogas produced by a herd in a year* percentage of methane in biogas * 1,000 British Thermal Units (BTUs) per cubic foot of methane * percent of methane going to engine generator * engine generator efficiency in percent * (1- percent of time that digester is down) =BTUs produced per year

5. BTUs produced per year * 1 kWh per 3,413 BTUs = kWhs produced by engine generator in a year

6. Electricity used per cow per year * herd Size = kWhs of electricity used by a farm each year

7a. If kWhs of electricity used by a farm each year > kWhs produced by engine generator in a year, then 0 = Electricity sold

7b. If kWhs of electricity used by a farm each year < kWhs produced by engine generator in a year, then (kWhs of electricity used by a farm each year - kWhs

108

produced by engine generator in a year)* price realized for sale of 1 kWh = Electricity sold

Section A.3 Bedding Savings

Inputs

• Herd size: 500; 1,000; or 2,000 cows • Bedding savings per cow per year (in dollars): Uniform distribution (50, 100)

Formula

1. Herd size * bedding savings per cow per year = Yearly bedding savings for a herd

Section A.4 Compost Sold

Inputs

• Cubic feet of composted solids produced per cow per day: 0.24 • Herd size: 500; 1,000; or 2,000 cows • Price (in dollars) for one cubic yard of compost: Uniform distribution (9, 16) • Time to dry compost in years: 0.513 • Length of windrow in feet: 150 • Width of windrow in feet: 15 • Height of window in feet: 3 • Volume of a windrow in cubic feet: 4,50014 • Space between windrow in feet: 20 • Cost (in dollars) to rent one acre of land in Pennsylvania: 46.50 • Cost (in dollars) to turn one cubic yard of compost: 1.33 for 500 cows; 1.22 for

1,000 cows; and 1.09 for 2,000 cows Formulas

Benefits Calculation

1. Cubic feet of compost produced per cow per day * 365 days in a year * herd size * 1 cubic yard per 27 cubic feet = cubic yards of compost produced by a herd in a year

13 Compost that consists of manure and is turned with a bucket loader takes, on average, 6 months to compost (Rynk et al., 1992) 14The volume of a windrow is calculated using this equation: V = 2/3 * length of windrow * width of windrow * height of windrow (Rynk et al., 1992).

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2. Cubic yards of compost produced by a herd in a year * price for one cubic yard of compost = yearly revenues

Opportunity Cost Calculation

1. Cubic feet of compost produced per cow per day * 365 days in a year * herd size * time to dry compost = volume of composting materials in cubic feet

2. Volume of composting materials in cubic feet / volume of windrow in cubic feet = number of windrows required 3. Number of windrows required *(width of windrows + space between windrows) = width of area needed for composting 4. Length of windrows + space between windrows = length of area needed for

composting 5. Width of area needed for composting * length of area needed for composting = square feet needed for composting 6. square feet needed for composting * one acre per 43,650 square feet = acres

needed for composting 7. Acres needed for composting * cost to rent one acre of land in Pennsylvania = opportunity cost of composting

Yearly Cost to Turn Compost Calculation

1. Cubic feet of compost produced per cow per day * 365 days in a year * herd size * 1 cubic yard per 27 cubic feet = cubic yards of compost produced by a herd in a year

2. Cubic yards of compost produced by a herd in a year * cost to turn one cubic yard of compost = yearly cost of turning compost

Profits from Composting

1. Yearly revenues- opportunity cost of composting - yearly cost of turning compost = Profits from composting

Section A.5 Carbon Credits

Inputs

• Herd size: 500; 1,000; or 2,000 cows • Price (in dollars) of carbon credits: 3.70 • Credits received per cow per year: 4.41 • Percentage of carbon credit revenues paid to aggregator: 50

110

Formula

1. Herd size * Credits received per cow per year = Yearly carbon credits 2. Yearly carbon credits * price of carbon credits = Yearly revenues from carbon

credits 3. Yearly revenues from carbon credits * (1- Percentage of carbon credit revenues

paid to aggregator) = Yearly profits from carbon credits

Section A.6 Renewable Energy Credits

Inputs

• Herd size: 500; 1,000; or 2,000 cows • Pounds of manure per cow per day: Triangular distribution (110, 150, 160) • Percentage of TS in 1 lb of manure: Triangular distribution (11, 13.33, 14) • Percentage of VS in 1 lb of manure: Percentage of TS in 1 lb of manure * 85% of

TS are VS • Cubic feet of biogas produced per 1 lb of VS: Triangular distribution (3, 5, 8) • Percentage of methane in biogas: Triangular distribution (55, 60, 80) • Percentage of methane going to engine generator: 100 • Percent efficiency of engine generator: 25 • Percentage of time that digester is down: 10 • RECs: 0.4 carbon credits per 1 mWh generated by engine-generator • Price (in dollars) of carbon credits: 3.70 • Percentage of carbon credit revenues paid to aggregator: 50

Formula

1. Pounds of manure per cow per day * herd size * 365 days per year = pounds of manure produced by a herd in a year

2. Pounds of manure produced by a herd in a year * percentage of VS in 1 lb of manure = pounds of VS produced by a herd in a year

3. Pounds of VS produced by a herd in a year * cubic feet of biogas produced per 1 lb of VS = cubic feet of biogas produced by a herd in a year

4. Cubic feet of biogas produced by a herd in a year* percentage of methane in biogas * 1,000 British Thermal Units (BTUs) per cubic foot of methane * percent of methane going to engine generator * engine generator efficiency in percent * (1- percent of time that digester is down) =BTUs produced per year

5. BTUs produced per year * 1 kWh per 3,413 BTUs = kWhs produced by engine generator in a year

6. RECs* kWhs produced by engine generator in a year * 1mWh per 1,000 kWh = Yearly renewable energy credits

7. Yearly renewable energy credits * price of carbon credits = Yearly revenues from renewable energy credits

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8. Yearly revenues from renewable energy credits *(1- Percentage of carbon credit revenues paid to aggregator) = Yearly profits from renewable energy credits

112

Section A.7 Operating and Maintenance Costs

Inputs

• Herd size: 500; 1,000; or 2,000 cows • Pounds of manure per cow per day: Triangular distribution (110,150,160) • Percentage of TS in 1 lb of manure: Triangular distribution (11, 13.33, 14) • Percentage of VS in 1 lb of manure: Percentage of TS in 1 lb of manure * 85% of

TS are VS • Cubic feet of biogas produced per 1 lb of VS: Triangular distribution (3, 5, 8) • Percentage of methane in biogas: Triangular distribution (55, 60, 80) • Percent of methane going to engine generator: 100% • Engine generator efficiency: 25% • Percent of time that digester is down: 10% • Cost of operating and maintenance per kWh (in dollars): 0.015

Formula

1. Pounds of manure per cow per day * herd size * 365 days per year = pounds of manure produced by a herd in a year

2. Pounds of manure produced by a herd in a year * percentage of VS in 1 lb of manure = pounds of VS produced by a herd in a year

3. Pounds of VS produced by a herd in a year * cubic feet of biogas produced per 1 lb of VS = cubic feet of biogas produced by a herd in a year

4. Cubic feet of biogas produced by a herd in a year* percentage of methane in biogas * 1,000 British Thermal Units (BTUs) per cubic foot of methane * percent of methane going to engine generator * engine generator efficiency in percent * (1- percent of time that digester is down) =BTUs produced per year

5. BTUs produced per year * 1 kWh per 3,413 BTUs = kWhs produced by engine generator in a year

6. kWhs produced by engine generator in a year * cost of operation and maintenance per kWh = Yearly cost of operation and maintenance

Section A.8 Financing Costs

Inputs

• Digester cost: Varies, see Table 3.1 • Cost of debt capital: 8% • Length of financing: 10 years

Formula

113

Yearly Payment

1. Digester cost * 80% financed = Amount of loan

2. factorPayment 1 - capital)debt ofcost (1

capitaldebt ofcost capitaldebt ofCost financing oflength =+

+

3. Amount of loan * payment factor = yearly payment

Interest Expense and Principal Payment

At time t, 1. Loan balance at time t-1 * cost of debt capital = Interest expense 2. Yearly payment- interest expense = Principal payment 3. Amount of loan – principal payment = Loan balance at time t

Section A.9 Depreciation

Inputs

• Depreciable cost of engine-generator: Varies, see Table 3.1 • Number of years engine-generator depreciated: 7 years • Depreciable costs of digester- Varies, see Table 3.1 • Number of years digester depreciated: 10 years

Formula

Depreciation of Engine-Generator

1. Depreciable cost of engine-generator/ number of years engine-generator depreciated = Yearly depreciation of engine-generator

Depreciation of Digester

1. Depreciable cost of digester/ number of years digester depreciated = Yearly depreciation of digester

Section A.10 Taxes

Inputs

• Avoided electricity purchase: See Section A.1

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• Electricity sold: See Section A.2 • Bedding savings: See Section A.3 • Compost sold: See Section A.4 • Carbon credits: See Section A.5 • Renewable energy credits: See Section A.6 • Operating and maintenance costs: See Section A.7 • Interest expense: See Section A.8 • Yearly depreciation of engine-generator: See Section A.9 • Yearly depreciation of digester: See Section A.9 • Marginal tax rate: 33%

Formula

At time t, 1. Avoided electricity purchase + electricity sold + bedding savings + compost sold

+ carbon credits + renewable energy credits = Taxable benefits 2. Operating and maintenance costs + interest expense + yearly depreciation of

engine-generator + yearly depreciation of digester = Taxable costs 3. Taxable benefits- taxable costs = Taxable Income 4. Taxable income * marginal tax rate = Tax expense