PEH:Petroleum Economics

Publication Information

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume I – General Engineering

John R. Fanchi, Editor

Chapter 16 – Petroleum Economics

John D. Wright, SPE, Norwest Questa Engineering Corp.

Pgs. 767-807

ISBN 978-1-55563-108-6
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Economics drives the entire oil/gas producing industry. Almost every decision is made on the basis of an economic evaluation. Economic evaluations are also performed to determine reserves and the "standardized measure of value" for reporting purposes for publicly held companies. In many cases, the goal of the company is to make decisions that have the best chance of maximizing the present day profit. This chapter discusses economic evaluation under two conditions. First, techniques that assume we know the future parameters with certainty are discussed. Later, methods of handling the inherent uncertainty involved in oil/gas operations are discussed.

Having stated a company goal in terms of profit, it behooves us to examine the definition of profit. There are at least three ways to calculate profit, each with its own set of assumptions and rules and each leading to a different answer. The three models are the net cash flow model, the financial net income model, and the tax model.

In the simplest analysis, profit for a period is the revenue received during the period less the costs incurred during the period. Note that profit is defined for some time period, which can be arbitrarily long. In the oil/gas business the period is usually one month or one year. The amount of revenue received during the period is usually similar for all three models, especially for yearly periods. There might be some timing differences in revenue recognition, but they are usually relatively minor. The three models differ considerably in the timing of the costs. Costs can be further subdivided into expenses, which benefit only the current period, and investments, which benefit more than one period. The cash flow model assumes that 100% of both the investment and the expenses are recognized when they occur. The financial net income model attempts to match the revenue with the costs to produce that revenue. This leads to recognizing the expenses in the current period and recognizing the investment over a longer time period—often the life of the project. The total cost over the life of the project is the same in the cash flow model and the financial net income model, but the portion of the costs allocated to each time period is significantly different. The concept of spreading the investment over the productive life of the project leads to depreciation.

Accountants may use various types of depreciation to match the revenues with the costs required to generate the revenues. Two major types of depreciation are "units-of-production" and "straight-line" depreciation. Dividing the production for a period by the total volume to be produced and multiplying that fraction by the total investment yields the units-of-production depreciation for a period. This results in the investments being allocated on a $/barrel basis. The straight-line method of depreciation allocates the investment on $/unit time basis. Table 16.1 shows an example of each of the different ways of allocating an investment including the modified accelerated cost recovery system (MACRS), which is currently used by the U.S. Internal Revenue Service for most oilfield investments.

Table 16.1

The federal and state governments use the tax model to determine the amount of profit that a company has made for each year so that a portion of that profit can be extracted to pay for government services. The tax model allocates the investments to the period under consideration in yet another manner. The Internal Revenue Service publishes a somewhat arbitrary table of allocation factors for different types of investments. This table is called a depreciation schedule and is currently the MACRS table shown in Table 16.1. The term "depreciation" is used in both the financial net income model and the tax model, but the calculation method is very different.

Economic Model for Oil and Gas Property Evaluation

The cash flow model is the most common model used to evaluate oil/gas projects. Normally, only very large acquisitions are evaluated by examining the impact of the acquisition on the financial net income. The tax model is used only if an after-tax analysis is done.

Cash flows for the project are forecast for each year or each month until the well or project is no longer economical. Because of the ready availability of powerful computers, evaluations are usually done on a monthly basis, and the results are reported on an annual basis. Monthly calculations are more detailed but not necessarily more accurate. (There is often a tendency to consider more detailed calculations as being more accurate. The use of finer time increments does not necessarily lead to projections that are more in conformity with truth.)

Whether the calculations are done on a monthly or yearly basis, the same process is followed. The profit for each period is defined as net cash flow and, for a Royalty-Tax system such as that used in the U.S. and about one-half the other countries in the world, is calculated as shown schematically in Fig. 16.1, adapted from Thompson and Wright,^[1] page 1-13. A sample calculation is shown in Table 16.2. The values in this table were calculated monthly and then accumulated on an annual basis. Several "terms of art" are used in the oil/gas business and are described next.

Fig. 16.1 – Calculation of periodic net cash flow for a royalty tax system (like the U.S.), adapted from Thompson and Wright.^[1]
Table 16.2
Table 16.2 (continued)

Gross Production

Gross production is the volume of oil/gas that is projected to be produced during the particular month or year being calculated. Gross production is one of the most important numbers entering the net cash flow calculation and, simultaneously, one of the most difficult to determine accurately. Much of the science and art of petroleum engineering is involved in estimating these numbers for future time periods.

Shrinkage

In the model defined in Fig. 16.1, "shrinkage" is used to reduce the volumes produced from a well to the volumes sold from a well. Usually the decline curves that are used to forecast future revenues are based on production rather than sales. If there is significant shrinkage, that should be taken into account before calculating the cash flows. Typical causes of shrinkage include lease use of gas for heater treaters or compressor fuel. Oil shrinkage might occur because of basic sediment and water (BS&W) corrections or because of temperature differences between the volume of oil measured at the tank and volume of oil measured at the refinery.

Gross Sales

Gross sales is the volume of oil/gas that is projected to be sold during the time period. If shrinkage is negligible, gross sales will equal gross production.

Typically, the people who drill and operate a well do not own the minerals they are extracting. For example, the U.S. Government, state governments, Indian tribes, or private citizens usually own minerals in the United States. In most other countries, the state usually owns the minerals. The producers lease the right to develop the minerals from the mineral owners. This leads to various kinds of interests in the property.

Working Interest

Working interest is a share of the costs. The total of all the working interests in a well must be equal to one. Along with the share of the costs comes a reduced (usually) share of the revenue. It is quite common for a company to own less than 100% of the working interest in a well or project. Owning smaller interests in many projects, called diversification, is one of the ways to manage the risk involved in drilling for oil/gas. Working interest may also change over time as a result of "oil field deals." Sometimes one party will pay a disproportionate share of the costs to drill the first well on a prospect to earn a share of a lease held by some other oil company. As deals become more and more complex, it becomes very difficult to determine ownership. One method of answering the "who, what, when" question is discussed in Thompson and Wright^[1].

Royalty

Royalty is a share of the revenue free and clear of all costs of development and production. The royalty is paid to the owner of the mineral interest under the land associated with the well. In the United States, the mineral interest can be "severed" from the surface ownership so that the person who owns the surface may not have any interest in the minerals and may not receive any income from a well. In rare cases the owners of the working interest will own the minerals and, in that case, there is no royalty. Typical royalty rates in the U.S. range from 1/8 (12.5%) to 25% of the production. Royalties in other countries can range from zero to more than 30%. Some royalties, such as Alberta’s, increase with increasing production and price.

Overriding Royalty

Overriding royalty is the same as a royalty, except it does not come about because of ownership of the mineral interest. An "override" is a classic way for a lease broker or geologist to be compensated for buying leases or putting a deal together. Overrides may range from 1% of large deals to 7.5% of really "hot" or promising prospects.

Net Revenue Interest

For net revenue interest, the working interest owners pay all of the costs. Because the royalty and overriding royalty interest owners share in the production, but not in the costs, the working interest owners as a group receive less than 100% of the revenue. In many cases, the working interest owners receive around 80% of the gross revenues, although sometimes it might be as high as 87.5% or as low as 70% (or less). The share of the gross production from the well is referred to as "net revenue interest." If there is a 12.5% royalty, a 7.5% overriding royalty, and you own a 50% working interest, then your net revenue interest is 40% (100% less 12.5% less 7.5% times 50%).

Net Sales

Net sales is the product of gross sales and net revenue interest. It is your share of the production after accounting for shrinkage, royalties, and splitting the proceeds with other working interest owners.

Price

Oil is usually priced in U.S.$/barrel except in some countries where it is priced by the tonne. Gas is priced either in $/million British Thermal Units (MMBtu) or by the cubic meter. Be careful to use the same volume units on the sales forecast and the price forecast. The price of oil/gas varies dramatically with time and less dramatically with the quality of the oil or gas. There are several "benchmark" crudes in the world, for which the price is reported on a daily basis. Benchmark crudes include Brent in the North Sea, Minas in Indonesia, Urals in Russia, Dubai in the Persian/Arabian Gulf, and others. The most commonly quoted number in the U.S. is West Texas Intermediate (WTI) delivered at Cushing, Oklahoma.

WTI is the underlying product for the New York Mercantile Exchange (NYMEX) futures prices. According to the NYMEX, "crude oil is the world’s most actively traded commodity. Over the past decade, the NYMEX division light, sweet (low-sulfur) crude-oil futures contract has become the world’s most liquid forum for crude oil trading, as well as the world’s largest volume futures contract trading on a physical commodity. Because of its excellent liquidity and price transparency, the contract is used as a principal international pricing benchmark."^[2] The physical specifications for WTI on the NYMEX are "crudes with 0.42% sulfur by weight or less, not less than 37°API gravity nor more than 42°API gravity." Since 1992, the spot price of WTI has been as low as $10.50/bbl in December 1998 and as high as more than $36.00/bbl in the fall of 2000.

The actual price received by a producer is usually set for several days or one month at a time and may include a transportation charge, which reduces the effective price, or a bonus depending upon the supply and demand conditions in the local area. Many refiners are posting prices online, and these postings change rapidly. Conoco, for example, posted 147 different prices between January 1, 2001 and August 31, 2001, while Enron had 191 different prices in the same period. (Crude oil price postings are another indicator of the volatility of the oil/gas industry. When this chapter was first drafted in September 2001, the links to the postings were Conoco, Enron, Tosco, and Phillips. Since then, Enron went out of business, while Phillips purchased Tosco and then merged with Conoco.) Websites that have crude oil price bulletins for various companies are [jttp://www.conocophillips.com/buy/postings ConocoPhillips]; Koch Supply and Trading; Shell Canada; and Phillips 66.

Projecting the future price of oil to use in an evaluation is quite difficult, and unfortunately, oil price is usually one of the most important factors in the evaluation. One popular, although not necessarily accurate, way of projecting future prices is to use a forward "strip" either from the NYMEX or from other crude oil traders. Fig. 16.2 shows the NYMEX oil strip as of September 1, 2001. A differential between the property being evaluated and the NYMEX is then applied to estimate prices at the property. More information can be obtained from the NYMEX website.

Fig. 16.2 – NYMEX light sweet crude oil futures prices as of September 1, 2001 (after NYMEX^[2]).

Natural gas prices are also quoted at more than 50 market centers throughout the United States. Examples include Opal, Wyoming; Katy Hub, Texas; and the southern California border. Sabine Pipe Line Company’s Henry Hub in Louisiana is the hub most often quoted. In April 1990 the NYMEX launched the world’s first natural gas futures contract with Henry Hub as the physical delivery point. In October 1992, the NYMEX began trading options on natural gas futures, which allowed traders and speculators to "play" the market. The natural gas market is even more volatile than the oil market with prices as low as $1.80 in early 1999 and over $10 for a short time in December 2000. Future gas prices for use in an economic evaluation are often forecast in the same manner as previously described for oil. The Henry Hub future price from the NYMEX is adjusted for "basis" differential. Fig. 16.3 shows the NYMEX gas strip as of September 1, 2001.

Fig. 16.3 – NYMEX Henry Hub natural gas futures prices as of September 1, 2001 (after NYMEX^[2]).

The NYMEX trades in "paper barrels," in which the seller of a contract either has to purchase an offsetting contract or deliver a specified volume of a specified quality of hydrocarbon at a specified location. A contract on the NYMEX consists of 1,000 bbl of crude or 10,000 MMBtu of gas. The forward strip shows the month when the crude or natural gas will have to be delivered; the prices at which the contract traded during the day; the number of contracts "open" (open interest) where the obligation to deliver the commodity still exists; and the number of contracts traded during the previous day. Fig. 16.2 shows that on September 1, 2001 there were open interests for 130 million barrels of crude for October delivery. This volume represents more than one-half the crude oil produced in the United States during October 2001. More information on these contracts can be obtained at NYMEX.

Many producers have chosen to use the NYMEX to "hedge" or set a price for their oil/gas in the future. This can be an effective strategy for managing risk, but it can also be extremely frustrating when gas is currently selling for $8/MMBtu on the spot market and it was agreed months ago to sell gas today for $2.80/MMBtu.

State and Local Taxes

In the U.S., most states levy a tax called a "severance" tax on all minerals extracted and sold from a property. This tax may range from 3 to 12.5% of the value of the minerals produced. Local taxing districts such as counties or taxing districts such as fire districts may also impose a tax on oil/gas production. This tax is often referred to as an ad valorem tax from the Latin for "according to value." In most cases the assessed value of the property is multiplied by the mill levy of the taxing district just like the property tax on houses and buildings. The method of calculating assessed value varies considerably from state to state. The two most common methods of calculating assessed value are to use some fraction of the revenue received and to use some fraction of the calculated net present value of the projected production.^[3] The methodology used varies widely from state to state and from time to time. Often, ad valorem taxes are approximated as a percentage of each owner’s revenue when calculating net cash flow. The percentage can range from less than 5 to 20%. Indian tribes and cities in some states may also collect a severance tax on oil/gas produced within their borders. In most cases, each party pays their own severance and ad valorem taxes. In other words, the working interest owners only pay state and local taxes on their share of the production, and the royalty owners pay the tax on their share of the production.

In countries other than the U.S., it is not so common for local governments to impose a severance tax, but it does occur. An example is Argentina, where the provincial governments impose a sales tax ranging from 1 to 2%. In most cases there are numerous other taxes that may need to be taken into account such as road taxes, airport taxes, or stamp duties. The oil company usually attempts to negotiate an agreement that exempts them from all these taxes in return for a royalty or a share of the production. These negotiations are sometimes successful and sometimes not.

Operating Costs

Operating costs are those costs that are necessary to maintain production from the well. They would include direct expenses, such as electricity for a pumping unit motor; hot oil treatments; payments to a pumper to monitor and do minor repairs on the well; replacement of pumps or rods; fixing flow line leaks; plowing roads; and a myriad of other expenses associated with owning an interest in an oil/gas well. Direct expenses might range from $250/well per month for a flowing gas well to $3,000/well per month for an onshore oil well producing a large amount of fluid. Offshore wells can have even higher operating costs.

Another component of operating costs is the Council of Petroleum Accountants Societies (COPAS) or fixed-rate overhead charge. (As stated from the COPAS website, "COPAS was created in 1961 to provide a forum for discussing and solving the more difficult problems related to accounting for oil/gas. These discussions frequently have resulted in the creation of guideline documents and educational materials. You can find these materials in the Products section of this site. COPAS has grown to 23 local Societies and over 2,700 members in the United States and Canada. COPAS has a strong emphasis in providing quality educational materials related to oil/gas accounting.")^[4] For producing wells this is a charge levied by the operator of the property to reimburse the operator for the costs of administering the payment of invoices, disbursement of monies, and filing of government forms associated with the lease. This charge is subject to negotiation between the operator and the non-operators in a property and can range from $100/well per month to $1,500/well per month or more. The accounting firm of Ernst & Young (E&Y) surveys operators for their costs of company operated wells annually. Even though the figures that E&Y solicit are internal numbers, they give an idea of what might be expected for COPAS charges. Their most recent report can be obtained from Overhead_Survey Ernst & Young.

Operating costs in other countries can vary dramatically. In most cases the operating costs will include facilities to house expatriate workers and their dependents as well as other normal operating expenses. Depending on the situation, these costs can be very significant in the cash flow calculations.

Net Operating Income

Sometimes called "cash generated from operations" or other names, net operating income is the cash flow to the working interest owner after operating costs and state and local taxes have been paid, but before investments have been made. This represents the cash generated during the period that is available for investment.

Income Tax

Almost all federal governments including the United States Government and most state governments levy a tax on income. The calculation of these taxes can be fairly straightforward in countries such as Indonesia or extremely difficult, especially when a single project is being evaluated for a reasonably large company in the U.S. Even when the appropriate software is available to aid in the evaluation, the input data necessary to accurately calculate income tax is often hard to obtain. For this reason and because income tax often has a relatively low impact on the final decision, it is common practice to calculate before federal income tax (BFIT) net cash flows when evaluating U.S. properties. Major oil companies are more likely to attempt to include the effects of income tax in their calculations, while independents seldom include it. The effect of using BFIT numbers on the ultimate decision is highly dependent on the individual case, but experience indicates that in the U.S. it seldom changes the decision.

Investment

Investments are costs that benefit future periods as opposed to operating costs that only benefit the current period. Examples include buying a lease, drilling a well, buying and installing a pumping unit, and building tank batteries. In all of these cases, the goods purchased are expected to help produce money far into the future.

Net Cash Flow

Net cash flow is the amount of money that flows into or out of the treasury during any one period. It is equal to the net operating income (either before or after income taxes) less the investments.

Each of these items is estimated for every future time period until the net operating income is no longer positive. At that time the well(s) is (are) usually assumed to be plugged and abandoned. There may be an additional expense at that time for abandonment costs, or the salvage value of the equipment may be equal to or greater than the abandonment costs.

Time Value of Money

Money has a time value. This means a dollar received today has more value to us than a dollar received far in the future. Other than a desire for instant gratification, there is a very rational reason for this phenomenon. If we have a dollar today, we can put it to work by making an investment and have more than a dollar at some future date. This concept of putting the money to work has important implications later in this section when discount rates are discussed.

Another important concept is the concept of equivalence between a current lump sum of money and a lump sum to be received in the future. Offering someone a choice between receiving $100 today and receiving $101 one year from today can demonstrate this. Most people will opt for the $100 today. If we increase the amount of future money to $115 or $125 or perhaps $200 and guarantee payment, there will be a point at which the future sum of money will become more attractive than the current $100. The amount of future money necessary to sway the person to choose the future sum is dependent upon many things—the inflation rate, current opportunities to invest the $100, and perceived risk, among others. No matter what the amount of money necessary to tip the scales, the concept that money has a time value is established.

In the case just discussed, if the person is indifferent to receiving $125 one year from now or $100 now, we say that the two sums are "equivalent." This concept of equivalence is fundamental to the evaluation of all engineering projects. We are often faced with the choice of having a certain sum of money now or receiving various sums of money in the future. By determining the equivalence between money received today and money received in the future, we can make an informed decision.

In the previous example, making the choice is relatively simple. There are only two sums to compare, and the time period is one year. This is usually not the case in oil/gas property evaluations, so we need a mechanism to handle complex choices. The mechanism that works best is interest.

We can define interest as the amount of money that must be added to our current sum to make an equivalent future sum. The amount of interest necessary to create equivalence is dependent upon the period under consideration. We may be indifferent to receiving $100 now, $112.50 six months from now, or $125 one year from now. In that case, the $12.50 or $25.00 is the amount of interest. To easily compare all three alternatives, interest is compared as a rate. It may be expressed as 12.5% per six months or 25% per year. The interest rate is calculated by dividing the amount of interest paid per period by the principal amount at the beginning of the period. Often, interest rate is expressed on an annual basis such as 25% per annum.

For loans or bank deposits that exceed one year, the interest is usually compounded. That is, the interest earned during the first period is added to the original principal to form the principal for the second period. The compound interest concept will be used when calculating the equivalence between a sum of money today and future sums of money.

Future Worth of a Lump Sum

If we have a present sum of money, P, and we put it to work at a compound interest rate, i, we will have a future lump sum of money, F, at the end of n periods. The relationship between these parameters is expressed in equation form as

....................(16.1)

The term (1 + i)ⁿ is called the single payment compound amount factor in many texts and is often tabulated.

Example 16.1

Future Value. $1,000 is placed in a bank paying 12% per compounding period. How much money will be in the account after five periods?

F = P(1 + i)ⁿ.
F = $ 1,000(1 + 0.12)⁵.
F = $ 1,762.

Present Worth of Lump Sum

Present worth of lump sum is by far the most important equation in discussing the time value of money. This one equation allows the creation of an equivalence between future projected net cash flows and current sums of money, which can then be compared to the amount to be invested to obtain those net cash flows.

If an amount, F, is going to be received n periods in the future, then its present value, P, can be calculated for a given interest rate, i, by rearranging Eq. 16.1.

....................(16.2)

This is the inverse of the single payment compound amount formula. The justification for this formula lies in the equivalence concept. The sums P and F are equivalent to each other because P could be invested at i for n periods to become F. The term (1 + i)^–n is often referred to as the "single payment present worth factor" or "discount factor." The value of n in the previous equations does not have to be an integer. Although there are some theoretical difficulties, it is quite practical to use a value of 2.5 for n to create an equivalence between a future lump sum received 2.5 periods in the future and a present lump sum. This technique is used quite often when calculating the present value of annual cash flow streams.

---

Example 16.2

Present Value. $1,762 will be received five periods from now. What is the present value of this amount at an interest rate of 12% per period?

When cash flows are calculated for several periods, as done in Fig. 16.1, Eq. 16.2 can be used repeatedly to find the equivalent present value of each of the future cash flows.

Annuities and Loans

There are a number of specialized equations that can be used when particular types of repetitive cash flows are projected. In particular, when the same cash flow, A, is received at the end of every period, the present value of the cash flow stream can be calculated from the equation,

....................(16.3)

Eq. 16.3 is often called the annuity equation because it can be rearranged to calculate the value of A, which is the amount of money one would receive at the end of every n period if one invested P at an interest rate of i. It is also used to calculate loan payments where P is the principal amount.

Example 16.3

Loan. What are the monthly payments on a $100,000 loan with a term of 360 months (30 years) at an interest rate of 1% per month compounded monthly?

....................(16.4)

There are a number of other specialized equations, but they are of limited use in today’s era of fast computers. See Thompson and Wright,^[1] Chap. 2, for examples.

Annual vs. Monthly Interest Rates

Interest rates are normally expressed on an annual basis or per annum. As the previous equations show, when working with monthly cash flows, it is necessary to convert the annual interest rate to a monthly interest rate. There are two ways to do this: divide the annual interest rate by 12, or calculate the equivalent effective monthly interest rate. These two methods will result in different answers.

Consumer lending groups use the divide-by-12 method to comply with the Truth-in-Lending Act. The Truth-in-Lending Act required disclosures from lenders, which include, among other things, the note interest rate, any points or origination costs, and most lender imposed fees, such as underwriting and processing fees. These fees are all rolled into the calculation of an annual percentage rate (APR) for the loan. As the name implies, this is an annual rate. Because most consumer loans are paid on a monthly basis, the monthly rate, used in Eq. 16.3 to determine the monthly payments, is obtained by dividing the annual rate by 12. Fig. 16.4 from the Federal Reserve website shows some sample monthly payments. The next example illustrates the calculation.

Fig. 16.4 – Comparison of monthly payments at different interest rates and terms from www.federalreserve.gov/pubs/consumerhdbk/cost.htm.

Example 16.4
Monthly Payment Calculation.

Loan principal = $6,000. APR = 15%. Monthly interest rate = 15%/12 = 1.25%/month. Loan term = 4 years (48 months).

The effective-monthly-rate method, although more complicated to calculate, has some advantages as discussed later in Sec. 16.5. In this method, the annual interest rate is converted to a monthly rate, which, when compounded 12 times, results in the annual interest rate. The derivation of the effective monthly interest rate begins with the relationship (1+i) = (1+i_m)¹².

Rearranging this equation gives

....................(16.5)

The divide-by-12 method leads to a higher monthly payment than the effective-monthly-rate method, as shown in the next example.

Example 16.5

Monthly Payments With Effective Monthly Interest Rates Using the data from Example Four, the effective monthly interest rate is calculated as

The loan payment is then calculated.

Key Economic Parameters

When the purpose of an economic analysis is to help make a decision, there are several key managerial indicators or economic parameters that are considered. Although there are many parameters that can be considered (see Thompson and Wright,^[1] Chap. 3), the most common decision criteria are net present value, internal rate of return, and profit-to-investment ratio (both discounted and undiscounted). Each of these criteria is discussed next.

Net Present Value

Net present value is the sum of the individual monthly or yearly net cash flows after they have been discounted with Eq. 16.2. In Table 16.2, the three columns labeled "Discounted Net Cash Flow" show this calculation at annual discount rates of 10, 20, and 34.3%. The net present values (NPV) at these discount rates are $99,368, $51,950, and $0, respectively. In this table, the NPV were calculated on a monthly basis using effective-monthly interest rates converted from annual rates with Eq. 16.5.

After the discounting method has been specified, there is still the question of what discount rate to use. The author recommends the company’s average investment opportunity rate (see Thompson and Wright,^[1] pages 3-7 and 3-8 and Newendorp and Schuyler,^[5] pages 9 through 12). The average investment opportunity rate is the interest rate that represents, on average, the return of the future investment opportunities available to the company. This is the rate at which the treasury will grow. An alternative interest rate is the weighted average cost of capital (WACC). This is an interest rate that, as the name indicates, is the average of the cost of each source of financing weighted by the fraction of the total financing that source represents. Sources of financing include debt, which has an explicit interest rate associated with it, and equity, which has an implicit cost associated with attracting and retaining investors. The average investment opportunity rate and the weighted average cost of capital are often very similar to each other and often much lower than the typical "hurdle rates" used in the industry.

The use of high discount rates to account for risk is not recommended. Much has been written about the fallacy of using high discount rates (see, for example, Capen^[6]). Later sections of this chapter deal with decisions under uncertainty.

The decision criterion using net present value is very simple. For project screening, all projects with a positive NPV at the company average investment opportunity rate are acceptable. If the projects with a positive NPV perform as projected, they will return more to the treasury than the average company project will return. In the case of mutually exclusive alternatives, where choosing one alternative precludes choosing another, the alternative with the highest NPV should be chosen. An example of mutually exclusive alternatives might be choosing between injecting CO₂ or high- pressure air as a secondary recovery method—only one or the other may be chosen, not both.

Internal Rate of Return

Internal rate of return (IRR) has been a popular managerial indicator since the 1950s, and it is still widely used today. IRR is defined as that interest rate which, when used in the calculation of NPV, causes the NPV to be zero. In Table 16.2 that interest rate is 2.488% per month or 34.30% per year. Notice that, once again, we are using the effective monthly interest rate and, therefore, must use Eq. 16.5 to convert to annual interest rate.

IRR can easily be used to screen projects. If the IRR is greater than the average investment opportunity rate, the project passes the screen. However, the unwary might be trapped in a situation where two mutually exclusive projects are being compared. Many evaluators have a tendency to think that the project with the larger IRR is the better project. This is not necessarily so. If IRR is used to compare two mutually exclusive projects, it is necessary to calculate the IRR on the incremental capital used for the project with the larger investment. Although this can lead to the correct decision, the procedure is tedious enough that it is easier to just compare NPVs at the average investment opportunity rate. Choosing the project with the higher NPV, at the average investment opportunity rate, leads to the same decision as calculating incremental IRR.

Under certain circumstances there may be more than one interest rate that will cause the NPV to be zero. This is referred to as multiple rates of return and occurs primarily in the evaluation of acceleration projects. As stated by Phillips,^[7] an acceleration project is "one in which money is invested, not necessarily to show a profit but to decrease the time required to obtain the income from a project. In fact, acceleration projects will generally result in a net loss." An example acceleration project might be a decision to downspace from 80 acres to 40 acres in a coalbed methane field. In this hypothetical case, virtually the same amount of gas is expected to be produced over a shorter time period, yet there is a large investment to drill the additional wells. When the infill project is evaluated on an incremental basis, the cash flow stream is negative then positive and then negative again, as shown in Table 16.3. On an undiscounted basis, the project loses money. The only justification for doing the project (in this hypothetical case) is to "accelerate" the cash flows, so the company can invest them elsewhere.

Table 16.3

The number of sign changes in the cash flow stream is the number of potential values for IRR. In Table 16.3, there are two sign changes (negative to positive in year one and positive to negative in year six), so there are two values of IRR.

The key to evaluating acceleration projects is again to examine the NPV of the project at the company average investment opportunity rate. The rationale for accelerating the cash flows is to invest them elsewhere, so you must know what you are going to do with them (on average). If the NPV of the project is positive at the company average investment opportunity rate, then you can profitably invest the accelerated cash flows elsewhere. If the NPV is negative, you are better off not accelerating the cash flows. Table 16.3 also illustrates how sensitive some of these projects can be to the company average investment opportunity rate. This project is only profitable at interest rates between 1.2 and 12.9%, as shown in columns E and F. The discounted net cash flow is zero at those interest rates. You would have to be very sure of the numbers to invest $2,500,000 to return $15,374 more than average projects.

Several years ago, a spirited discussion appeared in the literature sparked by E.L. Dougherty’s paper on discounted cash flow rate of return.^[8] This discussion presents a good analysis of different points of view.

Discounted Profit-to-Investment Ratio

Discounted profit-to-investment ratio has been touted by R.D. Seba^[9] as "the only investment selection criterion you will ever need," in his paper of the same name. This paper and the various discussions of it present a good discussion of the method. Mechanically, profit-to-investment ratio is calculated by dividing the sum of either the net operating income or the net cash flow from a project by the sum of the investments. If undiscounted numbers are used, the result is an undiscounted profit-to-investment ratio; if discounted numbers are used, the result is a discounted profit-to-investment ratio. If net operating income is used in the numerator, a value of 1.0 is a breakeven value where the investment is just recovered. If net cash flow is used in the numerator, a value of 0.0 is a breakeven value. Either definition is appropriate for the numerator, as long as it is clearly stated which definition has been used.

Discounted profit-to-investment ratio at the company average investment opportunity rate is indeed a powerful selection and ranking tool, as stated by Seba. As a selection tool, all projects with a value greater than 1.0 (or 0.0) would be selected. In the presence of limited capital, the projects are ranked in decreasing order of discounted profit-to-investment ratio and selected until the capital available for investment is exhausted. This very simple tool results in the portfolio of projects that causes the treasury to grow at the fastest rate, if the projects perform as expected. Erdogan et al.^[10] pointed out that "this approach maximizes expected value but ignores risk. In fact, funding projects with the highest discounted P/I will tend to produce a high-risk portfolio." This is a valid criticism and is addressed at length in Sec. 16.8.

The example in Table 16.2 can be used to demonstrate the calculation of profit to investment ratio.

Example 16.6 Profit-to-Investment Ratio from Table 16.2.

Total undiscounted net operating income = $585,369. Total undiscounted investment = $425,000. Total undiscounted net cash flow = $160,371. Total investment discounted at 10% = $425,000 (because only one investment was made and that was at time 0). Total net cash flow discounted at 10% = $99,368. Total net operating income discounted at 10% = $99,368 + $425,000 = $524,368.

....................(16.6)

Alternatively,

....................(16.7)

Using discounted values,

....................(16.8)

Again, alternatively,

....................(16.9)

Recommended Practices for Economic Calculations

More than 70 different people have calculated the simple problem shown in Table 16.2 in three phases. Phase One consisted of 30 runs with different programs during the 1990s. In Phase Two, the problem was run by 12 vendors of commercially available economic evaluation software in late 1999 and early 2000. Phase Three included runs by various oil company and vendor personnel during late 2000 and early 2001. In each phase, the problem statement was identical, except for the effective and production dates, which were always January 1 and February 1, respectively, of the year when the case was run. The problem is an evaluation of a drilling prospect assuming a single "time 0" investment, exponential decline, and escalating prices and costs. The problem was originally designed to be solved by hand, so it has a five-year life.

There was a surprising diversity in the reported answers. For example, the NPV at 20% was expected to be $51,950, yet the answers received ranged from about $3,000 to almost $100,000. In fact, the problem was solved 74 different times, which resulted in 68 different answers. (More details are available in Wright and Thompson.^[11]) In 2000, a large project conducted by the Society of Petroleum Evaluation Engineers (SPEE) found that the differences arise from a combination of unstated assumptions; differences in interpretation of various parameters; different, but equally valid, treatment of factors such as discounting or escalation; and apparent misunderstanding of the problem statements. The results of that study indicated that there was a great need for standardization and communication regarding upstream economic calculation. To begin that communication process, the SPEE formed committees to draft recommended evaluation practices. Ten of those recommended evaluation practices were approved by the members attending the annual meeting of SPEE in June 2001 and are in the process of being approved by the SPEE membership as a whole. These recommended engineering practices (REP) address issues such as the elements of a report (REP 1) and how to discount cash flows (REP 5). The REPs are available on the SPEE website, which is SPEE. The SPEE REPs are not intended to be required practices but are suggested ways to present the problem when you do not have compelling reasons to do otherwise.

REP 5 on discounting is of particular interest in the context of this chapter. It is recommended that end-month discounting be used for calculations using monthly cash flows and that mid-year discounting be used for calculations using yearly cash flows. Additionally, it is recommended that the effective monthly interest rate, as defined by Eq. 16.4, is used. When monthly cash flows are discounted using the effective monthly interest rate, the results are very similar to that obtained by discounting annual numbers using mid-year discounting.

Risk Analysis for Oil and Gas Property Evaluation

In the previous discussions, we assumed to know the model parameters with certainty. This is clearly not the case, so some method of handling uncertainty is appropriate. There have been a number of textbooks written on the subject, but one of the best is Decision Analysis for Petroleum Exploration, originally written in 1975 by Paul Newendorp and updated in 2000 by Newendorp and John Schuyler. The following discussion briefly covers some of the topics contained in their book.

Risk and Uncertainty

When making a decision in the oil/gas business, we are seldom certain of the results of that decision. This is both the curse and the attraction of the industry. Great fortunes can be made or squandered on the basis of a single decision. Some authors use the terms "risk" and "uncertainty" interchangeably, and some authors make a great distinction between them. We shall use the term uncertainty to express the concept that we do not know the outcome of a decision when we make it. We shall use the term risk to mean that in any decision we make, there is a possibility of an unpleasant outcome—losing money in the context of this chapter. Further, we will assume rational decision making.

The practice of "risk analysis" or "decision analysis" is a way to analyze the potential results of decisions objectively and consistently. Risk analysis does not eliminate dry holes or even bankruptcy, but applied properly it helps keep you in the game. One of the most important aspects of "playing the game" is to try to make sure you are in a winning game. Risk analysis can provide the information to keep you from playing a known losing game, such as being on the wrong side of the roulette wheel in a gambling casino, but it may not help you much if you are unknowingly playing a losing game while thinking you have a chance of winning. Judgment (and luck) still count.

Expected Monetary Value Concept

Expected monetary value (EMV) is the foundation of risk analysis as described in this chapter. Newendorp and Schuyler^[5] (page 82) state that "the expected value concept is more nearly a strategy, or philosophy for consistent decision making than an absolute measure of profitability." In applying the strategy, the decision maker should be playing a winning game, should have sufficient money for repeated trials, and should apply the concept consistently over a large number of decisions. Numerically, expected value is the return on average given repeated trials. That is, over the long haul, we would obtain an average result equal to the expected value. When making a decision, each alternative has an expected value associated with it. Having made a decision, there are a number of potential outcomes. The expected value of a decision alternative is obtained by summing the product of the probability of occurrence of a potential outcome and the payoff for each potential outcome. This is done for each decision alternative, and we then choose the one with the highest expected monetary value.

As an example, assume you have 100 prospects to drill: 10 prospects contain oil worth $10 million (each), and 90 contain nothing but heartbreak and cost you $100,000 (each) to drill and abandon. All numbers are net present values at the company average opportunity investment rate and, therefore, include the costs to drill and produce the oil. The costs include additional development wells. If you drill all 100 prospects, you will have a present-day profit of 10 × $10 million less 90 × $100,000 or $1 million. This is certainly a winning game. [(Now, if we could run a three-dimensional (3D) seismic survey to highgrade the prospects and only drill 20 wells to get our $100 million of oil, we would be playing a winning game. That analysis is beyond the purview of this chapter but is a topic under the category of "decisions to purchase imperfect information."] We can divide the $1 million profit by the 100 wells we drilled and see that each well is worth $10,000 on average. Let us use expected value to investigate the drilling of one well.

Out of 100 possible wells, ten will result in field discoveries. Therefore, the probability of success is 10%. If we are successful in discovering a field, the conditional value of that success is $1 million. The probability of failure (dry hole) is 90%, and the conditional value of failure is –$100,000. We can analyze this example in a table such as Table 16.4. The expected value of the decision alternative is +$10,000, found at the bottom of column D. This is the same value we calculated (on average) assuming we drilled all 100 prospects. So what happens if our model truly represents nature and we drill the well? We probably drill a dry hole and lose $100,000. In fact, that will happen nine times out of ten (on average). This is where the repeated trials and consistent application previously mentioned come in. If we apply the same methodology a large number of times, the odds are in our favor, and we will prevail. We expect the odds to catch up with us somewhere around 30 trials.

Table 16.4

In this table, we are modeling what we expect to find in nature. That model can be as simple as shown in Table 16.4, or it can be excruciatingly complex with hundreds or even thousands of potential outcomes. However, simple or complex, some conditions must be met. First, the probabilities in column B must sum to exactly one. That means our model includes all possible outcomes or states of nature. This may be uncomfortable at first, but it is necessary. Second, each conditional value for an outcome must include all costs to achieve that outcome and all revenues from that outcome. Third, the net present values must be calculated using the company average opportunity rate. We are always comparing a decision alternative with the "do nothing" alternative. If we "do nothing," we are implicitly saying we are going to invest the money in average projects for the company, which will earn money at the company average opportunity rate. The expected monetary value (EMV) for that decision alternative is zero. In order for our analysis to make any sense at all, that rate must be used in calculating expected values.

Expected value theory assumes that the decision maker is impartial to money. This means that a gain of $10,000 brings the same amount of positive "utility" as a loss of $10,000 brings negative utility. If the gain of $1 million does not bring ten times as much pleasure as the loss of $100,000 brings pain, then the expected values should be expressed in a "currency" such as utility that linearizes the problem. Newendorp and Schuyler (Chap. 5) discuss utility theory as do others.^[12]^[13]^[14] In practice, the decision maker is usually impartial to money because the individual decisions are small, relative to the size of the treasury, or can get that way by joint venturing with other companies.

EMV provides a means to screen projects, compare two mutually exclusive projects, and rank projects in the presence of uncertainty. When screening projects, all projects with positive EMVs pass the screen. When comparing mutually exclusive projects, choose the one with the largest EMV. When ranking projects, rank them by the EMV to expected investment ratio.

Ranking projects by EMVs to expected-investment ratio suffers from the same criticism that discounted profit-to-investment ratio suffers. Namely, this ranking may lead to the highest aggregate NPV, but it also leads to the highest risk portfolio. It may be possible to choose a different mix of projects that will significantly reduce the overall risk without significantly reducing the overall EMV. That is the goal of the emerging field of "portfolio analysis," which is discussed in Sec. 16.8.

EMV calculations lend themselves to sensitivity analysis. It is a simple matter to change probabilities and/or payoffs and recalculate the results. One very enlightening graph is a plot of expected value vs. probability of success. For a two-outcome scenario (or a scenario that can be reduced to two outcomes), the relationship between EMV and probability of success is linear. You only need to know the NPV given failure and the NPV given success to prepare the plot. The plot for the example in Table 16.4 is shown in Fig. 16.5. Note that at a probability of success of 10%, the EMV is $10,000. As the probability of success increases, the EMV increases rapidly, reaching $450,000 per well at a 50% success ratio. The plot can be used to estimate the breakeven probability of success, which in this case is 9%. If the considered opinion of the explorationists is that the probability of success lies between 15 and 30%, the decision is easy. If the probability of success lies between 5 and 15%, more analysis may be indicated.

Fig. 16.5 – Example of the effect of probability of success on expected monetary value.

Decision Trees

Decision trees are useful tools to analyze a series of sequential decisions, although they can be used for single decisions as well. They are a way to graphically represent the principles discussed in the section about expected monetary value. Traditionally, decision trees are constructed with "time" flowing from left to right and are made up of "nodes" that are connected by "branches." There are three types of nodes: decision nodes represented by squares, chance nodes represented by circles, and terminal nodes represented by triangles. Fig. 16.6 illustrates the previous sample EMV problem in a decision tree format.

Fig. 16.6 – Example decision tree.

The first square node shows two possible decisions: drill a well, or drop the acreage. In a real-world example, we may have more options, such as release the property, bring in a partner, and others, but that just adds to the number of branches emanating from the decision node. If we drop the acreage, we arrive at a terminal node, and it will cost us nothing. If there were abandonment costs, they would be shown as negative numbers on that terminal node. The round chance node, if we drill, shows all the possible outcomes just like the decision table (Table 16.4). Notice that if we drill, the probabilities and payoffs are the same as those used in the EMV problem.

The probabilities attached to a chance node must sum to exactly 1.0. This means we believe we have modeled all the possible outcomes. If a two-outcome decision tree is too simple for your problem, as it probably should be, then you can add as many branches as you wish to the chance node. There are two options in assigning costs or payoffs to get to the terminal node. One is to only show the incremental costs (payoffs) to get from one node to another. The other is to show all the costs and payoffs from the root to the terminal node. The incremental method makes sensitivity runs easier, while the total cost/payoff method is often easier to explain. Done properly, both methods arrive at the same answer.

A decision tree is solved from right to left by "rolling" it back. The expected value of each chance node closest to the terminal nodes is determined just as shown in the expected value table (Table 16.4). The chance node may now be replaced with its expected value and the tree to the right "trimmed" to simplify the presentation. This procedure is repeated with the next line of chance nodes, if any, until a decision node is reached. At this node the decision rule is to make the decision with the highest EMV. Note that the highest EMV may still be negative but less negative than the alternatives. Again, the decision node can be replaced with the value of the highest EMV. The procedure is repeated until there is only one decision node remaining at the left side of the tree. The decision rule is to make the decision with the highest EMV.

The solved decision tree for the example is shown in Fig. 16.7. The chance node on the upper right, which had shown a 10% chance of a +$1,000,000 outcome and a 90% chance of a –$10,000 outcome has been replaced by the "certainty equivalent" of +$10,000. (0.1 times +1,000,000 plus 0.9 times –100,000 equals +10,000.) Notice that the rejected decision alternative "drop" has two slashes through its branch. This is the traditional method of showing a rejected alternative. Once a decision tree has been solved, it is a simple matter to run sensitivity analyses. If we wish to evaluate the difference a $150,000 dry hole cost has on our example, we could revise the terminal value for dry hole and resolve the decision tree. In that case, the expected value of the decision alternative drill is –$35,000, and we should drop the lease without drilling. Fig. 16.8 shows the solved decision tree for that case. But what if the probability of success is really 20% with a $150,000 dry-hole cost? Fig. 16.9 shows that the decision is now to drill with an EMV of $80,000. This analysis can be continued as long as desired, and it is possible to plot sensitivities to any of the parameters.

Fig. 16.7 – Solved decision tree.
Fig. 16.8 – Solved decision tree with dry-hole cost of $150,000.
Fig. 16.9 – Solved decision tree with dry-hole cost of $150,000 and a probability of success of 20%.

Decision trees are quite useful for analyzing sequential decisions because all the possible courses of action can be laid out with probabilities and payoffs before the first decision is made. As the project proceeds, the tree can be modified to remove the decisions that have already been made and update the remaining decisions, probabilities, and payoffs. They can be as simple or as complex as desired and may even use Monte Carlo simulation to assign values to the terminal points. Additional discussions can be found in Newendorp and Schulyer (Chap. 4), as well as many others.^[15]^[16]

Monte Carlo Simulation

Monte Carlo Simulation is a calculation technique that uses distributions for uncertain input variables rather than single point estimates. It results in a distribution of potential outcomes with associated probabilities and has a number of advantages:

The possibility to describe uncertainty in the input variables using distribution of possible values rather than a single average or most likely value.
All of the parameters that are not known with certainty can be correctly modeled (limited by our ability to understand the distribution of the input values).
It can be used to model any system or process that can be described with mathematical relationships.
The model can be very simple or extremely complex as necessary. With the current computer power available, it is possible to run very complex models in minutes or hours.
Any type of distribution can be used to describe a particular input variable.
It allows for the blending of the expertise of the entire company. Geologists can describe the uncertainty of geological parameters. Engineers can describe the uncertainty of engineering parameters.
The cost of doing a simulation model is typically small, especially in comparison to a pilot project.
The method lends itself to sensitivity analysis. It is easy to change one or more of the parameters and rerun the simulation.

There are also a number of pitfalls in Monte Carlo Simulation:

It requires an attempt at quantifying uncertainty. Depending upon the corporate culture, it may be very difficult to accept that there is more than one possible answer.
It requires expertise to build the computer model and debug it. This is true even with current available spreadsheet models.
Like any computer calculation, it is subject to the garbage in/garbage out (GIGO) problem. If the input data distributions are wrong, the answers are almost sure to be wrong.
There is a tendency to believe the answers because of the sophistication of the calculation technique.
It is sometimes difficult to convey the results of the simulation to management in a manner they can understand. Often, management is looking for the answer rather than a range of answers with associated probabilities of occurrence.

A Monte Carlo simulation study consists of the following steps:

Determine the objectives of the study. This is often a simple step with an answer such as "determine remaining reserves" or "determine whether or not to take this deal." Occasionally, however, this can lead to intense discussions about the goals of the corporation and "what we are really about." Normally, we try to use simulation to help make a decision. Keep the objective in mind as you design the study.
Determine the mathematical relationships between variables. Again, sometimes this can be very simple and sometimes the system being simulated can be extremely complex.
Separate variables that are known with certainty and variables that are subject to uncertainty. There is a real tendency to say, "We know so little about this variable that we can’t come up with a distribution, so let us just assume it is known." An example might be a price forecast where there is so much uncertainty that we just use the current oil price and hold it constant. Of course, this is totally contrary to what we are trying to do. Important uncertain variables should always be simulated with a distribution. It is possible to simplify the problem and still get a valid result. At this stage, it would be appropriate to perform an analysis to see what variables have the greatest effect on the outcome. If a variable has little effect on the outcome or decision, even if it is uncertain, then a cost-effective solution can be obtained by fixing the value of that variable and treating it as known.
Determine whether variables are independent or if partial (or full) dependencies or correlations exist. In some cases variables are independent of each other. For instance, in calculating volumetrics, the area of the trap almost never depends on the value of oil formation volume factor. However, the average water saturation often depends on the average porosity, and that relationship should probably be taken into account. A great deal of thought should be put into this stage of modeling. An example of dependencies between offshore gas fields is contained in the paper by van Elk et al.^[17]
Choose distribution types and parameters for the independent variables. The types of distribution, which can be used in currently available simulation programs, are virtually unlimited. Examples include uniform distributions where any value of the input variable has the same chance of occurring as any other value, and triangular distributions where values close to the mode, or the most likely value, are much more likely to occur. Although the choice of the type of distribution may have a significant effect on the outcome, we often do not have sufficient data to discern a log-normal distribution from a triangular distribution with any degree of certainty. It is recommended that the distribution shape be chosen from theoretical considerations and the data we have used to determine parameters for that type of distribution.
Model total or partial dependencies. Dependencies and correlations can be modeled in a number of ways. One of the most popular methods uses the bounding envelope method, as described in Murtha,^[18] and Newendorp and Schuyler (pages 436 through 457). This has the advantage of allowing the user to fully control the type of dependency at the expense of some programming effort. The other method uses the rank correlation coefficient available in commercially available software programs. (See Murtha, page 89.)
Perform the simulation. A number of software programs are available to perform the calculations. Some are add-ins to spreadsheet programs, and some are stand-alone programs. A simple example, shown later in this chapter, steps through the calculation method. Depending on the complexity of the problem and the effect of low probability events, it might be necessary to run as few as 1,000 passes or as many as 1,000,000 passes. Typically, 5,000 or 10,000 passes is sufficient.
Calculate the results and display the answers. The results are usually presented as tables and graphs such as histograms and cumulative frequency curves. Values of interest, such as the mean or expected value of the outcome, are reported. In economic evaluation, the user is also usually interested in the probability that the project will lose money or exceed a certain minimum rate-of-return, so these values are examined.
Perform a sensitivity analysis. Once the model has been set up and verified, it is relatively easy to alter some of the critical assumptions and see what effect that has on the outcomes. Assumptions can be examined in greater detail, especially assumptions that have the greatest effect and might cause a different decision to be made.

A simple example of the use of Monte Carlo simulation would be to use the hyperbolic decline equation to calculate remaining reserves. The equation is

....................(16.10)

For the purposes of illustration, assume we have the deterministic or single-value estimates of the parameters: q_i = 100 B/D, q_el = 5 B/D, D_i = 0.6/year, and n = 0.3.

With these values the estimated remaining reserves (ERR) are calculated to be 76,231 bbl. However, if we do not know all the values with certainty, we can use Monte Carlo simulation to calculate the expected value for ERR, as well as a range of values and their associated probabilities. Let us assume that the uncertainty in all the values can be represented by triangular distributions and that the mean of the triangular distributions are the previously stated values. (The mean of a triangular distribution is the sum of the minimum, mode, and maximum values divided by three.) Let us also assume that the variables are independent; for instance, q el is not a function of q_i. An example set of distribution parameters is shown in Table 16.5. When this set of distributions is run through a Monte Carlo Simulator 10,000 times, the results are shown in Table 16.6.

Table 16.5
Table 16.6

There are several important results. Note first that the mean value of ERR from the Monte Carlo Simulation (79,088 bbl) is not the same as the deterministic value (76,231 bbl), even though the means for the inputs to the Monte Carlo simulation were the deterministic values. Also note that the value of ERR, using the "most likely" values for each variable (68,847 bbl), is also considerably different. These results occur because the nature of the equation for calculating ERR is complex; uncertain variables are raised to powers of other uncertain variables; and input distributions are not symmetrical.

The minimum and maximum values calculated for ERR are not particularly meaningful because they can vary considerably with the number of simulation passes. However, note that 90% of the values lie between 57,027 bbl and 111,288 bbl. This is a very wide range and gives an indication of the magnitude of uncertainty. Table 16.7 contains the complete distribution of ERR as calculated in this simulation. If another set of calculations was run, the 0 and 100% numbers could change considerably, but the numbers near the center of the distribution would not change significantly. A second simulation run was made, as shown in Table 16.7. In the "Pass 2" column, the median (50%) values vary by less than 0.2% and, in this case, even the 0 and 100% values do not change significantly from those in the "Pass 1" column.

Table 16.7

If one were to decide to use the 10% value as the official value for remaining reserves, that value would be 59,903 bbl, which is 21% less than the value calculated deterministically. (The presentation of cumulative frequency shown here is a percentage less than presentation, which is quite common. If a percent greater than presentation is used, the table will show that 90% of the calculated values are greater than 59,903 bbl.) A graph of the data in the cumulative frequency table is shown in Fig. 16.10.

Fig. 16.10 – Cumulative frequency of example estimated recoverable reserves.

Fig. 16.10 and its associated table are very common outputs from a Monte Carlo simulation. Another common output is a histogram in which the relative frequency of a particular range of outcomes is plotted, as shown in Fig. 16.11. The values plotted on the x-axis are the midpoints of the bars. The most common result of the calculation is a value between 75,000 and 85,000 bbl, which occurs about 26% of the time.

Fig. 16.11 – Histogram of example estimated recoverable reserves.

The sensitivity of the results to the various input parameters is often presented in one form of a "Tornado" diagram, such as Fig. 16.12. In these diagrams, the input parameter with the largest regression correlation coefficient is plotted at the top of the figure, and the other input parameters are plotted at the bottom in descending order; thus, resulting in a tornado shape. This same information can also be presented in tabular form, as shown in Table 16.8.

Fig. 16.12 – Tornado diagram for estimated recoverable reserves.
Table 16.8

The regression correlation coefficient, reported by the commonly available Monte Carlo simulation software, is simply the square root of the r² value from a linear fit of ERR to each of the input variables. In Table 16.8, n has a correlation coefficient of +0.73. The positive correlation coefficient means as n increases, ERR increases. The square of the correlation coefficient (0.53) means that 53 percent of the variability of ERR can be explained by the variation in n. Fig. 16.13 is a crossplot of ERR and n. The linear least squares fit of the data has an r² of 0.53, which results in an r of 0.73, as shown in Fig. 16.12 and Table 16.8. Similarly, D_i has a large effect on ERR, as shown in Table 16.8. However, because the correlation coefficient is negative, increases in D_i result in decreases in ERR. The estimate of q_el has little effect on the remaining reserves for this example. Only 2% (–0.146²) of the variability in ERR is explained by the variability in q_el.

Fig. 16.13 – Correlation of estimated recoverable reserves (ERR) with hyperbolic exponent, b.

Interpretation of Results

One of the more important numbers is the mean or expected value of the distribution. We can make a decision with this number alone, just as we have done with EMV and decision trees. However, much more information is available, such as the probability of the project losing money (assuming EMV was calculated in the simulation) and the chances of the project making a large amount of money. Ideally, the entire cumulative frequency graph should be presented and compared with other projects, so the decision maker can see the full spectrum of anticipated possibilities.

The Next Frontier

There are two methods of advanced decision analysis that are slowly making their way into the petroleum project evaluation process. One of these is known as portfolio analysis, and the other is the real options analysis. The portfolio analysis quantifies the effect of interactions between projects, and the real options analysis attempts to value the fact that a company has several options in developing projects, such as the ability to abandon a project early or defer a decision to make an investment until the financial climate is more beneficial.

Portfolio Analysis

Portfolio analysis is based on the Nobel Prize-winning work of Harry Markowitz in the early 1950s^[20]^[21] in which he showed that the variance in results from a portfolio of stocks could be reduced by choosing stocks with a negative correlation. If two stocks are correlated negatively, when one stock is down the other stock will be up, and the portfolio will grow with very few wild swings. This concept has been introduced into the petroleum literature by several authors^[22]^[23]^[24]^[25]^[26]^[27]^[28]^[29]^[30] with some modifications. The following paragraphs from Brashear, Becker, and Faulder^[23] give an overview of the methodology. (Superscripts denoting the references have been added by the current author.)

Harry Markowitz (1957)^[21] demonstrated in the stock market that risk and return are usually correlated. Achieving higher yields generally necessitates taking greater risks. Further, he pointed out the risk-reducing effects of diversification were reduced if multiple investments are positively correlated but amplified if the investments are negatively correlated. He posited that a rational investor would seek the mix of investments (portfolios) for which no other combination would have a higher return without increased risk or lower risk without loss of return. He coined the phrase "efficient frontier" for the set of portfolios that meets these conditions. The choice of a single portfolio along this frontier depends on the decision-maker’s tolerance for risk.

David Hertz (1968)^[25] extended these concepts from investments in financial assets to investments in "real" assets. An efficient frontier could be composed of capital projects that reflected both economic value and risk (measured by standard deviation). Newendorp^[26] recognized Hertz’s work and speculated about a 2D display (expected value and expected loss) to illuminate individual project selections but did not propose a full portfolio optimization approach. Ball (1983)^[27] applied Hertz’s insights specifically to the upstream oil business. This idea was later proposed also by Hightower and David (1991)^[28] and Edwards and Hewitt (1993)^[29] and updated by Howell et al. (1998),^[24] Ball and Savage (1999)^[30] and Brashear et al. (1999).^[31]^[32]^[33]

While these conceptual advances were being made, increases in the speed of commonly available computers and the efficiency of the required solution algorithms have made the approach practical at field, division, and corporate levels.

The method is conceptually simple but computationally complex. The algorithm is a mathematical programming solution that evaluates all combinations of investments that yield a specific "target" expected value to define the one combination (portfolio) with the lowest risk at a given capital constraint. Other constraints can be added. The process is repeated for all other specific target values, each time finding the specific portfolio with the lowest risk. The locus of the minimum-risk points, the efficient frontier, is the set of all portfolios that satisfies the criterion that no increase in value is possible (given the constraints) without greater risk and no reduction in risk is possible without loss in value. Other algorithms find the maximum value at each risk level; either way the result is the same.

Fig. 16.14 is an example of the results of a portfolio analysis presented as a graph of reward vs. risk showing the efficient frontier. In some references, the axes are reversed. Reward is often represented by expected monetary value (EMV), while risk has several possible definitions. Markowitz used variance of the expected portfolio return as a metric for risk (or its square root, standard deviation). The petroleum industry is more concerned about downside risk, so oftentimes, semistandard deviation (the downside) or mean loss is used as a measure of risk. Whatever the metrics, the objective is to select a portfolio of projects that maximizes reward for an acceptable level of risk. A quantitative example is used to demonstrate the methodology.

Fig. 16.14 – Reward vs. risk diagram showing the efficient frontier.

Table 16.9 contains the parameters for estimated distributions of net present value (NPV) for nine different projects (labeled A through I), along with the present value cost of each project and the ratio of EMV to investment. Net present value is the result of one possible outcome, while EMV is the expected value or average of all of the possible outcomes. In this example, the NPV is assumed to be normally distributed for each project. Using the ranking criterion from Sec. 16.6.2, the projects are ranked in order of decreasing EMV to investment ratio and chosen until the budget is exhausted. For a budget of $2,500 (which applies throughout the examples that follow), the project mix is I, D, G, and H, which results in an expected value of $1,591. Correlation between the projects, such as those caused by oil price, rig rates, geological concepts, or pipeline constraints, is ignored with this ranking method. Later in this section, correlations are included. While this portfolio has the highest EMV, it also has the highest risk, whether risk is defined as variance or as mean loss. For the purposes of illustration, the projects are assumed to be positively correlated, as shown in Table 16.10. If risk is defined as variance or standard deviation, an analytical solution can be used to calculate the variance of the correlated projects.^[29] When the efficient frontier is calculated with the methods shown by Winston,^[34] the result is as shown inFig. 16.15. The standard deviation of the expected returns can be reduced from $681 to $300, if the decision maker is willing to accept a reduction in expected results from $1,591 to $1,200. This might be a good trade because the company can still capture 75% (1,200/1,591) of the expected value while reducing its exposure to variance by 56% (381/681). Deviations about the mean for net present value at various levels of risk (defined as standard deviation about EMV) are shown in Fig. 16.16. There is about a 16% chance (–1 standard deviation) that the NPV will be less than approximately $900 for risk levels from $300 to $681. Notice that the less risky portfolios are very unlikely (

<

2.3%) to return less than about $575 (–2 standard deviations), while the most aggressive portfolio could return less than $228. Of course, there is a corresponding decrease in the potential upside if a more conservative portfolio is chosen.

Fig. 16.15 – Efficient frontier for example projects with standard deviation as risk for a $2,500 budget.
Fig. 16.16 – Ranges of net present value for various levels of risk and a $2,500 budget.
Table 16.9
Table 16.10

The optimal portfolio for each point on the efficient frontier can be calculated. Fig. 16.17 graphically illustrates the project mix for several points on the efficient frontier. At an expected value of $1,200 (standard deviation of $300), the project mix is 7% of A, 100% of D, 77.7% of E, 100% of F, 15.3% of G, and 100% of I. Project H, selected using EMV/investment, is not selected at all, while 100% of F was taken in which only two projects were ranked lower using EMV/investment. This is because F is almost a sure thing with a standard deviation of 22 about a mean of 147, while the return on H is quite uncertain with a standard deviation that exceeds its expected value.

Fig. 16.17 – Project mix on the efficient frontier using the example correlation matrix and a $2,500 budget showing the working interest invested in Project A, Project B, etc. at each value of expected monetary value.

This analysis assumes that a continuous range of interests from 0 to 100% is available in each project. For those instances in which the available interests are available only in increments such as 15, 25, 35, 50, 75, or 100%, the same methodology can be applied. However, the resulting efficient frontier will be jagged and discontinuous.

The effect of correlation between the projects is shown in Fig. 16.18. The highest EMV is independent of the correlation between the projects, but the variance (or standard deviation) for a given NPV is quite dependent on the correlation. If all projects are perfectly correlated (+1), the standard deviation of the NPV at maximum EMV is $950. If the projects are totally independent of each other (unlikely in the oil industry because of price, if nothing else), the standard deviation of NPV at maximum EMV is $519. If the projects are all negatively correlated with correlation coefficients of –0.1, the standard deviation of NPV at maximum EMV is reduced to $454. This illustrates the power of finding projects that are not correlated with each other or that have a negative correlation.

Fig. 16.18 – The effect of different correlation matrices on the efficient frontier and a $2,500 budget.

If the metric of risk is mean loss rather than variance, the calculations become more tedious. Ball and Savage^[30] discuss the methodology and give an Internet link to a spreadsheet with a sample calculation. When their methodology is applied to the example (with the example correlation matrix), the results are as shown in Fig. 16.19. Again, the portfolio with the maximum EMV of $1,591 has the highest risk. In this case, however, the decision maker can reduce the risk from $277 to $48 (83% reduction) with only a 25% reduction in EMV. The project selection, using mean loss as a metric of risk, is somewhat different. Now, we would select 100% of D, 85% of E, 100% of F, 15% of G, and 100% of I. Metrics other than EMV can be managed as shown by Howell et al. They give an example of a "generic" E&P portfolio in which the metrics include earnings, production, net cash flow, and reserves.

Fig. 16.19 – The efficient frontier in which risk is defined as expected mean loss ($2,500 budget).

A portfolio analysis has considerable value, but it is not easy to implement. Not only do we have to establish parameters describing the uncertainty on a project-by-project basis, we also have to determine the correlations between the various projects, which requires considerable skill. On a positive note, the computer capabilities currently available can certainly solve the problems once they are formulated. As more decision makers become educated in the methods, it is expected that portfolio analysis will become more common.

Real Option

In the last quarter-century, financial options such as "calls" and "puts" on publicly traded stocks have become an integral part of managing stock portfolios. The seminal work on financial options was done by Black and Scholes,^[35] published in 1973, and Merton,^[36] also published in 1973. Merton and Scholes shared the 1997 Nobel Prize in economics for their work. Black, Scholes, and Merton all worked on attempting to determine the value of an option. In recent years, the concepts of valuing options have been expanded from financial options to what are called "real" options in project evaluation.

Financial options include "calls" in which the owner of the option has the right, but not the obligation (thus, an option), to purchase a stock at a specified strike price. If the option can only be exercised at the end of a specified period of time, the option is referred to as a European option. If the owner of the option can exercise the option at any time up to the expiration date, the option is referred to as an American option. A financial put option is the right, but not the obligation, to sell a stock at a specified strike price. Again, there can be European put options that are exercisable only on a specific date or American put options that can be exercised anytime prior to the expiration date.

The proponents of real options valuation (ROV) or real options analysis (ROA) argue, for example, that "option pricing methods are superior to traditional DCF (discounted cash flow) approaches because they explicitly capture the value of flexibility."^[37] Copeland and Antikarov^[38] even go so far as to assert, "...the net present value technique systematically undervalues everything because it fails to capture the value of flexibility." There is a certain amount of irony in comparing the assertions of ROV proponents in which traditional methods undervalue everything with the assertions of portfolio analysis proponents in which "conventional treatments of uncertainty contribute to overestimation of returns..." (See Brashear, Becker, and Faulder,^[23] page 21.)

There are many types of real options that can be modeled as calls and/or puts or combinations thereof. In the oil/gas production business, the option to develop a field is similar to a call option. The producer has the option to invest the development costs and receive the value of the reserves. An example of a put option is the case in which the producer has the ability to abandon or sell the property. A property sale differs from a stock put in that the price of the sale might be unknown, while the exercise price of a stock put is usually known with certainty. There are a number of other types of real options. Trigeorgis^[39] lists several types of real options including the option to defer investment; the option to default during staged construction; the option to expand; the option to contract; the option to shut down and restart operations; the option to abandon for salvage value; the option to switch use; and the corporate growth option. Copeland, Koller, and Murrin (pages 400 through 402) list similar real options along with compound options, which are options on options, and "rainbow" options in which there are multiple sources of uncertainty. As they state, the exploration and development of natural resources is an example of a compound rainbow option. All of these various types of options can be valued if several parameters are known. The simplest place to begin is with a call option.

The most famous equation in option valuation is the Black-Scholes equation for a European call option. The following equations use the algebraic symbols of Black and Scholes rather than the more modern symbols.

....................(16.11)

where N(d) is the cumulative normal density function and

....................(16.12)

and

....................(16.13)

In Eq. 16.11, w(x,t) is the value (at any time, t) of the call option on a stock with a current price of x. The strike or exercise price is c; the risk-free interest rate is r; the maturity date is t*; and v² is the variance rate of the return on the stock. The variance is one of the most important and interesting parts of the equation. Black and Scholes assume, among other things, that "the distribution of possible stock prices at the end of any finite interval is log-normal" and that the stock price on any day is independent of the price on the previous day. This assumption of a "random-walk" in stock price is premised on the existence of an efficient market in which the stock is fairly valued on any given day, and all the information available concerning the stock has been taken into account by the market.

While the Black-Scholes formula revolutionized the financial markets, it has had little direct application in the oil/gas business because the assumptions used in its development are not particularly appropriate for oil/gas properties. As Davis* explains, "real options are different from financial options because the exercise price (in both calls and puts) is not known with certainty; exercise is not instantaneous; and the stochastic process for the underlying asset is not the same as it is for financial options. All of these aspects make the calculation of real option value considerably more complicated than calculating the value of a financial option."

Lohrenz and Dickens^[40] present a real-world oil/gas example using the Black Scholes formula in their comparison of option theory and discounted cash flow methods for an actual field in the offshore Gulf of Mexico. They discuss many options that are available during the "lifetimes of searchable, developable, and producible oil/gas assets." In their analysis, they showed that the value of the development option (like a call option on a stock) increased dramatically (by a factor of greater than 3) as the variance in the oil/gas asset value increased from zero (perfect knowledge) to 1.0/year. They end their paper with the warning "...we should always temper results from uncertainty analyses [both option theory and decision trees] and their use with the understanding that the real world and its real uncertainties have not been captured—only modeled by necessarily flawed and incomplete practice and practitioners." Another good example of the Black-Scholes model is presented by Copeland and Antikarov^[38] (pages 106 through 110).

The mathematics and application of real options can quickly become very complex. The value of the asset underlying the option is assumed to vary in time in a stochastic manner. That means at least part of the price varies in a random and unpredictable fashion. Dixit and Pindyck^[41] discuss several potential mathematical models for the price of the asset beginning on page 59 of their book. The most common of these in financial options is the Wiener or Brownian motion process. In this process, the change in value from one period to the next is assumed to follow a normal or log-normal distribution. (The standard assumption for stock prices is that the change in price over time is log-normally distributed.) This process is one of the underlying assumptions in the Black-Scholes model. As Dixit and Pindyck point out, the process has some interesting properties—one being that all future values depend only on the current value and not on any historical value. In other words, there is no memory of past prices. One consequence of the assumptions in a Wiener process, as stated in Dixit and Pindyck (page 65), is that "the variance of the change in a Wiener process grows linearly with the time horizon" and that "over the long run its variance will go to infinity." This is problematical when applying this process to the value of oil/gas properties. There is certainly an upper limit to the value of an oil/gas property no matter what the time frame. Those who lived through the "boom" in the late 1970s will remember the commonly accepted forecasts showing average annual oil prices of $100/barrel or more before the end of the 21^st century.

Dixit and Pindyck discuss a number of other potential mathematical models including "Brownian motion with drift" (a process with an increasing or decreasing trend and randomness superimposed), "mean-reverting processes," and jump processes (sometimes called Poisson processes). When these processes are applied to value-an-option on an underlying asset of a second-order partial differential equation results. These equations, just like the diffusivity equation common in fluid flow in porous media, only have analytical solutions for certain simple boundary conditions (such as those used by Black and Scholes). The equations can be solved by a number of techniques including finite difference techniques, as those discussed by Trigeorgis (pages 305 through 320) or by a "binomial lattice" technique, as discussed by Copeland and Antikarov in Chap. 7 of their book where they present a spreadsheet model for a binomial lattice. Winston^[42]^[43] presents a number of examples of valuing options using the Black-Scholes method and simulation. Winston’s book contains a CD with spreadsheets for all examples. Trigeorgis (pages 320 through 329) presents a log-transformed binomial lattice approach and gives references to other approaches from polynomial approximation to numerical integration. Pickles and Smith^[44] also discuss the binomial lattice method and present a numerical example for producing oil/gas properties.

Paddock, Siegel, and Smith^[45] applied real option valuation techniques to 21 tracts in the federal lease sale number 62, held in 1980. This study is also used as an example for oil/gas in the text by Dixit and Pindyck (pages 396 through 403) and is cited by Trigeorgis as "the first empirical evidence that option values are better than DCF-based bids in valuing offshore oil leases."

Valuing a developed reserve, even at a particular time, is quite difficult. Paddock, Siegel, and Smith used the work of Gruy, Garb, and Wood^[46] to estimate developed reserve prices as one-third of crude oil prices. In practice, the value of developed reserves is highly dependent on the perceptions of future prices, as well as the level of operating costs and the fiscal terms. Johnston^[47] (pages 13 and 14) estimates that "proved, developed, producing reserves are worth from one-half to two-thirds of the wellhead price times the contractor’s take." The United States has a contractor’s take of about 50%, so producing reserves (in this case, working interest reserves) would be worth about one-fourth to one-third of the wellhead price using Johnston’s rule of thumb. If contractor’s take is lower, such as the approximately 10% contractor’s take in Venezuela, then, obviously, the working interest reserves would be worth much less. Data on actual transactions, as reported by Cornerstone Ventures, L.P.^[48] for the period of 1991 through 1998, indicate that the median price of U.S. proved developed producing reserves, in the ground and net of royalty, during that time period ranged from $4.08/bbl to $5.26/bbl, while the yearly average spot price of oil (WTI) ranged from $14.37/bbl to $22.20/bbl. Table 16.11, from the Cornerstone "Annual Reserves Report," shows the annual averages for WTI and the median price for oil-dominated transactions from 1991 to 1998. The ratio of the median price to the WTI price ranges from 19 to 28% with an average of about 25%. Paddock, Siegel, and Smith calculate a standard deviation for the "real (CPI deflated) refiner cost of imported crude oil" of 0.142/year. They then assumed that this standard deviation would apply to the change in value of the underlying asset (reserves in the ground). An analysis of the Cornerstone figures indicates that, for the period of 1991 through 1998, the standard deviation of the change in median acquisition price is 0.15/year on either a nominal basis or on a CPI adjusted basis. This is in remarkable agreement with that calculated by Paddock, Siegel, and Smith covering the period of 1974 through 1980. Pickles and Smith calculated a standard deviation of 0.22/year for the period of 1985 through 1989 based on quarterly median reserve prices as reported by Strevig and Associates, the predecessor to Cornerstone Ventures. These values for standard deviation would appear to be in good agreement, but one of the underlying assumptions is that property prices follow a Wiener or Brownian motion process.

Table 16.11

One of the more disconcerting aspects of this assumption is the discussion of the underlying process for oil prices as discussed by Dixit and Pindyck (pages 403 through 405). They discuss a "unit root test" to determine "whether a price series is mean reverting or is a random walk." In their words, "this is a weak test that for short time series (for example, 30 years or less) will often fail to reject the hypothesis of a random walk, even if the series is in fact mean reverting." Considering the different processes underlying the price of oil/gas such as the Texas Railroad Commission in the 1950s and 1960s, the rise of OPEC’s power in the 1970s, the Federal Power Commission’s regulation of natural gas prices—all overlying the fundamental supply and demand relationship, it is difficult to imagine a meaningful process lasting much longer than 30 years. Dixit and Pindyck report that Wey used a 100-year series for the real price of crude oil and found that oil prices are mean reverting and not a random walk. They report that Wey** shows "ignoring mean reversion can lead one to undervalue the reserve by 40 percent or more" when the development cost in $/bbl is one-half the mean developed reserve price, while it will have little effect when the development cost is close to the developed reserve price.

Chorn and Carr^[49] discuss option pricing principles and then apply those principles to the purchase of information. Their advice is to "purchase information that will impact the upcoming decisions, if the value increase justifies the cost of the information. Secondly, adhere rigorously to the converse, i.e., invest now or abandon the project if there is no information to be gained (or it’s [sic] expense is too great) that will significantly change the project’s outcome or impact the investment decision process." This is sound advice with or without real option analysis, but they show how real options can be used to value that information—a value which is often difficult to quantify.

Davidson^[50] presented an excellent paper on benefits and difficulties with real options. He states that "the primary contribution of ROA is to produce a frame shift. Instead of thinking about a project from a do-it or don’t-do-it frame, ROA promotes thinking from a what-are-all-the-possibilities frame. The frame shift leads to a richer assessment of the opportunity." He criticizes the approach by stating that "the ROA valuation methodology is not only inaccurate for E&P projects, it is needlessly complicated. The methodology leads to procedures and presentations that can inhibit insightful discussions for key assumptions and choices." He then presents a method of getting the benefits of both real options analysis and present value analysis. Hooper and Rutherford^[51] also discuss the benefits of real options in framing the problem and the questions.

Real options analysis is slowly working its way into the E&P business. The mathematics are daunting; the terminology is foreign; the underlying assumptions are shaky; and communicating the results in an easily understood manner is difficult, but the method does show promise. The breach between the theoreticians and the practitioners in the E&P business needs to be bridged. This breach is well illustrated by the following quote from Copeland and Antikarov’s practitioner’s guide^[38] (page 164) in which they discuss compound options. "Exploration and development for natural resources (oil, natural gas, gold, copper, and coal) have multiple phases. Oil, for example, has sonic testing [sic] (2D and 3D), drilling, and development via construction of refineries [sic], pipelines, and storage facilities." On the other hand, the frame shift mentioned by Davidson and Hooper and Rutherford is a real benefit. It remains to be seen how widely the process will be applied. In his book, Trigeorgis (page 375) lists ten points of future research. The first two (listed next) are very relevant.

Analyzing more actual case applications and tackling real-life implementation issues and problems in more practical detail.
Developing generic options-based user-friendly software packages with simulation capabilities that can handle multiple real options as a practical aid to corporate planners.

If these points are done, along with educating the managers, real options could become a useful tool.

* Personal communication with Graham Davis, Colorado School of Mines, Golden, Colorado (2002).
**

Wey, L.: "Effects of Mean-Reversion on the Valuation of Offshore Oil Reserves and Optimal Investment Rules," unpublished undergraduate thesis, MIT, Cambridge, Massachusetts (May 1993).

Nomenclature

A	=	constant periodic payment
b	=	hyperbolic exponent
c	=	strike or exercise price
d₁, d₂	=	intermediate variables
D_i	=	initial decline rate, 1/year
e	=	the base of natural logarithms, 2.718...
f	=	factor to cause the time units to cancel, 365 days/year
F	=	future lump sum of money
i	=	the periodic interest rate
n	=	the number of periods for interest calculations or the hyperbolic exponent for decline curve equations
N(d)	=	cumulative normal density function
P	=	present lump sum of money
q_el	=	rate at the economic limit, B/D [m³/d]
q_i	=	initial rate, B/D [m³/d]
r	=	risk free interest rate
R²	=	the square of the sample correlation coefficient
t*	=	maturity date
t	=	time
v²	=	variance rate of the return on the stock
w(x,t)	=	value of a European call option at any time, t
x	=	current price of the stock

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↑ Gruy, H.J., Garb, F.A., and Wood, J.W. 1982. Determining the Value of Oil and Gas in the Ground. World Oil 194 (4): 105.
↑ Johnston, D. 1994. International Petroleum Fiscal Systems and Production Sharing Contacts, 325. Tulsa, Oklahoma: PennWell Publishing.
↑ Cornerstone Ventures. 1999. Annual Reserves Report No. 5, Cornerstone Ventures, L.P., Irvine, California (26 February 1999).
↑ Chorn, L.G. and Carr, P.P. 1997. The Value of Purchasing Information to Reduce Risk in Capital Investment Projects. Presented at the SPE Hydrocarbon Economics and Evaluation Symposium, Dallas, 16-18 March. SPE-37948-MS. http://dx.doi.org/10.2118/37948-MS
↑ Davidson, L.B. 2001. Practical Issues in Using Risk-Based Decision Analysis. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 30 September-3 October. SPE-71417-MS. http://dx.doi.org/10.2118/71417-MS
↑ Hooper, H.T. III and Rutherford, S.R. 2001. Real Options and Probabilistic Economics: Bridging the Gap. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 30 September-3 October. SPE-71408-MS. http://dx.doi.org/10.2118/71408-MS

SI Metric Conversion Factors

acre	×	4.046 873	E + 03	=	m²
°API		141.5/(131.5 + °API)		=	g/cm³
bbl	×	1.589 873	E – 01	=	m³
Btu	×	1.055 056	E + 00	=	kJ

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