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Monte Carlo: An Alternate Approach to Efficient Frontier
Balancing portfolio risk and return with efficient frontier
By: Jason McVean
Petroleum companies must continually decide which projects to invest in from an array of possibilities. A risk analysis technique called efficient frontier is one way to make these decisions easier. This article will explore the traditional, but sometimes inappropriate efficient frontier approach to selecting portfolios, along with a Monte Carlo-based approach which is more applicable to the oil and gas industry.
Competitive edge, more value, maximum profitability, efficiency, and improved performance are increasingly common terms facing decision-makers in today’s petroleum industry. They all drive toward the same goal: earn as much as possible using as little capital as possible. While terms like these apply to all levels of the industry, near the top of the corporate ladder the term portfolio optimization comes into play.
For this discussion, a fictional company called Oily Projects Ltd. (OPL), must make decisions regarding the optimization of its petroleum portfolio. They have hundreds of opportunities in which to invest, but only enough capital and/or resources to pursue a fraction of these opportunities. How can they select the best portfolio of projects while balancing the constraints and goals of the company?
One way to make these decisions is to use the efficient frontier theory developed by Harry Markowitz. This theory is based on three major precepts:
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A theoretical "rational investor"
Markowitz asserts that a rational investor is not indifferent to risk. He will choose more value over less value, but will also prefer less risk over more risk.
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There is more than one optimal portfolio
It is sometimes possible to find a better expected return by accepting more risk, but this is not always acceptable.
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The portfolio as a whole is more optimal than its individual projects
Each project or investment must be considered in the context of what it contributes to the entire portfolio: an optimal portfolio of ten investments does not always contain the ten "best" investments.
Based on this understanding, Markowitz defines a portfolio as being efficient if two conditions are met: no other portfolio has a greater expected return with no more risk, and no other portfolio has less risk with no less expected return. If one or both of these conditions are not true, a portfolio is said to be inefficient. When all portfolios are plotted on a graph of value vs. risk, the efficient portfolios form on a line called the “efficient frontier.,” There are no viable portfolios above this line. (See Figure 1) Once all the efficient portfolios are plotted, OPL can make informed decisions regarding their portfolio development, balancing acceptable risk with the highest possible return.
It is possible to find this efficient frontier using the approach described by Markowitz, but an alternate, Monte Carlo-based approach may be more applicable to the intricacies and complications found in the petroleum industry. Both of these approaches, including their setup and results, will be examined.
The Traditional Approach
Markowitz outlines a mathematical technique for deriving the efficient frontier in his 1959 book Portfolio Selection: Efficient Diversification of Investments. It should be noted that the description of the efficient frontier was originally formed around a discussion of securities investments, but it can, at least in principle, be applied to the petroleum industry.
In many analysis/optimization techniques, there is as much or more work involved in collecting the required information for each project and converting it into a usable format as there is in performing the actual analysis. This is especially true for Markowitz’ technique.
Stated briefly, there are five steps that must be completed to determine the efficient frontier:
- The expected value of each project under consideration is estimated.
- The variance in the potential values of each project is estimated as a measurement of risk.
- The correlation between each project and every other project is estimated.
- The constraints that limit which portfolios are acceptable is expressed in the form of simple linear equations.
- Once all of this information has been collected, an analytical expression for the efficient frontier is determined.
A detailed explanation of this method is not appropriate for this discussion; however, the end result, after considerable linear algebra and matrix manipulation, is the efficient frontier line on a value vs. risk graph. Any point on this line has a corresponding equation that represents a portfolio of projects. As one moves up the efficient frontier line, the result is generally an increase in both risk and return. (See Figure 1)
The Monte Carlo Approach
It is possible, and generally desirable, to take a different approach to efficient frontier analysis in the petroleum industry. This approach has been encompassed in a new software application developed by Merak called "Efficient Frontier."
As is the case with the traditional approach, effort must be invested in the initial setup before the actual analysis can begin. The steps involved in the Monte Carlo technique are as follows:
- Probability distributions are assigned to uncertain variables (see the "Monte Carlo Simulation" section for a brief explanation of this technique).
- Monte Carlo calculations are performed to determine individual risk profiles.
- Risk profiles for each project are combined in a way that preserves their intrinsic correlation information.
- Portfolios are generated and plotted on the efficient frontier graph.
By randomly generating hundreds or thousands of portfolios from the set of possible projects, a different type of efficient frontier graph is created where each point represents one possible portfolio. After enough portfolios are plotted, an upper boundary, above which no portfolios are found, will begin to emerge. This is the efficient frontier. Unlike the traditional efficient frontier, there is no explicit line, but rather a collection of points representing individual portfolios. (See Figure 2.)
Setting up the Efficient Frontier Analysis
Before Oily Projects Ltd. can perform its portfolio optimization, it must invest a considerable amount of effort in the setup of the problem. However, as will be explained below, much of the setup effort required for the Monte Carlo approach may already be done to one degree or another. The setup required for the traditional approach, however, requires information that may be difficult or impossible for OPL to attain accurately.
The Traditional Setup
In order to perform Markowitz’ efficient frontier analysis, some key information must first be collected: the expected value of each opportunity; the risk of each opportunity as measured by the variance of the possible outcomes; and the correlation in performance between each project and all the other projects.
Where this information comes from is not always clear. Historical data can give some clues to future performance if the project has a history to consult. Comparison with similar projects may also yield some information. Individuals involved with the project can provide some information, but there may be more subjectivity and errors of opinion than is likely in the Monte Carlo method.
The correlation between a project’s performance and that of other projects may be the most difficult to ascertain. There is generally some degree of correlation between projects, if for no other reason than oil and gas prices affect virtually everything in the petroleum industry. However, this understanding is of little help when asked whether the correlation coefficient between OPL’s Texas Oil Well 39-B and M-Gulf 6-D is closer to 0.2 or 0.4. For a set of 100 potential projects, nearly 5,000 questions like this need to be answered. This may be an almost intractable problem, but the information is nevertheless required for a meaningful analysis.
The Monte Carlo Setup
In the Monte Carlo approach, a considerable number of Monte Carlo simulations must be performed to generate the risk profiles for the individual projects. Before this can be done, uncertain input variables in every project (such as operation costs, price and production) must be evaluated to determine if they are significant contributors to the uncertainty in the expected value of the project, and if they are, suitable probability distributions must be chosen.
This represents a significant amount of time and effort; fortunately, the workload can be shared by a number of people at OPL. The engineers or geophysicists that work most closely with the individual projects will be able to specify how most of the uncertain variables should be set up. The remaining variables, such as price, are usually specified at the company level. Much of this data is already routinely collected by OPL in an effort to manage risk, so the requirements of the Monte Carlo approach to efficient frontier analysis are less of an additional burden than the traditional method.
Once the Monte Carlo setup is complete, the calculations can be performed. This stage requires a great deal of brute force number crunching. For example, OPL has 100 potential projects from which their optimal portfolio is to be drawn, and have decided that 500 simulations are needed to achieve a sufficient level of accuracy from the Monte Carlo results. This means that 50,000 economic cases need to be calculated. This may require on the order of a day to do, depending on their complexity and on the speed of their computer; however, little human intervention is required so the calculations can be left to run more or less autonomously.
After the risk profiles have been generated, the creation of the efficient frontier graph takes virtually no time. With the Merak Efficient Frontier software application, 10,000 random portfolios can be generated and plotted on a graph in under a minute. This speed makes it possible for OPL to explore many possibilities using different value measures and constraints until a satisfactory result is found.
Method Evaluation
The two approaches described above have their own strengths and weaknesses, and deciding on which method to use may depend on the priorities of the individuals performing the analysis. However, based on the requirements of the petroleum industry, the Monte Carlo method provides a more robust analysis of petroleum projects.
Assumptions
The traditional approach uses variance as a measure of risk for the individual investment opportunities. The implicit assumption is that the risk profile for an opportunity is fully specified by its mean and variance. Essentially, all risk profiles are being approximated by a normal distribution. This may often be an acceptable approximation, but petroleum economics can be very complicated. For instance, the level of operation costs affects when most of OPL’s wells cease to be economically viable; the amount of oil produced affects the duration of the project; the duration of the project affects the operation costs; and the operation costs affect how much oil is produced by changing the economic limit. With these kinds of relationships between variables, risk profiles can be notably different from normal distributions. Using a normal distribution to approximate a bimodal distribution, or a lognormal distribution with a lower cut-off, will lead to inaccuracies. The use of an approximation for a risk profile means that the line resulting from the traditional efficient frontier analysis may not be perfectly accurate. In other words, the actual location on the efficient frontier graph of any given portfolio will be somewhat different from the predicted location given by the traditional approach. (See Figure 3.)
With the Monte Carlo approach, the complete risk profiles of OPL’s projects are used throughout the analysis, and the approximation used in the traditional approach is not required. As a result, there is no difficulty handling the unusual risk profiles that can result from petroleum economics.
Results
The result of the traditional Markowitz approach to efficient frontier analysis is a mathematical expression for the efficient frontier line. The portfolio that corresponds to any point on the line can be found from this result. Because this is an analytical solution, one can be assured that there are no portfolios above this line.
This certainty is not possible with the Monte Carlo approach. Each point on the graph is a random portfolio chosen from a near infinite number of possible portfolios (depending on the size of the project set). While a point may be the highest one on the graph for a given amount of risk, it is impossible to rule out that another unfound portfolio with more value exists. However, several thousand (again, depending on the size of the project set) randomly chosen portfolios is usually enough to be quite confident that a portfolio at the upper boundary of the graph cannot be dramatically improved.
It should also be noted that the continuous line resulting from the traditional approach implies that any point on the line corresponds to a valid portfolio. In practice, this is not likely to be the case. For example, it simply may not be possible for OPL to participate at the 12.7 percent level of Texas Oil Well 39-B and the 59.2 percent level of the M-Gulf 6-D project, even though this might be the efficient portfolio suggested by the traditional approach. This real-life limitation may mean that many of the points on the traditional efficient frontier line require adjustment to change them from theoretically optimal but invalid portfolios, to sub-optimal but valid portfolios. In contrast, the Monte Carlo efficient frontier graph contains only valid portfolios that have passed the constraints that OPL has set up, and therefore adjustments do not need to be made.
Flexibility
In the traditional approach, any constraints that limit what is to be considered a valid portfolio must be expressed in the form of simple linear equations. Constraints such as minimum and maximum capital expenditure, or minimum and maximum investment in particular projects can be specified in this format. However, it is not uncommon to face more complicated constraints that cannot be expressed in the required format. For instance, OPL may consider investing in an off-shore platform in a new field. For this project to be practical, they would want to drill at least three wells, but they may want as many as six wells. The constraint is, therefore, that if the platform project is completed, at least three associated wells must also be drilled. This type of logical rule constraint cannot be expressed in the form of a simple linear equation.
This is not a limitation of the Monte Carlo approach. Constraints of any complexity can be set up, and randomly generated portfolios can easily be checked against them. This allows a great deal more user interaction. Once the Monte Carlo simulations are complete, it is an easy matter to explore the effects of changing constraints or goals. For instance, OPL may have modeled several cases representing different developments of the same opportunity. The company can then apply a rule to the efficient frontier analysis stating that if one of these cases is included in a portfolio, the others must be excluded (as they represent the same case). Alternatively, OPL could explore the effects of removing all opportunities with a rate of return less than 10%, or demand a constraint forcing at least 30% of the total revenue of the portfolio to be gas revenue.
By using efficient frontier analysis, OPL has the information to make the best decisions regarding which portfolios in which to invest its limited capital. While the traditional approach does have its advantages, the Monte Carlo approach is better suited for the complicated decisions facing professionals in the petroleum industry.
Notes: Monte Carlo Simulation
Monte Carlo simulation is a technique that uses random numbers and repetitive calculations to statistically analyze problems that are not otherwise easily solvable. The economics involved in calculating the value of petroleum opportunities are often of this nature as there are many complications, like changing economic limits, that make it difficult to assess the effect of a change in an input variable without redoing the entire calculation from the start.
Since two projects with identical expected values can have completely different risk profiles, a single number that represents the best estimate for the value of an opportunity is often insufficient, as the uncertainty in this estimate must also be evaluated. Monte Carlo simulation is a convenient and accurate method of doing this. Instead of applying a single number to each input variable (e.g., gas price, oil volume, capital expenditure, operation costs, etc.) and receiving a single number back from the calculation engine (e.g., After Tax NPV, Rate of Return, Profit to Investment Ratio, etc.), a probability distribution is applied to each input variable. This probability distribution represents the uncertainty in the input variable. A random value is then drawn from each probability distribution and the output value measures are calculated. By applying this procedure repeatedly (perhaps thousands of iterations) and plotting a histogram of the output value measure, a risk profile for the project is built. The accuracy of this risk profile is dependent on how many simulations were performed; the more simulations, the more accurate the profile. (See Figure 4.)
Jason McVean is the Project Manager for Merak's Risk Team. He has two degrees in the field of astrophysics, and now applies his mathematical and analytical skills to the theory and design of Merak decision and risk analysis software.
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