MATH METHOD AIDS EXPLORATION RISK ANALYSES
Richard A. Box
British Gas
Houston
Combinatorial mathematics has generally not been used in the petroleum business-other than one oversimplification known as "Gambler's Ruin"-because its complexity made it impractical for humans and computers.
The present method is quite practical on mainframe computers, if it is possible to cluster the data as it is in most real life situations. The method calculates a histogram showing the distribution of financial outcomes possible from any portfolio of investments under consideration.
The user will find it similar to Monte Carlo techniques, but it is a complete solution rather than an approximation. Iterative adjustment of the portfolio ends when a histogram satisfying strategic goals is reached.
Thus, upper management gets away from the technical aspects of each prospect and back to the formulation of overall corporate goals.
Tradeoffs between short term and long term desires can be studied. Conflicts between various investor attitudes toward risk can be analyzed.
After upper management has agreed on an investment philosophy, an exploration manager can seek the mix of prospects necessary to achieve the goal (or can report the likelihood that the goals are unattainable). Artificial limits on risk and minimum prospect size are rendered unnecessary. It is possible to have a long-range plan and stick to it, to assign workloads efficiently amongst exploration staff.
Prospect generators are no longer required to judge how large an investment in each prospect is consistent with corporate goals. The chance that they will spend months developing a prospect only to have management dismiss it as "technically sound, but not what we need," is greatly reduced. Communication between management levels is greatly facilitated.
THE CHALLENGE
Professional people in the oil and gas industry have never had a good way of answering their investors (or upper management) who ask, "How should we distribute our spending between prospects of various risk in such a way that the total portfolio of investments is sound, given my company's goals?"
The most common theoretical answer given is some variant of a gambler's ruin analysis (Arps and Arps, 1974), which warns us to embark on a combination of ventures only if the probability that every well fails is acceptably small. A plan that is far too conservative for a particular company's objectives will always look excellent with respect to gambler's ruin, which only provides an upper limit on aggressiveness. Success is more than just avoiding instantaneous failure, however.
Experienced managers already know so much about "spreading the risk" that, in practice, a gambler's ruin analysis usually just approves each of the plans under consideration and cannot distinguish between them.
The most common practical solution is a "social darwinism" analysis: management goes out and studies the policies of successful firms of similar size, then mimics them, thinking that those surviving decades of competition deserve imitation. This method is of doubtful accuracy because mergers and serependipity and the like complicate the picture. At best, it results in a policy of equality with the competition, not superiority. Besides, lessons from the past may not apply very well to a future full of rollercoastering prices and other nasty surprises. What we need is a direct, simple way to calculate an answer to the root financial question so that we may recalculate when conditions change.
Monte Carlo techniques have been the best answer but are infrequently used because they are difficult to implement and understand.
Another possible approach would be to obtain the proposed drilling budget, with the economic analysis of each prospect therein, and use combinatorial mathematics to calculate an overall risk profile. Given the odds of several events individually, straightforward formulas exist to calculate the odds of various sequences of events (for example, combinatorial mathematics is what statisticians use to calculate the odds of a bridge hand containing exactly three kings). However, this method has not been judged useful in the oil and gas business because the number of possible combinations to be calculated can get unmanagably large.
Even a small independent might wish to consider drilling 200 wells during 5 years, a problem astronomically more involved than anything involving just 52 bridge cards; the analysis for a large active corporation is so vast that it defies human comprehension. Computers would seem to offer no help, because even if people were willing to accept the sizable memory requirements and enormous computation-times, computer memory (even double-precision) is insufficiently accurate: the roundoff errors exceed the numbers being studied.
THE SOLUTION
A new method is presented here of grouping wells by class, calculating a risk-profile for each class, then combining the classes.
The result is a histogram showing the likelihood of various outcomes for the company. The calculation time on modern mainframe computers for a 200 well case is less than 1 min. The requirement that the wells be sorted into classes is not a major hardship; in fact, the data are likely to be naturally clustered by play or basin or other geologically- or politically-relevant units.
The biggest single advantage of the portfolio-analysis method is that it focuses upper management on the long-term, strategic planning issues that are its rightful concern and which are so difficult to study by other methods. They need not get involved in whether a particular prospect or play is good per se but only with whether certain patterns of investment will get the company where it wants to go in the long run. They answer specific strategic questions such as, "Should we use a strategy with a 2.3% chance of bankrupting the company this year but with a 6.6% chance of tripling our reserves; or should we use a strategy with essentially no chance of bankruptcy or tripling?"
Having decided on strategic goals, they can leave it to middle management to find the types of petroleum prospects necessary to achieve the goals.
EXAMPLE: PRELIMINARIES
The RPA (risk portfolio analysis) method, in practice, is a trial-and-error search: One designs a series of possible budgets, and for each the program calculates the spectrum of possible results, expressing it as a histogram. One then weighs the advantages and disadvantages of each spectrum, chooses the one that best fits corporate goals, and concludes that the budget it represents is the best of those tried. The fact that the program executes in seconds makes the searching easy.
The examples shown below are what might be run by a small, fictitious independent oil and gas firm called "Freedom and Justice Petroleum," abbreviated "FJP" that wishes to design a budget to meet corporate objectives.
CORPORATE OBJECTIVES
The lion's share of FJP's revenue comes from one oil and gas field that will produce about 1,000,000 bbl of oil equivalent (BOE) in the next 2 years, then be abandoned.
Consequently, the company wishes to design a 2 year plan that will replace these reserves. The company is quite risk-averse.
After considerable discussion, FJP management has decided not to accept any 2 year plan that carried with it:
- More than a 2% chance of having no successes at all (this is the classic gambler's ruin, a small subset of RPA), or
- More than a 7% chance of getting less than 70% of its investment back (a loss of 30%, in other words), or
- Less than a 67% chance of breaking even, or
- Less than a 25%
chance of getting 150% of its investment back, or
- Less than a 4% chance of getting double its investment back.
FJP's version of the RPA program prints warnings whenever any of these objectives is not met, making it easier to reject faulty plans. One other quirk FJP has: it does its budgeting in BOE, rather than dollars, pesos, or yen; the program is dimensionless, so it accommodates this nicely.
EXAMPLE: RUNNING A RISK-PORTFOLIO ANALYSIS
TEST RUN
First, the exploration manager of FJP wishes to run a simple, abstract example through the program to understand how it works and check for accuracy. The input is shown in the first five columns of Fig. 1. The question the exploration manager asks the program amounts to, "If my uncle offered me the opportunity to pay $1 for each of two coin-flips, and promised me $2.10 for each win, what does the histogram of possible outcomes look like?"
The output, shown in Fig. 2, can be checked against what we know the correct answers must be.
We know that there is a one-quarter chance that we will lose both flips, getting nothing for our $2 investment, a return of 0%; a one-half chance that we will win one flip and lose the other, getting $2.10, a return of 105% (a 5% profit, in other words); and a one-quarter chance that we will win both flips, getting $4.20, a return of 210%. The program correctly calculates and tabulates these results, along with various statistics. Note that the outcome 105% goes into a bin with all outcomes from 100% to 109.999%, and so on.
This is done to conserve memory and time in the computer. Note also that this short cut makes the calculation of the mean subject to small errors, which tend to get smaller as the number of ventures (two in this case) gets larger. The manager ignores the warnings at the bottom, because this was just a drill.
FIRST BUDGET
The exploration manager now wishes to scrutinize the 2 year plan.
A list, well by well, of all planned drilling is made. For each well, the estimated chance of success, the amount (net) to be invested, and the (net) value assigned (most often this should be the number-of-times-investment-returned, although rate-of-return, or other measure of value, may be more appropriate in certain situations) are listed.
All values must be converted to a common unit (typically dollars, but in this case BOE. Wells with similar attributes are grouped and descriptive names assigned to each class (Fig. 3).
In the example, FJP always drills in its five favorite exploration plays, and this year has one field development project, so the exploration manager naturally creates six categories.
Because category 1 represents a 10% working interest in an expensive frontier play, while category 2 represents a 100% interest in an inexpensive frontier play, the manager chooses to keep the categories separate even though this analysis would run faster if they were merged.
Besides, some accuracy would be lost because the value ratings are not quite equal. Notice that it would be quite wrong to merge category 2 with category 3, because the chance of success numbers are not the same. All three variables-amount risked, chance, and value-must match for merging to be fully appropriate. Of these three, the value parameter is the least critical, but when in doubt, it is best not to combine categories.
The results, shown in Fig. 4, are disquieting, although they begin with good news. There is a 0.0008 chance (0.08%, in other words) of a gambler's ruin (bin #1, 0-9% of the investment returned) result, which is tremendously better than management specified under Rule #1.
It is nearly impossible for all the wells to be dry holes because there are so many wells (47) and because seven are so likely to hit (the development wells). The chart also shows a negligible (0.0031) chance of results between 10% and 19%. However, the values grow frightening thereafter. More than 4.5% of the time, results fall between 30% and 49%.
A look at the "running total" column shows that 16.75% of the results are worse than 69% return, violating management's Rule #2. Also, Rule #3 is violated, because there is a 36.02% chance of losing money. All the upside numbers are very good: the 13.24% chance of at least doubling the company (shown as a 86.76% chance of not doubling) is far better than is required by Rule #5.
What is wrong with this plan, causing two of the five rules to be violated? There are two possibilities, which may be happening together: either the ventures themselves are not profitable enough, or the amount of risk taking, the way they are combined into a budget or portfolio, is wrong.
Examination of Fig. 5, the histogram equivalent to Fig. 4, shows that it is a nearly bell-shaped curve, leaning to the left, as are most RPA histograms for the petroleum industry.
The statistics at the bottom of Fig. 4 bear consideration, also. After some contemplation, the exploration manager concludes that the ventures themselves are fine but that the plan as it stands has too much risk taking in it; it achieves more on the upside than management requires while risking too much on the downside. It needs to cluster more towards the middle.
What changes in the plan would correct this histogram? Qualitatively, obviously what ares needed are more conservative ventures and fewer risky ones. Quantitatively, we must find out by trial and error.
SECOND BUDGET
FJP is already committed to eight ventures: three offshore, two in the Charlemagne Play, and three in the Merlin Play. Besides, these are the most profitable opportunities they have, so the exploration manager would be very reluctant to forgo them.
The most practical way available to shift toward conservatism would be to drill more development wells and fewer wildcats in the Musasi Deep play. It just so happens that seven development wells was the minimum figure and 14 would have been a more reasonable one.
Only two of the Musasi Deep prospects must be drilled any time soon. After penciling in these changes, the exploration manager calculates that if the number of Musasi Shallow wells is increased to 33 from 30, the total expenditure is about the same as in the original plan.
Accordingly, the "number of ventures" column in the budget is changed to that shown in Fig. 6, which is then input to the RPA program, producing Fig. 7, which in turn is graphed to produce Fig. 8. This histogram looks just right for the philosophy of FJP's management. All five rules are satisfied.
Fig. 9 is an overlay of Figs. 5 and 8. The revised plan has less chance of extremely good or extremely bad results; it is more bunched in the middle, more conservative, as upper management has stated it prefers. Also, the exploration manager is pleased to note that the revised budget is achievable, being composed of realistic quantities of prospects that should actually exist in the future.
The exploration manager now has a plan for recommendation to higher management. The budget for the first year can be any appropriate portion of the 2 year plan. If there were single year goals in addition to the 2 year goals, a separate run would be done on each single year plan.
SECOND BUDGET EXTENDED
Fig. 10 was created because the exploration manager was curious about where 6 consecutive years (three repetitions of 2 years each) of such budgets might carry the company.
To make this histogram, all the inputs under "number of ventures" in Fig. 6 were simply tripled. As usually happens with long term risk portfolio analyses, the results are heavily clustered towards the middle because the infamous "law of averages" is taking over. The exploration manager at FJP concludes, however, that Fig. 10 does not conflict with anything known about the wishes of FJP's upper management. There is no reason to reject the second budget.
ANSWERS TO QUESTIONS ABOUT USING RISK-PORTFOLIO ANALYSIS
Q: How does this method differ from the best Monte Carlo simulation techniques? The input and output look just the same to me.
A: A properly run Monte Carlo analysis will indeed address the same questions as will RPA, and it may get the same answers. Practically speaking, the disadvantage of Monte Carlo techniques is that users not sophisticated in statistical analysis will tend to burn up vast amounts of computer resources needlessly. Since the idea behind Monte Carlo techniques is to simulate real-world statistical situations by rolling dice and keeping track of the results, it is obvious that using too few iterations can produce incorrect results (a run of good or bad luck pollutes the small sample).
Therefore, the programs usually default to an outrageously high number of repetitions.
Computer and human time is squandered.
Q: But if /am willing to ignore the waste, /can trust the answers from my Monte Carlo program?
A: Yes, but some valuable detail may be missing. Notice that the bin 130%-139% in Fig. 12 is much smaller than the 100%-110% bin or the 150%-159% bin. Is this real? With Monte Carlo programs, the answer is never yes unless the number of iterations is astronomical. RPA, on the other hand, works by considering the probability of every possible permutation. It goes through a set of nested loops, exhaustively. Therefore, the detail in the histogram is always correct: allowing interpretation of the lumps within the histrogram, not just the overall shape.
Q: Are the VALUE numbers adjusted for inflation of the currency?
A: It does not matter. As long as all the input numbers are consistent, the RP program will output correct numbers in the same units. The histograms are then read accordingly. FJP did its in BOE (and ignoring inflation altogether). Taxation questions are answered in a similar vein.
Q: What if management cannot or will not supply a defined investment strategy? How did FJP get such well-defined limits? It seems like the RPA method requires management to thoroughly understand RPA before they can get any experience using it.
A: The first few risk-profile analyses may be done "backwards." The exploration manager simply takes the budget designed by whatever process, condenses it to a form like that of Fig. 3, lets the RPA program calculate the equivalent of Fig. 4, makes a histogram plot like Fig. 5, and attaches this plot to the completed budget with illuminating remarks but no conclusions. Often, the multiyear projection is the most thought provoking. The histogram plot is easy to understand: if it makes members of upper management uneasy, they will ask the exploration manager for clarification. If this causes them to reject the entire budget, they usually will disclose the reason for rejection. This now becomes Rule #1 for the company. The RPA is redone, then resubmitted, and so forth. Eventually, the exploration manager winds up with a list of negatives to be avoided, such as FJP's five rules, above. The only real difference is that each iteration of the RPA takes the full blown process of rejection by management, rather than just a minute's consideration by the exploration manager.
Q: What if no combination of available ventures can produce a histogram acceptable to investors?
A: This is a difficult situation, but resolving it may be the most valuable thing RPA can do to help a company survive. Unrealistic goals must be scrapped. Conflicting goals must be renegotiated or prioritized.
It is important to remove any element of "right" versus "wrong" from the discussion. However an investor wishes to gamble existing money in an effort to make more is "right" by definition. One investor is willing to take risks in an attempt at big results, another prefers lesser risk and is satisfied with smaller gains; both are right.
Nevertheless, if they are partners in the same enterprise, it may be impossible to satisfy both. For example, an entrepreneur may have entered the petroleum exploration business with the dream of big success but finds himself financed by lots of smaller partners who would be scared off by one dismal year. To keep them happy, he must drill a conservative portfolio of prospects that offers him little or no chance of fulfilling his dream.
RPA cannot magically erase these conflicts. What it can do is quantify and display the issues, thereby clarifying the entire discussion.
Q: There is an old rule of thumb: "Spend 80% of the budget in bread and butter projects and 20% looking for the big one;" Does RPA confirm or deny this maxim?
A: Any given manager should do his own RPA to fit the particular situation, because the correct percentage to spend aggressively is often closer to 1 0% than 20%; however, important benefits do accrue from splitting the budget this way. For example, Fig. 11 is such a budget, showing 16% invested in two high stakes plays and the remaining 84% in more average risk plays. Fig. 12 is the histogram of possible outcomes.
Note the skew, almost triangular shape. In many ways, this is the best type of shape possible: almost no bad results, lost of okay results, and a few good results. Something akin to this is almost always achievable for all but the largest oil companies.
Q: Many articles in the literature have pointed out what a poor job most companies do in assigning risk and predicting reward (Rose, 1987). Since these factors are the data RPA calculates its histograms from, how valid can RPA really be?
A: Other methods of budgeting are based on the same flawed inputs but compound the error in their calculations.
We must remember that billions of dollars are spent by the petroleum industry every year based on such figures; however flawed they might be, their value clearly exceeds their tendency to mislead.
Many companies have systems to statistically analyze their ability to predict each geologic variable in a prospect by comparing pre-drilling predictions to post-drilling estimates (Rose, 1987). As long as this is not misused to judge individual performance (Lohrenz, 1986), this practice is a valuable adjunct to RPA. RPA will be more precise when the inputs are accurate: for example, when drilling close in wildcats in long established plays. But RPA is most valuable when considering risky portfolios: for example, the first several wells in a frontier basin.
BIBLIOGRAPHY
Arps, J.J., and J.L. Arps, 1974, Prudent risk taking: Journal of Petroleum Technology, Vol. 26, pp. 711-715.
Lohrenz, J., 1986, Can Exploration Performance be Measured?: OGJ, Mar. 10, pp. 72-74.
Rose, P.R., 1987, Dealing with Risk and Uncertainty in Exploration: How Can We Improve?: AAPG Bulletin, Vol. 71, No. 1, pp. 1-16.
Copyright 1990 Oil & Gas Journal. All Rights Reserved.