HOW TO ESTIMATE WORTH OF MINOR VALUE OIL, GAS PRODUCING PROPERTIES AT PUBLIC AUCTION

Dec. 31, 1990
Bruce L. Randall Unit Corp. Tulsa The purpose of this paper is to evaluate the divestiture of minor value working and royalty interests (worth less than $20,000) in producing oil and gas properties through the transaction medium of no-minimum, English "open outcry" public auctions. Specifically, the paper seeks to answer the question, "What can the seller expect to receive for his minor value properties at a public auction, knowing only how he values those properties to himself?"
Bruce L. Randall
Unit Corp.
Tulsa

The purpose of this paper is to evaluate the divestiture of minor value working and royalty interests (worth less than $20,000) in producing oil and gas properties through the transaction medium of no-minimum, English "open outcry" public auctions.

Specifically, the paper seeks to answer the question, "What can the seller expect to receive for his minor value properties at a public auction, knowing only how he values those properties to himself?"

To answer this question, a mathematical model that predicts the seller's expected present worth (EPW) as a function of the seller's Securities & Exchange Commission-case "book value" (Xs), and the winning bid value (XB) is derived from classical auction theory.

The model predicts that the probability density functions of both the seller's book values (Xs) and the winning bid values (XB) will be defined by exponential probability distributions, as represented by the coefficients lambda s and lambda B, respectively.

It is shown that by invoking a "common values" assumption; i.e., that the distribution Of XB will be proportional to the distribution of the Xs, the general model can be reduced to yield a specific relationship that predicts the seller's EPW.

To facilitate evaluation of the predictive model for EPW, a data base of actual Xs and XB values was compiled from auction data representing the sale of 526 properties from four different auctions that occurred in 1988 and 1989.

Analysis of this data suggest that the actual present worth (APW) values received by sellers indicate three distinct APW-vs.-Xs behavioral regimes.

For the first regime of Xs values less than $2,000, it is concluded that the "winner's curse" phenomenon is essentially always operative, and that the EPW values predicted by the model were within a cumulative .51% degree of accuracy when compared to values of APW.

For the second regime of Xs values greater than $2,000 but less than $6,000, it is concluded that the winner's curse phenomenon may or may not be operative, and that the predicted EPW values are within a cumulative 1% degree of accuracy when compared to APW values.

Lastly, for the third regime of Xs values greater than $6,000, it is concluded that the fundamental assumption of XB being proportional to Xs may no longer apply, and that sellers of properties in this regime could suffer from deep discounts in a no-minimum auction.

INTRODUCTION

The purpose of this paper is to evaluate the evolving practice of the sale and purchase of working and royalty interests in producing oil and gas properties through public auctions.

Specifically, this work focuses upon the divestiture of minor value (less than $20,000) properties through no-minimum, English "open outcry" auctions.

The perspective taken throughout the analysis is that of the potential seller, who seeks to answer the question, "What value should I expect to receive for my minor value producing properties at a no-minimum public auction, given only my own valuation of each property?"

To answer this question, a mathematical model that predicts the seller's expected present worth (EPW) as a function of the seller's S.E.C.-case "book value" (Xs) and the winning bid value (XB) is derived from classical auction theory.

Predicted results from the model are then compared to the actual present worth (APW) of profits received by sellers who sold more than 500 properties in four different auctions in 1988 and 1989.

A solution to the potential seller's question posed above seems quite timely in light of the ever increasing volume of properties being sold at auctions. Since the first of these auctions in 1987, both the frequency of property auctions and the number of industry sellers using them has literally ballooned, with about 20 such auctions having occurred in 1990.

Survey results of the top three oil and gas property auctioneering firms, representing an estimated 90% of both volume and value of properties auctioned through first quarter 1990, are presented (Table 1).

It is interesting to note that, though the first Miller & Miller/Davis Oil Co. auction involved properties having an average sales value of $37,500/property, the industrywide average has rapidly gravitated towards a much smaller value of $2,260/property.

Also noteworthy is that no single auctioneer's per property average varies by more than $660 from the mean of $2,260.

Of considerable significance is the fact that the data in Table 1 represent the activities of approximately 300 different sellers in about 40 different auctions during 2 1/2 years since the first (1987) auction.

Hence, it is quite apparent that the petroleum industry has begun to recognize the no-minimum public auction as the medium of choice for divesting marginal value producing properties.

LITERATURE SURVEY

A survey of auction literature1-8 reveals that the subject case of the no-minimum, English "open outcry" auction can be classified as a common value auction for which the assumption of .1 public" (i.e., symmetric) information generally applies.

Once the symmetry of information is assumed, and if the assumption of risk-neutral bidders is also applied, then it necessarily follows that auction outcomes of the subject case should conform to Bayesian-Nash equilibrium (between bidders).

However, as observed by Milgrom3, because the English "open outcry" auction provides for both "positive correlation" and "affiliation" of competing bids, one should expect to see sellers realize greater premiums as the number of bidding participants increases.

Milgrom's theoretical results were in fact observed in experiments conducted by Kagel and Levin2, who noted that "in auctions involving a limited number of competitors (three to four bidders), average profits were consistently positive and closer to the Nash equilibrium bidding outcome than to the winner's curse..."

The authors further claimed that, as the number of bidders was increased, there existed a strong tendency towards positive correlation in the bidding "...in clear violation of risk neutral Nash equilibrium bidding theory...", thus contributing towards "...a reemergence of the winner's curse ... in auctions with large numbers (six to seven) of bidders."

Of all the works reviewed in the auction literature, probably the most pertinent to the subject case is the classic paper by Capen, Clapp, and Campbell1 exposing the competitive bidding phenomenon of the winner's curse.

The authors' fundamental conclusion is that "in competitive bidding, the winner tends to be the player who most overestimates true (property) value ... only in a noncompetitive environment, can he counter his over-evaluated (properties) with his under-evaluated (properties) and expect to do well on average. In bidding, however, he has a poor chance of winning when he has underestimated value and has a good chance of winning when he has overestimated it. So we say the player tends to win a biased set of (properties)-namely, those on which he has overestimated value..."

The authors then rather colorfully surmise that "when it was all over, we concluded that the competitive bidding environment is a good place to lose your shirt."

The most beneficial aspect of Capen et al.'s work to the subject case is in their derivation of an analytical model.

Though their actual data was best described by a log-normal probability distribution, the authors found that they "...could not carry out the necessary integrations if (they) used the log-normal distribution." As an approximation, they therefore derived their analytical model based upon an exponential probability distribution.

The goal of their efforts was to "...derive an equation that will tell us EPW (expected present worth) as a function of our bid level, the opponent's bid levels, and the number of opponents."

As it turns out, this is somewhat analogous to the model derivation for the subject case as a function of the seller's book value (Xs), the winning bid value (XB), and the probability density functions (pdf) of each.

EPW MODEL DERIVATION

Similar to the general expression presented by Capen et al., the general EPW relationship for the subject case can be expressed as the product of the following three terms:

  1. The present value difference between the winning bid value (XB) and the seller's book value (Xs); i.e.,

    [SEE FORMULA (1)]

  2. The probability that the value of a given property is Xs. For an exponential pdf,

    [SEE FORMULA (2)]

  3. The probability that the winning bid value is XB. Again, for an exponential pdf,

    [SEE FORMULA (3)]

    Hence, if both Xs and XB are considered to be entirely independent variables; i.e., if XB has no minimum and is in no way "affiliated" or "correlated" to Xs, the general EPW relationship is expressed as

    [SEE FORMULA (4)]

    which reduces to

    [SEE FORMULA (5)]

    where CA, CB, and Cc are constants of integration.

To solve for the constants of integration, the following two boundary conditions are noted:

  1. For the special case of lambda B = lambda S, XB will always equal Xs, and EPW must always equal zero. For the special case of Xs equals zero, EPW equals XBo.

  2. As will be demonstrated from the statistical analysis of auction-data, when Xs equals zero, XBo has a mean value of $688.24.

From the first boundary condition, it necessarily follows that Cc must equal zero. Hence, what remains at this point is a relationship with two unknowns governed by only one boundary condition.

To further reduce equation 5 to a useable form, one other boundary constraint must be defined.

As will be demonstrated later from statistical analysis, actual auction results do in fact reveal that Xs and XB each have exponential pdfs that are defined by lambda s and lambda B, respectively.

Mathematically then, it follows that XB must be proportional to Xs, as stated by the relationship

[SEE FORMULA (6)]

By using equation 6 as the third boundary condition, equation 5 can then be reduced to the more useable form

[SEE FORMULA (7)]

Obviously, all that is needed to apply the EPW model expressed as equation 7 for any given value of Xs (the only remaining independent variable) are values for the exponential pdf coefficients, lambda B and lambda s.

Before proceeding to the statistical analyses from which lambda B and lambda s will be determined, it is important to note two fundamental characteristics regarding equation 7:

  1. The third boundary condition, equation 6, which enabled the reduction of the general relationship equation 5, requires that XB be proportional to Xs.

    This is, in fact, invoking a type of "common values" constraint, which therefore mandates that the winning bidders' property valuation procedures be at least similar to the seller's procedure in determining Xs.

    Obviously, in cases where bidders are willing to give consideration for value related to factors other than the discounted cash flow from remaining proved reserves (salvage value of production equipment, operating overhead fees, "possible" or probable" reserves, etc.), the model will not be applicable.

  2. The EPW model expressed as equation 7 predicts that, as Xs approaches zero, EPW approaches( XBo = $688.24). Obviously, this is a result of invoking the second boundary condition.

It follows then that as Xs approaches zero, the seller should receive increasingly larger premiums.

As will be shown in the next section of this discussion, this is in fact what is observed in actual property auctions, thereby explaining why "...the industrywide average (of Xs values) has rapidly gravitated towards $2,260" per property.

DATA BASE ANALYSIS

In order to use the predictive EPW model derived above equation 7, estimates of the following three values are required:

  1. The average winning bid value for properties having an Xs value of zero; i.e., XBo.

  2. The value of lambda s defining the exponential pdf of the non-zero Xs values of the properties to be sold.

  3. The value of lambda B defining the exponential probability density function of the winning bid values corresponding to the non-zero Xs values in (2) above.

To obtain estimates of XBo, lambda s, and lambda B representing a "typical" minor value property auction, a data base representing the individual sale of 526 properties was compiled and statistically analyzed.

Table 2 presents some of the more pertinent characteristics of this data base. It is estimated that these properties, sold via four different auctions, represent approximately 2% of the total volume of properties auctioned through 1989.

Also, there appeared to be no chronological correlation of seller's premium with time.

Fig. 1 presents a histogram of XBo values for the 173 properties in the data base having Xs values equal to zero.

The last XBo value of $24,000 significantly skews the value of XBo corresponding to the exponential pdf depicted in Fig. 1.

If this point is ignored, lambda Bo, is increased from 0.00122 to 0.00145, and XBo becomes

[SEE FORMULA (8)]

Fig. 2 presents a histogram of all non-zero Xs values for the 353 remaining properties.

It is important to note that the distribution of Xs values could only be characterized by an exponential pdf-shown as the white curve in Fig. 2-and that all attempts to define the pdf with other distributions (normal, rectangular, gamma, log-normal, Chi square, binomial, Poisson, geometric, and Bernoulli) were unsuccessful or unsatisfactory.

The lambda s value defining the exponential pdf depicted in Fig. 2 is 0.00055.

Fig. 3 is the histogram of XB values corresponding to the 353 non-zero Xs value properties. Once again, the exponential pdf was the only distribution yielding satisfactory results. The exponential pdf depicted by the solid curve in Fig. 3 represents a lambda B value of 0.00049.

To illustrate the accuracy with which the above values of XBo, lambda s, and lambda B characterize the data, the following comparison of EPW to APW can be made, where

[SEE FORMULA]

Thus, the expected present worth for all 526 properties is $220,393.26. From Table 2, note the actual present worth of profits received by the sellers was $220,877, for a percentage error of only .22%.

PREDICTIVE EPW MODEL

Once the observed values of XBo = $688.24, lambda s 0.00055, and lambda B = 0.00049 are substituted into equation 7, predictive values of EPW can be computed as functions of Xs only.

These values have been computed for the entire 526 property data base and are presented graphically with corresponding actual present worth (APW) values in Fig. 4.

Note that the cumulative APW values simply represent the difference of cumulative XB minus cumulative Xs, both of which are also shown in Fig. 4. A magnified presentation of cumulative APW and cumulative EPW versus Xs is presented as Fig. 5.

Before analyzing the results of predictive EPW model, the reader should note from Fig. 5 the following three observations regarding the behavior of cumulative APW as a function of Xs:

  1. Over the range of O Xs < $2,000, cumulative APW is constantly increasing at an exponentially decreasing rate.

    Hence, because the slope of cumulative APW versus Xs is always positive in this range, one can conclude that, on average, the seller should anticipate receiving a positive EPW when selling properties having Xs values less than $2,000.

    This fact alone implies, according to auction theory, two important conclusions:

    1. The winner's curse phenomenon is essentially always operative over the interval O Xs < $2,000, and

    2. By extension, according to Kagel and Levin2, one should expect on average that the number of competing bidders will exceed three to four.

      Note that these observations are based on data from 429 of the 526 total cases; i.e., 82% of the data base.

  2. Over the range of $2,000 < Xs < $8,000, the behavior of cumulative APW is almost sinusoidal, with the slope changing direction no less than three times.

    This would suggest that, over this particular range of Xs values, the seller is just as apt to receive a negative APW as he is a positive APW.

    Thus, over this particular range of Xs values, one can conclude that:

    1. The winner's curse phenomenon is not always operative, and

    2. According to Kagel and Levin2, the number of competing bidders is just as likely to be less than three to four as it is greater than three to four.

      Note that these observations are based upon data representing the sale of 84 properties (approximately 16% of the data base), and are therefore obviously less certain than those stated in (1) above.

    Of these 84 transactions, 55 represent negative APW sales totaling -$108,572, with the remaining 29 positive APW sales totaling $103,019, for a net negative APW of - $5,553 over the interval.

    Given the aggregate Xs value of $317,470 for these 84 properties, the net negative value of -$5,553 represents a negative premium (i.e., discount) to the seller of 1.75%.

    Hence, one can conclude for values of Xs greater than $2,000 and less than $8,000, the seller should expect, on average, to simply break even.

  3. Lastly, note the behavior of cumulative APW corresponding to Xs $8,000. For this interval, one can observe that, for all practical purposes the slope is generally negative, never to become positive again. Once again invoking the common values assumption, auction theory suggests that:

    1. The winner's curse is essentially absent over this range, and

    2. The number of competing bidders is always less than three to four.

      Note that these conclusions are based upon activity representing only 13 transactions(1), or only 2.5% of the data base, and the level of assurance of these conclusions is correspondingly less than those preceding.

Notwithstanding, this data does suggest the possibility of the seller suffering deep discounts when auctioning properties of Xs values greater than $8,000.

Thus, it necessarily follows that the "common values" assumption, particularly between buyer and seller, is no longer valid in this interval.

Bearing in mind these observations of cumulative APW versus Xs behavior, the analysis is now focused upon the results of the predictive EPW model (equation 7).

From Fig. 5, note that the EPW model generally fits the cumulative APW data quite well when utilizing values of XBo, lambda s, and lambda B ($688.24, 0.00055, and 0.00049, respectively) obtained from the statistical analysis of the entire 526 property data base.

However, because of the apparent breakdown of the fundamental "common values" assumption for (Xs $8,000) properties noted previously, the percentage error of cumulative EPW versus cumulative APW for the entire data base is intolerably high. Specifically, cumulative EPW = $291,675 and cumulative APW = $220,877, representing a percentage error of 32%.

If the analysis is confined to the Xs interval over which the model's fundamental assumptions obviously hold, i.e., Xs < $8,000, the results become vastly improved.

For these 513 properties, the cumulative APW value is $278,888 and the predicted cumulative EPW is $288,750, for a percentage error of only 3.5%.

Fig. 6 is presented as a "blown-up" representation of Fig. 5 for all Xs < $8,000.

Other refinements in accuracy can be made for specific upper limits of values for Xs. Of particular note, as is observable from Figs. 5 and 6, is the fact that at an Xs value of approximately $6,000, the general trend in cumulative APW becomes negative, apparently never to become positive again.

Rigorously speaking, this observation suggests that at Xs $6,000, the probability Of XB being greater than Xs is always zero; i.e., XB is always less than Xs. This, of course, violates the fundamental presupposition that XB has an exponential pdf (equation 3) similar to that of Xs (equation 6).

Hence, it follows that the accuracy of the predictive model might be further enhanced by excluding the final 23 data points of Fig. 5 corresponding to all Xs $6,000. The remaining 503 observations (96% of the original data base) yield refined values of lambda s = 0.00080 and lambda B = 0.00057.

Substituting these values into equation 7 yields a predicted cumulative EPW value of $317,577 that, when compared to the corresponding cumulative APW value of $314,634, suggests a percentage error less than 1%.

Similarly, by confining the data to an upper limit of Xs < $1,800 (429 properties representing 82% of the original data base) the results can be even further refined. This is because, in this specific regime, the behavior of cumulative APW Vs Xs always behaves in accordance with equations 3 and 6.

Recalculation of lambda s and lambda B for this specific group of property sales yields values of 0.00164 and 0.00083, respectively.

Substituting these values into the predictive model (equation 7) yields a value of cumulative EPW of $282,987 that, when compared to the corresponding cumulative APW value of $284,441, represents a negligible percentage error of only .51%.

Table 3 has been provided as a summary of observed values of lambda s and lambda B, corresponding to upper value limits of Xs.

ACKNOWLEDGMENTS

This work is derived from an MBA thesis by Bruce Randall, submitted in partial fulfillment of the requirements of the Master of Business Administration Degree from Oklahoma State University. The author expresses his sincere appreciation for the contributions of the thesis adviser, Dr. James F. Jackson. In addition, the author acknowledges with special gratitude the contributions of auction data by Dyco Petroleum Corp. and his employer, Unit Corp., both of Tulsa, without which this study would not have been possible.

REFERENCES

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  2. Kagel, J.H., and Levin, D.: "The Winner's Curse and Public Information In Common Value Auctions," American Economic Review, December 1986, Vol. 76, pp. 894-920.

  3. Milgrom, P.: "Auctions and Bidding: A Primer," Journal of Economic Perspectives, Vol. 3, No. 3, Summer 1989, pp. 3-22.

  4. Milgrom, P.R., and Weber, R.J.: "A Theory of Auctions and Competitive Bidding," Econemetrica, Vol. 50, No. 5, September 1982, pp. 1,089-1,122.

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  8. Hendricks, K., and Porter, R.: "An Empirical Study of An Auction With Asymmetric Information," American Economic Review, December 1988, pp. 865-883.

  9. Bryan, J.G., and Wadsworth, G.P.: "Introduction to Probability and Random Variables," McGraw-Hill Book Co., Inc., New York, 1960.

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  11. Kendall, M., and Stuart, A.: "The Advanced Theory of Statistics," Vol. 1, MacMillian Publishing Co., Inc., New York, 1977.

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