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I. Introduction

An economics classic [14] was restated by Dasgupta and Heal [6]: the net price of a mineral must rise at the rate of discount. But econometric analysis failed to verify this, according to Miller and Upton [15].

The Hotelling Principle is often couched in terms of flow equilibria in output markets. The Rule can also be derived [27] as a condition of stock equilibrium in asset markets. Miller and Upton followed the asset framework in defining the Hotelling Rule, a formulation they called the Hotelling Valuation Principle (HVP): the market value of a mineral in the ground is equal to its current net price. This follows from the fact that if the net price rose at the rate of interest, the present value of the net price would be the same whenever the resource is extracted, given asset value maximization.

The HVP expresses the Hotelling Rule in a form which lends itself easily to empirical testing. Miller and Upton compared a sample of companies’ estimated oil and gas reserve values directly with net prices, and concluded that they were indeed equal. But estimates they made later [16] showed the in-situ value as only about half the net price; they ascribed this to unsatisfactory data.

Later papers by Adelman [1] and Watkins [30] rejected the HVP for reasons both of theory and data. The issue is important because long run changes in the prices of minerals are important to the world economy and to many governments. For example, Boskin et al. [4] estimated the value of the U.S. government’s mineral holdings at nearly 20 times what they would be worth if the 1989 price were assumed constant, and revenues discounted at the conventional industry rate [1].

We show the Hotelling Valuation Principle to be a special case, which can be tested for, of a more general formula. The estimated in-situ values then give useful insights into expected oil and gas prices.

[TABULAR DATA OMITTED]

II. The Data

The data consist of amounts paid for 34 developed oil and gas reserves in Alberta over the period February 1989 to March 1991. Only six observations were for single properties; three were for three to four properties. The remaining 25 observations were for corporate reserve assets (12) or for transactions relating to 20 or more properties. An absence of price and cost data for individual reservoirs precludes analysis on a property basis. But since the bulk of the transactions are for multiple properties, reliance on average cost and price data is not distortive.

[ILLUSTRATION OMITTED]

Miller and Upton inferred in-situ reserve values from stock market values, after adjusting for liabilities and the value of non-reserve properties. They would avowedly have preferred to use the kind of data we employ here [15, 15]. The transaction values relate solely to the purchase of reserves and thus are uncontaminated by other property values.

III. Analysis of the Reserve Transaction Data

The data in Table I immediately suggest regressing the transaction values on the respective volumes of oil and gas reserves to estimate unit reserve prices. What is the appropriate specification of the function to be estimated?

Conventional cash-flow capital budgeting methods for valuing mineral properties suggest a valuation expression ([V.sub.T]) of the form:

[V.sub.T] = [integral of] ([p.sub.ot] — [c.sub.ot])[f.sub.ot][e.sup.-it] dt between limits [n.sub.o] and t=0 + [integral of] ([p.sub.gt] — [c.sub.gt])[f.sub.gt][e.sup.-it] dt between limits [n.sub.g] and t=0 (1)

where

[p.sub.ot], [c.sub.ot] are prices and costs of oil, respectively, times t

[p.sub.gt], [c.sub.gt] are prices and costs of gas, respectively, times t

[f.sub.ot], [f.sub.gt] are quantities of oil and gas production, respectively, times t

t is (continuous) time

[n.sub.o] is remaining reservoir life for oil

[n.sub.g] is remaining reservoir life for gas.

Costs represent all costs, i.e. extraction, future development (if any), income taxes, royalties, etc. If future development costs were anticipated, unit costs could be quite lumpy. All RHS variables are of course expected values.

The linear separation inherent in (1) implies no interaction between oil and gas production and costs. This would certainly hold if the transaction related to an oil reservoir and a non-associated gas reservoir.

However, it is more likely that the transactions relate to oil reservoirs with some associated gas, or to natural gas reserves containing some liquids. If so, there could be some interaction between oil and gas production, individual costs, and joint costs. This would not matter if such relationships were linear, as they would be if production gas-oil ratios (GORs) were constant. But if there were strong non-linear relations between, say, associated gas production and oil production, then a reduced form of expression (1) might thwart a linear regression of asset values on reserve volumes and prevent interpretation of the reserve coefficients in a straightforward way.

It is doubtful that any non-linearities among the oil and gas production and cost data would be sufficient to violate the essentially additive nature of the conventional reserves evaluation procedure: estimate the net present value of expected flows of oil production plus the net present value of expected natural gas production from each property under review.

Thus the basic regression expression can be written as:

[V.sub.T] = [b.sub.o] + [b.sub.1][R.sub.o] + [b.sub.2][R.sub.g] (2)

where

[R.sub.o] = [integral of] [f.sub.ot]dt between limits [n.sub.o] and 0

and

[R.sub.g] = [integral of] [f.sub.gt]dt between limits [n.sub.g] and 0

If reserves were zero, [V.sub.T] would be zero and the intercept term should be constrained as zero. Then the regression equation becomes

[V.sub.T] = [b.sub.1][R.sub.0] + [b.sub.2][R.sub.g]. (3)

Note that the expression (2) can still be useful since the statistical significance of the intercept can indicate the degree of cleanliness of the data.

IV. Regression Results

When the constant term was suppressed, the data in Table I yielded:

[Mathematical Expression Omitted].

These results suggest an in-situ value for oil reserves of $5.37/barrel and $0.44/Mcf for natural gas reserves over a two year period (February 1989 to March 1991).

The degree of (linear) fit is reasonable, both coefficients are highly significant and there is no evidence of autocorrelation. The Goldfeld-Quandt test (and visual inspection of the data) suggested no heteroscedasticity.

Imposition of the intercept yielded:

[Mathematical Expression Omitted].

The intercept term is insignificant — which indicates the data are quite clean. And the statistical properties of expression (5) are nearly identical to (4), the expression we employ hereafter.

The equation specification excludes a variable for sulphur reserves. Some of the transactions did provide information on sulphur. This enabled us to test the impact of the inclusion of sulphur reserve data (see Appendix). The sulphur variable lowers the oil and gas reserve coefficients (as would be expected). To the extent, then, that the transactions we use include sulphur reserves — which some undoubtedly do — the oil and gas reserve regression coefficients are biased upwards. This has some bearing on our later conclusions.

V. Discussion of Regression Coefficients

First, we test the HVP against the coefficients from expression (4) in a straightforward way by comparing them with representative net prices. Second, we broaden the analysis by making a few simplifying and non-distortive assumptions which provide an interpretation of what the reserve coefficients would represent, if viewed in reduced form. This enables us to derive implicit growth rates in prices that might underlie the empirical results.

In particular, suppose expected nominal prices and unit costs were constant, production were set by a fixed production/(remaining) reserves ratio (thus defining the decline rate), and the reserve life were infinite.

Then expression (1) becomes

where [a.sub.0], [a.sub.g] are the production/reserves ratios, respectively, for oil and gas [1, 6].

Reverting now to (3), the implication of (6) is that the coefficient [b.sub.1] represents

[[p.sub.0] — [c.sub.0]][[a.sub.0]/([a.sub.0] + 1)]

and the coefficient [b.sub.2] represents

[[p.sub.g] — [c.sub.g]][[a.sub.g]/([a.sub.g] + i)].

Given values for the reserve coefficients [b.sub.1] and [b.sub.2], implicit prices for oil ([p.sub.0]) and natural gas ([p.sub.g]) can be derived, predicated on assumed values for [a.sub.0], [a.sub.g], i and [c.sub.0] and [c.sub.g]. Table II sets out Alberta representative data to provide a check on the structural relations.

The reference oil price (in Canadian dollars) is the average of Alberta wellhead prices for February 1989 to March 1991 (the period of observed transactions), excluding the months August 1990 to January 1991 when the Gulf crisis noticeably influenced the market, but possibly not longer term price expectations.(1) The reference gas price is the average Alberta field price, February 1989 to March 1991 [20].

Unit costs for both oil and gas include (average) operating costs, royalties, and taxes [7; 8; 17, 126, 209; 24]. The P/R ratio for oil is based on recent data from Alberta’s Energy Resources Conservation Board [11]; that for gas is predicated on the long-term Western Gas Marketing Limited (WGML) contracts which run from 1988 to 2003.(2) The discount rate of 15 percent is the nominal after-tax rate suggested by previous work on Canadian hurdle rates [9].

Further discussion of the data is given in the Appendix.

We deal first with oil. The Hotelling Valuation Principle implies that the oil reserve coefficient would be $13.36/barrel (the net price), about ten standard deviations away from the regression coefficient of $5.37. This degree of deviation could tolerate substantial variations in the imputed average net price and still remain overwhelming.

We also insert the values in Table II for c, a, and i in expression (6) and derive a value for the implicit price of oil of $21.43/barrel. Since this derived value is very close to the representative value of $21.36 in Table II, the interpretation of the oil reserves coefficient is straightforward. It suggests oil price expectations underlying the various purchases of oil reserves were flat, at around $21/barrel (CDN $).

The case of natural gas is similar, up to a point. The HVP value is $0.74/Mcf, compared with the $0.44/Mcf regression value (nearly six standard deviations away from it). Again the HVP is strongly rejected.

The implicit constant value for price derived from expression (6) is $2.23/Mcf, in contrast to the contemporaneous average of $1.54/Mcf.

This result suggests that the gas price expectations underlying the valuation of natural gas reserves were considerably above prevailing short-term prices. The implication is that buyers of gas reserves were anticipating future price appreciation. This accords with our knowledge of market expectations, at that time [18].(3)

We can test this suggestion by asking what results imply about growth rates (g) in prices. Adapting expression (6) above, and Adelman [1, 8], we have a general expression for the reserve price, b, as:

b = [a(p — c)]/[a + i — g]

and hence

g = i + a<1 — [(p — c)/b]>. (7)

Our results for oil above were ([p.sub.0] — [c.sub.0]) = 2.49[b.sub.0]; for gas, ([P.sub.g] — [c.sub.g]) = 1.68[b.sub.g].

For further refinement, we try to measure the P/R ratio, a, more precisely. In general, initial reserves are cumulative expected output, declining at a constant exponential rate:

R = P [integral of] [e.sup.-at]dt = P(1 — [e.sup.-aT])/a between limits T and t = 0 (8)

where T is the reservoir life.

As T becomes large, the limiting value of a is P/R. But for shorter lives, we need a correction. Since terminal output [P.sub.f] = [Pe.sup.-aT], it follows that:

a = (P/R)(1 — (Pf/P)). (9)

We approximate [P.sub.f]/P by P/R because the higher is the initial rate of output relative to reserves, the higher the level of operating expenditures, hence the higher the abandonment output level.(4) Then a = (P/R) — [(P/R).sup.2]. We adjust from Table II:

[a.sub.0] = 0.1 — 0.01 = 0.09; [a.sub.g] = 0.067 — 0.0045 = 0.0625.

The implicit growth rates for oil ([g.sub.0]) and natural gas ([g.sub.8]) are:

[g.sub.0] = 0.15 + 0.09(1 — 2.49) = 0.016

[g.sub.g] = 0.15 + 0.0625(1 — 1.68) = 0.108.

We caution that derivation of the g’s in this way impounds errors in any of the five contributing parameters. However, the contrast between the oil and gas expected growth rates does generally agree with what we know of representative industry forecasts.

Expression (7) takes the Hotelling Valuation Principle as a special case, for which one can test. If the net price (p — c) were equal to the in-situ value b, then a becomes irrelevant, g = i, and the net price grows at the rate of discount [1]. Moreover, when (p — c) = 2b, as has been approximately true for many years in the United States [2], then g = i — a. Since i and a were historically near equality, then g would approximate zero. If so, it was a good forecast.

Insofar as our calculations contain any biases, nearly all tend to favor the Hotelling Valuation Principle. The representative price excludes the price hump during the Gulf crisis. Yet Gulf instability might have imparted an upward shift to price expectations.(5) Representative operating costs may not reflect technological progress. Exclusion of sulphur reserves biases oil and gas reserve coefficients upward.

Some bias in the other direction might arise if the average unit operating costs assumed were lower than the levelized operating costs of the reservoirs involved in the various transactions. This would typically occur if the stage of depletion of the reserves sold were more advanced than for the average. We can get a handle on the maximum degree of any such bias by assuming that all operating costs were fixed in relation to production and that the unit operating cost used applied to initial production volumes and would therefore rise at the decline rate over the production life. If so, the levelized unit operating cost would exceed the initial value by a factor of 1.64. If this factor were applied to the assumed operating costs (see Appendix) of $3/barrel for oil and $0.40/ Mcf for gas, and the HVP values recalculated, they would still lie some 7.6 (oil) and 6.8 (gas) standard deviations away from the regression coefficients of expression (4). Note this adjustment does not allow for any offsetting reduction in income tax with higher operating costs.

We have couched the analysis in nominal terms. If we treat expected prices and costs in real terms the calculations can be repeated after adjusting the discount rate to real terms as well. We set the real discount rate for developed properties at 10 percent.(6) If so, the implicit growth rates for oil ([g.sub.0]) and for gas ([g.sub.g]) become -3.6 percent and 5.8 percent, respectively.

VI. Concluding Remarks

We have examined some pristine transaction data on the sale of developed oil and gas reserves in Alberta over a recent two year period (February 1989 to March 1991). The Hotelling Valuation Principle (HVP) suggests that the unit values of reserves in the ground will be in the neighborhood of the prevailing net price.

Our estimate of reserve values cannot be reconciled with the HVP since they are considerably below the net wellhead prices. The analysis also suggests that transactors expected little change in oil prices but sustained growth in natural gas prices. The latter expectations would seem to be partly confirmed by the rise in gas prices in 1992.

Crude oil is very different from natural gas in respect of wellhead price, in-situ values, and costs. Their prices have diverged significantly over time. It is erroneous to assume a priori that they can be added together as oil equivalent or gas equivalent. If we know nothing about the two commodities, this addition would be suspect; knowing what we do, it is unacceptable. Possibly such aggregation problems in the independent variables explain the results in Miller and Upton [15].

Appendix: Details on Data and on Other Results

This appendix provides additional detail on the data employed.

Transactions

The 34 oil and gas reserve purchase transactions were made over the period February 1989 to March 1991. Transactions labeled 1 through 27 were examined in Adelman and Watkins [3] and in Watkins [30]. The additional seven transactions are labeled 28 through 34; all but one (August 1990) were made in the last quarter of 1990. For all transactions, land values were eliminated. Hence the data relate solely to the reserves.

[ILLUSTRATION OMITTED]

[ILLUSTRATION OMITTED]

The breakdown by type of transaction was:

Note that transactions 7 through 19 seemingly are properties for which buyers paid significant premia in relation to competing bids [30, footnote 22].

[ILLUSTRATION OMITTED]

Regression Results for Transactions Identifying Sulphur Reserves

The seven observations labeled 28 through 34 in Table I had information on sulphur reserves ([R.sub.s]). Inclusion of their data yielded the following:

[Mathematical Expression Omitted].

Without the sulphur reserves data, the regression results for the seven observations are:

[Mathematical Expression Omitted].

This result tells us that the sulphur reserves variable ([R.sub.s], in millions of long tons) is significant and as would be expected the oil and gas reserve coefficients fall with its inclusion. Their overstatement in the absence of sulphur reserve information is not negligible (oil is some 19 percent lower, gas 14 percent lower).

Prices

The reference price is based on averages of Edmonton posting of the Alberta Par price of 40 [degrees] API oil. Prices of heavier crudes would be lower. As mentioned in the text, prices affected by the Gulf crisis, August 1990 to January 1991, were excluded. The number of transactions during this period of our set of 34 observations was 14. Hence the representative price was the average monthly postings from February 1989 to July 1990, plus February and March 1991. A deduction was made from the Edmonton price to reflect gathering costs of $0.50/barrel, to arrive at field or wellhead prices.

[ILLUSTRATION OMITTED]

Royalties

Alberta oil royalties distinguish between old and new oil (when old oil refers to pre-1974 reservoirs) and are price and well production sensitive; also the Government does enjoy discretion in setting royalty reference prices [13, AR 248/90, Schedules 1 and 2]. The reference price adopted of $21.36/barrel would generate unadjusted royalties of 21 percent for new oil and 29 percent for old oil (assuming $21.36 were the par price for oil in the royalty context). However, various credits for royalty holidays and other factors effectively reduce royalty rates.

Alberta’s natural gas royalties are also price and well production sensitive [13, AR 246/90]. Unadjusted royalties for the reference price of $1.54/Mcf would be some 23 percent (assuming well production was greater than 6 Mcf/day). However, again these royalties are subject to reduction for various factors, including a gas processing cost allowance.

A better handle on representative royalties is provided by the ratio of royalties paid to (upstream) production values. Canadian data for 1989 and 1990 show oil and gas royalties as a proportion of aggregate field values (including NGLs and sulphur) were about 17 percent in both these years [25, Tables B-15, B-16]. Applying this average percentage to the representative wellhead price of $21.36 suggests a royalty of $3.63/barrel.

As a compromise between the royalty formula calculations and the average percentages based on production values, we adopt a representative oil royalty of $4/barrel or close to 19 percent of the representative wellhead price. In the case of natural gas, 17 percent of the field price of $1.54/Mcf would be $0.26/Mcf. Again, as a compromise between this value and the royalty formula calculation we adopt $0.30/Mcf as a representative natural gas royalty.

Operating Costs

Data on oil and gas well operating costs are quite sparse. Detailed estimates of operating costs made by the Canadian National Energy Board in 1981 indicate costs of $2.22/barrel in 1980$ [19, Table L-2]. If this estimate were inflated by the GDP price index, the 1990$ figure would be $3.60/barrel [28, Table 1-16]. However, this figure could be high, given technological progress in well operating efficiencies over the past decade, other things equal.

Canadian Petroleum Association data on operating costs in Alberta for 1990 suggest a cost of $4.23/barrel [5, Tables III-I A, III-10, IV-3A].(7)

DataMetrics Limited estimated operating costs for a medium sized oil reservoir at some $2.18/barrel in 1982$ [7, 14]. If this were conveniently inflated by the GDP price index [28], the 1990 operating cost figure would be $2.94/barrel. But again this may well be an overestimate, given technological progress.

Given this albeit scanty evidence we adopt a figure of $3/barrel for representative levelized oil operating costs, since undoubtedly costs of oil and gas operations have been squeezed in the 1980s.

Operating costs for natural gas are derived from recent NEB in-situ estimates of $0.21/Mcf [17, 126, Table 6-3]. This figure is converted to a production basis by a factor of 1.96 yielding some $0.40/Mcf, assuming a real discount rate for production operations of 7.5 percent [9, 19].(8) An operating cost of $0.40/Mcf is of the same order of magnitude as estimates in DataMetrics Limited [8, 121,123].

Income Taxes

Petroleum Monitoring Agency data show income tax of $4.01/[m.sup.3] on Alberta hydrocarbon production in 1989 and $4.88 in the first half of 1990 [24, Table B-8]. Second half 1990 figures were excluded to avoid including much higher income taxes generated by higher oil prices associated with the Gulf crisis. This is consistent with our treatment of prices.

Multiplication of unit tax values by hydrocarbon output provides estimates of total income taxes paid in 1989 and the first half of 1990. These aggregate taxes were allocated between oil and gas on the basis of production values [24, Table B-15], which indicate a split of 30 percent to natural gas and 70 percent to oil (including NGLs). Division of the resulting taxes allocated to oil and gas by respective volumes of production indicates taxes per unit of oil production of $0.91/barrel in 1989 and $1.18/barrel in the first half of 1990; corresponding figures for natural gas are $0.08/Mcf in both periods. These data suggest representative income taxes for oil production of $1/barrel and for natural gas of $0.10/Mcf.

1. Prices from Shell Canada [26]. Note that 3 month futures were consistently below spot prices during this period [12].

2. Source is typical WGML contracts. We thought it more appropriate to use a commercial P/R ratio here to avoid bias attributable to surplus natural gas reserves; the latter occur when gas enters the reserves inventory above and beyond that required for normal commercial backing. Historically, in Alberta such surplus gas has mainly been policy induced [29, 13].

3. In the United States, the industry had expected a dramatic rise in prices for 3 to 5 years [21, 71].

4. Decline rates are rarely published. However, Prudhoe Bay was reported in 1992 to have a natural underlying decline rate [of] about 9 percent [10, 29]. Average Prudhoe Bay 1992 crude oil reserves were 4230 thousand barrels, and 1992 crude production was 436.5 [22, 74; 23, 80]. Hence P/R — [(P/R).sup.2] = .103 — .0106 = .092.

5. However, there would have been a partly offsetting shift in income taxes and royalties associated with higher prices. And to the extent that the average gravity of transaction reserves was below 40 [degrees] API (the reference crude gravity), prices would be modestly high. No information was available on this aspect.

6. This reflects expected annual inflation of about 5 percent [9].

7. Natural gas converted on a thermal basis to oil equivalence.

8. The National Energy Board estimates relate to future reserves additions in the Western Canadian Sedimentary Basin and thus may well be subject to some upward bias in relation to recent historical costs, given the NEB’s upward sloping supply function for new reserves.

References

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2. ——, H. De Silva and M. F. Koehn, User Cost in Oil Production . Resources and Energy, December 1991, 217-40.

3. Adelman, M. A. and G. C. Watkins. Reserve Asset Values and the ‘Hotelling Valuation Principle.’ MIT Working Paper, MIT-CEEPR 92-004WP, March 1992.

4. Boskin, M. J. Marc S. Robinson, T. O’Reilly and P. Kumar, New Estimates of the Value of Federal Mineral Rights and Land. American Economic Review, December 1985, 923-36.

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8. ——. Overview of the Canadian Natural Gas Industry. Gas Research Institute, November 1987.

9. —— and Coles Gilbert Associates. Costs of Natural Gas Supply: Differentials Associated with Contracts, Reserve Types, and Contracts. Mimeo, March 1989.

10. DOE/EIA. US Crude Oil, Natural Gas, and Natural Gas Liquids Reserves: 1992 Annual Report.

11. Energy Resources Conservation Board. Alberta’s Reserves of Crude Oil, Oil Sands, Gas, Natural Gas Liquids, and Sulphur, Calgary, Alberta, Canada, December 1990.

12. Globe and Mail, various issues.

13. Government of Alberta. Natural Gas Royalty Regulation.

14. Hotelling, Harold, The Economics of Exhaustible Resources. Journal of Political Economy 39 (2), 1931, 137-75.

15. Miller, M. H. and C. W. Upton, A Test of the Hotelling Valuation Principle. Journal of Political Economy 93 (1), 1985, 1-25.

16. —— and——, The Pricing of Oil and Gas: Some Further Results. Journal of Finance 40 (3), 1985, 1009-20.

17. National Energy Board. Canadian Energy Supply and Demand 1990-2010. Ottawa: June 1991.

18. ——. Canadian Energy Supply and Demand 1982-2005. Ottawa: September 1988.

19. ——. Canadian Energy Supply and Demand 1980-2000. Ottawa: June 1981.

20. Natural Gas Focus, various issues.

21. Oil and Gas Journal, December 2, 1991.

22. ——, January 27, 1992.

23. ——, January 25, 1993.

24. Petroleum Monitoring Agency. 1990 Monitoring Report, First Six Months.

25. ——. 1990 Canadian Petroleum Industry Monitoring Report, 1991.

26. Shell Canada. Crude Oil Price Bulletin, various issues.

27. Solow, Robert M. The Economics of Resources or the Resources of Economies. American Economic Review, May 1974, 1-14.

29. Watkins, G. C. The Development of Canadian Controls on Natural Gas Exports. International Research Center for Energy and Economic Development, Occasional Paper Number Three, 1987.

30. ——, The Hotelling Principle: Autobahn or Cul de Sac? The Energy Journal, January 1992, 1-24.