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When to drill? Trigger prices for the Arctic National Wildlife Refuge
Resource and Energy Economics 27 (2005) 273–286 www.elsevier.com/locate/ree
When to drill? Trigger prices for the Arctic National Wildlife Refuge Jon M. Conrad 1,*, Koji Kotani 1 Department of Applied Economics and Management, 455, Warren Hall, Cornell University, Ithaca, NY, 14853, USA Received 31 March 2004; received in revised form 25 August 2004; accepted 12 January 2005 Available online 13 April 2005
Abstract The social net benefit of drilling for oil in the Arctic National Wildlife Refuge (ANWR) has been a contentious policy issue since 1998. This paper applies real option theory to the issue and asks ‘‘What price for crude oil would justify the investment in field development and the loss of an amenity (wilderness) dividend?’’ Trigger prices are identified for two stochastic processes; when crude oil prices evolve according to geometric Brownian motion (GBM) and when crude oil prices are meanreverting (M-R). # 2005 Elsevier B.V. All rights reserved. JEL classification: D81; Q32 Keywords: Resource development; Real option theory; ANWR
1. Introduction and overview The decision on whether to allow exploration and production of oil suspected to lie beneath the Arctic National Wildlife Refuge (ANWR) in Alaska has been a contentious policy issue since 1998. During the first term of President George W. Bush, Democrats in the U.S. Senate were joined by a few ‘‘moderate’’ Republicans to block attempts by the administration to remove the existing ban on drilling in the ANWR. With his * Corresponding author. Tel.: +1 607 255 7681; fax: +1 607 255 9984. E-mail address: [email protected] (J.M. Conrad). 1 Authorship is equally shared. 0928-7655/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.reseneeco.2005.01.001
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re-election, and Republican majorities in the U.S. Senate and House of Representatives, it seems likely that the question of drilling in the ANWR will be revisited in 2005. In this paper we evaluate the prospect of exploring for oil in the ANWR within a realoption framework, as described in Dixit and Pindyck (1994) and Trigeorgis (1996). This framework is especially well-suited to evaluate investments which are risky and impossible, or very costly, to reverse. If an investment can be viewed as a binary (0–1) variable, then the real-option approach becomes a ‘‘stopping-rule problem.’’ In a stopping-rule problem one is continuously trying to determine whether it is optimal to switch a choice variable, say B(t) = 0, denoting no oil field development, to B(t) = 1, when oil field development is initiated. In stopping-rule problems, the decision-maker is monitoring the stochastic evolution of a state variable and waiting until the state variable reaches a threshold which says ‘‘Costly and irreversible investment is now optimal.’’ In our models of the ANWR, the stochastic state variable will be the cost to a refinery of buying a barrel of crude oil ($/barrel). This price is referred to as ‘‘refiner acquisition cost.’’ We wish to determine the price which would optimally trigger oil field development. We assume if oil is extracted from the ANWR that, in the minds of many people, the amenity dividend is irrevocably lost. Specifically, we assume that a switch from B(t) = 0 to B(t) = 1 incurs not only the cost of development, production, and transport, but also the irreversible loss of amenity value. We will vary the size of the amenity loss from $ 200 million to $ 300 million per year to determine the sensitivity of the trigger price.1 A key question which must be addressed before determining the price which would trigger exploration and development of any oil beneath the ANWR is ‘‘How do oil prices evolve over time?’’ We will determine trigger prices for two alternative price processes. The model of geometric Brownian motion (GBM) assumes that the price of crude oil, P = P(t), evolves according to dP ¼ mP dt þ sP dz
(1) pffiffiffiffi where m is called the mean drift rate, s > 0 is the standard deviation rate, and dz ¼ eðtÞ dt is the increment of a Wiener Process, with e(t) being a standard normal variate. This price process is referred to as a ‘‘continuous-time random walk with drift.’’ The other plausible price process is a mean-reverting (M-R) process which assumes that the price of crude oil evolves according to dP ¼ hðP¯ PÞ dt þ s dz
(2)
where P¯ > 0 is the mean or average price which prices tend to revert to over pffiffiffiffi time, h > 0 is the speed of reversion, s > 0 is the standard deviation rate and dz ¼ eðtÞ dt is, once again, the increment of a Wiener Process.2 1 There are approximately 100 million households in the U.S. If each household were willing to pay $ 2 or $ 3 per year, to preserve a pristine ANWR, the amenity dividend would be $ 200 million or $ 300 million per year. Welsh and Poe (1998) report a median willingness-to-pay for modifying water releases from the Glen Canyon Dam to improve environmental conditions in the Grand Canyon to be at least $ 9.92 per household, per year. Thus, we do not regard an amenity dividend of $ 200 million to $ 300 million to be unrealistic. 2 Alternative forms of the mean-reverting process include dP ¼ hðP¯ PÞP dt þ sP dz and dP ¼ hðP¯ PÞ dtþ sP dz. The specification in Eq. (2) is used by Pindyck and Rubinfeld (1998) in their analysis of crude oil prices.
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GBM is a tractable stochastic process, and in our model, assuming that field development of the ANWR is irreversible, it will allow us to derive an analytic expression for the trigger price. This expression will depend on five parameters, describing field development and production, the discount rate, d, m and s from Eq. (1), and the size of the annual foregone amenity value, A.3 For the M-R process there is no analytic expression for the trigger price, but it is possible to numerically solve for the value function, which will depend on the current price for oil, ¯ P, the five field and production parameters, the discount rate, d, amenity value, A, and h, P, and s from Eq. (2). By examining the numerical values for the value function, one can identify the price that triggers field development. We will vary amenity value, re-solve for the value function and the price trigger, and compare the results under the M-R process to those obtained under GBM. With GBM, it is also possible to allow amenity value to grow exponentially, and/or to allow production from the ANWR to decline exponentially, and still obtain an analytic expression for the trigger price. This more complex analytic expression contains the earlier expression, where amenity value and production were constant, as a special case. Papers that view oil field development as a real option include Paddock et al. (1988) and Cortazar and Schwartz (1997). In both papers the perspective is that of an oil company currently holding or about to bid for the option to explore and develop oil underlying a tract of land or the continental shelf. In these papers the oil company earns no dividend while waiting, may face an expiry date (where the option reverts back to the government), and will pay royalties and a corporate income tax on the oil it extracts. Our perspective is that of the U.S. government trying to decide when it is in the country’s interest to open the ANWR for leasing, development and production. We view the amenity flow from the pristine ANWR as a legitimate ‘‘social dividend.’’ In the next section we provide some background on the ANWR, identifying the technical parameters common to both the GBM and M-R models. In Section 3 we present ¯ and s, for the M-R the crude oil price data and estimate m and s, for GBM, and h, P, process. In Section 4 we compute and compare the trigger prices for GBM and the M-R process. We offer some conclusions and a critique of our models and analysis in Section 5.
2. The Arctic National Wildlife Refuge (ANWR) The ANWR is located in the northeastern corner of Alaska and is bounded by the Canadian Yukon to the east and the Beaufort Sea to the north (see Fig. 1). The ANWR coastal plain, where oil and gas development would take place, is about 100 miles across and 30 miles wide, an area slightly larger than the State of Delaware. In 1998, the U.S. Geological Survey issued a report assessing the oil and gas potential for what is referred to 3 A reviewer asked why we did not use a contingent claims approach. This would avoid the need of specifying and defending a risk-adjusted discount rate, but would require us to estimate the convenience yield for crude oil. We adopt a discount rate of d = 0.1, which from the U.S. Government’s perspective, might reflect a social rate of time preference of 0.025 and a risk premium of 0.075 for extracting and transporting oil in a harsh but ecologically fragile environment.
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Fig. 1. A map of Alaska and the ANWR.
as Federal Area 1002 of the ANWR. Area 1002 includes most of the ANWR coastal plain. The report states that the amount of oil beneath Area 1002 has a 0.95 probability of being at least 11.6 billion barrels and a 0.05 probability of being at least 31.5 billion barrels. The mean, or expected, oil in situ was estimated to be 20.7 billion barrels. The amount of economically recoverable oil will ultimately depend on recovery technology and the price of crude, both of which will be evolving during the economic life of the field. Attanasi and Schuenemeyer (2002), in their Fig. 4B, compute three incremental costs curves based on three subjective distributions for the amount of oil discovered. For the triangular distribution the marginal cost of production from a field with 2 109 to 8 109 barrels of oil would appear to range from $ 13.50 to $ 21.00/barrel. In the model where price evolves according to GBM, we will vary marginal cost, c, from $ 10 to $ 25/barrel. Once developed, the ANWR field would have an expected life of t = 65 years. Our estimate of average annual production is Q = 100,000,000 (barrel/year), for a cumulative production of tQ = 6.5 billion barrels. This estimate is smaller than the 9.12 billion barrels which the oil industry expects to produce if oil prices are in excess of $ 24/barrel. These parameter values are, however, consistent with those reported by the McDowell Group (2002). While the productive life for an ANWR oil field is expected to be t = 65 years, there will be a construction or field development phase expected to be l = 5 years. During the construction phase, a present-value fixed cost of K = $ 2.9 billion is expected (McDowell Group, 2002). We adopt an annual discount rate of d = 0.10 as the appropriate risk-adjusted annual rate for this project. This rate of discount may seem high, but from the U.S. Government’s perspective, it should reflect the social rate of time preference and the risk of
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oil spills in the ANWR Coastal Plain, along the length of the Trans-Alaskan Pipeline, and in Prince William Sound, site of the Exxon Valdez spill. Six parameters, c, t, Q, l, K, and d, will be common to both the GBM and the M-R models. The other variable common to both models is amenity value, A ($/year) which is assumed to be irrevocably lost if oil development takes place. We have no idea what this amount should be, so we will vary it between $ 200 million and $ 300 million per year to determine the sensitivity of trigger prices in both the GBM and M-R models. As noted earlier, this range for amenity value might correspond to a willingness-to-pay of $ 2 to $ 3 per household per year. If crude oil prices evolve according to a M-R process, we will also need estimates of m and s, the mean drift and standard deviation rates in Eq. (1). If crude oil prices evolve ¯ and s, the speed of reversion, according to a M-R process, we will need to estimate h, P, mean to which prices revert, and standard deviation rate in Eq. (2).
¯ and s for the M-R process 3. Estimation of m and s for GBM and h, P, Fig. 2 shows the price of crude oil paid by refiners for the years 1968 through 2001. These prices are in real 1996 dollars. The data may be found at the U.S. Department of Energy web site: http://www.eia.doe.gov/emeu/aer/txt/ptb0519.html. This series contains 34 observations. Dixit and Pindyck (1994), in their discussion of mean-reverting processes, present a time-series of real crude oil prices spanning 120 years, from 1870 through 1990. Running a unit-root test for the full 120 years of data they reject GBM and fail to reject that the data were generated by an M-R process. They note, however, that if one does unit-root tests for the most recent 30 or 40 years, one cannot reject the hypothesis that the data were generated by GBM. This turned out to be the case for the data shown in Fig. 2. We could not reject GBM nor the M-R process, so we keep both models ‘‘in play,’’ and present estimates ¯ and s for the M-R process in Eq. (2). of m and s for GBM in Eq. (1) and for h, P,
Fig. 2. The real price of crude oil, 1968–2001 (in 1996 dollars).
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Table 1 Parameter values Parameter
Value
c t Q l K d m ¯ sˆ hˆ Pˆ¯ sˆ aˆ bˆ se Adjusted R 2
15 (marginal cost, $/barrel) 65 (years of production) 100,000,000 (average annual production) 5 (construction phase, years) $ 2.9 billion (present value investment) 0.1 (annual discount rate) 0.04 [GBM, Eq. (1)] 0.05 [GBM, Eq. (1)] 0.18 [M-R, Eq. (2)] 25.09 [M-R, Eq. (2)] 4.5 [M-R, Eq. (2)] 4.07 (1.71) [M-R, Eq. (3)] 0.16 (1.75) [M-R, Eq. (3)] 5.79 [M-R, Eq. (3)] 0.06
The Maximum Likelihood Estimates of m and s for GBM can be obtained by calculating the mean, m, and standard deviation, s, of the series ln(Pt+1/Pt) [see Reed and ˆ ¼ m þ s2 =2 and sˆ ¼ s. Clarke, 1990, p. 149, for a discussion]. Then, m For the parameters of the M-R process, we ran the regression Ptþ1 Pt ¼ a þ bPt þ et
(3) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ 2 1 (see ˆ hˆ ¼ lnð1 þ bÞ, ˆ and sˆ ¼ sˆ e hˆ =½ð1 þ bÞ and then calculated P¯ ¼ ˆa=b; Dixit and Pindyck, 1994, pp. 76–77). The regression results giving rise to the M-R parameter estimates, the estimates of m and s for GBM, and the base-case parameter values for c, t, Q, l, K, and d are summarized in Table 1.
4. Trigger prices under GBM and the M-R process We first consider the case where oil prices follow GBM. We need to determine two value functions; the value function for the ANWR while optimally waiting to develop, which will be denoted by VW(P), and the value function when field development is started, which will be denoted by VF(P). Recall that there is a construction phase of l = 5 years before oil production begins and that production is expected to last t = 65 years, with an average output of Q = 100,000,000 barrels per year. (We will subsequently allow oil production to exponentially decline from a higher initial level, but for now we assume Q is constant at 100 million barrels per year.) Suppose when construction is initiated that the price of crude oil is P. With prices evolving according to GBM we expect the price to be P eml when production starts l years from the start of construction. Recall that m is the expected drift rate in the price of crude oil. (We assume that the discount rate, d, exceeds the expected drift rate for crude
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oil prices, so that d > m.) Then, the expected price at instant t l would be P emt. With the present value for the fixed cost of field development at K = $ 2.9 billion, the expected present value of the ANWR field, when construction is initiated with an oil price of P, is given by Z lþt VF ðPÞ ¼ K þ Q ½P emt c edt dt (4) l
Integrating yields the analytic expression ðdmÞt ÞP eml dl ð1 e dt ð1 e Þðc=dÞ VF ðPÞ ¼ K þ Q e ðd mÞ
(5)
As part of the smooth-pasting condition, to be discussed in a moment, we will need the expression for the derivative VF0 ðPÞ which is easily seen to be ml e 0 dl ðdmÞt Þ VF ðPÞ ¼ Q e ð1 e (6) dm The value function while optimally waiting to develop, VW(P), must satisfy the Hamilton–Jacobi–Bellman (H–J–B) Equation which requires 2 s 0 00 dVW ðPÞ ¼ A þ mPVW ðPÞ þ ðPÞ (7) P2 V W 2 where A is the constant amenity dividend provided by an undisturbed ANWR. The solution to this second-order ordinary differential equation is VW ðPÞ ¼
A þ BPb d
(8)
where B > 0 is an unknown constant, which will be determined using the value-matching and smooth-pasting conditions, and where b > 1 is the positive root of a quadratic. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m 1 m 2 2d 2 þ 2 þ 2 (9) b¼ 2 s 2 s s The two terms on the right-hand side of Eq. (8) have the following interpretation. A/d is the present value of the amenity dividend in perpetuity, and is thus the value of never developing the ANWR’s oil potential. BPb is the value of the option to develop the ANWR’s oil potential when the price of crude oil is currently P. At the trigger price, where one is indifferent between preserving a pristine ANWR and developing its oil potential, the value-matching condition requires VW(P) = VF(P) or ð1 eðdmÞt ÞP eml ð1 edt Þðc=dÞ (10) A=d þ BPb ¼ K þ Q edl ðd mÞ To ensure continuity when one switches from B(t) = 0 (no development) to B(t) = 1 (oil 0 ðPÞ ¼ VF0 ðPÞ at the trigger development), the smooth-pasting condition requires that VW
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price, or bBPb1 ¼ Q edl ð1 eðdmÞt Þ
eml dm
(11)
The value-matching condition, Eq. (10), and the smooth-pasting condition, Eq. (11), constitute a two-equation system in the unknown constant B > 0, and the trigger price, P*. Some algebra leads to the following analytic expression for P*: P ¼
½A þ dK þ edl Qð1 edt Þcbðd mÞ dðb 1Þ eðdmÞl Qð1 eðdmÞt Þ
(12)
Knowing P*, you can determine B according to B¼
eðdmÞl ðP Þð1bÞ Qð1 eðdmÞt Þ bðd mÞ
(13)
For GBM it is possible to allow amenity value to grow exponentially and/or production to decline exponentially and to solve the revised value-matching and smooth-pasting conditions for an analytic expression for P*. This expression contains Eq. (12) as a special case. Suppose AðtÞ ¼ A0 egt , where d > g 0 and g is the rate at which amenity value is increasing over time. Suppose further that QðtÞ ¼ Q5 eht , where d > h 0 and h is the rate at which production declines from some initial level Q5 at t = l = 5. Given a value for h, we solve for Q5 so that cumulative production (with exponential decline at rate h) equals 6.5 billion barrels; the same cumulative production as in the base case, when amenity value and the rate of production were constant. This results in Q5 > Q = 100 million barrel/year. The expression for P* is given by P ¼
bðd þ h mÞ½Aðd þ hÞ þ Kðd gÞðd þ hÞ þ Q5 cðd gÞðedl eðdþhÞtdl Þ ðb 1Þðd gÞðd þ hÞQ5 ðeðdmÞl eðdþhmÞtðdmÞl Þ (14)
With mean-reverting oil prices, if development is initiated at price P, VF(P) will take the form dt ¯ ¯ ehl ÞðP cÞ ð1 eðdþhÞt ÞðP PÞ dl ð1 e þ VF ðPÞ ¼ K þ Q e (15) d dþh ¯ eht . VF(P) uses the result, when prices are M-R, that EfPðtÞjPð0Þg ¼ P¯ þ ðPð0Þ PÞ While optimally waiting to develop the ANWR, the value function, VW(P), must satisfy an H–J–B Equation which now takes the form 2 s 0 00 dVW ðPÞ ¼ A þ hðP¯ PÞVW ðPÞ þ ðPÞ (16) VW 2 While there is no analytic solution for the VW(P) which satisfies Eq. (16), we can solve a finite-time analogue using the Implicit Finite Difference Method (IFDM), as discussed in Hull (2003), Wilmott et al. (1995) and Tavella and Randall (2000). The IFDM provides a numerical approximation for the value function that satisfies Eq. (16). The
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problem is posed as a finite-horizon problem and the domain for time and price are in the bounded intervals t 2 [0, T] and P 2 [0, PMAX]. T must be sufficiently large so that the finite-horizon value function, evaluated at t = 0, yields a good approximation to the infinite-horizon value function. PMAX must be sufficiently high so that field development is optimal. For a finite-horizon, the H–J–B Equation, corresponding to Eq. (16), becomes 2 s W dV W ðP; tÞ ¼ A þ VtW ðP; tÞ þ hðP¯ PÞVPW ðP; tÞ þ ðP; tÞ (17) VPP 2 where the superscript W now indicates that this is the value function while optimally waiting and the subscripts denote partial derivatives with respect to time, t, or price, P, since VW(P,t) depends on both price and time in the finite-horizon approximation. Because there is no analytic solution, the value-matching and smooth-pasting conditions cannot be imposed directly. Instead, this problem may be formulated as a linear complementarity problem (LCP) (Wilmott et al., 1993; Insley, 2002). Define 2 s W W W W ¯ HV dV ðP; tÞ A þ Vt ðP; tÞ þ hðP PÞVP ðP; tÞ þ (18) VPP ðP; tÞ 2 The conditions for the LCP require (a) HV 0, (b) VW(P,t) VF(P) 0, and (c) HV[VW(P,t)–VF(P)] = 0. Condition (a) says that dVW(P,t) must be greater than or equal to the sum of the dividend and the expected capital gain of the value function while waiting to develop. Condition (b) says that the value while waiting should not go below the expected net revenue from exercising the option now, VF(P). Condition (c) says that either (a) or (b) must hold as a strict equality. This implies that if HV = 0, then VW(P,t) VF(P) > 0, and it is optimal to hold the option. If HV > 0, then VW(P,t) VF(P) = 0, and it is optimal to exercise the option. Conditions (a)–(c) would be satisfied by a rational social planner holding the option to drill in the ANWR. When (a)–(c) hold, Friedman (1988), has shown that the valuematching and smooth-pasting conditions are satisfied. Numerical solution, using the IFDM, requires that certain boundary and terminal conditions hold at P = 0, P = PMAX, and t = T, respectively. These conditions require V W ð0; tÞ ¼
A d
8 t 2 ½0; T ðBoundary Condition OneÞ
V W ðPMAX ; tÞ ¼ VF ðPMAX Þ 8 t 2 ½0; T ðBoundary Condition TwoÞ A W V ðP; TÞ ¼ max VF ðPÞ; 8 P 2 ½0; PMAX ðTerminal ConditionÞ d Boundary Condition One says that when the price of oil is zero, the value function while waiting is the present value of the amenity dividend in perpetuity.4 Boundary Condition 4 Boundary Condition One is not strictly correct for the M-R process given by Eq. (2). A zero oil price at instant t would support a positive conditional probability that P(t + dt) > 0. The value function at P = 0 may be greater than A/d, reflecting the option to develop when price becomes sufficiently positive. If, however, the option value at P = 0 is small relative to A/d, which is likely the case, Boundary Condition One will be approximately correct in the numerical context of our solution algorithm.
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Table 2 Trigger prices for GBM when A(t) = A0 eyt and Q(t) = Q5 eht A or A0 ($/year)
P* (g = h = 0)
P* (g = 0.005, h = 0)
P* (g = 0, h = 0.005)
200000000 210000000 220000000 230000000 240000000 250000000 260000000 270000000 280000000 290000000 300000000
19.84 19.99 20.13 20.27 20.41 20.55 20.69 20.84 20.97 21.12 21.26
19.99 20.14 20.29 20.44 20.59 20.74 20.89 21.04 21.19 21.34 21.49
19.62 19.75 19.88 20.02 20.15 20.28 20.41 20.54 20.67 20.80 20.93
Notes: (a) When g = h = 0 and A = 0, P* = $ 17.01/barrel. (b) When g > 0, A0 varies from $ 200 million to $ 300 million. When h > 0, Q5 is calculated so that cumulative production remains at 6.5 billion barrels for t = 65 years. These trigger prices are for the base-case marginal cost, c = $ 15/barrel. For sensitivity analysis with regard to c, see Table 3.
Two says that at PMAX, it is optimal to develop. The Terminal Condition says that at the expiry date (t = T) the value function is the argument that maximizes [VF(P), A/d]. The IFDM then seeks to determine the values of VW(P,t) that simultaneously satisfy the LCP, and the boundary and terminal conditions. The domain of the LCP is divided into a finite grid, {0, DP, 2DP, . . ., NDP}{0, Dt, 2Dt, . . ., MDt}, such that N = PMAX/DP and M = T/Dt. There are several possible algorithms that might be used within the IFDM. We employ the projected successive over-relaxation method (PSOR) as discussed in Wilmott et al. (1995) and Tavella and Randall (2000). Our MATLAB program, or faster running code in C++ or C#, may be obtained by emailing the second author.5 In this paper, we choose PMAX = $ 50/barrel and T = 100 years, which are high and long enough to approximate the infinite-horizon problem. With DP = $ 0.01 and Dt = 1 (year) there are 505,101 interior, boundary, and terminal values that must be computed. The values for VW(P,t) are determined for A ranging from $ 200,000,000 to $ 300,000,000. The trigger prices for the M-R process are determined where VW(P,t) first coincides with VF(P), thus satisfying the LCP conditions.6 In Table 2 we report the trigger prices for GBM. Eq. (14) was used for the base-case parameter values in Table 1 and for the three cases, where [g = h = 0], [g = 0.005, h = 0], and [g = 0, h = 0.005]. When [g = h = 0], the trigger price varies from P* = $ 19.84/barrel when A = $ 200 million per year, to P* = $ 21.26/barrel when A = $ 300 million per year. We also note that when A = 0, P* = $ 17.01, so an annual amenity value of $ 200 million raises the trigger price by $ 2.83. 5 We owe a deep debt of gratitude to Linda Buttel and John Zollweg of the Cornell Theory Center who reprogrammed our original MATLAB code, reducing the run time from days to less than 3 min in the C# version. 6 The values for PMAX, T, DP, and Dt would appear to provide a very close approximation to the values for the infinite-horizon problems when evaluated at t = 0. We modified our M-R program to numerically solve for the trigger prices when oil prices follow GBM. We were curious to see how the numerical prices obtained using the PSOR algorithm would compare to those obtained from the analytic expression given by Eq. (12). The numerical values, for GBM, using the PSOR algorithm were identical, to the penny, to those obtained from the analytical expression for P*.
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Table 3 Trigger prices for GBM when marginal cost takes the values c = $ 10/barrel, c = $ 20/barrel, and c = $ 25/barrel, and when g = h = 0 A ($/year)
P* (c = $ 10/barrel)
P* (c = $ 20/barrel)
P* (c = $ 25/barrel)
200000000 210000000 220000000 230000000 240000000 250000000 260000000 270000000 280000000 290000000 300000000
15.55 15.69 15.83 15.97 16.11 16.26 16.40 16.54 16.68 16.83 16.97
24.14 24.28 24.42 24.57 24.71 24.85 24.99 25.13 25.28 25.42 25.56
28.44 28.58 28.72 28.86 29.01 29.15 29.29 29.43 29.57 29.71 29.86
Notes: (a) The values for P* when c = $ 15/barrel and g = h = 0 are found in the second column of Table 2. (b) For a given value of c, the change in P* is slightly in excess of $ 0.14 for each $ 10,000,000 increase in A.
When amenity value grows exponentially but oil production is constant, the trigger price increases for each level of A0. For the case where [g = 0.005, h = 0], the trigger price ranged from $ 19.99 when A0 = $ 200 million per year to $ 21.49 when A0 = $ 300 million per year. When amenity value is constant, but oil production declines, the trigger price is lower than that when [g = h = 0]. The reason for this is that Q5 is increased above the constant Q = 100 million barrels per year in order to keep cumulative production constant at 6.5 billion barrels over the life of the field. This has the effect of moving production closer to the present and thus lowering the trigger price. For [g = 0, h = 0.005], the trigger price ranges from $ 19.62 when A = $ 200 million per year to $ 20.93 when A = $ 300 per year. In Table 3, when prices evolve according to GBM with [g = h = 0], we vary the marginal cost parameter, c, from $ 10/barrel to $ 25/barrel, in increments of $ 5, omitting the case where c = $ 15/barrel, since it is reported in Table 2. This range for marginal cost is likely to contain the ‘‘true’’ marginal cost should the ANWR be developed. For c = $ 10/barrel and A = $ 200 million per year, the trigger price drops to P* = $ 15.55. At the other extreme, if c = $ 25/barrel and A = $ 300 million per year, P* = 29.86. For a given value of c, P* increases by slightly more that $ 0.14 for every $ 10,000,000 increase in A. In Table 4 we compare P* under GBM with [g = h = 0] to the M-R process for the basecase parameters with c = $ 15/barrel. As expected, the trigger prices under the M-R process are higher, ranging from P* = $ 25.41 when A = $ 200 million per year to P* = $ 31.42 when A = $ 300 million per year. Under GBM, the trigger prices were not particularly sensitive to the variation in A. The trigger prices under the M-R process were more sensitive to the variation in amenity value and, for A > $ 250 million per year, development of the ANWR would not be optimal unless the price of oil was greater than $ 28.29/barrel.7 7 We performed additional sensitivity analysis for the M-R process. For c = $ 18 per barrel, P* ranged from $ 37.29 per barrel (A = $ 200 million) to $ 46.20 per barrel (A = $ 300 million). For d = 0.135, P* = $ 38.31 per barrel (A = $ 200 million) and P* = $ 47.70 per barrel (A = $ 300 million). A higher marginal cost or discount rate raises the trigger price under both GBM and the M-R process. See Table 5.
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Table 4 Trigger prices for GBM and the M-R process A ($/year)
P* (GBM, g = h = 0)
P* (M-R process)
200000000 210000000 220000000 230000000 240000000 250000000 260000000 270000000 280000000 290000000 300000000
19.84 19.99 20.13 20.27 20.41 20.55 20.69 20.84 20.97 21.12 21.26
25.41 25.97 26.54 27.12 27.70 28.29 28.90 29.51 30.14 30.77 31.42
Note: The P* values for both GBM and the M-R process assume c = $ 15/barrel.
Table 5 Comparative statics of P* under GBM and the M-R process Price process
c
d
h
K
l
m
P¯
Q
s
t
GBM M-R
+ +
+ +
n.a
+ +
+ +
n.a
n.a
+ +
Note: n.a. = not applicable.
The comparative statics of P* under GBM or the M-R process, for constant amenity and production [g = h = 0], are summarized in Table 5. For GBM, an increase in c, d, K, l, or s cause P* to increase, while an increase in m, Q, or t cause P* to decrease. For the M-R ¯ Q, process, an increase in c, d, K, l, or s also cause P* to increase, while an increase in h, P, * or t cause P to decrease.
5. Conclusions and caveats This paper has determined, from the perspective of the U.S. Government, the per barrel price for crude oil which would optimally trigger the exploration for oil in the coastal plain of the ANWR. The problem was posed as an optimal stopping problem, where the price of crude oil evolved according to geometric Brownian motion or a mean-reverting process. For geometric Brownian motion, it was possible to derive an analytic solution for the trigger price. For [g = h = 0], and for variations in A ranging from $ 200 million to $ 300 million per year, the trigger price ranged from $ 19.84/barrel to $ 21.26/barrel. With no amenity loss (A = 0) the trigger price was $ 17.01/barrel. If amenity value is exponentially growing and/or oil production is exponentially declining (but with cumulative production constant over the life of the field), it is still possible to obtain an analytic expression for P*, if oil prices follow a process of GBM. If amenity values were instantaneously growing at a rate g = 0.005 from A0 = $ 200 million, while oil production was constant at Q = 100 million barrels, P* = $ 19.99, only slightly greater than P* = $ 19.84 when g = 0. For GBM, increases in amenity value did not significantly increase the trigger price.
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If crude oil prices follow a mean-reverting process, it is not possible to derive a closedform solution for P*, but it is possible to numerically solve for the trigger price after approximating the value function. When A ranges from $ 200 million to $ 300 million per year, the price which would trigger exploration ranges from $ 25.41/barrel to $ 31.42/ barrel, respectively. The expected present net revenue of an ANWR field producing 6.5 billion barrels of oil over a 65-year horizon will depend on the stochastic price process and the price when development is initiated. The expected present net revenue of the ANWR field is higher under geometric Brownian motion than under a mean-reverting process. For example, under geometric Brownian motion, and with an initial price of $ 24/barrel, the expected present net revenue is $ 17.049 billion. Under a mean-reverting process, for the same initial price of $ 24/barrel, the 6.5 billion barrel field is only worth $ 3.115 billion. The expected present net revenue under the mean-reverting process is smaller since our estimate of P¯ was $ 25.09/barrel, based on data for the period 1968–2001. Our models might be criticized as being oversimplified and subjective in at least three respects. First, production from the ANWR, if it ever occurs, would likely start below 100 million barrels per year, increase to over 100 million barrels per year for some number of years, and then decline. This shift in the extraction profile is likely to decrease the trigger price compared to constant production in the basic GBM and M-R models. Second, we do not know what the annual amenity value is for an undisturbed (pristine) ANWR. We assumed that whatever its value, that it would be irrevocably lost if oil development takes place. Some claim that the caribou and polar bear will be unaffected by production facilities and pipelines, and point to our experience with the production facilities at Prudhoe Bay and along the Trans-Alaska Pipeline. The Central Arctic Caribou Herd, which calves in the Prudhoe Bay and Kuparuk oil fields is currently estimated at 23,400 animals, having increased from 3000 animals during the four decades of development and production at Prudhoe Bay. While the incremental risk of extinction for any extant arctic species as a result of ANWR would seem vanishingly small (given the extent other undisturbed arctic areas), we opted to take the perspective of a conservationist who would view the production facilities and connecting pipeline as an irreversible violation of the arctic wilderness, even if the caribou, bear, and other wildlife might be more tolerant. Third, we argued earlier that a discount rate of d = 0.1 might appropriately reflect the government’s risk premium when extracting and transporting oil in a harsh but ecologically fragile environment. This is a subjective judgment, and some might regard a risk premium on the order of 0.075 as being too high. The effect of all three of these simplifications or subjective assessments (constant production, high amenity value, and a high discount rate) would work in the same direction; that is, to raise the trigger price under either GBM or the M-R process. Despite this, the trigger prices, as of the Fall of 2004, were below the current spot prices. It has been argued that the amount of oil likely to be recovered from the ANWR does not justify the environmental risk, and that the United States would be better served by adapting to high oil prices through conservation. It is worth remembering that high oil prices in the mid-1970s did induce a more energy efficient stock of appliances, houses, and automobiles, and that both prices and oil consumption declined in the early 1980s (Fig. 2).
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If and when the issue of drilling for oil in the ANWR is revisited, it will be interesting to see if conservation, the global economy, or the political situation in the Middle East will have caused oil prices to revert to lower levels, so that drilling the ANWR is less attractive, or whether they will remain in the $ 30–$ 40/barrel range and the U.S. Congress and President decide to exercise the ANWR option.
Acknowledgements We specifically want to thank Linda Buttel, Toshihiro Oka, Takahiro Tsuge, Kenji Takeuchi, Masanobu Ishikawa, Hiroshi Takamori, Hiroshi Yamaguchi, John Zollwig and two anonymous reviewers. Earlier versions of the paper were presented at Cornell University, the 2004 ASSA Meetings in San Diego, California, the 2004 Japanese Economics Association Meetings held at Meiji-Gakuin University, the 2003 Environmental Economics Workshop at Kobe University, and the 2003 Real Options Workshop held at Aoyama-Gakuin University.
References Attanasi, E.D., Schuenemeyer, J.H., 2002. Some aspects of resource uncertainty and their economic consequences in the assessment of the 1002 area of the Arctic National Wildlife Refuge. Natural Resources Research 11 (2), 109–120. Cortazar, G., Schwartz, E.S., 1997. Implementing a real option model for valuing an undeveloped oil field. International Transactions in Operational Research 4 (2), 125–137. Dixit, A.K., Pindyck, R.S, 1994. Investment Under Uncertainty. Princeton University Press, Princeton, NJ. Friedman, A., 1988. Variational Principles and Free Boundary Problems. Krieger Publishing Company, Melbourne, FL. Hull, J.C, 2003. Options, Futures and Other Derivatives, 5th ed. Prentice-Hall, Upper Saddle River, NJ. Insley, M., 2002. A real options approach to the valuation of a forestry investment. Journal of Environmental Economics and Management 44 (3), 471–492. McDowell Group, 2002. ANWR and the Alaska Economy: An Economic Impact Assessment. Juneau, Alaska. Paddock, J.L, Siegel, D.R., Smith, J.L., 1988. Option valuation of claims to real assets: the case of offshore leases. Quarterly Journal of Economics 103 (3), 479–508. Pindyck, R.S., Rubinfeld, D.L, 1998. Economic Models and Economic Forecasts. Irwin/McGraw-Hill, Boston/ Massachusetts. Reed, W.J., Clarke, H.R., 1990. Harvest decisions and asset valuation for biological resources exhibiting sizedependent stochastic growth. International Economic Review 31 (1), 147–169. Trigeorgis, L., 1996. Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT Press, Cambridge, MA. Tavella, D., Randall, C., 2000. Pricing Financial Instruments: The Finite Difference Method. Wiley, New York. Welsh, M.P., Poe, G.L., 1998. Elicitation effects in contingent valuation: comparisons to a multiple bounded discrete choice approach. Journal of Environmental Economics and Management 36 (2), 170–185. Wilmott, P., Howison, S., Dewynne, J., 1995. The Mathematics of Financial Derivatives. Cambridge University Press, Cambridge. Wilmott, P., Howison, S., Dewynne, J., 1993. Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford.
When to drill? Trigger prices for the Arctic National Wildlife Refuge Jon M. Conrad 1,*, Koji Kotani 1 Department of Applied Economics and Management, 455, Warren Hall, Cornell University, Ithaca, NY, 14853, USA Received 31 March 2004; received in revised form 25 August 2004; accepted 12 January 2005 Available online 13 April 2005
Abstract The social net benefit of drilling for oil in the Arctic National Wildlife Refuge (ANWR) has been a contentious policy issue since 1998. This paper applies real option theory to the issue and asks ‘‘What price for crude oil would justify the investment in field development and the loss of an amenity (wilderness) dividend?’’ Trigger prices are identified for two stochastic processes; when crude oil prices evolve according to geometric Brownian motion (GBM) and when crude oil prices are meanreverting (M-R). # 2005 Elsevier B.V. All rights reserved. JEL classification: D81; Q32 Keywords: Resource development; Real option theory; ANWR
1. Introduction and overview The decision on whether to allow exploration and production of oil suspected to lie beneath the Arctic National Wildlife Refuge (ANWR) in Alaska has been a contentious policy issue since 1998. During the first term of President George W. Bush, Democrats in the U.S. Senate were joined by a few ‘‘moderate’’ Republicans to block attempts by the administration to remove the existing ban on drilling in the ANWR. With his * Corresponding author. Tel.: +1 607 255 7681; fax: +1 607 255 9984. E-mail address: [email protected] (J.M. Conrad). 1 Authorship is equally shared. 0928-7655/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.reseneeco.2005.01.001
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re-election, and Republican majorities in the U.S. Senate and House of Representatives, it seems likely that the question of drilling in the ANWR will be revisited in 2005. In this paper we evaluate the prospect of exploring for oil in the ANWR within a realoption framework, as described in Dixit and Pindyck (1994) and Trigeorgis (1996). This framework is especially well-suited to evaluate investments which are risky and impossible, or very costly, to reverse. If an investment can be viewed as a binary (0–1) variable, then the real-option approach becomes a ‘‘stopping-rule problem.’’ In a stopping-rule problem one is continuously trying to determine whether it is optimal to switch a choice variable, say B(t) = 0, denoting no oil field development, to B(t) = 1, when oil field development is initiated. In stopping-rule problems, the decision-maker is monitoring the stochastic evolution of a state variable and waiting until the state variable reaches a threshold which says ‘‘Costly and irreversible investment is now optimal.’’ In our models of the ANWR, the stochastic state variable will be the cost to a refinery of buying a barrel of crude oil ($/barrel). This price is referred to as ‘‘refiner acquisition cost.’’ We wish to determine the price which would optimally trigger oil field development. We assume if oil is extracted from the ANWR that, in the minds of many people, the amenity dividend is irrevocably lost. Specifically, we assume that a switch from B(t) = 0 to B(t) = 1 incurs not only the cost of development, production, and transport, but also the irreversible loss of amenity value. We will vary the size of the amenity loss from $ 200 million to $ 300 million per year to determine the sensitivity of the trigger price.1 A key question which must be addressed before determining the price which would trigger exploration and development of any oil beneath the ANWR is ‘‘How do oil prices evolve over time?’’ We will determine trigger prices for two alternative price processes. The model of geometric Brownian motion (GBM) assumes that the price of crude oil, P = P(t), evolves according to dP ¼ mP dt þ sP dz
(1) pffiffiffiffi where m is called the mean drift rate, s > 0 is the standard deviation rate, and dz ¼ eðtÞ dt is the increment of a Wiener Process, with e(t) being a standard normal variate. This price process is referred to as a ‘‘continuous-time random walk with drift.’’ The other plausible price process is a mean-reverting (M-R) process which assumes that the price of crude oil evolves according to dP ¼ hðP¯ PÞ dt þ s dz
(2)
where P¯ > 0 is the mean or average price which prices tend to revert to over pffiffiffiffi time, h > 0 is the speed of reversion, s > 0 is the standard deviation rate and dz ¼ eðtÞ dt is, once again, the increment of a Wiener Process.2 1 There are approximately 100 million households in the U.S. If each household were willing to pay $ 2 or $ 3 per year, to preserve a pristine ANWR, the amenity dividend would be $ 200 million or $ 300 million per year. Welsh and Poe (1998) report a median willingness-to-pay for modifying water releases from the Glen Canyon Dam to improve environmental conditions in the Grand Canyon to be at least $ 9.92 per household, per year. Thus, we do not regard an amenity dividend of $ 200 million to $ 300 million to be unrealistic. 2 Alternative forms of the mean-reverting process include dP ¼ hðP¯ PÞP dt þ sP dz and dP ¼ hðP¯ PÞ dtþ sP dz. The specification in Eq. (2) is used by Pindyck and Rubinfeld (1998) in their analysis of crude oil prices.
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GBM is a tractable stochastic process, and in our model, assuming that field development of the ANWR is irreversible, it will allow us to derive an analytic expression for the trigger price. This expression will depend on five parameters, describing field development and production, the discount rate, d, m and s from Eq. (1), and the size of the annual foregone amenity value, A.3 For the M-R process there is no analytic expression for the trigger price, but it is possible to numerically solve for the value function, which will depend on the current price for oil, ¯ P, the five field and production parameters, the discount rate, d, amenity value, A, and h, P, and s from Eq. (2). By examining the numerical values for the value function, one can identify the price that triggers field development. We will vary amenity value, re-solve for the value function and the price trigger, and compare the results under the M-R process to those obtained under GBM. With GBM, it is also possible to allow amenity value to grow exponentially, and/or to allow production from the ANWR to decline exponentially, and still obtain an analytic expression for the trigger price. This more complex analytic expression contains the earlier expression, where amenity value and production were constant, as a special case. Papers that view oil field development as a real option include Paddock et al. (1988) and Cortazar and Schwartz (1997). In both papers the perspective is that of an oil company currently holding or about to bid for the option to explore and develop oil underlying a tract of land or the continental shelf. In these papers the oil company earns no dividend while waiting, may face an expiry date (where the option reverts back to the government), and will pay royalties and a corporate income tax on the oil it extracts. Our perspective is that of the U.S. government trying to decide when it is in the country’s interest to open the ANWR for leasing, development and production. We view the amenity flow from the pristine ANWR as a legitimate ‘‘social dividend.’’ In the next section we provide some background on the ANWR, identifying the technical parameters common to both the GBM and M-R models. In Section 3 we present ¯ and s, for the M-R the crude oil price data and estimate m and s, for GBM, and h, P, process. In Section 4 we compute and compare the trigger prices for GBM and the M-R process. We offer some conclusions and a critique of our models and analysis in Section 5.
2. The Arctic National Wildlife Refuge (ANWR) The ANWR is located in the northeastern corner of Alaska and is bounded by the Canadian Yukon to the east and the Beaufort Sea to the north (see Fig. 1). The ANWR coastal plain, where oil and gas development would take place, is about 100 miles across and 30 miles wide, an area slightly larger than the State of Delaware. In 1998, the U.S. Geological Survey issued a report assessing the oil and gas potential for what is referred to 3 A reviewer asked why we did not use a contingent claims approach. This would avoid the need of specifying and defending a risk-adjusted discount rate, but would require us to estimate the convenience yield for crude oil. We adopt a discount rate of d = 0.1, which from the U.S. Government’s perspective, might reflect a social rate of time preference of 0.025 and a risk premium of 0.075 for extracting and transporting oil in a harsh but ecologically fragile environment.
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Fig. 1. A map of Alaska and the ANWR.
as Federal Area 1002 of the ANWR. Area 1002 includes most of the ANWR coastal plain. The report states that the amount of oil beneath Area 1002 has a 0.95 probability of being at least 11.6 billion barrels and a 0.05 probability of being at least 31.5 billion barrels. The mean, or expected, oil in situ was estimated to be 20.7 billion barrels. The amount of economically recoverable oil will ultimately depend on recovery technology and the price of crude, both of which will be evolving during the economic life of the field. Attanasi and Schuenemeyer (2002), in their Fig. 4B, compute three incremental costs curves based on three subjective distributions for the amount of oil discovered. For the triangular distribution the marginal cost of production from a field with 2 109 to 8 109 barrels of oil would appear to range from $ 13.50 to $ 21.00/barrel. In the model where price evolves according to GBM, we will vary marginal cost, c, from $ 10 to $ 25/barrel. Once developed, the ANWR field would have an expected life of t = 65 years. Our estimate of average annual production is Q = 100,000,000 (barrel/year), for a cumulative production of tQ = 6.5 billion barrels. This estimate is smaller than the 9.12 billion barrels which the oil industry expects to produce if oil prices are in excess of $ 24/barrel. These parameter values are, however, consistent with those reported by the McDowell Group (2002). While the productive life for an ANWR oil field is expected to be t = 65 years, there will be a construction or field development phase expected to be l = 5 years. During the construction phase, a present-value fixed cost of K = $ 2.9 billion is expected (McDowell Group, 2002). We adopt an annual discount rate of d = 0.10 as the appropriate risk-adjusted annual rate for this project. This rate of discount may seem high, but from the U.S. Government’s perspective, it should reflect the social rate of time preference and the risk of
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oil spills in the ANWR Coastal Plain, along the length of the Trans-Alaskan Pipeline, and in Prince William Sound, site of the Exxon Valdez spill. Six parameters, c, t, Q, l, K, and d, will be common to both the GBM and the M-R models. The other variable common to both models is amenity value, A ($/year) which is assumed to be irrevocably lost if oil development takes place. We have no idea what this amount should be, so we will vary it between $ 200 million and $ 300 million per year to determine the sensitivity of trigger prices in both the GBM and M-R models. As noted earlier, this range for amenity value might correspond to a willingness-to-pay of $ 2 to $ 3 per household per year. If crude oil prices evolve according to a M-R process, we will also need estimates of m and s, the mean drift and standard deviation rates in Eq. (1). If crude oil prices evolve ¯ and s, the speed of reversion, according to a M-R process, we will need to estimate h, P, mean to which prices revert, and standard deviation rate in Eq. (2).
¯ and s for the M-R process 3. Estimation of m and s for GBM and h, P, Fig. 2 shows the price of crude oil paid by refiners for the years 1968 through 2001. These prices are in real 1996 dollars. The data may be found at the U.S. Department of Energy web site: http://www.eia.doe.gov/emeu/aer/txt/ptb0519.html. This series contains 34 observations. Dixit and Pindyck (1994), in their discussion of mean-reverting processes, present a time-series of real crude oil prices spanning 120 years, from 1870 through 1990. Running a unit-root test for the full 120 years of data they reject GBM and fail to reject that the data were generated by an M-R process. They note, however, that if one does unit-root tests for the most recent 30 or 40 years, one cannot reject the hypothesis that the data were generated by GBM. This turned out to be the case for the data shown in Fig. 2. We could not reject GBM nor the M-R process, so we keep both models ‘‘in play,’’ and present estimates ¯ and s for the M-R process in Eq. (2). of m and s for GBM in Eq. (1) and for h, P,
Fig. 2. The real price of crude oil, 1968–2001 (in 1996 dollars).
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Table 1 Parameter values Parameter
Value
c t Q l K d m ¯ sˆ hˆ Pˆ¯ sˆ aˆ bˆ se Adjusted R 2
15 (marginal cost, $/barrel) 65 (years of production) 100,000,000 (average annual production) 5 (construction phase, years) $ 2.9 billion (present value investment) 0.1 (annual discount rate) 0.04 [GBM, Eq. (1)] 0.05 [GBM, Eq. (1)] 0.18 [M-R, Eq. (2)] 25.09 [M-R, Eq. (2)] 4.5 [M-R, Eq. (2)] 4.07 (1.71) [M-R, Eq. (3)] 0.16 (1.75) [M-R, Eq. (3)] 5.79 [M-R, Eq. (3)] 0.06
The Maximum Likelihood Estimates of m and s for GBM can be obtained by calculating the mean, m, and standard deviation, s, of the series ln(Pt+1/Pt) [see Reed and ˆ ¼ m þ s2 =2 and sˆ ¼ s. Clarke, 1990, p. 149, for a discussion]. Then, m For the parameters of the M-R process, we ran the regression Ptþ1 Pt ¼ a þ bPt þ et
(3) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ 2 1 (see ˆ hˆ ¼ lnð1 þ bÞ, ˆ and sˆ ¼ sˆ e hˆ =½ð1 þ bÞ and then calculated P¯ ¼ ˆa=b; Dixit and Pindyck, 1994, pp. 76–77). The regression results giving rise to the M-R parameter estimates, the estimates of m and s for GBM, and the base-case parameter values for c, t, Q, l, K, and d are summarized in Table 1.
4. Trigger prices under GBM and the M-R process We first consider the case where oil prices follow GBM. We need to determine two value functions; the value function for the ANWR while optimally waiting to develop, which will be denoted by VW(P), and the value function when field development is started, which will be denoted by VF(P). Recall that there is a construction phase of l = 5 years before oil production begins and that production is expected to last t = 65 years, with an average output of Q = 100,000,000 barrels per year. (We will subsequently allow oil production to exponentially decline from a higher initial level, but for now we assume Q is constant at 100 million barrels per year.) Suppose when construction is initiated that the price of crude oil is P. With prices evolving according to GBM we expect the price to be P eml when production starts l years from the start of construction. Recall that m is the expected drift rate in the price of crude oil. (We assume that the discount rate, d, exceeds the expected drift rate for crude
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oil prices, so that d > m.) Then, the expected price at instant t l would be P emt. With the present value for the fixed cost of field development at K = $ 2.9 billion, the expected present value of the ANWR field, when construction is initiated with an oil price of P, is given by Z lþt VF ðPÞ ¼ K þ Q ½P emt c edt dt (4) l
Integrating yields the analytic expression ðdmÞt ÞP eml dl ð1 e dt ð1 e Þðc=dÞ VF ðPÞ ¼ K þ Q e ðd mÞ
(5)
As part of the smooth-pasting condition, to be discussed in a moment, we will need the expression for the derivative VF0 ðPÞ which is easily seen to be ml e 0 dl ðdmÞt Þ VF ðPÞ ¼ Q e ð1 e (6) dm The value function while optimally waiting to develop, VW(P), must satisfy the Hamilton–Jacobi–Bellman (H–J–B) Equation which requires 2 s 0 00 dVW ðPÞ ¼ A þ mPVW ðPÞ þ ðPÞ (7) P2 V W 2 where A is the constant amenity dividend provided by an undisturbed ANWR. The solution to this second-order ordinary differential equation is VW ðPÞ ¼
A þ BPb d
(8)
where B > 0 is an unknown constant, which will be determined using the value-matching and smooth-pasting conditions, and where b > 1 is the positive root of a quadratic. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m 1 m 2 2d 2 þ 2 þ 2 (9) b¼ 2 s 2 s s The two terms on the right-hand side of Eq. (8) have the following interpretation. A/d is the present value of the amenity dividend in perpetuity, and is thus the value of never developing the ANWR’s oil potential. BPb is the value of the option to develop the ANWR’s oil potential when the price of crude oil is currently P. At the trigger price, where one is indifferent between preserving a pristine ANWR and developing its oil potential, the value-matching condition requires VW(P) = VF(P) or ð1 eðdmÞt ÞP eml ð1 edt Þðc=dÞ (10) A=d þ BPb ¼ K þ Q edl ðd mÞ To ensure continuity when one switches from B(t) = 0 (no development) to B(t) = 1 (oil 0 ðPÞ ¼ VF0 ðPÞ at the trigger development), the smooth-pasting condition requires that VW
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price, or bBPb1 ¼ Q edl ð1 eðdmÞt Þ
eml dm
(11)
The value-matching condition, Eq. (10), and the smooth-pasting condition, Eq. (11), constitute a two-equation system in the unknown constant B > 0, and the trigger price, P*. Some algebra leads to the following analytic expression for P*: P ¼
½A þ dK þ edl Qð1 edt Þcbðd mÞ dðb 1Þ eðdmÞl Qð1 eðdmÞt Þ
(12)
Knowing P*, you can determine B according to B¼
eðdmÞl ðP Þð1bÞ Qð1 eðdmÞt Þ bðd mÞ
(13)
For GBM it is possible to allow amenity value to grow exponentially and/or production to decline exponentially and to solve the revised value-matching and smooth-pasting conditions for an analytic expression for P*. This expression contains Eq. (12) as a special case. Suppose AðtÞ ¼ A0 egt , where d > g 0 and g is the rate at which amenity value is increasing over time. Suppose further that QðtÞ ¼ Q5 eht , where d > h 0 and h is the rate at which production declines from some initial level Q5 at t = l = 5. Given a value for h, we solve for Q5 so that cumulative production (with exponential decline at rate h) equals 6.5 billion barrels; the same cumulative production as in the base case, when amenity value and the rate of production were constant. This results in Q5 > Q = 100 million barrel/year. The expression for P* is given by P ¼
bðd þ h mÞ½Aðd þ hÞ þ Kðd gÞðd þ hÞ þ Q5 cðd gÞðedl eðdþhÞtdl Þ ðb 1Þðd gÞðd þ hÞQ5 ðeðdmÞl eðdþhmÞtðdmÞl Þ (14)
With mean-reverting oil prices, if development is initiated at price P, VF(P) will take the form dt ¯ ¯ ehl ÞðP cÞ ð1 eðdþhÞt ÞðP PÞ dl ð1 e þ VF ðPÞ ¼ K þ Q e (15) d dþh ¯ eht . VF(P) uses the result, when prices are M-R, that EfPðtÞjPð0Þg ¼ P¯ þ ðPð0Þ PÞ While optimally waiting to develop the ANWR, the value function, VW(P), must satisfy an H–J–B Equation which now takes the form 2 s 0 00 dVW ðPÞ ¼ A þ hðP¯ PÞVW ðPÞ þ ðPÞ (16) VW 2 While there is no analytic solution for the VW(P) which satisfies Eq. (16), we can solve a finite-time analogue using the Implicit Finite Difference Method (IFDM), as discussed in Hull (2003), Wilmott et al. (1995) and Tavella and Randall (2000). The IFDM provides a numerical approximation for the value function that satisfies Eq. (16). The
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problem is posed as a finite-horizon problem and the domain for time and price are in the bounded intervals t 2 [0, T] and P 2 [0, PMAX]. T must be sufficiently large so that the finite-horizon value function, evaluated at t = 0, yields a good approximation to the infinite-horizon value function. PMAX must be sufficiently high so that field development is optimal. For a finite-horizon, the H–J–B Equation, corresponding to Eq. (16), becomes 2 s W dV W ðP; tÞ ¼ A þ VtW ðP; tÞ þ hðP¯ PÞVPW ðP; tÞ þ ðP; tÞ (17) VPP 2 where the superscript W now indicates that this is the value function while optimally waiting and the subscripts denote partial derivatives with respect to time, t, or price, P, since VW(P,t) depends on both price and time in the finite-horizon approximation. Because there is no analytic solution, the value-matching and smooth-pasting conditions cannot be imposed directly. Instead, this problem may be formulated as a linear complementarity problem (LCP) (Wilmott et al., 1993; Insley, 2002). Define 2 s W W W W ¯ HV dV ðP; tÞ A þ Vt ðP; tÞ þ hðP PÞVP ðP; tÞ þ (18) VPP ðP; tÞ 2 The conditions for the LCP require (a) HV 0, (b) VW(P,t) VF(P) 0, and (c) HV[VW(P,t)–VF(P)] = 0. Condition (a) says that dVW(P,t) must be greater than or equal to the sum of the dividend and the expected capital gain of the value function while waiting to develop. Condition (b) says that the value while waiting should not go below the expected net revenue from exercising the option now, VF(P). Condition (c) says that either (a) or (b) must hold as a strict equality. This implies that if HV = 0, then VW(P,t) VF(P) > 0, and it is optimal to hold the option. If HV > 0, then VW(P,t) VF(P) = 0, and it is optimal to exercise the option. Conditions (a)–(c) would be satisfied by a rational social planner holding the option to drill in the ANWR. When (a)–(c) hold, Friedman (1988), has shown that the valuematching and smooth-pasting conditions are satisfied. Numerical solution, using the IFDM, requires that certain boundary and terminal conditions hold at P = 0, P = PMAX, and t = T, respectively. These conditions require V W ð0; tÞ ¼
A d
8 t 2 ½0; T ðBoundary Condition OneÞ
V W ðPMAX ; tÞ ¼ VF ðPMAX Þ 8 t 2 ½0; T ðBoundary Condition TwoÞ A W V ðP; TÞ ¼ max VF ðPÞ; 8 P 2 ½0; PMAX ðTerminal ConditionÞ d Boundary Condition One says that when the price of oil is zero, the value function while waiting is the present value of the amenity dividend in perpetuity.4 Boundary Condition 4 Boundary Condition One is not strictly correct for the M-R process given by Eq. (2). A zero oil price at instant t would support a positive conditional probability that P(t + dt) > 0. The value function at P = 0 may be greater than A/d, reflecting the option to develop when price becomes sufficiently positive. If, however, the option value at P = 0 is small relative to A/d, which is likely the case, Boundary Condition One will be approximately correct in the numerical context of our solution algorithm.
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Table 2 Trigger prices for GBM when A(t) = A0 eyt and Q(t) = Q5 eht A or A0 ($/year)
P* (g = h = 0)
P* (g = 0.005, h = 0)
P* (g = 0, h = 0.005)
200000000 210000000 220000000 230000000 240000000 250000000 260000000 270000000 280000000 290000000 300000000
19.84 19.99 20.13 20.27 20.41 20.55 20.69 20.84 20.97 21.12 21.26
19.99 20.14 20.29 20.44 20.59 20.74 20.89 21.04 21.19 21.34 21.49
19.62 19.75 19.88 20.02 20.15 20.28 20.41 20.54 20.67 20.80 20.93
Notes: (a) When g = h = 0 and A = 0, P* = $ 17.01/barrel. (b) When g > 0, A0 varies from $ 200 million to $ 300 million. When h > 0, Q5 is calculated so that cumulative production remains at 6.5 billion barrels for t = 65 years. These trigger prices are for the base-case marginal cost, c = $ 15/barrel. For sensitivity analysis with regard to c, see Table 3.
Two says that at PMAX, it is optimal to develop. The Terminal Condition says that at the expiry date (t = T) the value function is the argument that maximizes [VF(P), A/d]. The IFDM then seeks to determine the values of VW(P,t) that simultaneously satisfy the LCP, and the boundary and terminal conditions. The domain of the LCP is divided into a finite grid, {0, DP, 2DP, . . ., NDP}{0, Dt, 2Dt, . . ., MDt}, such that N = PMAX/DP and M = T/Dt. There are several possible algorithms that might be used within the IFDM. We employ the projected successive over-relaxation method (PSOR) as discussed in Wilmott et al. (1995) and Tavella and Randall (2000). Our MATLAB program, or faster running code in C++ or C#, may be obtained by emailing the second author.5 In this paper, we choose PMAX = $ 50/barrel and T = 100 years, which are high and long enough to approximate the infinite-horizon problem. With DP = $ 0.01 and Dt = 1 (year) there are 505,101 interior, boundary, and terminal values that must be computed. The values for VW(P,t) are determined for A ranging from $ 200,000,000 to $ 300,000,000. The trigger prices for the M-R process are determined where VW(P,t) first coincides with VF(P), thus satisfying the LCP conditions.6 In Table 2 we report the trigger prices for GBM. Eq. (14) was used for the base-case parameter values in Table 1 and for the three cases, where [g = h = 0], [g = 0.005, h = 0], and [g = 0, h = 0.005]. When [g = h = 0], the trigger price varies from P* = $ 19.84/barrel when A = $ 200 million per year, to P* = $ 21.26/barrel when A = $ 300 million per year. We also note that when A = 0, P* = $ 17.01, so an annual amenity value of $ 200 million raises the trigger price by $ 2.83. 5 We owe a deep debt of gratitude to Linda Buttel and John Zollweg of the Cornell Theory Center who reprogrammed our original MATLAB code, reducing the run time from days to less than 3 min in the C# version. 6 The values for PMAX, T, DP, and Dt would appear to provide a very close approximation to the values for the infinite-horizon problems when evaluated at t = 0. We modified our M-R program to numerically solve for the trigger prices when oil prices follow GBM. We were curious to see how the numerical prices obtained using the PSOR algorithm would compare to those obtained from the analytic expression given by Eq. (12). The numerical values, for GBM, using the PSOR algorithm were identical, to the penny, to those obtained from the analytical expression for P*.
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Table 3 Trigger prices for GBM when marginal cost takes the values c = $ 10/barrel, c = $ 20/barrel, and c = $ 25/barrel, and when g = h = 0 A ($/year)
P* (c = $ 10/barrel)
P* (c = $ 20/barrel)
P* (c = $ 25/barrel)
200000000 210000000 220000000 230000000 240000000 250000000 260000000 270000000 280000000 290000000 300000000
15.55 15.69 15.83 15.97 16.11 16.26 16.40 16.54 16.68 16.83 16.97
24.14 24.28 24.42 24.57 24.71 24.85 24.99 25.13 25.28 25.42 25.56
28.44 28.58 28.72 28.86 29.01 29.15 29.29 29.43 29.57 29.71 29.86
Notes: (a) The values for P* when c = $ 15/barrel and g = h = 0 are found in the second column of Table 2. (b) For a given value of c, the change in P* is slightly in excess of $ 0.14 for each $ 10,000,000 increase in A.
When amenity value grows exponentially but oil production is constant, the trigger price increases for each level of A0. For the case where [g = 0.005, h = 0], the trigger price ranged from $ 19.99 when A0 = $ 200 million per year to $ 21.49 when A0 = $ 300 million per year. When amenity value is constant, but oil production declines, the trigger price is lower than that when [g = h = 0]. The reason for this is that Q5 is increased above the constant Q = 100 million barrels per year in order to keep cumulative production constant at 6.5 billion barrels over the life of the field. This has the effect of moving production closer to the present and thus lowering the trigger price. For [g = 0, h = 0.005], the trigger price ranges from $ 19.62 when A = $ 200 million per year to $ 20.93 when A = $ 300 per year. In Table 3, when prices evolve according to GBM with [g = h = 0], we vary the marginal cost parameter, c, from $ 10/barrel to $ 25/barrel, in increments of $ 5, omitting the case where c = $ 15/barrel, since it is reported in Table 2. This range for marginal cost is likely to contain the ‘‘true’’ marginal cost should the ANWR be developed. For c = $ 10/barrel and A = $ 200 million per year, the trigger price drops to P* = $ 15.55. At the other extreme, if c = $ 25/barrel and A = $ 300 million per year, P* = 29.86. For a given value of c, P* increases by slightly more that $ 0.14 for every $ 10,000,000 increase in A. In Table 4 we compare P* under GBM with [g = h = 0] to the M-R process for the basecase parameters with c = $ 15/barrel. As expected, the trigger prices under the M-R process are higher, ranging from P* = $ 25.41 when A = $ 200 million per year to P* = $ 31.42 when A = $ 300 million per year. Under GBM, the trigger prices were not particularly sensitive to the variation in A. The trigger prices under the M-R process were more sensitive to the variation in amenity value and, for A > $ 250 million per year, development of the ANWR would not be optimal unless the price of oil was greater than $ 28.29/barrel.7 7 We performed additional sensitivity analysis for the M-R process. For c = $ 18 per barrel, P* ranged from $ 37.29 per barrel (A = $ 200 million) to $ 46.20 per barrel (A = $ 300 million). For d = 0.135, P* = $ 38.31 per barrel (A = $ 200 million) and P* = $ 47.70 per barrel (A = $ 300 million). A higher marginal cost or discount rate raises the trigger price under both GBM and the M-R process. See Table 5.
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Table 4 Trigger prices for GBM and the M-R process A ($/year)
P* (GBM, g = h = 0)
P* (M-R process)
200000000 210000000 220000000 230000000 240000000 250000000 260000000 270000000 280000000 290000000 300000000
19.84 19.99 20.13 20.27 20.41 20.55 20.69 20.84 20.97 21.12 21.26
25.41 25.97 26.54 27.12 27.70 28.29 28.90 29.51 30.14 30.77 31.42
Note: The P* values for both GBM and the M-R process assume c = $ 15/barrel.
Table 5 Comparative statics of P* under GBM and the M-R process Price process
c
d
h
K
l
m
P¯
Q
s
t
GBM M-R
+ +
+ +
n.a
+ +
+ +
n.a
n.a
+ +
Note: n.a. = not applicable.
The comparative statics of P* under GBM or the M-R process, for constant amenity and production [g = h = 0], are summarized in Table 5. For GBM, an increase in c, d, K, l, or s cause P* to increase, while an increase in m, Q, or t cause P* to decrease. For the M-R ¯ Q, process, an increase in c, d, K, l, or s also cause P* to increase, while an increase in h, P, * or t cause P to decrease.
5. Conclusions and caveats This paper has determined, from the perspective of the U.S. Government, the per barrel price for crude oil which would optimally trigger the exploration for oil in the coastal plain of the ANWR. The problem was posed as an optimal stopping problem, where the price of crude oil evolved according to geometric Brownian motion or a mean-reverting process. For geometric Brownian motion, it was possible to derive an analytic solution for the trigger price. For [g = h = 0], and for variations in A ranging from $ 200 million to $ 300 million per year, the trigger price ranged from $ 19.84/barrel to $ 21.26/barrel. With no amenity loss (A = 0) the trigger price was $ 17.01/barrel. If amenity value is exponentially growing and/or oil production is exponentially declining (but with cumulative production constant over the life of the field), it is still possible to obtain an analytic expression for P*, if oil prices follow a process of GBM. If amenity values were instantaneously growing at a rate g = 0.005 from A0 = $ 200 million, while oil production was constant at Q = 100 million barrels, P* = $ 19.99, only slightly greater than P* = $ 19.84 when g = 0. For GBM, increases in amenity value did not significantly increase the trigger price.
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If crude oil prices follow a mean-reverting process, it is not possible to derive a closedform solution for P*, but it is possible to numerically solve for the trigger price after approximating the value function. When A ranges from $ 200 million to $ 300 million per year, the price which would trigger exploration ranges from $ 25.41/barrel to $ 31.42/ barrel, respectively. The expected present net revenue of an ANWR field producing 6.5 billion barrels of oil over a 65-year horizon will depend on the stochastic price process and the price when development is initiated. The expected present net revenue of the ANWR field is higher under geometric Brownian motion than under a mean-reverting process. For example, under geometric Brownian motion, and with an initial price of $ 24/barrel, the expected present net revenue is $ 17.049 billion. Under a mean-reverting process, for the same initial price of $ 24/barrel, the 6.5 billion barrel field is only worth $ 3.115 billion. The expected present net revenue under the mean-reverting process is smaller since our estimate of P¯ was $ 25.09/barrel, based on data for the period 1968–2001. Our models might be criticized as being oversimplified and subjective in at least three respects. First, production from the ANWR, if it ever occurs, would likely start below 100 million barrels per year, increase to over 100 million barrels per year for some number of years, and then decline. This shift in the extraction profile is likely to decrease the trigger price compared to constant production in the basic GBM and M-R models. Second, we do not know what the annual amenity value is for an undisturbed (pristine) ANWR. We assumed that whatever its value, that it would be irrevocably lost if oil development takes place. Some claim that the caribou and polar bear will be unaffected by production facilities and pipelines, and point to our experience with the production facilities at Prudhoe Bay and along the Trans-Alaska Pipeline. The Central Arctic Caribou Herd, which calves in the Prudhoe Bay and Kuparuk oil fields is currently estimated at 23,400 animals, having increased from 3000 animals during the four decades of development and production at Prudhoe Bay. While the incremental risk of extinction for any extant arctic species as a result of ANWR would seem vanishingly small (given the extent other undisturbed arctic areas), we opted to take the perspective of a conservationist who would view the production facilities and connecting pipeline as an irreversible violation of the arctic wilderness, even if the caribou, bear, and other wildlife might be more tolerant. Third, we argued earlier that a discount rate of d = 0.1 might appropriately reflect the government’s risk premium when extracting and transporting oil in a harsh but ecologically fragile environment. This is a subjective judgment, and some might regard a risk premium on the order of 0.075 as being too high. The effect of all three of these simplifications or subjective assessments (constant production, high amenity value, and a high discount rate) would work in the same direction; that is, to raise the trigger price under either GBM or the M-R process. Despite this, the trigger prices, as of the Fall of 2004, were below the current spot prices. It has been argued that the amount of oil likely to be recovered from the ANWR does not justify the environmental risk, and that the United States would be better served by adapting to high oil prices through conservation. It is worth remembering that high oil prices in the mid-1970s did induce a more energy efficient stock of appliances, houses, and automobiles, and that both prices and oil consumption declined in the early 1980s (Fig. 2).
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If and when the issue of drilling for oil in the ANWR is revisited, it will be interesting to see if conservation, the global economy, or the political situation in the Middle East will have caused oil prices to revert to lower levels, so that drilling the ANWR is less attractive, or whether they will remain in the $ 30–$ 40/barrel range and the U.S. Congress and President decide to exercise the ANWR option.
Acknowledgements We specifically want to thank Linda Buttel, Toshihiro Oka, Takahiro Tsuge, Kenji Takeuchi, Masanobu Ishikawa, Hiroshi Takamori, Hiroshi Yamaguchi, John Zollwig and two anonymous reviewers. Earlier versions of the paper were presented at Cornell University, the 2004 ASSA Meetings in San Diego, California, the 2004 Japanese Economics Association Meetings held at Meiji-Gakuin University, the 2003 Environmental Economics Workshop at Kobe University, and the 2003 Real Options Workshop held at Aoyama-Gakuin University.
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