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__ ___ _ _ __ _OP _4~
POLICY RESEARCH WORKING PAPER 1655
Uncertainty and the Price Returns to storage for crude
oil reserves contain both a
for Crude Oil Reserves costereducing component
(consistent with Kaidor's
original notion of
Tiinotby J. Considine oiia oino
Td"convenience') and often
I)ona[d F. Larson
sizable premiums associated
with the dispersion of
petroleum prices.
The World Bank
International Economics Department
Commodity Policy and Analysis Unit
September 1996
POLICY RESEARCH WORKING PAI'ER I 65S
Summary findings
Innovations in futures, options, and derivative They specify optimal production and inventory
instruments permit active trading, speculating, and conditions using a third-order cost function and estimate
hedging - linking markets for physical petroleum them using monthly observations. Their inventory
products with financial markets. These derivative arbitrage condition embodies the Hotelling principle and
markets continuously value petroleum delivered today Kaldor's convenience yield, and includes a premium on
and for future dates, thus providing a market price for the dispersion in crude oil prices.
inventories. Underground petroleum reserves are also an The empirical results suggest that returns to storage
inventory defined by exploration surveys and contain both a cost-reducing component (consistent with
development drilling. As a result. observable market Kaldor's original notion of "convenience") and often
information can be used to value rhese reserves. sizable premiums associated with the dispersion of
Option-valuation models can be used to price reserves petroleum prices. Their findings suggest that crude oil
using observable markets, but are dependent on markets differentiated by quality and location provide
unexplained convenience yields revealed by the term similar premiums.
structure of futures prices. Considine and Larson apply a The premiums associated with the dispersion of
general model of inventory pricing to petroleum petroleum prices may account for persistent
inventories and generate an empirical model of the backwardation in crude oil prices. This finding may also
returns to storage for petroleum markets. They examine explain the wide discrepancies between Hotelling values
the determinants of the convenience Yield for crude oil and transaction prices found in previous studies.
using a stochastic control model.
This paper - a product of the Commodity Policy and Analysis Unit, International Economics Department - is part of a
larger effort in the department to further the understanding of resource pricing and commodity markets. The study was
funded by the Bank's Research Support Budget under the research project "Uncertainty and the Price of Crude Oil Reserves"
(RPO 679-23). Copies of this paper are available free from the World Bank, 1818 H Street NW, Washington, DC 20433.
Please contact Pauline Kokila, room N5-030, telephone 202-473-3716, fax 202-522-3564, Internet address
pkokila@worldbank.org. September 1996. (18 pages)
The Policy Researcb Working Paj Series disseminates the findings of work in progress to encourage the exchange of ideas about
developnment issues. An objective, c,'the series is to get the findings out quickcly, even if the presentations are less thanl fully polished. The
papers carry the names of the aUth-,rs and skh,uld be tused and cited accordingly. The findings, interpretations, and conclusions are the
authors' own and should not be ati-ibuted tc, the WXorld Bankz, its Executive Board of Directors, or any of its member countries.
Produced lDv the Policv Research Dissemination Center
Uncertainty and the Price for
Crude Oil Reserves
by
Timothy J. Considine
and
Donald F. Larson
The authors are Associate Professor in the Department of Mineral Economics at The Pennsylvania
State University and Economist in the International Economics Department at the World Bank. The
authors would like to thank M. A. Adelman, T. Akiyama, T. Malliaris, and G. C. Watkins for their helpful
comments. The research was funded under RPO 679-23.
I
Table of Contents
Abstract ....
1. Introduction ......................I
II. Valuation of Crude Oil Reserves .....................I
III. Crude Oil Markets .....................3
IV. Empirical Model ......................5
V. Econometric Results ..................... 10
VI. Conclusions ..................... 13
VII. References ..................... 17
Tables
Table I Summary Statistics for Petroleum Markets, 1988 to 1994 .4
Table 2. Parameter Estimates .11
Table 3. Elasticities and Scaled Derivatives .11
Table 4. Estimated Dispersion Premiums for Various Crude Oils .13
Figures
Figure 1. Prices for West Texas Intermediate Crude Oil, 1988-1994 .14
Figure 2. Crude Oil Inventories in the United States, 1988-1994 .14
Figure 3. Implied Volatility in Crude Oil Options, 1988-1994 .15
Figure 4. Inventories and the Four Week Forward Spread .15
Figure 5. Estimated Cost Reducing Effect of Stocks .16
Figure 6. Estimated Dispersion Premium .16
I. Introduction
Innovations in futures, options, and derivative instruments permnit active trading, speculating, and
hedging linking markets for physical petroleum products with financial markets. These derivative markets
continuously value petroleum delivered today and for future dates, thereby providing a market price for
inventories. Underground petroleum reserves are also an inventory defined by exploration surveys and
development drilling. As a result, observable market information can be used to value these reserves.
Option-valuation models can be used to price reserves using observable markets, but are dependent
upon unexplained convenience yields revealed by the term structure of futures prices. This paper applies a
general model of inventory pricing to petroleum inventories and generates an empirical model of the returns
to storage for petroleum markets. The results suggest that the returns contain both a cost-reducing
component consistent with Kaldor' s original notion of "convenience" as well as a frequently sizeable
premium associated with the dispersion of petroleum prices. Further, our findings suggest that crude oil
markets differentiated by quality and location provide similar premia.
The valuation of crude oil reserves raises a number analytical issues, which we identify and explore
in the following section. To set the stage for our empirical analysis, we then provide an overview of crude
oil markets. Our empirical results appear in section five after presentation of the empirical model. The
paper concludes with a summary of our main findings and suggestions for further research.
HI. Valuation of Crude Oil Reserves
Hotelling (1931) proposed that the equilibrium rate of return on an exhaustible natural resource
will rise at the rate of interest. The logic is that in equilibrium the producer is indifferent between holding a
barrel of oil in reserve and producing one barrel now when the rate of appreciation on the reserve is just
equal to the rate of return from pumping now and investing the proceeds.
There have been several empirical studies to test the validity of the Hotelling Principle. Time series
studies by Heal and Barrow (1980) and Srnith (1981) generally have not been supportive although the tests
and data have serious difficulties. Miller and Upton (1985a) test a related relation, called the Hotelling
Valuation Principle (HVP), in which the value of mineral reserves depends upon prices net of extraction
costs. They find that the Hotelling values account for a significant portion of the variation in market values
but in a later study Miller and Upton (1985b) find less support for the HVP.
Adelman (1990) argues that the valuation principle is not useful because it implies unrealistically
high depletion rates. Moreover, Adelman (1990) asserts that there is really no fixed economic stock of
petroleum resources, only flows into a reserve inventory. Later, Adelman and Watkins (1995) use actual
transactions prices for oil and gas properties and find that their estimates of reserve values are less than half
2
those suggested by the HVP. These results support Adelman's (1990) previous suggestion that such a
discrepancy might arise because the buyer is paying for the option to more intensively develop a known oil
pool sometime in the future.
An alternative approach to pricing petroleum assets has its roots in the literature on the cost of
storage. Kaldor (1939), Working (1934), Brennan (1958) and Telser (1958) argue that inventory holding
is profitable even during price backwardation due to a convenience yield that offsets the negative returns
from storage. Starting with Kaldor, a number of authors attributed the convenience yield to cost-reducing
features of inventory holding. However, more recently, the discussion on the value of inventories has
emphasized other attributes. Lovell (1961) suggested that inventories serve to smooth production. This
notion has been tested empirically with mixed results by Blinder (1982, 1986), Fair (1989), Krane and
Braun (1991), and Blinder and Maccini (1991). Williams and Wright (1990) emphasize the returns to
storage due to potential stockouts and Considine (1991) and Pindyck (1994) stress the non-linear
relationship between production and storage costs.
Brennan and Schwartz (1985) and later Gibson and Schwartz (1990) sought to derive the market
price for pricing petroleum assets based on the observable term structure of petroleum futures markets.
Although motivated by the time series properties of the forward convenience yield of crude oil (Gibson and
Schwartz, 1989), Gibson and Schwartz (1990) developed an empirical model of pricing assets which
explicitly modeled a risk-premium based on stochastic price movements. Later, Larson (1993) provided a
stochastic model and derived a price arbitrage equation from first principals to explain the negative retums
to storage for refined copper.
In this paper, we stress that measures of the convenience yield contain two separate components.
First, inventories may be cost-saving in the tradition of Kaldor. Second, the value of inventories embodies
a premium related to the stochastic nature of prices, similar to the value of an option. The value of this
premium can be derived from observed market information. We conclude that, on average, 40% of the
value of US petroleum inventories can be attributed to the volatile nature of petroleum prices. The premia
are also substantial for UK Brent-38b and Dubai Fateh Prompt crude oil and smaller for Malaysia Tapis
and Indonesia Minas-34 crudes.
3
III. Crude Oil Markets
Once crude oil is extracted from the earth, it is collected by pipelines at the oil field and then
shipped either by pipeline or tanker to refineries where it is refined into an array of petroleum products,
including gasoline, distillate fuel, residual fuel oil, jet fuel, and petrochemical products. Despite the
perceived dominance of the Organization of Petroleum Exporting Countries (0. P. E. C.), many countries
around the world now produce crude oil. The United States is the most mature producing area with
extensive trading in crude oil futures and options. Many developing countries, such as Columbia, Mexico,
and Malaysia, have become significant crude oil producers in recent years.
Inventories of crude are held at various points in the petroleum distribution network. A small
amount is held by oil lease holders at the oil field. The largest stocks, nearly two-thirds of total private
stocks in the U. S., are in pipelines and tank farms. The remaining third of private stocks are held at
petroleum refineries. In the U. S. almost 50 percent of total stocks are held by the federal government in
the strategic petroleum reserve. In this study, we ignore these stocks because they do not vary with market
conditions. Moreover, we examine total private stocks because Blanchard (1983) shows that basic
arbitrage relations are unaffected by aggregation over the distribution chain. We measure stocks and flows
in physical units unadjusted for seasonal variation, which avoids the measurement errors identified by Fair
(1989).
Although extensive price data exist for several regions around the world, monthly inventory and
production data for developing countries is much more difficult to obtain. As a result, we confine our
initial investigation of the determinants of the convenience yield for crude oil to the United States. We then
use these results to estimate dispersion premium for other crudes traded around the world.
Crude oil production in the United States averages 7.36 mnillion barrels per day (see Table 1).
Total private crude oil inventories are on average about 340 million barrels with a standard deviation of
fourteen million barrels. Variability in domestic production is slightly less than the variability in domestic
crude oil sales, which may reflect production smoothing.
Spot prices for West Texas Intermediate (WTI) and several other major crudes appear in Table 1.
Forward contracts are actively traded in oil so we use the four week forward price in computing
convenience yields.' The convenience yields are identical reflecting highly integrated global markets.
Time series plots for crude oil prices and inventories appear in the following two figures. Both
plots reveal no clear trend in prices and inventories over this period. However, both display sharp swings
' As Pindyck (1994) notes, unlike forward contracts, futures require a settlement and transfer of funds at the close
of each trading day. As a result, oil futures prices exceed forward prices if the risk free interest rate is stochastic
and positively correlated with spot crude oil prices.
4
before and after August 1990, when the Persian Gulf crisis started. Before the crisis, spot crude prices
were falling and inventories were increasing. Spot sales were selling at a deep discount relative to forward
sales so it was cheap to hold inventories. With the onset of the crisis, spot prices shot above forward
prices, holding inventories became very expensive and, as a result, stock levels declined sharply (see
Figures 1 & 2). Also during this period, implied volatility on crude oil options increased sharply as Figure
3 illustrates.
The relationship between storage levels and the forward price spread is illustrated in Figure 4.
Here we see that large negative spreads occur when the market is in backwardation with spot prices
exceeding the forward prices. These negative returns occur at relatively low stock levels. In contrast, large
positive forward spreads occur with high inventory levels, which depress spot prices. The scatter diagram
also reveals that the crude oil market is often in price backwardation.
Table 1: Summary Statistics for Petroleum Markets, 1988 to 19942
.Rtand.Rrd
Mean Deviation
U. S. Crude Oil (Millions of Barrels)
Dailv Production 7.36 0.45
Daily Sales 7.37 0.52
Endina Stocks 341.00 14.17
Snot Prices (Dollars Der Barrel)
West Texas Intermnediate Crude 19.77 3.75
Malavsia TaDis 19.61 4.26
Dubai Fateh PromDt 16.16 3.64
UK Brent-38 b 18.63 4.19
Indonesia Minas-34 18.48 4.09
Four Week Forward Prices
West Texas Intermediate Crude 19.63 3.56
Dubai Fateh 16.02 3.39
UK Brent-38 b 18.47 3.90
Convenience Yields* (Percent)
West Texas Intermediate Crude 5.96 2.24
Dubai Fateh 5.96 2.31
UK Brent-38 b 5.96 2.27
2 These yields are computed as yield, = r+12[1n(Pj,,1)-ln(Pf,)] where r is the yild on a one-month certificate of
deposit at an annual rate and Pj, is the spot price of oil and Pi,., is the forward price.
5
IV. Empirical Model
In this section, the formal model is presented3. A generalized price-arbitrage condition is derived
from the first-order conditions of the optimization problem which is consistent with inventory-holding
during an anticipated price fall. The problem is characterized as a continuous two-cycle problem with
uncertain future demand. In the current period, the producer knows the current sales price. By deciding
how much to produce and sell, he determines how much inventory he will bring into the next period. The
expected marginal value, or the shadow price, of the inventory in the next period is not known, but contains
a stochastic element since demand is uncertain. The effects of random demand shocks on the shadow price
of inventories may be asymmetric -- that is, a positive random shock may increase prices by more than an
equally sized negative random shock. In such a case, the shadow price of inventory will carry a dispersion
premium so that the shadow price of inventories increases with the variance of the stochastic component of
sales. Such a premium is analogous to the volatility premium in an options price and can result in positive
inventory levels even when price declines are expected.
When sales contain a random element, inventory levels will also be stochastic and changes in
inventories will contain a planned and unplanned component. The difference between planned and actual
inventories will be the difference between expected and actual production minus sales. For the moment,
assume that the change in inventories can be expressed as the following process:
-x, -Et l [xt -x,=£ =x, -x,-l -Et l [Y, -st] (1)
where £ has an expected value of zero and a variance a2. Rewriting the constraint on inventories in
continuous-time notation, the value of ending inventories at time t, is the solution to the following infinite-
horizon problem:
e'nt V(x,,)=Maxsr EJ{[psT-C(y,x)]e (f~I)}dt,s.t.dx=E(y-s)dt+cdv. (2)
ti
The term dv u(t)dte2 is a Wiener process, where u(t) -N(O, 1). Because inventory changes include a
random component, dx/dt does not exist in the usual sense and the rules of stochastic calculus apply.
Evaluated at t, , the solution to (2) gives the value of ending stocks, that is, J(x,, tr) = V(x,) e"', . In the
language of optimal control theory, the firm's problem is a stochastic infinite-horizon problem, stretching
from t, onwards. As a result, the shadow price for the end-of-period inventories in the first stage of the
' For a more detailed description of the model derivation, see Larson (1993).
producer problem is based on expectations of an on-going process of production amid uncertain demand.
Generally, the solution for J(x.,ti) can be found by solving Bellman's equation, a partial differential
equation: -aJ(xi, t,) /at = rV(xd)e e- . Arbitrarily setting t, = 0 simplifies the equation somewhat so that the
solution to the inventory problem from t, onward can be represented by the Hamilton-Jacobi equation of
dynamic programming:
rV(X,) = MaxSE[p -C (y, )]+V(Y-) + I Va2 (3)
The first-order conditions for the maximization of (3) are:
i) E(p - VJ = (4)
ii)E(-Cy - Vx)=o
iii) E (dx) = E (y - s) dt
iv)x(t, = O)=xl
The producer solves for planned production, sales and inventories by setting expected marginal costs
equal to expected price equal to the shadow price of inventories, v, -- which is itself an expected value.
It is worth noting that the distribution of the error term, especially a 2, is independent of the decision
variables. The variance of the error is regarded as a state of nature and is not subject to choice on the part
of the producer. This assumption is implicit throughout the paper and works well empirically.
Several additional assumptions must be made to guarantee that the first-order conditions do indeed
provide a maximum. V must be concave in x; the solution values of x, y, and s must be positive4
(otherwise border solutions must be considered); and the transversality-at-infinity condition must hold.5
4For the empirical problem at hand, refiner inventories of refined copper are all positive as are quantities sold
and produced. Inventories at both the COMEX and LME have remained positive throughout the history of those
institutions as well. However, to be complete, stock-outs need to be considered in the theoretical model and non-
negativity constraints introduced to the maximization problem. These are given in Annex 2.
'For the infinite-horizon autonomous problem given above, the transversality condition is:
lim V (t)Z() e`(-to) = lim J.(t)z(t)e-r = 0.
Benveniste and Scheinkman (1982) showed that the condition is necessary and sufficient for the solution of Al.6
to be optimal. The logic is that any positive stock level must have no value as the problem approaches infinity.
Otherwise, the firm could further increase profits by either producing less or selling more in the last period.
Inventories have value because, ultimately, they can be sold. If some price exists at which no copper can be sold,
then some upper bound must exist for the shadow price of inventories. If so and if stock levels, x, are limited by
physical storage or natural endowments, then discounting will assure that the transversality condition holds. See
Brock (1987) for further details on the general condition.
7
An expression for the marginal value of inventories is found by applying the envelope theory (Dixit,
1990) to the Hamilton-Jacobi equation given in (3):
rV, = E[-q, + V,= (y - s) +- Y, Vc2 ](5
~~~~~~~~~~~ (5)
At the solution, E[p] = VI, (as part of the first-order conditions in (4) so that:
E[dp / dt]= Vu E[dx / dt] (6)
Rewriting part iii of (4) provides Eldx /dtJ = Ely - s] . These results can be combined to form a price-
arbitrage condition. First, from rearranging (6) and using iii from (4):
V E[y-s]=V E[dx/dt]=rV +4E[C ]-_+v 2 (7)
Combining (6) with (7) provides:
E [dpd/d]=rE [p]+Er[r ] V a, for x5,y>0 (8)
E 1~j-r iPJ'~1'XI 2 xx (8
Equation (8) is a generalization of price-arbitrage conditions given in cost-of-carry models such as
Williams and Wright (1991, p. 27). The arbitrage condition states that the expected change in price will be
equal to rEip) -- interest on investing the money elsewhere -- plus C. -- the costs of physical storage and
any amenity from storage -- minus i V, a . If V.,, is positive, then this last term constitutes a dispersion
premium that increases with the variability of the stochastic component of inventories2. The last two
components of (8) have important implications for holding inventories in the face of less-than-full carrying
charges. According to the condition, it still may be optimal to hold inventories when the market is in
backwardation -- E[dpIdt] < 0 -- if inventories provide a cost-reducing Kaldor-convenience (that is, if C1
is sufficiently negative) and/or the dispersion premium, i v. cr, is sufficiently positive. The two
components are not mutually dependent. Kaldor-convenience alone can potentially explain inventory-
holding in backward markets, as can a dispersion premium. When Vm = 0 and C, is positive, (8) can be
intrepreted either as the cost-of-carry arbitrage condition from the literature on inventories, or as the
Hotelling Principle without extraction costs.
8
Two of the results of the model can be used to derive an econometric estimate of the dispersion
premium. Combining i) and ii) from (4) states that expected price is equal to marginal cost:
E(p)= (C Y))
This result along with the price-arbitrage equation (8), are given functional form to derive empirical results.
To implement the model of crude oil prices and inventories given by equations (8) and (9), we
approximate the cost function with a third order quadratic function:
5 5 5 5 5 5
C=L+) oX +jXDjXiXj + , XIXJXk (10)
i=l e., j=l i=1 j=I k=I
where X = wages, w,
X2 = field production of crude oil, y,
X3 = month ending private stocks of crude oil, x,
X4 = oil wells drilled, k, and
X5 = technical change index, z.
We assume symmetry in Pi and Ygk. In addition to production and ending inventories, we assume the cost
function depends upon wage rates, production capacity, and technical change. This function represents the
cost of producing, transporting, and delivering crude oil to the wholesale market, where the buyers are
crude oil traders and refiners. We assume that production and inventories jointly determine costs. Due to
data unavailability, drilling activity is used as a proxy for production capacity because drilling typically
increases sharply as production capacity limits are approached. Likewise, we must use a proxy for
technical change. Since the late 1980s there have been major advances in oil exploration and development,
including seismic imaging, automatic positioning of rigs in deep water, and horizontal drilling. As a result,
the efficiency of oil and gas production has increased substantially. The only readily available proxy for
these changes is the success rate in drilling measured by the ratio of discoveries per well drilled, which
serves as our proxy for technical change, z.
The third order cost function implies that marginal cost, C,, and the partial derivative of cost with
respect to inventories, C., are quadratic functions:
Dc 5 5 5
a a + X Py Xj + XQ, Y Xj Y 4,kX i = ,x
axi J=l j=I k=l
9
Our cost function implies that the third order derivatives in the dispersion premium are parameters. The
expected marginal cost equation takes the following form:
P = + 0,.w+ p,y++ Pyx + Ikk+ PI,z
+(Y2) y w2 + y.,, wy + yx wx +y,, wk + y vz
+(Y2) y, y2 + y YX +y iyk yk+Y, yz
+(Y2) 'yx X2 + yk xk+ y , xz
+(Y2) yy kk +y kk+(Y2)y z2, (12)
where we have replaced the numbered subscripts with letters used in (3) above. The estimating equation
for the arbitrage relation is as follows:
pf-p-rp=at +Pf. w+par y+p. x+pxt k+p, z
+(y2) -y"W W 2+Y KY vY+ywXr wx+y.x wk+ + 1t Wz
+(y2).Y. y2 + -y' yx+yk yk+y yz
+(/2)Y. x2 +yk xk + y,, xz
+(Y2)y k2+ kZ+(Y2) ye Z2
-(Y2) y)(Y)2 + 2'y Y, hdxdyx f 2] (13)
Using results from Cootner (1964) and Merton (1992), Larson (1993) shows that the volatility of
inventories can be expressed in terms of the associated price volatility. We, therefore, use implied price
volatility from the crude oil options market for as t.. We use the difference between the 4 week forward
price and the spot price for the expected price change in (6).
We append random error terms to (5) and (6) to represent expectational errors and estimate the two
equations as a system using the Generalized Method of Moments (GMM) estimator. The GMM estimator
is potentially superior to conventional instrumental variable estimators, yielding consistent and efficient
estimators when past or future values of the instruments are corTelated with the error terms. Our sample
period starts in February 1989 and ends in June 1994. The petroleum production, sales, and inventory data
are from the Energy Information Administration. Spot and forward prices are collected from Petroleum &
Energy Intelligence Weekly, Inc. Implied price volatility is from the Commodity Research Bureau.
The information set available to the oil market provides a basis for selecting instrumental variables.
The instruments include lagged variables from the model, including spot and forward prices, wages,
10
production, stocks, technical change, short term interest rates, drilling activity, and volatility. We
supplement this list with demand shock instruments that include heating and cooling degree days and
lagged, seasonally unadjusted values for the Standard and Poor 500 stock price index, housing starts, and
industrial production. We also include dummy variables for fixed monthly effects, the Exxon Valdez
tanker accident, the invasion of Kuwait in August 1990, "Desert Shield" from September through
December 1990, and for "Desert Storm" during January and February 1991. These last three dummy
variables represent the impact of political events on expectations in the oil market. All continuous
variables are normalized except prices and the returns to storage to avoid scaling problems in estimating the
model and in calculating elasticities.
V. Econometric Results
We first test the maintained restrictions of the model, such as dynamic optimization and the
quadratic approximation of the cost function. The value of the objective function for the GMM estimator is
distributed as a Chi-squared statistic with degrees of freedom in this study equal to the 30 instruments times
2 equations minus the 36 parameters. The value of the objective function is 13.4, which is less than the
critical value of 36.45 at the 5 percent level of significance with 24 degrees of freedom. Consequently, we
cannot reject the over-identifying restrictions of the model. The fit of the two equations is reasonably good
with the R2 coefficient 0.55 for marginal cost and 0.53 for the returns to storage equation. The estimated
residuals do not have a unit root, which suggests the absence of serious dynamic mis-specification.
In Table 2, we present the parameter estimates. The first order effects of wages, production,
drilling, and technical change are significant in the marginal cost equation. The "own" second-order
coefficients for wages, inventories, drilling, and technical change in the marginal cost equation are also
significant. Seven interaction parameters are also significant in the marginal cost equation: wages -
production, drilling with production and wages, and technical change with wages, production, inventories
and drilling. The first order effects for stocks, wages, and technical change in the returns to storage
equation are significant. Like marginal cost, the parameters on squared wages, stocks, drilling and
technical change in the returns to storage relation are significant. Three second-order effects in C, are
significant: wages with drilling and technical change, and drilling with technical change. The dispersion
term is a function of three parameters, Y,,,,, . Y,, . The last parameter, 'Yyxx, is significantly positive
and when multiplied by implied volatility is large enough so that the dispersion premium is also positive.
11
Table 2: Parameter Estimates
Parameter F.stinmate t S1tafistic Parameter Fctimate t !Rtatistic.
a., -3311.53 -3.95 'Y,,1,% -86.14 -3.11
2557.83 3.86 Yvvkz -86.33 -3.91
kv 2472.79 2.67 Y,zz -96.97 -4.14
p^,x -258.13 -1.01 aX 240.05 1.14
Pvk 1098.17 5.01 owx -280.06 -1.84
IV7 728.93 3.67 IYY 175.76 2.32
'Ywv -1303.71 -4.38 Ok -25.08 -0.75
Y,VV ,\, -844.02 -1.71 j3,, -73.85 -2.56
l^, . 158.30 1.63 y . 121.22 2.50
Y,^,,,, -424.78 -5.50 YW,. -56.51 -1.38
Y,,,,, -158.09 -2.04 8k 29.51 2.72
'y ,\,, -683.01 -1.33 'Y%VX7 25.17 1.91
Y ,-., 39.15 0.24 y -82.09 -2.04
'Y wvk -486.13 -3.48 Yx -19.94 -1.33
SY wz -418.78 -3.07 x,z -11.34 -1.50
Rye vx1.08 3.50 ,Y 14.75 2.76
'Y vxk -5.50 -0.26 Y xkz 5.67 2.06
'Y vxz 44.33 2.22 1 xzz 8.70 2.79
The model has variable elasticities of marginal cost and of returns to storage, which are presented
in Table 3. Since the benefits of stock holding can be negative, we do not divide these elasticities by
predicted C.,. Instead, we multiply the derivatives by the level of the variable in the partial equilibrium
differentiation.
Table 3: Elasticities and Scaled Derivatives
Marginal Cost, Cy Elasticities "t" Ratios
Wages 0.22 0.36
Production -1.10 -1.10
Inventories -0.44 -2.84
Drilling 0.16 1.00
Technical Change -0.55 -3.22
Stock/Cost Benefit, Cx Scaled Derivatives "t' Ratios
Wages -1.03 -0.62
Production -8.81 -2.84
Inventories 5.33 2.93
Drilling 0.09 0.19
Technical Change 0.19 0.35
The inventory elasticity of marginal cost is -0.44 with a relatively high "t" statistic suggesting that
marginal costs shift up (down) with lower (higher) inventories. This finding in conjunction with our
positive and significant estimate for 'Yyx. suggests that the dispersion premium arises because marginal cost
decreases (increases) with higher (lower) inventories at an increasing rate. We also find that recent
12
advances in drilling and exploration technology have significantly reduced marginal cost. Our negative
production elasticity of marginal cost suggests increasing returns but its relatively low t statistic suggests
that this may not be significant.
The scaled derivative of the estimated C1 function with respect to inventory levels is positive and
significant, consistent with the theory of storage. The predicted values of C,: are plotted against inventory
levels in Figure 5. The graph illustrates that at low inventory levels, the cost reducing effect of holding
inventory is large and at high stock levels C, is positive indicating that costs actually rise with additional
stocks. We also find that the C, function shifts up (down) with lower (higher) production. The C1
function does not seem to be significantly affected by wages, drilling, or technical change. Overall, our
estimates reveal that at least a portion of the returns to storage reflects the cost-reducing benefits from
holding stocks.
The dispersion premium, which constitutes the other portion of the retums to storage, is plotted
below in Figure 6. We express the dispersion premium as a percent of the crude oil spot price at an annual
rate. Our estimate is on average nearly 40 percent with a standard deviation of almost 18 percent. Notice
in Figure 6 that the premium reached over 100 percent at the height of the Gulf War crisis and fluctuates
considerably. This uncertainty premium may help explain the gap discovered by Adelman and Watkins
(1995) between transactions prices for oil and gas properties and prices implied by the Hotelling Valuation
Principle.
These dispersion premium may vary with political, economic, and geophysical uncertainties. To
assess this possibility, we estimate the same model for crude oil sold in the United Kingdom, Dubai,
Malaysia, and Indonesia. Due to data limitations we only estimate the intercept terms, t,, and CX, and YKy.
in equations (5) and (6) and assume the remaining parameters for C,. and C, are the same as those
estimated above. The forward prices for Malaysian Tapis and Indonesia Minas-34 are unavailable. As a
proxy, we use the forward price for the Dubai Fateh (DF) crude since both crudes are traded in Asia.
We find the dispersion premium for U. K. Brent oil nearly identical with West Texas Intermediate
while those for Dubai, Malaysia, and Indonesia are somewhat lower. All premiums, however, are
substantial, suggesting that a significant dispersion premium exists for crude oil assets. As a result, we
conclude that, for the countries examined here, country risk is not a factor in explaining the premium in oil
inventory pricing. 6
6 Using a different time period and different methodology, Gibson and Schwartz (1990) also found evidence of a
premnium related to the stochastic component of petroleum price movements. However for their study, the
premium ranged lower, averaging around 4.5%.
13
Table 4: Estimated Dispersion Premiums for Various Crude Oils
Annual Percent
Moan Standard Deviation
West Texas Intermediate Crude 39.81 17.78
Dubai Fateh Prompt 37.56 17.28
UK Brent-38 b 40.71 17.30
Malaysia Tapis 25.37 11.67
Indonesia Minas-34 27.16 12.51
VI. Conclusions
This paper provides a theoretically consistent empirical model of the convenience yield. Using a
stochastic control formulation, we show how production and inventories are related to spot and forward
prices for crude oil. Our results show that apparent losses from holding crude oil stocks are offset by the
benefits derived from inventories and by a significant dispersion premium that rises sharply during market
disruptions. This finding is consistent with the lack of empirical support for Hotelling-based theories of
natural resources pricing. Moreover, our results suggest that current methods of pricing reserves and
related assets significantly undervalue those assets. Further, this result holds across several markets for
petroleum differentiated by location and type of crude.
Evaluating an investment in an oil project involves a daunting array of unknowns. Traditional
discounted cash flow analysis projects a stream of discounted net revenues. Petroleum engineering
relations provide a basis for projecting expected production levels with a reasonable degree of accuracy.
Crude oil prices, however, are much more difficult to predict. Frequently, available future prices are used
to a point and some version of the Hotelling Principle is used to value more distant production. The
practical consequence is that oil reserves are undervalued since such methods ignore the dispersion premia
and Kaldor-convenience effects which jointly determine the observed term structure for forward and future
prices.
Further research could reveal the robustness of this estimate. Our preliminary analysis yields
plausible dispersion premia, but more complete information is needed. A cross sectional time series
analysis for several oil producing countries could be useful. Finally, an examination of the demand for
inventories could result in more efficient estimates.
million barrels Dollars pcr barrel
W o- w o B o W o W A
Feb-88 - . Feb-88 -
Jun-88 Jun-88
Oct-88 - Oct-88
Feb-89 1 a Feb-89 * ..
Jun-89 - _ Jun-89
Oct-89 ~ ~~Oct-89:
Feb-90 - Feb-90 -
Jun-90 * -.,e ~ ~Jun-90 -
Oct-90 - ~~~~~~Oct-90 *
Feb-91 rA ~~~~~~~~~~~~~~~Feb-91 . .1W
Jun-91 P9~ ~ ~~~~~. JunI91
Oct-91 Oct-91
2~~~Fb9
Feb-92 ~ ~ ~ ~ ~ ~ ~ ~ Fb92.
Jun-92~~~~~~- . Jn92~
Oct-92 2 ~~~~~~~~~~~~~~~~~Oct-92 . .4-»-
Feb-93 ~~~~~~j~~> * ~~~~A.F"Fe-9
"~~~~~A~~~~ ~~~~~ ..u.9 T;.
Jun-93Jn-3 >
Oct-93 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ AJ~~ct9
Feb-94 ~~~~~~~ ~~~~~~~~Feb-94
Jun-94 - - ~~~~~~~~~~~~~~Jun-94
forward less spot (S per barrel percent
>~~~ ' S o o &
l.A 0 ~~~~~~~~Feb-88 A
0
May-88 I
Nov-88
Feb-89
May-9 . -'9
"* - -. -. il.4- C Aug-89
o _ _ 4 ^~M'>; f Nov-89
. , 2 2 h ? w - . ,z', ^ A t Feb-93Fe-9 :-.v p h.t.
.. . .; .... ...... . -:. v<,^..:. .w q ..... May-90 t
Kr- ~~~~~~~~~~~~~~~Aug-91
Nov-90 < .,-
, - ' P 'svw3 -91-+e2SR ....... .4,.-,
Feb-92 '
Aug-91
-~~~~ " -' ~~~~~~~~Nov-921
Feb-9320
__ ........ . .. . .. : < . >May-94 2
Aug-92
16
Figure5: Nirgnal effect of stocks on costs
1.50-Hg g |X
°Q(00 A 0*1N|
-,-50: X|X
310 32D 330 340 350 360 370 38) 30
inrAmies (niilirbureis)
Figure 6: Estimated Dispersion Premium
120
* 80
60
20~ ~ ~ ~~~~ -0~
z I
,~~~~~~~~~~ O
I~~~~~~~~L Cs a 0
| .c :< ez 7- < D z < E6z E < z <; E
17
VII. References
Adelman, M. A. 1990. Mineral Depletion, with Special Reference to Petroleum. The Review of
Economics and Statistics. 72(1): 1-10.
Adelman, M. A. and G. C. Watkins 1995 Reserve Asset Values and the Hotelling Valuation principle:
Further Evidence. Southern Economic Journal 61(3) 664-672.
Blanchard, Oliver J. 1983. The Production and Inventory Behavior of the American Automobile Industry.
Journal of Political Economy: 365-400, June.
Blinder, Alan. 1982. Inventories and Sticky Prices. American Economic Review 72: 334-48.
Blinder, Alan. 1986. Can the Production Smoothing Model of Inventory Behaviour Be Saved? Quarterly
Journal of Economics 101: 431-53.
Blinder, Alan S., and Louis J. Maccini. 1991. Taking Stock: A Critical Assessment of Recent Research
on Inventories. Journal of Economic Perspectives 5: 73-96.
Brennan, Michael J. 1958. The Supply of Storage. American Economic Review 47: 50-72.
Brennan, Michael J., and E. S. Schwartz. 1985. Evaluating Natural Resource Investments. Journal of
Business 58: 135-57.
Considine, Timothy J. 1991. A Short-Run Model of Petroleum Product Supply. The Energy Journal 13:
61-9 1.
Cootner, Paul 1960 Retums to Speculators: Telser vs. Keynes. Journal of Political Economy 68: 396 -
404.
Dixit, Avinash 1990 Optimization in Economic Theory. Oxford: Oxford University Press.
Fair, Ray C. 1989. The Production Smoothing Model is Alive and Well. Journal of Monetary Economics
24: 353-70.
Gibson, Rajna, and E. S. Schwartz. 1990. Stochastic Convenience Yield and the Pricing of Oil Contingent
Claims. The Joumal of Finance 45(3): 959-75.
Gibson, Rajna, and E. S. Schwartz. 1989. Valuation of Long Termn Oil-Linked Assets, Anderson
Graduate School of Management, UCLA, Working Paper, #6-89.
Heal, Geoffrey, and Michael Barrow. 1980. The Relationship between Interest Rates and Metal Price
Movements. Review of Economic Studies 47: 161-81
Hotelling, Harold. 1931 The Economics of Exhaustible Resources. Joumal of Political Economy 39 April:
137-75.
Kaldor, Nicholas. 1939. Speculation and Economic Stability. Review of Economic Studies 7: 1-27.
18
Krane, Spencer D., and S. N. Braun. 1991. Production Smoothing Evidence from Physical-Product Data.
Journal of Political Economy 99(3): 558-77.
Larson, Donald F. 1993. Copper and the Negative Price of Storage, Policy Research Working Paper
1282, World Bank.
Lovell, Michael C. 1961. Manufacturers' Inventories, Sales Expectations, and the Acceleration Principle.
Econometrica 24: 293-314.
Merton, Robert C. 1992. Continuous Time Finance, Cambridge, Massachusetts: Blackwell.
Miller, Merton H. and Charles W. Upton. 1985a A Test of the Hotelling Valuation Principle. Journal of
Political Economy 95(1): 1-25.
Miller, Merton H. and Charles W. Upton. 1985b The Pricing of Oil and Gas: Some Further Results. The
Journal of Finance 40(3): 1009-1020.
Pindyck, Robert S. 1994. Inventories and the Short-Run Dynamics of Commodity Prices. Rand Journal of
Economics 25(1): 141-159.
Smith, V. Kerry. 1981. The Empirical Relevance of Hotelling's Model of Natural Resources. Resources and
Energy 3: 105-17.
Telser, Lester G. 1958. Futures Trading and the Storage of Cotton and Wheat. Journal of Political Economy 66:
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