POLICY RESEARCH WORKING PAPER 1282n
Copper and the Just as the price of a call
option contains a premium
Negative Price of Storage based on price variability, so
the shadow price of
Donald Frederick Larson inventories contains a
dispersion premium
associated with the
unplanned component of
inventories. When inventory
levels are low, the value of
the premium increases to the
point where inventories will
be held even in the face of a
fully anticipated fall in price.
The World Bank
International Economics Deparmnent
International Trade Division
April 1994
[OI.cY RESEARCH WORKING PAPER 1282
Summary findings
Commodities are often stored during periods in which world level, or more narrowly defined as the New York
storage returns a negative price. Further, during periods Commodities Exchange or the London Metal Exchange.
of "backwardation," the expected revenue from holding Larson argues that although inventories may provide a
inventories will be negative. Kaldor cost-reducing convenience yield, inventories also
Since the 1930s, the negative price of storage has been have value because of uncertainty. just as the price of a
attributed to an offsetting "convenience yield." Kaldor, call option contains a premium based on price variability
Working, and later Brennan argued that inventories are a so the shadow price of inventories contains a dispersion
nece-ssary adjunct to business and that increasing premium associated with the unplanned component of
inventories from some minimal level reduces overall inventories.
costs. This theory has always been criticized by Larson derives a generalized price-arbitrage condition
proponents of cost-of-carry models, who argue that a in which either a Kaldor-convenience and/or a dispersion
negative price for stor2 e creates arbitrage opportunities. premium may justify inventory holding even during an
Proponents of the cost-of-carry model have asserted that expected price fall. He uses monthly observations of U.S
storage will occur only with positive returns. They offer producer inventories to estimate the parameters of the
a set of price-arbitrage conditions that associate negative price-arbitrage condition. The estimates and simulations
returns with stockouts. Still, stockouts are rare in he presents are ambiguoLs with regard to the existence
commodity markets, and storage appears to take place of a Kaldor-convenience but strongly support the notion
during periods of "backwardation" in apparent violation of a dispersion premium for copper. And although the
of the price-arbitrage conditions. averagt value of such a premium is low, the value of the
For copper, inventories have always been available to premium increases rapidly during periods when
the market regardless ot the price of storage. This is true inventories are scarce.
whether the market is broadly defined at the U.S. or
This paper-a product of the International Trade Division, International Economics Department- is part of a larger effort
in the departriient to understand international commodity markets. Copies of the paper are available free from the World
Bank, 1818 H Street NW, Washington, DC 20433. Please contact Anna Kim, room S7-038, extension 33715 (78 pages)
April 1994.
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Produced by the Policy Research Dissemination Center
Copper and the Negative Price of Storage
by
Donald Frederick Larson
I wou!d like to thank Ramon Lopez and Ron DuncRn for their guidance and support during
this project. I would also like to thank Howard Leathers, Nancy Bockstael, Bruce Gardner,
and Karla Hoff for useful comments.
Table of Contents
1.0 Introduction 1
2.0 Refied Copper Prices and the Negative Price of Storage 3
3.0 The Optimization Model and the Price-Arbitrage Condition 19
4.0 Empirical Results 29
5.0 Conclusion 58
References 60
Annex 1: The Derivation of the Price-Arbitrage Condition 65
Annex 2: Necessary Border Conditions 76
Annex 3: The Quadratic-Plus Model 77
List of Figures
2.1 Copper prices, August 1978 to December 1989 3
2.2 Future price profile for high-grade copper 6
2.3 Future prices over time with constant future price expectations 12
2.4 The marginal cost-of-storage function 18
3.1 The effect of an increase in price on inventories and production 21
3.2 Convex shadow price for inventories 24
4.1 Monthly constant US producer prices and producer-held inventories
for October 1979 through December 1989 29
4.2 Discounted and deflated near-by spreads on the last day of the
month mapped against closing producer inventory levels for
October 1979 to December 1989 30
4.3 Discounted and deflated second-position spreads on the last day of
the month mapped against closing producer inventory levels for
October 1979 to December 1989 31
4.4 Simulated dispersion premiums for March 1980 to December 1989 40
4.5 Simulated marginal production costs 42
4.6 Simulated marginal storage costs 43
Lit of Tables
2.1 Spreads, stocks, and producdon of refined copper in the United States
on the first business day of January, 1974-1983 10
2.2 Implicit interest from lending refned copper stocks using September 24, 1991
closing COMEX prices for refined copper 16
4.1 Distributional characteristics of monthly producer prices and
COMEX end-of-month copper futures prices 32
4.2 Instruments used for fitted values of the log of inventories
and the log of sales 36
4.3 Estimated parameters and t-scores 37
4.4 Estimated parameters of marginal cost functions expressed as elasticides,
and estimated averaged dispersion premium 38
4.5 Simulated values for the marginal cost of production (C.), the marginal cost
of storage (Cz), and the dispersion premium 39
4.6 Average spreads and simulated dispersion premiums during periods of backward
and forward COMEX markets 41
4.7 Summary statistics on the estimated dispersion premium under alternative
estimadon methods and alternative lag-lengths for the price-variance
measure for the log-linear model 44
4.8 Tests on the effects of excluding the price of electricity from the marginal
cost-of-production function 46
4.9 Summary statistics on the estimated dispersion premium under alternative
estimation nethods and alternative lag-lengths for the price-variance measure
for the quadratic model 48
4.10 Estimated parameters and t-scores (in parentheses) for base model and model
treated for assumed errors-in-variables problem 50
4.11 Hausman test for misspecification based on LIML, 2SLS and 3SLS estimates
for varying moving-variance calculations 52
4.12 Tests related to the marginal cost of storage 53
4.13 Tests on convexity of the shadow price for refined copper inventories under
non-joint costs and constant marginal storage costs 55
4.14 Summary of the empirical results 57
A3.1 Parameter estimations for the quadratic-plus model 78
I,.
1.0 Introduction
The purpose of this paper is to explain producer-held inventories of refined copper
durirg an anticipated fall in prices. Commodities are often stored during periods in which
storage returns a negative price -- that is, when the expected price change (as indicated by
futures prices) does not cover the time-value of money plus storage expenses. In fact, during
periods of backwardation (when the settlement price for a near-by future contract is greater
than the price for contracts with more distant settlement dates), the anticipated revenue from
holding inventories will be negative. In the case of copper, inventories have always been
available to the market regardless of the price of storage. This is true whether the market
is broadly defined at the US or world level, or more narrowly defined as the New York
Commodities Exchange, or the London Metal Exchange. Since the 1930s, the negative price
of storage has been attributed to an off-setting "convenience yield". Kaldor (1939), Working
(1948) and, later, Brennan (1958) argued that inventories are a necessary adjunct to carrying
out business and, at some minimal level, increasing inventories reduces overall costs. This
theory has always been criticized by proponents of cost-of-carry models, who argue that a
negative price for storage creates arbitrage opportunities. More recently, Williams and
Wright (1991) have correctly argued that "convenience yields" have not been derived from
a formal optimization model. Proponents of the cost-of-carry model have asserted that
storage will occur only with positive returns and offer a set of price-arbitrage conditions.
Williams and Wright have augmented the cost-of-carry argument by stressing the effects of
stock-outs on price. Nonetheless, Williams and Wright (1991, p.140) note there have been
no stock-outs in the Chicago wheat markets during the last 120 years, and concede that
storage appears to take place duiring periods of backwardation, in apparent violation of the
I
price-arbitrage conditions.
This paper argues that while inventories may provide a Kaldor cost-reducing
convenience yield, inventories also have value because of uncer.ainty. Just as the price of
a call option contains a premium based on price variability, the shadow price of inventories
conte;ns a dispersion premium associated with the unplanned component of inventories. A
goneralized price-arbitrage condition is derived in which either a Kaldor-convenience and/or
a dispersion premium may justify inventory-holding even during an expected price fall.
Monthly observations of US producer inventories are used to estimate the parameters of the
price-arbitrage condition. The estimation and simulations presented provide little evidence
for supporting or dismissing the existence of a Kaldor-convenience, but strongly support the
notion of a dispersion premium for copper. Further, while the average value of such a
premium is low, the value of the premium increases rapidly during periods in which
inventories are scarce.
Following this introduction, the remainder of the paper is organized as follows:
Section 2 reviews the behavior of prices for the spot and futures markets in refined copper
and discusses backwardation and a negative price for storage in the context of the economic
literature. In this section, the arguments of Working, Brennan, Keynes, and Williams and
Wright are more fully explained. Section 3 develops the formal model of inventory-holding
in terms of a stochastic-control problem for the ending-value of inventories. The generalized
price-arbitrage conditions are derived from the optimization problem's first-order conditions.
Section 4 contains a description of the data and an estimation of the parameters of the
generalized price-arbitrage condition. Section 5 concludes.
2
2.0 Reflned Copper Prices and the Negative Price of Storage
Refining copper is a risky business and refined copper prices are characterized by volatile
price movements. Price swings can be large and sudden while low price levels can linger
persistently. The movement in refined coppek is especially significant given the small profits
obtained through refining -- typically 6 to 10 per cent of the final cost of refined electrolytic
copper (Brook Hunt & Associates, 1986). Since the refiner must buy scrap or blister copper,
the spread between the two prices contains the implicit fee for processing copper (Figure 2.1).
Yet, because of the time it takes to process the copper, rapid price movements can
dramatically inflate profits or generate losses even when the spread between inputs and output
Prices for US copper
180
to-
40-
201
Au-79 JunI Apr-63 Feb-5 Dec-66 Oct-68
Jul-6 May-62 Mr-64 kzi-66 IOy-67 %-6
- Rfmwe comr -+- Scrp oopper
Figure 2.1: Copper prices, August 1978 to December 1989.
3
price remains constant. There are several options available to the refiner. He can contract
the processing services directly, charging a fee but returning the refined copper to the original
owner of the blister or scMp copper (tolling); he can implicitly make the same arrangement
by buying spot in the blister or scrap market and simultaneously selling (shorting) a futures
contract in New York or London; he can contract with a semifabricator for future delivery
while buying in the spot market, etc.
The value of finished inventories held by refiners is subject to the same volatility.
Gains or losses genemted by wide swings in the value of held inventories may affect the
firm's profitability more greatly than returns from flows (production and sales). Options open
to the refiner include rcducing inventories to near-zero levels or contracting to sell inventories
either explicitly, or implicitly through the futures market. In addition, even when forward
contracts have been writLan, further revenue can be generated by implicitly "lending"
inventories. The mechanics of such an arrangement are discussed later; however the practice
has obvious and certain positive returns during periods of market backwardation.
Markets for refined copper and other metals are somewhat distinctive in their proclivity for
backwardation in futures pricing. Backwardation occurs when the price of futures contracts
with a more distant delivery date are discounted with respect to near-by contracts. A more
common and more general condition is when inventories are held at less-than-full carrying
charges, which Working (1949) termed a negative price for storage. A negative price for
storage occurs when the difference between the expected rate of change in the price fcr
immediate delivery and full carrying costs (interest, insurance, warehousing fees, and
spoilage) is negative. As a practical matter, a situation in which the difference between the
price for near-by and more distant futures prices falls below full carrying charges is taken as
4
Closug prioenc NM Coppr Futurs
for two dates in 991
.1L- -~
9110 _ *
105 + * _ ,
85
0 . . .. . . . . . 6 .s .;m .in.. . . . . . . ..
lbw It Mm* 1_31b IlIb
deliey dates
Figure 2.2: Future price profile for high-grade copper.
a negative price for storage. Backward markets necessarily carry a negative price fo: storage
since revenues from et rage are negative.
Figure 2.2 gives the profile of closing future prices on high-grade copper for two
separate dates in 1991. From July I to September 23 the entire price curve rose, while the
futures market remained in backwardation. While the situation depicted in Figure 2.2 cannot
be characterized as "normal", it is not unique for copper. From 1980 to 1989 the last-day-of-
the-month spreads on COMEX copper futures were backward 29 times -- or roughly 24%
of the time. A negative price for storage has been even more common. Jeffrey Williams
(1986) writes:
In nine of the ten years from 1974 through 1983 at least some of the spreads' in
copperfailed to displayfully carrying charges, sometimes, as in 1974 and 1980,
by a considerable margin. That any spreads in copper were less than full
'A spread, in this context, is the price of a futures contract with a more distant delivery date minus
the price of a futures contract with an earlier delivery date.
5
carrying charges is surprising in itself Because copper is a natural resource,
having little seasonality in production or demand, and large reserves of scrap, the
presumption would be strong that i:s price should Increase to cover interest ana
storage costs, as presumed in most models of natural resources.
Fama and French (1988), looking at daily interest-compensated spreads between spot and 3-
month, 6-month, and 12-month forwaid copper contracts on the LME between 1972 and 1983
reported negative average spreads for copper.
The general failure of the spreads in futures prices to cover full carrying charges is
well known to the literature. Holbrook Working noted the phenomenon in 1934, and
Nicholas Kaldor wrote about a "convenience yield" to stocks in 1939. In 1948, Working
provided a theory on "inverse carrying charges" which included a supply of storage at a
negative price:
Another important condition is thatfor most of the potential suppliers of storage,
the costs are joint; the owners of large storage facilities are mostly engaged
either in merchandising or in processing, and maintain storage faciliti2s largely
as a necessary adjunct to their merchandising or processi;.g business. And not
only are the facilities an adjunct; the exercise of the storing function itself is a
necessary adjunct to the merchandising or processing business. Consequently,
the direct costs of storing over some specified period as well as the indirect costs
may be charged against the associated business which remains profitable, and so
also may what appear as direct losses on the storage operation itself. For any
such potential supplier of storage, stocks of a commodity below some fairly well
recognized level carry what Kaldor has aptly called a convenience yield. This
convenience yield may offjet what appears as a fairly large loss from exercise of
the storage function itself
The concept of a convenience yield has become a popular and enduring part of the
literature on inventory behavior. (See Howell, 1956; Brennan, 1958; Telser, 1958; Weymar,
1974; Gray and Peck 1981, Thompson, 1986; Williams, 1986; Tilley and Campbell, 1988;
Thurman, 1988; Fama and French, 1988; and Gibson and Schwarz, 1990.) However, as with
many economic terms, there is a conflict in the literature over its definition. Kaldor and
Working, for example, apply convenience yield to mean an implicit return to holders of
6
inventories whose market value equals the difference between futures price spreau;s and full
carrying costs. The value to the holder derives from jointness between inventory holding and
related merchandising or processing. More recently, "the" convenience yield, has taken %
more operational definition, coming to mean the evaluation itself or, more precisely, the array
of spreads between futures prices adjusted for carrying costs. Often warehouse fees and
transaction costs are ignored so that carrying charges are exclusively composed of interest
charges. (See, for example, Fama and French, 1988). in such a case, the convenience yiold
is an empirical entity, constantly changing much like a tond yield curve. In this paper, I will
retain the definition given by Kaldor and Working. A similar idea, in keeping with
Working's definiti-:. of convenience, is that inventories are essential to production and
therefore inventory-holding cf inputs is joint with production as well as with marketing.
Ramey (1989) used such an argur..ent to justify inclusion of inventory levels in modeling
production technology.
Jeffrey WVilliarnr and Brian Wright (Wright and Williams, 1989; Williams and
Wright, 1991) have been especially critical of literature concerning a convenience yield.
Williams and Wright justifiably argue that convenience yields are not derived from "first
principles" (i.e. optimization conditions).2 They go on to argue that the "observation of
storage under backwardation is an aggregation phenomenon" and that "a spread below full
carrying charges can emerge only when there is no storage of that commodity.... Profit-
maximizing storage takes place only at full carrying charges, properly calculated."3 (Wright
2Ramey is an exception not noted by Williams and Wright.
'Williams (1986) credits Higinbotham (1976) with suggesting that the paradoxes of spreads result
from misleading averaging.
7
and Williams, pp. 3-4). By including the costs of transporting inventories from one location
to another, or the costs of transforming one close substitute to the relevant commodity (say
dirty wheat to clean wheat), Wright and Williams argue, the convenience yield disappears.
Backwardation in futures markets arises when the probability of a stock-out for the near
period is greater than for a later period.4 In William. and Wright (1991, p. 140) they soften
their stance, stating that "Empirically it does seem that storage takes place when the spot
spread is in backwardation, if for no other evidence than the disquieting fact that stock-outs
are never observed." At the same time, in their simulation model, they maintain a set of
price-arbitrage conditions that precludes storage at less-than-full carrying charges.
Since inventories are often reported aggregated over location, if not over type and
grade of commodity as well, it is difficult to test the Wright and Williams assertion in many
instances. However, the inventory-holding behavior of certified warehouse inventories of
refined copper provides a clear counter-example of the type which Williams and Wright
recognized in their 1991 statement. At numerous times throughout their history, both the
LME and the COMEX have reported abundant supplies of certified warehouse stocks while
simultaneously reporting backwardation in futures pricing. The added value of certification
is that certified inventories can be used to fulfill commitments resulting from futures
contracts. Should a holder of an open short futures contract in copper decide to deliver
copper rather than close his position, he need only transfer a receipt for inventories held in
any certified warehouse. No transportation or transformation is involved. Table 2.1
reproduces data reported by Williams (1986) on copper spreads on a single day in January
4Bresnahan and Suslow (1986) make a similar argument specificaUy in the case of copper and
backwardation in the Lordon Metal Exchange.
8
cver a ten-year period and supplements it with data on inventory levels. For the observations
reported, it was more common than not for refined copper inventory to be held at a negative
price, that is, below full carrying costs. At no time did a "stock-out" occur, although
inventories were low in January 1974. At the same time, however, it is difficult to argue that
effective stock-outs occurred in 1979 and 1980 but did not in 1975 and 1976. And copper
is not the only commodity to retain certified stocks in backward markets. Williams (1986)
notes that, despite the fact that there is no reason why elevators cannot be completely
emptied, certified wheat and soybean stocks, precisely those eligible for delivery on futures
contracts, have never fallen to zero. For wheat in Chicago, this includes s period of over 120
years.5
While Williams and Wright recognize the tension between the cost-of-carry model
and observed periods of negative storage with positive inventory levels, a negative price for
storage is routinely excluded from the possible range of solutions in the literature on
commodity policy issues. The general inter-temporal arbitrage conditions are often given as:
VWith regard to the 1973 soybean crop, Williams writes (pp. *6-37):
In 1973, the cost of keepina the more than 3 million bushels in store in Chicago was more than
$2.1 million. This expense was even larger nationwide. The total stock of old-crop soybeans as
of I September 1973 was 60 million bushels, the smallest carryover of the decade (although it was
still some 4% of the crop being then harvested.) As of I August 1973, the spread between the spot
price and the Augustfutures contract, on which deliveries were eligible until the end of the month,
was -$1.30 per bushel. This suggests that the holders of those 60 million bushels paid on the order
of $78 million, quite apartfrom physical storage costs, for the privilege of keeping them in store
that one extra month.
9
Table 2.1: Spreads. stocks, and producion of refined copper in the United States on the first business day of January. 1974-1983.
Prisd of spread 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983
iaauaay-March 4.60 .45 .40 .40 .40 .75 1.45 1.30 .80 .60
march-may -1.40 .55 .50 .45 .50 .75 .15 1.30 .85 .55
May-July -1.00 .60 .45 .45 .50 .70 .10 .95 .85 .55
July,Sepem*ber -.60 .60 .45 .45 .45 .55 .15 .85 .85 .60
Seplomber-Decembaer -.60 .65 .35 .45 .45 .45 .15 .80 .80 .70
Price of December
conract 80.60 58.50 59.60 67.40 54.50 75.30 112.70 99.40 33.40 75.75
Spreads afker removal of carrying charges
janry-Maub -5.20 .00 .00 .00 -.05 .00 .00 .00 -.10 .00
March-May -2.10 .00 .00 .00 -.05 .00 -1.30 .00 -.10 -.05
May-July -1.65 .00 .00 .00 -.05 -.05 -1.30 -.20 -.10 -.0S
July-Sepsember -1.20 .00 .00 -.05 -.05 -.25 -1.25 -.30 -.10 -.05
Sephmbe-December -1.15 .00 -.05 -.05 -.05 -.35 -1.20 -.40 -.15 .00
Stcks and mventory of refined copper expresaed m short tous
Comex Warehouse stocks 5,873 43,214 100,102 200,953 184,390 179,572 98,856 179,770 186,920 272,999
Total US stos 49,098 194,851 360,700 473,800 471,100 367,900 186,300 253,000 338,600 484,500
US production in Januay 157,700 146,000 130,300 140,900 129,000 135,600 161,000 133,500 117,500 97,300
Note: Spreds in copper ae based on the closing prices as of the first business day in January and are expressed m cenls per pound per month, taken from Wllias
(1986). Inventory vahles corresponig to January are December 3 t ;evels. Inventory and producton da were compiled from various issues of Meal Slatdcs.
10
p, s p,',/(l r) - k t- 2 0 (1.1)
pt > pe,'11(l +r) - k- zt - °-
where z is inventory level, p is price, r is a discount rate, k is a constant positive per unit
storage cost, and a superscript c denotes expectations. Recent examples include Miranda and
Helmberger (1988) and Glauber, Helmberger, and Miranda (1989).
Another argument,
related to a negative price
for storage, is Keynes's
(1931) notion of normal
backwardation. Keynes
argued that commodity I ,= : =
markets could remain
backward in equilibrium (at _
Con~e Corm
constant price expectations), NO I
implying negative reveizue
for storage, because of the Figure 2.3: Futures prices over time with constant future
price expectations.
way in which risk is
transferred from commodity producers, who are naturally long in the commodity, to
speculators. Hedging producers take a short position in the market (to offset their naurally
long positions) while speculators take the opposite position. Keynes argued that speculators
would decline taking an opposing long position unless the expected rate-of-retum on long
positions exceeded the riskless rate-of-return. In order for this to be the case, the expectd
rate-of-return on the futures position would have to be less than the expected spot price and
11
rise as the contract matured; that is, futures prices would have to be discounted to expected
prices. Keynes called this price-time relationship normal backwardation. (See Figure 2.3.)
Conversely, if hedgers are, on net, long in futures contracts, then speculators would require
a higher-than-riskless rate of return on short positions. In this instance, the futures price
would have to contain a premium over the expected price, which declines as the contract
matures. Keynes referred to this relationship as normal contango. Despite its implications
of a negative price for storage, Keynes's theory of normal backwardation has little to say
about inventories and arbitrage possibilities. This omission has formed the basis of criticism
by cost-of-carry proponents (see, for example, Figlewski, 1986).
Regardless of the validity of Keynes's arguments, the theory set off a number of
empirical studies aimed at verifying the presence of a downward bias in futures (that is, a
tendercy for the price of a futures contract to "rise up" to expected price levels as the
contract approaches maturity). Houthakker (1957), using trade statistics on corn and wheat,
concluded that naive small traders could benefit by blindly following a long-side trade
position. Tessler (1958, 1960) found no evidence of normal backwardation. Cootner (1960,
1967) then presented several cases in support of normal backwardation, while Dusak (1973),
using a portfolio approach, noted that even gains from arbitrating backward markets were
small when compared to investments in the stock market which carried similar levels of risk.
Two more recent studies, however, have emerged which provide support for Houthakker's
original findings. In the first, Carter, Rausser, and Schmitz (1983) modify Dusak's portfolio
approach to allow for systematic risk and found non-zero estimates of systematic risk for
most of the speculative return series examined. More recently, using non-parametric
techniques, Eric Chang (1990) found statistical support for normal backwardation in a study
12
based on wheat, com, and soybean futures. In studies relating directly to the market tor
metals, Hsieh and Kulatilaka (1982), using monthly data on LME forward contracts from
January 1970 to September 1980, reported average risk premia of 2.8% for copper, 17% for
tin, 12.7% for zinc, and 16% for lead. In a later study, MacDonald and Taylor (1989)
reported evidence for a "time-varying premium" in the forward prices for tin and zinc.
While cditicisms of Keynes's theory of normal backwardation and Kaldores
convenience yield remain valid, the empirical studies offered in their support are often at
odds with the simple cost-of-carry price-arbitrage conditions given in 2.1.
When inventories are being held at less than full carrying charges or when the price
on futures contracts is below expected prices, incentives are created to inter-temporally
arbitrage the market under the cost-of-carry model. The most obvious course of acdon would
be for holders of inventories to reduce their costs by selling immediately into the spot market,
bringing near-prices down relative to more distant delivery dates. Such inter-temporal
arbitrating should continue until the expected returns to arbitrage equal the expected returns
to inventory holding, or a "stock-out" occurs, when inventories are reduced to some near-zero
minimum level.
Similar arbitrage opportunities are available even when inventories have been
contracted for future delivery. For example, consider a holder of inventories (say a producer)
who has sold copper forward for delivery in 60 days. Regardless of the terms of the forward
contract, the seller can generate revenue in a backward market, or reduce holding costs when
storage returns a negative price, by "lending" inventories to the market.
Although direct loan markets are currently relatively rare for commodities, they do
exist for uranium, and a brokerage firm, the Nuclear Exchange Corporation (NUEXCO),
13
exists to facilitate such ioans. In addition, there are active explicit loan markets for equity
shares on many stock exchanges.6 Williams (1986) documents an active loan market in grain
warehouse receipts in the United States in the 1860s.
While no explicit loan market exists for refined copper, loan transactions can be
accomplished in an equivalent manner using futures and spot markets, obviating the need for
an explicit market. Once the decision has been made to hold inventories for a span covering
the delivery date of at least one futures contract, lending inventories can be accomplished in
a straightforward manner. Holders of inventories can sell into the spot market while
simultaneously contracting to repurchase the copper (by purchasing a futures contract) at a
future date for a fixed price. In a backward market, this sum will be positive and constitutes
a positive interest payment on an implicit copper loan. The effect on the copper market is
to increase supplies available for immediate delivery and increase the demand for future
deliveries, arbitrating the backwardation. It should be noted that not only does the supplier
of inventories receive a payment, but he also eliminates the need to store. As a result, even
when the futures market does not exhibit backwardation but does return a negative price for
storage, a holder of inventories can reduce his costs by lending supplies to the market.
Therefore, in cost-of-carry models, a negative price for storage generates arbitrage
opportunities as well, encouraging supplies to be freed from inventories.' As with the direct
6The London Exchange provides the most straight-forward example. In London the exchange
settles every fortnight rather than every day as in New York. Buyers of stocks have contracted to
receive the stocks on a particular day, but, if compensated, may agree to delay taking delivery of the
stock. If there is sufficient pressure for immediate delivery, the person agreeing to postpone a
contracted delivery may receive either a concessionary rate on margin loans or a fee from the seller.
The fee is called a "backwardation" and is equivalent to interest on a loan of deliverable stock.
7The flip-side of lending is, of course, borrowing. Because of the !wo-step nature of the implicit
loan procedure given above, first selling near while buying long, each "lender" need not correspond
with a single borrower. For example a fabricator may be purchasing the copper from the spot market
14
inter-temporal arbitrage, holders of inventories can be expected to lend into markets
exhibiting backwardation or negative storage prices until the expected return on lending
equals the expected return on holding inventories.
The implicit returns to loans from inventories of refined copper were calculated from
futures price spreads for September 24, 1991 and are given in Table 2.2. The returns are
probably underestimated, as warehouse fees, which would increase the returns to lending, and
transaction costs, which would decrease the returns slightly, have been omitted. Recalling
that total profits from refining may equal less than 10% of the price of refined copper, the
incentives to lend are significant. Put another way, the price of borrowing inventories is
high.
In the following section, it is argued that a convex shadow price for inventories,
combined with uncertain demand, gives rise to a dispersion premium for inventories. When
inventories are low, the effect of a higher-than-expected sales level will have a greater effect
on price than when inventories are more plentiful. In addition, when inventories are low, the
range of possible price outcomes become more skewed toward higher prices. By carrying
inventories into the period, the producer has the option of taking advantage of higher prices,
should they materialize, without increasing production and incurring increasingly expensive
marginal costs. Because of the asymmetry, the drop in the value of the carried inventories,
when sales are correspondingly lower-than-expected, is not as severe. This relationship gives
rise to a generalization of the price-arbitrage conditions in which it is rational to hold
while a toller may be locking in a future sales price for refined copper. However, if desired, the
mechanism is readily available for a refiner to borrow inventories which are replenished from future
production. This mechanism allows individual firms to hold a negative inventory of copper, even
though stocks must be non-negative in the aggregate.
15
inventories in the face of expected price declines.
In its emphasis on nonlinearities, the model is similar to Gardner's (1979) model of
optimal grain storage. Gardner used recursive programming techniques to demonstrate that
non-linearites in the sh&tdow price for inventories can result, in part, from nonlinearities in
Table 2.2: Implicit interest from lending refined copper stock using September 24, 1991 cloing COMEX pos
for refined copper.
Delivery Sottlement
Date Price 1-month 2-months 3-months
------------------- cents per pound ------
(% annualized return)
September '91 110.25 3.03 4.94 6.69
(33.0) (26.9) (23.9)
October 107.75 1.92 3.68 5.42
(21.4) (20.5) (20.1)
November 106.35 1.77 3.52 4.56
(19.9) (19.9) (17.2)
Decmber 105.10 1.76 2.81 4.10
(20.1) (16.0) (15.6)
January '92 103.85 1.06 2.35 3.44
(12.3) (13.6) (13.2)
February 103.30 1.30 2.40 3.48
(15.1) (13.9) (13.5)
March 102.50 1.10 2.19 3.13
(12.9) (12.8) (12.2)
April 101.90 1.10 2.04 2.97
(13.0) (12.0) (11.7)
May 101.30 0.94 1.88 2.81
(I11. 1) (I11. 1) (I11. 1)
June 100.85 0.94 1.88 2.36
(11.2) (11.2) (9-4!
July 100.40 0.94 1.43
(11.2) (8.5)
August 99.95 0.49
(5.9)
September 99.95
Note: Implicit interest calculated on an annualized 5.98% discount rate where:
interest = pf(t) - p'(t+n)em, n = 1,2,3
and where p' represent settlement prices for futurs contracts deliverable at t and t+n.
16
the underlying objective function. Since Gardner's topic was the optimal storage of grains,
the primary source of asymmetry was the fact that supplies of grairn can be withheld for
future consumption while a symmetric transfer from the future to tlie present is impossible.
Nonetheless, Gardner also showed that nonlinearities in demand, for example, would affect
the valuation of the shadow prices as well.
As will be seen later, the convex relationship between price and inventories is
suggested by simply plotting the two series and is a standard feature throughout a long
history of economic literature. It is
often stated as a stylized fact that the C
marginal value of inventory increases
with scarcity, declining in a nonlinear !
manner to zero as inventory levels 0 o Nz
z >~~~~ le~vol of inventories
increase. The marginally high value l of nenoe
of inventories at low levels of lnt
physical stocks is usually ascribed to
convenience and is often the Figure 2.4: The marginal cost-of-storage function.
departure point in textbooks and applied studies (for example, Stein, 1987; Fama and French,
1988; Gibson and Schwartz, 1990). Conversely, when holders of inventories appear willing
to retain inventories despite arbitrage opportunities, as in backward markets or other markets
with a negative price for storage, the holders are presumed to receive a higher "convenience
yield" as compensation. Working (1948) and Brennan (1958) provide a number of reasons
why the value of inventories above carrying charges might rise to very high values at scarce
levels and fall to zero at sufficiently high stock levels. Working argues primarily that stocks
17
are often an adjunct of business and provide convenience and cost savings through a
reduction in restocking costs and an ability to quickly meet orders. As stock levels increase,
the marginal contribution of additional stocks goes to zero. As inventories build, storage
facilities reach capacity levels and marginal storage becomes increasingly expensive.
Generally, the marginal cost-of-storage function is depicted as shown in Figure 2.4,
where z' marks the inflection point. However, in addition to the notion of convenience,
Working also offered some arguments based on expectations and probability, noting that
"Merchants who deal in goods that are subject to whims of fashion, or to sudden
obsolescence for other reasons, must lay in stocks and carry them in expectation that some
part of the stocks will have to be sold at a heavy loss." Additionally, he points out that as
stocks become low the probability of a "squeeze" in the futures market increases. Brennan
also argues that the "convenience yield" derives primarily from fewer delays and lower costs
(because of less frequent ordering and stocking) in delivering goods to consumers. However,
Brennan also discusses what he calls a "risk-aversion factor", noting that the larger the level
of inventories, the greater the effect of a price change and the revaluation of held stocks.
In the next section a formal model is developed to show that convex inventory-
shadow prices for copper can arise regardless of whether inventories are cost-reducing. The
model does allow for a cost-reducing Kaldor-convenience yield as well. A generalized set
of price-arbitrage conditions is then derived from the first-order conditions of the model. The
model is then used to test r ipirically for convexity in the shadow price of inventories which
gives rise to an estimatable dispersion premium. The existence of a cost-reducing effect for
inventories is also examined.
18
3.0 The Optmizaton ModW and the Price-Arbitrage Condition
In this section, the formal model is summarized from the detailed derivation
contained in Annex 1. 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 copper-refining 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 th;t 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.
The solution to the copper refiners profit maximization problem can be found by
solving the Hamiltonian:
Max,,,H a [ps - C(y,z)]e * + A(y-s) (3.1)
where p is the sales price; s is the sales level; C is a joint cost function for storage and
production where z is level of inventories and y is the production level; r is the discount-
rate. A is the change in profit due to a marginal change in the inventory level, or the
19
shadow price of inventories. The producer maximizes profits by setting the marginal cost
of production and the narginal revenue of sales equal to the value of a marginal change in
the level of inventories at the end of the period.
Prices are known in the current period, and inventory levels are deternined once
sales and production levels are decided, but the ending value of inventories is not knowr.
exactly. Rather, the value of stocks, and the shadow price of inventories, X, are based on
expectations about future prices, production, and sales.
When sales contain a random element, inventory levels will also be stochastic and
changes in inventories will contain a planned and unplanoed component. The difference
between planned and actual inventories will be the difference between expected and actual
production minus sales. For the momnent, assu,ne that the change in inventories can be
expressed as the following process:
;z;_z ,s Et [y,-s, l ] + (3.2)
where E 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"'V(z,l) * Max, EF{[ps - C(yz)]t. d = E(y-)d + .)
The term dv m u(t)dt112 is a Wiener process, where u(t) - N(O, 1).
The solution to the problem makes use of stochastic calculus, the mechanics of which
are somewhat tedius and have been relegated to Annex 1. However the relevant first-order
conditions can be summarized graphically. The producer solves for planned production,
20
sales and inventories by setting erpected marginal costs equal to expected price equal to the
shadow price of inventories, V; - which is itself an expected value. Because the producer
has the option of either producing more oL storing less, he must solve on two margins. This
is shown graphically in Figure 3.1 where marginal costs we drawn convex in y and the
shadow price of inventories is drawn convex in z. Whir. tS ere is an increase in the expected
price, inventories are reduced until the shadow price is equal to the new expected price
Elpj, V E[p]. V
z zi
-mz al cost
i Io
zl Z iniventories y y productionl
Figure 3.1: The effect of an increase in price on inxentories and production.
freeing inventories for current consumption. At the same time, planned pioduction increases
and expecte,d marginal costs increase until marginal cost equals expected price.
The first-order conditions can be manipulated to express the optimization conditons
in terms of expected price:
21
E[dp/dt] = rE(p] + E[Ct] - 2 V02, for z,s,y > 0 (3.4)
Equation 3.11 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 rE[p] -- interest on investing the money elsewhere -- plusC,
- the costs of physical storage and any amenity from storage -- minus lV_2 If Va is
positive, then this last term constitutes a dispersion premium that increases with the
variability of the stochastic component of inventories o2. The last two components of 3.4
have importart 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[dp/dt] < 0 - if inventories provide a cost-reducing
Kaldor-convenience (that is, if Cz is sufficiently negative) and/or the dispersion premium,
5V=C2, 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 V. = 0 and C. is positive, 3.4 reduces to the cost-of-carry
price-arbitrage condition.
For the cost-of-carry model, no inventories are held when the sum of the current
price plus a constant physical storage cost is greater than the expected discounted future
price. Putting this constraint into a continuous-time counter-part:
E [/dpt - rp +k, fir z >0. (3.5)
The two arbitrage conditions differ in two respects. In 3.12, storage costs are
treated as a constant positive marginal cost, and separate from other activities such as sales
or production. Cost-of-carry models are usually based on the activities of professional
22
speculators who presumably receive no convenience from holding inventories and do not
participate in production. Generally, however, there is nothing fundamental to the derivation
of cost-of-carry models which requires fixed marginal storage costs and the marginal
storage-cost function could be written in a more flexible manner. The second difference
between the two arbitrage conditions is the presence of a dispersion premium in the
generalized price-arbitrage conditions. This comes from treating the value of the inventories
as stochastic. Cost-of-carry models use expected prices (or futures prices representing
expected prices) but do not treat the price changes themselves as a stochastic process. This
differs from option-pricing models where the variance of the underlying commodity price
enters explicitly into the evaluation of the option.
The dispersion premium, 2 V o2, can be interpreted as the expected difference
between the stochastic and deterministic value of inventories. To see this, start with the
marginal-value function of inventories defined in terms of a deterministic component
(planned inventories) plus a random element, and calculate a Taylor-series approximation:
V(Zd+E) aVz) + V4E + ! V6e2 + I V,e (3.6)
zz 2 6 = E
When e has an expected value of zero, and is symmetrically distributed, where
E[e] = E[e3] = 0 and E[e2] = o,2, the expected difference between the stochastic and
deterministic component of the shadow price for inventories is approximately the dispersion
premium:
EEVC(zd+e) - V(zd)] _ IV,n2 (3.7)
Even when demand and inventories are treated as stochastic, there is nothing in the
23
first or second-order mnaximization conditions that would require V= to be positive. In
fact, V could certainly be quadratic in . so that V. need not exist. However, in much of
the literature on inventories the shadow price of inventories is often described as convex in
z - at least imnplicitly.' Usually it is stated that there is some type of "pipeline" mninimum
stock level (Figure 3.2). As stock-levels are drawn down and approach pipeline levels,
larger and larger price increases are required to deplete diminished inventories. Usually the
convexity is attributed to a
convenience yield. According to this
argument, pipeline levels are Vz
required to carry out business in an
orderly fashion, while additional
stocks, up to a point, can still j
facilitate transactions or minimize
costs such as re-orders, deliveries, or
restocking. p
pipel ine z
As it turns out, V= will
exists if either of the marginal cost Figure 3.2: Convex shadow price for inventories.
functions, C,, the cost of physical storage, or C , the cost of production, are nonlinear.
This is true whether or not the cost function is joint or whether stocks are cost-reducing at
any level. A fundamental question is why the marginal cost function should be convex in
either storage or production levels. Copper storage is simplicity itself, and requires only a
secure and dry area. However, if storage and production are joint and storage, at some
'For example, Working (1948), p. 19 and Brennan (1958), p. 54.
24
level, is cost-reducing, then there is probably some range over which the convenience
diminishes rapidly as inventories grow.
In addition, marginal production costs are also likely to be increasing at an
increasing rate over some production range, leading to nonlinearities in the cost function.
The refining industry is constantly changing and there is a wide diversity to the scale on
which refiners operate. For example, one of the newest refining plants in the US opened
in 1982 in South Carolina (AT&T Nasaue Recycle Corp) with a relatively small capacity of
70,000 refined tons per year. Since then capacity has grown to 87,500 tons. For these
small plants, which attempt to control costs by runmning near peak capacity, there remains
little room for production increases. Alternatively, the country's largest plant, ASARCO's
Texas plant, expanded its huge 420,000 ton/year capacity to 456,000 in 1987. At the same
time, parts of the facility are quite old. Certainly under normal circumstances ASARCO
could probably increase production by 20,000 tons more readily than AT&T Nasaue. Still,
for the industry as a whole, increasing production means bringing on line older and
increasingly less efficient facilities, operating with double shifts or otherwise using
increasingly more expensive inputs, both of which are likely to lead to convex marginal
costs.
Earlier it was stated that stochastic demand will give rise to stochastic inventories;
however, little was said about the relationship between the means and variances of the two
distributions. As it turns out, the problem can be cast in terms of either stochastic demand
or stochastic price. This result comes from the finance literature (for example, Cootner,
1964, and especially Merton, 1992). There is long history of decomposing inventories into
expected and random components (planned and unplanned inventories), particularly in
25
Keynesian macroeconomnics. Stochastic prices are often employed in finance and capital
literature (for example, Black and Scholes, 1973, or Abel, 1983).
For the copper-refining problem at hand, either interpretation is appropriate. Since
there are no real restrictions on trade or the recovery of scrap copper, US producers are
subject to events and decisions made around the world. On the demand side, copper is used
in products such as plumbing components, wiring, and brass. Not only does the derived
demand for these goods fluctuate with fluctuations in those associated industries world-wide
(for example, the construction of new homes), but demand is also subject to cross-price
effects from competing materials (for example, plastic plumbing components, or fiber-
optics). In addition, US refiners are not the only agents who hold inventories. Speculators
hold inventories at both major exchanges (COMEX and LME), fabricators (for example,
brass-mills) often hold inventories at their plants, and non-US refiners hold inventories
overseas. Domestic market conditions are also influenced by the flow of new supplies,
either in the form of imports or in the supply of recycled copper.
Refiners would prefer to know exactly about events in other sectors which influence
price and sales in their own market. Ultimately, this is impossible and some components
of final demand and price will not be known with certainty. In an abstract sense, the refiner
can be thought of as knowing the deterministic component of his demand schedule and the
general properties, but not the value of the stochastic component. The production function
presents less of a problem for the producer since, unlike agriculture where weather, disease,
and pests create uncertainty, "ie yield of refined copper from a given set of inputs is known.
Still, the final level of production - after adjustments to unanticipated changes in demand
and price - is not known ahead of time, but is conditional on final demand and price. The
26
initial stochastic term therefore originates in the demand schedule but will ultimately
influence sales as well. Normalizing the demand schedule with respect to quantity, sales are
stochastic; normalizing with respect to price, price is stochastic.
In the next section, estimation results are presented based on monthly data for US
copper refiners. After calculating the money return to storage,
m(t) * p(t) -p(t-l)-r(t-l)p(t-l), the following two equations are estimated jointly:
m(t) - Cp(t)-* X =Op(t)+u,(t) (3.8)
3(t) = C,(t) + up(t)
The first equation in 3.8 is a discrete version of the price-arbitrage condition given
in 3.11 plus an error term u,. The second equation is taken from the first-order conditions
given in Annex 1, and simply states that the producer will produce to the point where
marginal cost equals piice.. Merton's result is especially handy in this application, since the
second of the two equations provides a formal model for up, the stochastic component of
price. By using instruments, consistent (in the econometric sense) estimated values ofup
can then be used to form a logically self-consistent estimate of 2 . As noted earlier,
expressing the constraint in terms of the unanticipated component of price rather than sales
or inventories is essentially a simple re-scaling of the constraint. The symbol P= is
therefore used to reflect the re-scaled value of V=. The term Pm is treated as a constant
and estimated directly as a parameter. This treatment is equivalent to assuming that the
underlying marginal cost function can be reasonably approximated by a quadratic.2
These results are employed in Section 4 of this paper to test for convexity of the
2Recall equation A1.15.
27
shadow price for copper and provide an estimate of the dispersion premium associated with
copper inventories. In addition, the model can also provide a test of jointness between
production and inventory-holding which forns the economic incentive for a Kaldor-
convenience benefit.
28
4.0 Empirical Results
In this section, results from estimating the price-arbitrage and marginal cost
functions, using monthly data on sales, production, and inventory levels for US copper
refiners, are presented. Before the estimating procedure and results are discussed however,
the underlying data is presented in terms of the price-arbitrage equation developed earlier
in Section 3.
Figure 4.1 plots the US producer price for refined copper against inventories held
by producers. The data readily suggests a convex relationship. This relationship is not
Copper prices and inventory levels
held by producers
1.6 -
1.5
0 1.3
0i.2 -2
c150 1 0 5
0. gI 2
11-
~0.6 .
0 idO 2600 25
producer inventories (thousand tons)
Figur 4.1: Monthly constant US producer prices and producer-held inventories for October
1979 through Decemnber, 1989.
29
unique for copper, and the convex relationship between price and inventories is discussed
in much of the literature on inventories. For authors such as Working and Brennan, or more
recently Stein and Famma, this relationship is based on a convenience yield. In most cost-
of-carry models the relationship is ignored, although an exception is Williams and Wright,
who argue that high prices result from the increasing potential for a stock-out.
Figure 4.2 maps the spread' between the closing prices of the two nearest COMEX
copper futures contracts on the last day of the month against monthly closing producer-held
inventories. Figure 4.3 maps the spread between the two futures prices with second and
third closest settlement dates. The spreads become negative when inventory levels are low,
rising to near-zero positive levels with larger inventory levels. Again, this relationship is
not unique to copper. Working (1948) described the same relationship using wheat data for
1926. Brennan's 1958 article
provides a number of similar
graphs for eggs, cheese, butter, b__w_n_two_c__ st_C_Xcontracts
wheat, and oats. Working, 0 O a 0 .9 a ,
Brennan and others attributed ;-s I
such negative spreads to - ,s5
convenience yields and -20
-20 100 260
subtracted estimates of the l 0 ,o2 250
physical storage costs from the I
Figure 4.2: Discounted and deflated near-by spreads on the last
spreads to quantify the day of the month mapped against closing producer inventory
levels for October 1979 to December 1989.
'The spread is expressed in constant 1985 cents per pound and was calculated as spread a
[F(t+ 1)/(1 +r) - F(t)i/PPJ., where the future with the closest settlement date is subtracted from the
more distant future's price. The difference is then deflated by the monthly US producer price index.
30
convenience yield. Williams and
Producr hsnvutorri end the sprd
Wright argue that backwardation e betwe 2nd d 3rd drce con.t.cte
I I & S - i I .I
in futures prices comes from 4',-Ed'I-
increased probability of a stock- j..i l l
out. Storage in such
1-20.
circumstances is not fully 25 ' 0
5 1 00 O 250
explained; however, Wright and ,0 so2
WAilliaist argue that observedI
Wlawre 4.3: Discounted and deflated second-position spreads on
the last day of the month mapped against closing producer
storage is, in some case, an inventory levels for October 1979 to December 1989.
aggregation phenomena. They
argue that while the market in New York may promise negative returns to storage, prices
are forward in the geographically diverse markets where the inventories are actually stored.
The generalized price-arbitrage condition developed in Section 3 suggests two
possible reasons for storage in backward markets. First, the condition formalizes the
arguments put forward by Kaldor and Working. As inventory levels are drawn down, the
remaining inventories could potentially confer greater and greater convenience yields -- that
is, the marginal cost-of-storage function may become increasingly negative. The second
reason is that the convex shape of the shadow price function, which arises from non-
linearities in the cost function, conveys a dispersion premium when demand is uncertain.
For example, when inventories are low and demand is unexpectedly high, marginal increases
in production become increasingly expensive and prices rise increasingly quickly. At the
same time, stocks will be drawn down. Following the reduction in stocks, prices will rise
even more dramatically should a consecutive period of high demand materialize.
31
Alternatively, while a drop in demand may be just as likely, the effects on price are not
symmnetric. From any point on the convex marginal cost curve, the absolute value of the
price drop required to lower production by a unit is less than the price increase required to
increase production by a unit. Likewise, a greater price change is required to draw down
inventories by a single unit than to increase inventories by a single unit. This asymmetry
skews the possible outcome of prices (and therefore the shadow price of inventories) toward
higher prices and generates a dispersion premium. As a result, not only should prices be
higher when inventory levels are low, but the distribution of prices in general should be
skewed toward higher values.
The distributional characteristics of the real monthly producer prices as well as the
real futures prices (as observed on the last day of the month) for January 1980 to December
1989 are given in Table 4. 1.2 The spot prices are US producer prices (which are used in
the empirical estimation
reported later in this Table 4.1: Distributional characteristics of monthly producer prices and
COMEX end-of-month copper futures prices.
section), while the
futures :,rices are from Standard
Mean Deviation Skewness
the COMEX in New
Producer price 0.91 0.24 0.97
York. Because of the Nearby future 0.83 0.25 1.06
One-ahead future 0.83 0.23 1.01
difference in market
Note: Futures prices are for COMEX high-grade copper while spot
locations, the means are prices are US producer prices. All prices are deflated by the producer
price index, 1985-100.
slightly different.
2The coefficient of skewness given in Table 4.1 is E[(p - pT)]l o, so the expected value for a
symmetric distr.ibtion is 0.
32
However, the spot and the futures series exhibit similar standard deviations, and all three
series are skewed with the long tail of the distribution extending toward higher prices.
In order to obtain the parameters of the price-arbitrage equation derived in Section
3, as well as the parameters of the marginal cost functions and the dispersion premium,
variations of the following two generalized equations were estimated:
M(t) * ¢,,(t) - t (t) + V'M (4.1)
p(t) *C( + 5(t)
where a. and u. are random errors, where m(t) *p(t)-p(t-1)-r(t-l)p(t-1) is the
money-return to storage, p is the producer price, C, and C, are marginal cost functions for
inventories, z, and production, y. Several functional forms were considered for C, andC,,
and these will be discussed in greater detail later.
As discussed earlier, the variance in the stochastic differential equation associated
with the change in copper inventories can be expressed in terms of the variance in the
unanticipated component of the price for copper. In turn, the unanticipated component of
price is given by u, in 4.1, which can be used to construct an estimate of 2(t). The
estination process involves several stages. First, to avoid simultaneity biases, instruments
were used to provide fitted values for endogenous sales and production. Next, substituting
fitted values for endogenous right-hand-side variables, the second equation of 4.1 was
estimated using least-squares. One result is an unbiased estimate of the residual vector, 3 .3
An Il-month moving-variance estimate, 2(t), of the variance of the unanticipated
3Since a2(t) may vary with time, u, will be heteroskedastic. Nonetheless, the 2SLS estimates
for il, alithough inefficient, will be unbiased. (See Kennedy, 1992, p. 114.)
33
component of producer price O2(t) was then constructed from 4, where
2 -l t-I
6,(t), m [4,(n)- p(n)] /(l-l) and i,(t)O () ,(n) is the moving-mean.
Following the construction of 2p(t), both equations of 4.1 were estimated together
simultaneously as the final stage of a three-stage least-squares procedure. As discussed
earlier, P. is treated as a constant and estimated directly as a parameter. The entit-
62 Qdconstitutes an estimate of the dispersion-premnium.
The choice of a functional-form for the underlying cost-function, C(z,y), had to
meet several criteria. First, the form had to be flexible enough to allow third-order
derivatives and jointness between inventories and production, that is CI,, Co, C, I Cm,
etc. had to be potentially non-zero. The following log-linear equations were used initially
to approximate the marginal cost functions C, and C, from 4.1, since they meet the
criterion with a paucity of estimated parameters:
C, = bo + blnz + b2ny + bx + b4lnw + b5nf (4.2)
C = Co + clnz + c2lny + c3lnx + c4inw + c5lnf
where the b1 and cl for i = 0,,..._ are fixed parameters. The choice of functional fonn has
implications -- especially for the underlying marginal cost-of-storage function. These
implications, along with more general and more restricted functional forms, are discussed
later in this section.
The process of refining copper, explained in Section 2, is relatively simple, which
helps limit the number of parameters in 4.2. In addition to inventory levels and production
levels, the price of electricity, x, the price of raw input copper, w, and the capacity of the
refining plants, f, are given as arguments of the cost function. All prices are deflated by the
US producer price index which serves as a proxy for omitted prices (primarily wages). The
34
cost-function C(y,z) is initially modeled as joint. The second-derivative functions, C,, and
CX are not constrained to be symmetric, although symmetry is later tested.
Monthly data covering a period from September 1978 through December 1989 was
used in the estimation. Production, sales, and producer-held inventories were taken from
various issues of World Metal Statisdcs. The price of refined copper was taken as the US
producer price for refined copper from various issues of Metl Statisdcs. The price of No.
2 Scrap (New York) served as the price of raw input copper and also came from Metal
Statstis. The price for industrial electricity was taken from various issues of the United
States Department of Energy publication Monthly Energy Review. The interest rate used
was the 30-day US Treasury Bill rate. All prices and the interest rate were converted to
constant 1985-dollar values, using the US Producer Price Index. The Treasury Bill and
Producer Price data was taken from the International Monetary Fund's Financial Statistics
data base. The data on copper refinery capacity is collected by the American Bureau of
Metal Statistics, Inc and reported in various issues of Non-Ferrous Metal Data. The
capacity is reported at the plant level, based on surveys of plant managers. For most periods
the combined capacity of all plants greatly exceeded production levels. Plant managers may
have an incentive to exaggerate capacity levels (to forestall new entrants) and have no real
incentive to be accurate. In addition, the sur"ey is only reported once a year. As a result,
the data was treated as suspicious. However, using instruments and a smoothing technique
designed to detect errors-in-variables problems did not affect the estimation results
materially. Results of this procedure are reported later in this section. During the period
a major strike by labor unions dramatically reduced refinery output from July 1980 through
October 1980. An intercept-dunmmy variable, k, was used to designate the observations
35
associated with the strike. These atypical observations were, however, illustrative of the
relationship between inventories and uncertainty, as will be seen later during the discussion
of the simulation results.
The initial modcl was estimated from:
m(t) - bo + b9,.t) + b2lh(t) + b,lx(t) + b.lnw(t) + bslqt) + bsk(t) - , V02(t) + ut) (43)
p(t) - co +1,ln(t) + c2ln(t) + c.lnx(t) + c4lnw(t) + CsW(t) + clk(k) + .(g)
In the first-stage of the estimation process, fitted values for Int(t) and In9(t) were obtained
by regressing the log of inventory and production levels on the instruments given in Table
4.2 using OLS. The regressions resulted in R2s of .93 and .69, respectively. In the
second-stage of the estimation process, the fitted values for the log of production and the log
of inventories were substituted into
the second equation of 4.3, and Table 4.2: Instruments used for fittee values of the log of
invenwories and the log of sales.
estimates of the random-component
of price, fi(t) were estimated by interest rate
price of electricity
least-squares. A six-month moving scrap-copper price
fixed capacity level
variance, d:, was constructed from log of the price of electricity
log of the scrap-copper price
log of fixed capacity level
the resulting residuals and month of the observation
year of the observation
included on the RHS of the first lagged production level
laued inventory level
equation in 4.3.4 Simultaneously lag of production level squared
lag of inventory level squared
estimating both equations as the
4Results from the six-month moving variance are emphasized here because the model providcd
the maximum likelihood estimate among the lag-length choices. At a practical level, however, models
using four, five, seven, or eight-month moving-variances were not statistically different from the six-
month model when subjected to a likelihood ratio test.
36
third-stage in a three- Tabb 4.3: Estinated paruneters and t-scores.
stage least-squares Parameter Estimate t-score
procedure yielded the DiserlSion coefficifl
results given in Table V. 8.94 4.97
Marginal cost of storge (C.) coefficients
4.3, which gives the
bo. intercept 1.81 4.09
estimated coefficients, bl, inventories 0.04 1.89
b,, production 0.06 1.25
b3, electricity price 0.43 3.12
and Table 4.4, which b,, scrap copper price 0,22 4.91
bs. capacity -0.11 -2.22
expresses the coefficients b5, strike 0.06 1.23
as elasticities. T'he Murtinal cost of production {C> coeffickjis
results, for the most co. intercept 0.39 0.68
c,, inventories 0.09 4.11
part, were insensitive to c2, production 0.16 2.21
C3, electricity price -0.46 -2.70
alternative choices for c4, sCMp copper price 0.66 10.82
cs, capacity 40.23 -3.66
c,, strikce 0.14 1.86
the length of the lag
used to construct 6, as
well as the estimation method. Alternative estimates are discussed later.
The estimate of the dispersion premium is positive and significant at a greater-than
99% level of confidence. All of the arguments to the marginal cost functions C, are
significant at 97% + confidence level, with the exceptions of the intercept and the coefficient
on the strike dummy. At the mean, marginal production costs are rising with additional
production and are reduced by additions to capacity. Evaluated at the mean, additional
inventory slightly increases the marginal cost of production. A 1 % increase in the price of
scrap copper increases marginal production costs by 739%. The elasticity of marginal
production costs with respect to the price of electricity is negative, implying the unlikely
37
Table 4.4: Estimated parameters of marginal cost hinctions expressed elasticities, and estimated averaged
dispersion premium.
Estimated hfctions evabuated at seris' mea:
'Me elsticity of C. C,
with respect to:
inventories 1.93 0.09
production 3.32 0.18
electricity price 22.75 0.51
scrap copper price 11.79 0.73
refinery capacity -5.82 -0.25
strike dummy 3.21 0.16
Average dispersion premium
In cents per pound: 1.41
Note: Elasticities and dispersion premium calculated at muns. a, is constructed by a six-month moving
variance of second-stage price-forecast residuals for refined copper.
conclusion that marginal production costs fall as utilities raise their rates. While a negative
elasticity is unlikely, it may be that, in the short-run for a given fixed set of plant and
equipment, energy costs are relatively fixed so that the elasticity is properly zero. As it
turns out, constraining the elasticity on the price of electricity to zero has little effect on
subsequent hypothesis testing or simulation results. This topic is discussed more fully later
in this section.
The estimates associated with the marginal cost of storage are more mixed. The
elasticity with respect to inventory levels has the expected sign, and is significant with a
93% + level of confidence. The elasticity with respect to production levels and the
coefficient on the strike dummy are not significant5. The electricity, scrap copper, and
refinery capacity elasticities are significant. The sign on the refinery capacity is negative,
5Although both strike dummies are not significant individually, when subjected t? a Wald-test they
are jointly significant with a 99% confidence level.
38
which is reasonable since larger refinery capacity is probably associated with a larger facility
with more storage space. The electricity elasticity is positive, which is also reasonable if
the storage facilities are either lighted or heated. Mysteriously, the elasticity associated with
the price of scrap copper is also positive and significant. Later, results are presented from
constrained versions of the marginal cost-of-storage function including constant marginal
storage costs, as asserted in cost-of-carry models, and non-joint costs. As it turns out, the
significance and the size of the dispersion premium varies little under alternative versions
of the model.
The estimated model was simulated in order to calculate average marginal production
costs, average marginal storage costs, and the average dispersion premium. The average
levels, along with the minimum and maximum values simulated for these functions, are
reported in Table 4.5. The marginal storage costs are on-average low but reasonable at 1.9
cents per pound. The
Table 4.5: Simulated values for the marginal cost of production (Cr), the
range, from around 5 marginal cost of storage (C), and the dispersion premium.
cents to 11 cents is Average Minimum Maximum
reasonable andl suggests ----------cents per pound ------------
C. 1.9 -4.8 11.0
some scope for a Cy 87.7 56.5 126.0
Premium 1.4 0.0 9.4
convenience-yield. The
average levels and
ranges of the marginal cost of production and the dispersion premium appeared reasonable
as well.
Figure 4.4 maps the simulated dispersion premium against actual producer inventory
levels. With the exception of about 15 observations, the simulated dispersion premiums
39
Simulated dispersion premiums
0.101-
0.09-
i0.08-
0.07 -
, 0.06 -
0.05-
-0.04
0.03 -i
0.01
0 1 00 200
50 150 250
producer Inventorles (thousomd tons)
Figure 4.4: Simulated dispersion premiums for March 1980 to December 1989.
behave as expected, climbing when inventories are low and dropping off quickly when
inventories grow. The outliers were generated by either a strike, the threat of a strike, or
war, all of which resulted in a higher dispersion premium. In the figure, the outliers marked
with a 1 represent the three months leading up to the anticipated 1980 copper strike, and
observations marked with an s are months during the strike. The observations marked with
a 2 represent a period in 1983 when another labor strike seemed likely, but which ended as
negotiations were successfully concluded, starting with Kennecott's provisional agreement
with its unions in April (Crowser and Thompson, 1984). Finally, the observations marked
with an f represent the Falkland crisis (April to June, 1982). For those months, price
variability and the dispersion premium grew despite moderate inventories of refined copper.
Although estimated from data on US producer production, price, and inventories,
the model behaves as expected vis-a-vis the futures market in New York. The simulation
results indicate that, in the case of copper, information about price spreads in New York can
be used to predict dispersion premiums for US producers scattered across the country. As
40
shown in Section 3, inventories are rationally held when a price fall is anticipated provided
the dispersion prenmum is sufficiently large. Further, in the absence of a dispersion
premium, backwardation in the futures market generates powerful incentives to arbitrage the
market intertemporally. As a result, the dispersion premium for inventory-holders, including
US producers, should be higher during periods of extended backwardation on the COMEX.
Otherwise, inventories would flow to New York to take advantage of arbitrage opportunities.
To test this aspect of the model's performance, the sample period was divided into two sub-
samples. Observations were categorized as "backward market" observations if the
discounted spread between the second and third nearest COMEX contracts (the spreads
shown graphically in Figure 4.3) was negative; the observation was categorized as a
"forward market" observation if the spread was positive. The spread between the second
and third closest contracts was chosen to categorize the observations since it would allow
time for the copper to be physically shipped. Of the observations, about 48% were
"backward-market" observations. Average spreads and dispersion premiums were calculated
from the two samples and are reported on the first line of Table 4.6. In addition,
observations associated with
Table 4.6: Average spreads and simulated dispersion premiums
wars, strikes, or anticipated during periods of backward and forward COMEX markets.
strikes were removed from both backward markets forward markets
average average average average
samples, and means were Sample spread premium spread premium
calculated for the "purged" per pound
Full -3.22 1.85 0.44 0.98
samples as well. In the Purged -3.38 1.80 0.40 0.48
"purged" sample, roughly 53% Note: In the purged sample, observations associated with
anticipated strikes, strikes, or wars were removed. Spreads
of the observations were are discounted and reported in constant cents per pound.
41
categorized as "backward market" observations. The means from this sample are reported
in the second line of Table 4.6. For the "backward market" sample, the spread averaged -
3.22 cents and the simulated value of the dispersion premium averaged 1.85 cents. For the
"forward market" sample, the spread averaged 0.44 cents and the average dispersion
premium dropped in half to 0.98 cents. Purging the samples of war and labor unrest, the
difference was more dramatic, with the average dispersion dropping from 1.80 cents in the
"backward market" sample to 0.48 cents in the "forward market" sample.
Figure 4.5 maps the simulated marginal costs against production levels. The outliers
again are the strike months, when US production was constrained. Interestingly, some of
the highest marginal costs occurred in the months prior to the strike as inventories were
accumulated to substitute for anticipated drops in production.
Figure 4.6 maps the simulated cost of storage against inventory levels. The
estimated elasticity of marginal storage costs with respect to inventory levels is positive but
insignificant, and it is difficult to discern any clear pattern from the plotted data. If refined
Simulated marginal production costs
1.30
1. 20 _ S _
_1.1 eo- , ' ,,' I
1.00 I I
S
I 0.90 Is T I,L1 I
U01J
U0.70-
If
0.50
40 80 120 160 200
60 100 140 1S0 220
production leveis (thousand tons)
Figure 4.5: Simulated marginal production costs.
42
Simulated marginol storage costs
0.12-
0.1 0
0.08
0.06 III -, , ,
0.04 gIl '-' l
-0.06
0 1 0 200
50 1 50 250
Inventory levels (thousand tons)
F!g 4.6: Simulated marginal storage costs.
copper inventories do indeed generate a Kaldor-convenience, then negative marginal storage
costs should be associated with low levels of inventories. Further, after inventory levels
reach some rninimal levels, the marginal costs should become positive as the marginal
"convenience" disappears. In simulation, however, iany of the most negative values for
the marginal cost of storage are associated with relatively high inventory levels. Later in
this section, the issues of jointness and Kaldor-convenience are examined under alternative
specifications.
The estirnation and simulation results presented to this point are not sensitive to the
lag-length chosen to generate the movir,g-variance for the random component of producer
prices. Nor are the results sensitive to the estimation technique chosen. In addition to the
three-stage least-squares estimaes presented earlier, the model was also estimnated using two
43
Tabl 4.7: Summary statistics on the estimated dispersion premium under alternative estimation methods and
alternative lag-lengths for the price-variance measure for the log-linear model.
maximum ..... 2SLS----- ----- LIML----- ----3SLS-----
lag on t-score premium premium t-score premiumpremium t-score premium premium
variance on V, mean std on V. mean std on V,, mean std
4 7.01 1.21 2.13 4.96 1.21 2.13 5.94 1.11 1.96
5 5.09 1.29 1.99 5.93 1.29 1.99 4.65 1.22 1.88
6 5.21 1.46 2.09 8.03 1.46 2.09 4.97 1.41 2.00
7 4.81 1.44 1.98 8.03 1.44 1.98 4.67 1.39 1.90
8 3.78 1.33 1.77 7.60 1.33 1.77 3.65 1.28 1.70
9 3.53 1.38 1.77 7.63 1.38 1.77 3.37 1.30 1.68
Note: the hypothesis that the shadow-price of copper inventories is not strictly convex could be rejected
with a 99% + level of confidence under all estimation methods and all moving-variance choices. The
mean and standard deviations reported in the table are from simulations of the estimated models.
alternative instrumental variable techniques - two-stage least-squares and limnited-
information-maximnum-likelihood. Because endogenous variables appear as regressors in4. 1,
instruments or full-information likelihood techniques are required to avoid biased estimates.
However, all instrumental variable techniques should be consistent. If the model is correctly
specified, three-stage least-squares should produce the most efficient unbiased estimates,
since the procedure includes information on the contemporaneous variance-covariance matrix
of the structural equations' dIsturbances. In practice, with limited sample sizes, the
superiority of one method over another becomes less clear. Kennedy (1992, pp. 166-167)
reports the results from several Monte Carlo studies that examine the sensitivity of
estimators to changes in sample size, specification errors, multicollinearity, etc. Kennedy
concludes that Monte Carlo studies consistently rank two- and three-stage least-squares
estimators quite high in terms of robustness. At a practical level, the estimates should be
similar. In fact, Hausman (1978) has argued that large differences between estimators that
are consistent is, in itself, a sign of misspecification.
44
Table 4.7 reports the t-score6 associated with estimates of P from the three
estimators under alternative specifications for d:(t). The table also reports the average
simulated premium calculated from the estimate, as well as the standard deviation of the
simulated premium. The estimated coefficient on the moving-variance, J=, was always
positi - and significant - that is, the hypothesis that the shadow price of inventories is not
strictly positive could be rejected regardless of the lag-length chosen in the variance
calculation and regardless of the estimation technique. Further, the distribution of the
simulated premiums was quite stable across variance-choices and estimation techniques. This
was less true for the extreme values of the various simulations. Minimum values for all
simulations were slightly greater than zero, while maximum values ranged between slightly
above 7 cents to slightly under 14 cents. Extreme values consistently declined as the lag-
length increased, regardless of estimation technique.
As discussed earlier, the unconstrained estimation of the model retumed the wrong
sign for the price elasticity for electricity. The model was re-estimated under the assumption
that energy costs are detem-ined primarily by plant and equipment and that the correct short-
run coefficient value should be zero. Since the t-score on the original estimate was
significant, the constraint is a binding one; however, imposing the constraint had little
practical consequence. Regardless of the estimation procedure or lag-length chosen for the
moving variance, P. remained positive and significant. The three-stage least-squares
estimates for a range of lag-length choices are reported in Table 4.8. Further, Wald tests
6The t-scores reported in Table 4.6 (and later in Table 4.7) for the two- and three-stage least-
squares results are computed by TSP Version 6 to be robust in the presence of heteroskedasticity
using White's method. This procedure tended to lower the t-score compared to alternative
calculations.
45
were constructed to see Tabb 4.8: Tests on the effects of excluding the price of electricity from the
marginal cost-of-production fmction.
if the values of the
maxium lag In t-score 4J test
remaining fourteen moving variance on V. score
parameters changed 4 6.51 1.04
5 4.56 0.48
when the price of 6 4.91 0.55
7 4.62 0.64
electricity was excluded 8 3.62 0.60
9 3.33 0.58
from the marginal cost-
Note: Under the maintained hypothesis that electricity use is a fixed
of-production un tion. cost, the hypothesis that shadow prices are not convex can be rejected
at a 99% + level of confidence for all moving-variance choices. The
The test was cf instructed hypothesis that excluding the prke of electricity from the marginal cost-
of-production function does not change the remaining coefficient
estimates cannot be rejected at any reasonable level of confidence. The
by pairing each of the Wald-test is distributed as X2 with I degree-of-freedom.
remaining fourteen
parameter estimates from the three-stage least-squares estimates of the two models - one
model including the price of electricity, and one model excluding the price of electricity.
When the exclusion does not affect the estimates for the fourteen parameters, the sum of the
differences between the parameters will be jointly zero. As can be seen in the second
column of Table 4.8, the hypothesis that the estimated parameters are not different could not
be rejected with any reasonable level of confidence.
As discussed earlier, the existence of the dispersion premium depends in part on
non-linearities in the marginal cost-of-storage and production function. The log-linear form
chosen for the marginal cost functions (4.2) has the advantage of being nonlinear while
requiring few estimated parameters. However, there is a tradeoff between the practical
advantage of having fewer parameters to estimate and fiexibility in the functional form. This
is particularly true given that the marginal cost-of-storage function is usually depicted as s-
46
shaped (recall Figure 3.4). For this particular functional form, a positive coefficient on
inventories in the cost-of-storage equation implies that the marginal cost-of-storage function
is increasing as inventories accumulate, but at a decreasing rate -- that is, C. = b2/z and
C= -b2/z2. Often inventory-storage costs are expected to rise first at a declining rate
(C= is at first negative), but then to rise increasingly rapidly (so that C. is positive) as
storage facilities reach full capacity. While storage capacity for some grains or frozen
commodities is strictly limited, refined copper is easily stored and not subject to the same
type of storage-capacity limits. Nonetheless, in order to test whether the results reported
above are dependent on the functional form chosen, a more general form, qtiadratic with
third-order terms for z and y, was estimated as well. More specifically, the following
marginal cost functions were substituted for 4.2:
5 5 s
Cs=bo+ b1Xi+ bXIX,j + b7t3
, 1 1 (4.2b)
C, - Co + cxi + C C jVZIX + C793
where Xi, for i 1,2,...,5 represents z, y, x, w, and f from 4.2. The estimation procedure
was repeated on:
5 55
1 11~~~~~~~~~
m(t) = bo + biX, + EbuX,Xj + b6 k(t) + b723 - 2 d2 + CI(t)
(4.3b)
p(t) - co + cX + rcuXX + ¢6k(t) + C79 + (t)
using the three estimation techniques and the six specification of 6(t)
The more general quadratic-plus form generated an additional thirty-two parameters
to estimate. For the base model (three-stage least-squares using a six-month lag to generate
47
6;(t), only seventeen of the forty-seven estimated parameters were individually significant
at a 90% confidence level. These parameters and their t-scores are given in Annex II.
Nonetheless, P. proved relentlessly significant regardless of the estimation technique or
lag-length choice. In simulation, the quadratic version produced lower average premiums,
primarily due to lower maximum values. The maximum simulated values across the
estimated models did not decline as the lag-length on the variance was increased. Rather,
the range was more limited, from 4.92 cents to 7.16 cents, and showed no clear pattern.
Table 4.9 provides the summary results for the quadratic-plus estimations.
Evaluated at the mean of the series, the linear function Cm turned out to be negative
at a 97% + level of confidence, although, in simulation, C= did take on small positive
values when inventories were extremely high. At the same time, C., which should be
positive, turned out to be negative on average, although not significantly so. In summary,
despite the fact that many of its parameters proved insignificant, the estimated quadratic-plus
Table 4.9: Summary statistics on the estimated dispersion premium under alternative estimation methods and
alternative lag-lengths for the price-variance measure for the quadratic model.
m.aximum ---- 2SLS -- ---- LIML -- 3SLS
lag on t-score premium premium t-score premiumpremium t-score premium premium
variance on V. mean std on V. mean std on V. mean std
4 2.72 0.42 1.12 2.47 0.42 1.12 2.39 0.39 1.02
5 1.65' 0.42 1.01 2.62 0.42 1.01 1.63' 0.42 1.01
6 2.43 0.54 1.26 3.28 0.54 1.24 2.43 0.54 1.24
7 2.69 0.52 1.19 2.61 0.52 1.19 2.68 0.52 1.19
8 2.71 0.51 1.14 2.42 0.51 1.14 2.71 0.51 1.14
9 2.85 0.55 1.18 2.42 0.55 1.18 2.83 0.55 1.18
Note: the hypothesis that the shadow-price of copper inventories is not strictly convex could be rejected
with a 99% + level of confidence under most estimation methods and moving-variance choices. The two
scores marked with an * were significant at a 94% + level of confidence. The mean and standard
deviations reported in the table are from simulations of the estimated models.
48
model supports the notion of a dispersion premium, while suggesting that the extreme-case
premiums are quantitatively smaller than those simulated by the log-linear model. At the
same time, while the marginal cost-of-storage function was given greater flexibility, it did
not trace out the classic path of rising slowly to an inflection point, and then rising more
rapidly.
Despite the consistency of the estimates and simulations, one potential problem is
shared by all of the specifications. Since industry-wide refinery capacity is only published
once a year, the annual figure was repeated for all twelve months of the year when
estimating the model. On the one hand, shadow prices for inventories are based on
expectations, and it could well be that the market expectations must be based on the
published refinery data since no better source of information exists. Alternatively, since the
number of refiners is small, industry participants may form better subjective estimates of
capacity. Regardless, there is certainly the potential for measurement error in the variable.
The practical consequences of an errors-in-variables problem cannot be readily known. If
the measurement error is distributed independently of the disturbance terms, then t"1e
estimates remain consistent. However, if not, the estimates may be biased even
asymptotically. Since the exact nature of an errors-in-variables problem can never be
known, the correct course of action cannot be known. In "correcting" the problem, the cuie
may be worse than the disease -- if indeed a disease exists. The general prescription for an
errors-in-variables problem is to use instruments for the problem regressor or to somehow
average out the effects of the measurement errors. (Kennedy, 1991 pp. 137-40.) The
rationale is that by averaging out the data, the measurement errors are averaged as well,
reducing their impact. Table 4.10 reports the parameter estimates after addressing the
49
Table 4.10: Estirnated parameters and t-scores (in parentheses) for base model and model treated for assumed
errors-in-variables problem.
Durbin in-year
base model rank method smoothing
In equation m(t)
coeffcient on:
intercept: 1.81 1.28 1.84
(4.09) (2.76) (4.04)
inventories: 0.04 0.02 0.03
(1.89) (1.22) (1.25)
production: 0.06 0.04 0.06
(1.25) (0.80) (1.06)
electricity: 0.43 0.48 0.42
(3.12) (3.58) (2.62)
scrap copper: 0.22 0.22 0.21
(4.91) (4.68) (4.52)
capacity: -0.11 0.01 -0.11
(-2.22) (1.47) (-1.49)
price variance: 8.94 8.10 8.91
(4.97) (4.96) (4.50)
strike dummy: 0.06 0.04 0.07
(1.23) (0.72) (1.12)
In equation p(t)
coefricient cn:
intercept: 0.39 -0.87 1.07
(0.68) (-1.44) (1.86)
inventories: 0.09 0.05 0.13
(4.11) (3.07) (5.00)
production: 0.16 0.14 0.14
(2.21) (2.03) 2.06
electricity: -0.46 -0.36 -0.65
(-2.70) (-2.24) (-3.51)
scrap copper: 0.66 0.66 0.64
(10.82) (10.62) (10.61)
capacity: -0.23 0.04 -0.39
(-3.66) (4.1) (-4.86)
strike dummy: 0.14 0.12 0.14
(1.86) (1.82) (2.04)
errors-in-variables problem two ways. The first column represents the coefficients from the
base model, repeated from Table 4.3. The second column lists parameter estimates
following a treatment for the errors-in-variables problem suggested by Durbin (see Kennedy,
50
p. 140, or Johnston, pp. 430-2). Following this procedure, the observations on refining
capacity were ranked from highest to lowest. Observations on the rank were then substituted
as an instrument for the observations on capacity. Since the capacity was ranked from
highest to lowest, the sign on the capacity coefficients switched and, of course, the scale of
the capacity coefficients changed. Otherwise, the effects of treating the estimation for
measurement errors in the capacity variable were negligible. Column three of Table 4.10
reports estimation results in which the annual observations were smoothed. The smoothing
procedure was to measure the difference between refining capacity for any two years and,
from the January observation for the first year, add one-twelfth of the difference to each
monthly observation. Again, the effects of the smoothing procedure on the estimation results
are quite marginal.
As mentioned earlier, Hausman has argued that estimates obtained from alternative
but consistent estimators should themselves be consistent. Based on this principal, Hausnian
proposed the following test for misspecification. Two sets of consistent estimates are
differenced and then standalrdized by the difference in the covariance matrices of the two sets
of estimates. The resulting quadratic form is asymptotically chi-squared, with the degrees
of freedom equal to the number of linearly independent rows in the differenced covariance
matrix. The model fails the test, signaling misspecification of the model, when the
hypothesis that the two estimate sets are the same can be rejected. The estimation package
TSP Version 6.0 provides a matrix procedure for calculating the Hausman test. Table 4.11
reports the results of the Hausman test based on the difference between the three-stage
(consistent and efficient) parameter estimates and both two-stage and limited-infonration
(consistent) parameter estimates. Since different algorithms are used to numerically compute
51
Tabk 4.11: Hausman test for misspecification based on LIML, 2SLS and 3SLS estimates for varying moving-
variance calculations.
- three-stage least-squares compared to alternative estimators -
twotage kast-squares Ihmited-lnformatlon maxinum-likelihood
maximum lag in degrees of degrees of
moving variance freedom test-score freedom test-score
4 2 0.34 6 0.02
5 2 0.21 5 0.01
6 1 0.06 7 0.01
7 1 0.05 8 0.02
8 1 0.01 8 0.02
9 1 0.00 8 0.00
Note: Test-scores ae distributed as X2. The hypothesis that the two sets of estimates are equal could not
be rejected with any reasonable level of confidence.
the three estimates, they will in practice yield different estimates. The Hausman test
measures whether the estimates are critically different. The test was repeated using various
lag-lengths in the moving-variance calculation. Under all versions of the model the
hypothesis that efficient and consistent estimates were equal could not be rejected with any
reasonable level of confidence.
Earlier simulation and estimation results cast some doubt on the existence of a
Kaldor-convenience for refined copper inventories. Kaldor and, later, Working and Brennan
argued that a convenience yield exists when storage is a necessary adjunct to business,
which, in the case of copper refining, implies a jointness in the cost of production and
storage. Further, if this convenience yield effectively explains periods of backwardation and
a negative price for storage, then the marginal cost of storage must be negative over some
range of low inventory levels. Contrasted against this is the assertion in many cost-of-carry
models that marginal storage costs are fixed and positive.
52
Table 4.12 reports the test results on three hypotheses about inventory costs. All
three tests are constructed as Wald tests so the test scores have a chi-square distribution.
The symmetry test tests the hypothesis that C. = C, I evaluated at the mean for the series.
If the true underlying cost function is joint, then Young's theorem states that the partial
second derivatives should be equal. The hypothesis of symmetry is rejected when the test-
score exceeds a critical value. The results of the symmetry test are mixed and are sensitive
to the moving variance chosen in the estimation model. For lag-lengths 4 and 9, the
hypothesis of symmetry could be rejected at a 90% confidence level.
The second column of Table 4.12 tests the hypothesis that the cost function is
actually non-joint. This test is constructed to be quite weak in the sense that only the
coefficients on production in the marginal cost of storage and on inventories in the marginal
cost of production (b2 and cl in equation 4.2) were constrained to zero. This left, for
Table 4.12: Tests related to the marginal cost of storage.
tests on hypotheses of ----------------------------
maimum lag hi constant
moving variance symmetry non-joint costs marginal costs
Wald test scores ----------------
4 5.210 2.19 15.24"
S 2.04 4.420 17.93"
6 1.37 7.87" 24.79"
7 1.32 10.86" 27.73"
8 2.30 10.50" 26.02"
9 279' 8.77" 26.41"
Note: All tests wer constructed as Wald tests distributed as X2 with one degree of freedom. Test scores
denoted with an * are significant at a 90% + confidence level. Test scores marked with an 0 are significant
at a 95%+ level of confidence and test scores marked with - are significant at a greater-than 99%
confidence level.
53
example, the price of scrap copper as an cxplanatory variable in the marginal cost of
storage. Despite the weak definition, the hypothesis of a non-joint cost-function was rejected
at a 95% + level of confidence for all but the four-month moving-variance models. Using
a stronger definition of non-jointness, which would exclude the price of scrap copper from
the marginal cost-of-storage function cl = b2 = b4 = 0, non-jointness was rejected for all models
with a 99% + level of confidence.
The third column reports the results on the test that marginal storage costs are
constant, as proponents of cost-of-carry models often assert. Constant storage costs imply
that bi - 0, for i = 1,2,3,4,5 (from equation 4.2) in the estimated model. This hypothesis
was rejected at greater-than 99% confidence levels for all moving-variance choices.
Kaldor argued that the adjunct nature of selling and storing provides the economic
rationale for convenience. For copper producers this means a jointness between the
activities of producing, selling, and storing copper. Others have argued that inventory
activity is less complex, and, in the case of most cost-of-carry models, storage costs can
simply be best described by a constant marginal cos;. The test results provide strong
evidence that marginal storage costs are not constant, as asserted in cost-of-carry models.
Further, the combined results on the tests of symmetry and jointness provide somewhat less
strong support to the notion that costs are joint. Still, a Kaldor-convenience yield relies on
more than jointness. Inventories are expected to yield their highest marginal convenience
as inventories are initially accumulated - that is, the marginal cost of inventories is expected
to be negative at low levels, but grow positive as additional inventories build. None of the
estimated models presented here displayed that characteristic. Rather, the evidence here
suggests that, although inventories are a necessary adjunct to business, they generate no
54
savings.
As stated earlier, the existence of a dispersion premium is sufficient to justify storage
during arn anticipated price decline and strong empirical evidence was presented to support
the existence of such a premium. The estimation results reported thus far have come from
joint-cost models. Table 4.13 reports tests on the hypothesis that the shadow price for
copper inventories is convex under two maintained hypotheses. The first column reports test
scores on the hypothesis
that the estimated Table 4.13: Tests on convexity of the shadow price for refined copper
inventories under non-joint costs and constant marginal storage costs.
parameter is less-
than or equal to zero ~ in~ ing ~ - maintained hypotheses ---
te tmaximum lag In non-joint constant
moving variance cost function marginal cost
under the maintained _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
--------- t-scores-------
hypothesis that the cost- 4 4.61 4.69
3.48 3.88
function is not joint. 7 3.892 3.93
8 2.73 2.41
The results here include 9 2.32 2.04
the assertion that Note: The hypothesis that shadow prices are not convex can be rejected
at a 99% + level of confidence under both maintained hypotheses for all
marginal storage costs moving-variance choices.
have nothing to do with
the cost of scrap copper, although similar results were obtained from weaker definitions.
The hypothesis that the shadow price of copper inventories is not strictly convex was
strongly rejected at a greater-than 99% level of confidence. The second column tests the
same hypothesis under the assertion that marginal costs are constant. Again the alternative
hypothesis was strongly rejected.
Generally, the estimation results provide strong support for the notion that the value
55
of inventories contains a dispersion premium. This finding is empirically robust under a
variety of maintained hypotheses and under a variety of estimation techniques. Further, the
model behaves in simulation as would be expected even when simulated through periods of
rare events, including a war and labor disputes. The existence of a dispersion premium is
not guaranteed by the theory developed in Section 3, so empirical tests are crucial in
explaining inventories when a fall in price is anticipated. The model does not strongly
support or contradict the possibility of Kaldor-convenience, which could also explain the so-
called negative price for storage. On the one hand, the hypothesis of jointness between
production and inventory-holding for US copper refiners, a pre-condition for Kaldor-
convenience, is generally supported by the model results. However, the explanations of the
convenience - savings in delivery costs, re-ordering costs, etc. -- imply that inventories are
likely to be marginally less cost-saving as they accumulate. In simulation, the model did
generate some negative marginal values for the cost of inventory-holding; however, those
values were not consistently associated with lower levels of inventories. Table 4.14
summarizes the empirical results.
56
Table 4.14: Summary of the empirical results.
Results relating to Kaldor-convenlence yields:
-Jointness between production and storage, which provides the economic rationale for a Kaldor-
convenience yield is generally supported. The result is not completely robust under alternative
models.
-Symmetry between marginal production and inventory-holding costs, an implication of
jointness, is generally supported, although the result is not completely robust under alternative
models.
-The hypothesis of constant marginal storage costs, a feature of cost-of-carry models which
excludes the possibility of jointness, is robustly rejected.
-In simulation, negative marginal storage costs are generated. While negative marginal storage
costs imply a Kaldor-convenience, the observations do not appear exclusively when inventories
are low, as Kaldor's theory would predict.
Results relating to the dispersion premium
-A dispersion premium will exist if the shadow-price function for inventories is strictly convex
in inventories. The alternative hypothesis of concavity is strongly rejected with a high degree
of confidence.
-In simulation, the dispersion premium for copper from 1980 to 1989 averaged around 1.4 cents
per pound. However, when inventories were low, when strikes appeared imminent, and, during
the Falkland crisis, the premium rose. In simulation, the premium ranged from near zero to
about 9 cents per pound.
-The estimated value of the premium as well as tests on the convexity of the shadow price for
copper inventories were extremely robust under alternative estimation techniques.
-Although the model was estimated exclusively from data on US producers, in simulation,
larger-than-average dispersion premiums were associated with periods of backwardation in
COMEX copper futures.
-Hausman argues that since two-stage least-squares, limited-information-maximum-likelihood,
and three-stage least-squares estimators are all consistent, estimation results from the various
estimators should be similar. The model passes Hausman's specification test even under a
variety of specifications for the price-error-variance term.
-The test on the convexity of the shadow price and the existence of a dispersion premium was
robust under a variety of specifications for the underlying marginal cost functions. Generalizing
the functional form greatly increased the number of parameters to be estimated and generated
a large number of individually insignificant parameter estimates. Nonetheless, the dispersion
premium proved relentlessly signifiant.
-The simulated dispersion premiums, as well as the tests regarding its existence, were robust
even when non-jointness or fixed marginal storage costs were imposed.
57
5.0 Conclusion
As with many primary commodities, the price of storage for refined copper is
sometimes negative and copper futures markets have remained in backwardation for extended
periods of time. These market characteristics are in direct violation of frequently used price-
arbitrage conditions which maintain that storage only occurs when the price of storage is
positive. In Section 3, a generalized price-arbitrage condition is developed which is
consistent with observed inventory-holding in the face of an anticipated fall in price.
Potentially, there are two reasons why inventories might be held under such circumstances.
First, the marginal cost of storage may indeed be negative at certain low levels of inventories
- that is, inventories may produce a Kaldor-convenience yield. In the case of copper, while
there was some evidence that inventory-holding are joint activities resulting in a joint cost
function, there is little evidence in the simulations of a Kaldor-convenience yield.
The second justification for inventory-holding in backward markets stems from
uncertainty and a convex relationship between the shadow-price of inventories and inventory
levels. When stock prices are low, an unanticipated positive shift in the demand schedule
will result in rapid price gains. While a negative shift may be just as likely, the effect on
price is not symmetric. This skews the distribution of potential price outcomes toward
higher prices and generates a dispersion premium.
Estimates and simulations in Section 4 provide strong evidence that the shadow price
function of refined copper inventories is indeed convex in inventories. Under a variety of
assumptions, the key parameter in testing for convexity proved statistically robust. Further,
the estimated value of the dispersion premium proved robust as well. The premium
remained low on average (around 1.4 cents per pound). Simulations revealed that the low
average premium masked the potential importance of the premium. Producer-held
58
inventories for the simulation period ranged from around 25,000 to 230,000 tons. For
inventory levels above 80,000 tons, the dispersion premium fell to positive but near-zero
levels in the absence of labor strife or war. However, as inventories fell further the
premium shot up rapidly to a maximum value of about nine cents per pound. In simulation,
the dispersion premium also jumped during the months leading up to the 1980 refiner's labor
strike and immediately fell at the conclusion of the strike. Similar movements occurred
during a 1983 labor dispute and the 1982 Falkland War. Further, the model simulated
larger dispersions during periods of backwardation in the copper futures market in New
York (COMEX) even though U.S. producer prices, production, and inventories were used
in estimating the model.
Just as the price of a call option contains a premium based on the underlying
variability of prices, the shadow price of copper inventories contain a premium based on the
variability of the unplanned component of inventories. When inventory levels are low, the
value of the premium increases to the point where certain levels of inventories will be held
even in the face of a fully anticipated fall in price.
59
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64
Annex 1: The Derivation of the Price-Arbitrage Condition
In this annex, the formal model is presented. 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 copper-refining 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 premiurn 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 premnium in an options price and can result in
positive inventory levels even when price declines are expected.
The first-period producer problem can be written as:
Max,> f[ps-C(y,z)]e-'dt + J(zl,t1) st. z = (y-s) (Al.1)
where p is the sales price; s is the sales level; C is a joint cost function for storage and
production where z is level of inventories 3nd y is the production level; r is the discount-
rate and J(zl,tl) is the value of ending stocks. The cost function is treated jointly to allow
for a potential cost-reducing Kaldor-convenience for inventories.
65
The solution can be found by solving the Hamiltonian:
Max,,H a [ps - C(y,z)]el + A(y-s) (A1.2)
where the first-order conditions are given by:
i) 8H1.3s - 0 - p - A el, for to ststl,
U) a811o8y - 0 - Cy - A el', for tostst1,
M) a8lH& - A - C, - A-e, for toits:1, (A1.3)
fv) A(t) = ar/aqt),
v) z(to) = Zo
The producer maximizes profits by setting the marginal cost of production and the marginal
revenue of sales epal to the value of a marginal change in the level of inventories at the end
of the first period - that is, the shadow price of inventories at time tS (from conditions i,
ii and iv in Al.3, evaluated at t,). Evaluating the solution at t, - O, p - C, X A * ai/8.
Prices are known in the current period, and inventory levels are determined once
sales and production levels are decided, but the ending value of inventories is not known
exactly. Rather, the value of stocks J, and the shadow price of inventories, X, are based
on expectations about future prices, production, and sales.
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:
66
;-Ztl - E,-1[zt;-Z-] - et = ;-;-I - Et d[yt-sg] (A1.4)
where e has an expected value of zero and a variance o2. 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 Max, Ef{tps - C(y,z)]e"K}dt. 51 A - E(y-s)t* + adv. 5)
The term dv u(t)dt112 is a Wiener process, where u(t) -N(O,1). Because inventory
changes include a random component, dz/dt does not exist in the usual sense and the laes
of stochastic calculus apply.
Evaluated at t,, the solution to A1.5 gives the value of ending stocks, that is,
J(.zt1) - V(zl)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 producer problemn is based on
expectations of an on-going process of production amid uncertain demand. Generally, the
solution for J(zl,tl) can be found by solving Bellman's equation, a partial differential
equation: -aJ(zj,tr)1&8 = rV(z1)e"'. Arbitrarily setting tO -0 simplifies the equation
somewhat so that the solution to the inventory problem from t1 onward can be represented
by the Hamilton-Jacobi equation of dynamic programming:
rV(z1) = Max,,E [Ps - C(y,z) + VZ(y-s) + V(A1.6
The first-order conditions for the maximization of A1.6 are:
67
i) E(p - VS) . 0
a) E(-Cy + V.) - 0 (A.7)
L) E(A) - E(y-S)dt
tv) z(t1 0) - Zi
The producer solves for planned production, sale and inventories by setting
expected marginal costs equal to expected pnce equal to the dsdow price of inventories, V,
- which is itself an expected value.
It is worth noting that the distribution of the error term, especially a2, 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
thoughout the paper and works well empirically. Further, the derivation of stochastic
differenial equations are premised on the fact that the distribution is dependent solely on
rraet-period information (see, for example, Malliaris nd Brock, 1982, p. 67, or Merton,
1992, p. 67). The distinction is a subtle one, since it is reasonable to think of the variance
as some function of demand or supply. However, in terms of the optimization problem, it
is the expected value of the function, not the function itself, that is relevant.
Several additional asuumptions must be made to guarantee that the first-order
conditions do indoed provide a maxinmu. V must be concave in z; the solution values of
z, y, ad s must be positivel (otherwise border solutions must be considered); and the
' For the empirical pblem at had, refner ientories of refined copper re all positive u are
quaties sold d produced. Inventories at both the COMEX ad LME have remained postive
thzoughout the bsty of thou insditutios well. However, to be complete, stock-outs need to be
onsidered in the thoretical model ad non-negativity consorints intoduced to the moximizaon
problem. Thee are given in Anex 2.
68
transversality-at-infinity condition must hold.2
An expression for the marginal value of inventories is found by applying the
envelope theory (Lopez, 1989) to the Hamilton-Jacobi equation given in Al.6:
rVg - E-C, + V,(y -S) + .!V 21. (A1.8)
At the solution, Etp] - Vz, (as part of the first-order conditions in A1.7) so that:
EBdpetkJ - V1[*k/d*j (Al.9)
Rewriting part iii of A1.7 provides E[dz/dkJ - Ely-asl. These results can be combined to
form a price-arbitrage condition. First, from rearranging A1.8 and using iii from Al.7:
V,xELy-sl = VYE[dzdd - rV, + E[C3J - V,c,a. (A1l.O)
Combining Al. 10 with A1.9 provides:
Edpl/d] - rE[p] + E[C,] - 1V02, for z,a,y > 0 (Al.ll)
2
Equation Al.11 is a generalization of price-arbitrage conditions given in cost-of-carry
models such as Williams and Wright (1991, p. 27). The arbitrge condition states that the
2For the infinite-horizon autonomous problem given above, the tverslity condition is:
lrm V,(t)Z(t)# * liin JS(t)Z(t)rf - 0.
Benveniste and Schnkmn (1982) owed that the condition is necoy mid sffiient for the
solution of A1 .6 to be opdmal. he logic is that any posidve sock level g hve no value a the
problem apoaes infinity. Otherwe, the firm could furhe Ices proft by either producing
less or selling more in the last period. Invene have value beuse, uldtmay, twy can be sold.
If some price exists at which no copper can be sold, thm me upper bound mu eist for the uhdow
price of inventories. If so and if sock levels, z, re limited by pysic sore or naural
endowmu, then dioun wil assure that the laaverality condition holds. See Brock (1987)
for ftrther detils on the geal condition.
69
expected change in price will be equal to rE[pl -- interest on investing the money elsewhere
plus C, - the costs of physical storage and any amenity from storage -- minus 2 V= 02.
If V= is positive, then this last term constitutes a dispersion premium that increases with
the variability of the stochastic component of inventories 02. The last two components of
Al.11 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[dplb] < 0 -- if inventories provide a cost-
reducing Kaldor-convenience (that is, if C, is sufficiently negative) and/or the dispersion
premium, 2 Vo2, 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 V, = 0 and C. is positive, Al. 11 reduces to the cost-
of-carry price-arbitrage condition.
For the cost-of-carry model, no inventories are held when the sum of the current
price plus a constant physical storage cost is greater than the expected discounted future
price. Putting this constraint into a continuous-tirne counter-part:
E[d,pldt] = rpk jb fr z > 0 (A1.12)
The two arbitrage conditions differ in two respects. In Al.12, storage costs are
treated as a constant positive marginal cost, and separate from other activities such as sales
or production. Cost-of-carry models are usually based on the activities of professional
speculators who presumably receive no convenience from holding inventories and do not
participate in production. Generally, however, there is nothing fundamental to the derivation
of cost-of-carry models which requires fixed marginal storage costs and the marginal
storage-cost function could be written in a more flexible manner. The second difference
70
between the two arbitrage conditions is the presence of a dispersion premium in the
generalized price-arbitrage conditions. This comes from treating the value of the inventories
as stochastic. Cost-of-carry models use expected prices (or futures prices representing
expected prices) but do not treat the price changes themselves as a stochastic process. This
differs from option-pricing models where the variance of the underlying commodity price
enters explicitly into the evaluation of the option.
The dispersion premium, 2 Va2, can be interpreted as the expected difference
between the stochastic and deterministic value of inventories. To see this, start with the
marginal-value function of inventories defined in terms of a detemiinistic component
(planned inventories) plus a random element, and calculate a Taylor-series approximation:
VZ(zd+e) s VZ(Zd) + Vege + V + 6 (A1.13)
When e has an expected value of zero, and is symmetrically distributed, where
E[e] = E[e3j = 0 and E[e21 = o2, the expected difference between the stochastic and
determiniktic component of the shadow price for inventories is approximately the dispersion
premium:
E|Vz+[ V(zd)] 0 I V 2, (A1.14)
Even when demand and inventories are treated as stochastic, there is nothing in the
first or second-order maximiza.on conditions that would require V= to be positive. In
fact, V could certainly be quadratic in z so that V= need not exist. However, in much of
the literature on inventories the shadow price of inventories is often described as convex in
71
z - at least implicitly.3 Usually it is stated that there is some type of "pipeline" minimum
stock level (Figure A1.2). As stock-levels are drawn down and approach pipeline levels,
larger and larger price increases are required to deplete diminished inventories. Usually the
convexity is attributed to a convenience yield. According to this argument, pipeline levels
are required to carry out business in an orderly fashion, while additional stocks, up to a
point, can still facilitate transactions or minimize costs such as re-orders, deliveries, or
restocking.
As it turns out, V= will exists if either of the marginal cost functions, Cs, the cost
of physical storage, or C,,, the cost of production, are nonlinear. This is true whether or
not the cost function is joint or whether stocks are cost-reducing at any level. To see this,
recall that E[p] = E[C,J = V. from the first-order conditions (Al.7) and that:
d2p = V Z2 = d2C, (A1.lS)
From equation A1.15 it is easy to see that if the joint cost function is linear or
quadratic in z and y so that d2C. is z. ro, then V= = E[d2p/dz2l will always equal zero.
Earlier it was stated that stochastic demand will give rise to stochastic inventories;
however, little was said about the relationship between the means and variances of the two
distributions. As it turns out, the problem can be cast in terms of either stochastic demand
or stochastic price. This result comes from the finance literature (for example, Cootner,
1964, and especially Merton, 1992). There is long history of decomposing inventories into
expected and random components (planned and unplanned inventories), particularly in
Keynesian macroeconomics. Stochastic prices are often employed in finance and capital
3For example, Working (1948), p. 19 and Brennan (1958), p. 54.
72
literature (for example, Black and Scholes, 1973, or Abel, 1983).
For the formal model, the only distinction between the two concepts involves a
normalization of the stochastic difference equation, which constrains the optimization
problem. This result is due to Merton (1992, pp. 57-75). Consider first the process:
v(t) r X(k)-X(k-I)-Ek_{X(k)-X(k-I)), k=l,...n (A1.16)
where k denotes observed values, so that the partial sums Sn1?jv(k) form a martingale.
Now let F(t) =f(X,t), where f is some non-linear function. Merton shows that
F(k)-F(k-1) will give rise to essentially the same stochastic difference equation as
X(k) - X(k-1). This result is used frequently in finance literature -- for example, when
valuing instruments such as warrants in terms of the variance of the underlying stock price.
For the problem at hand, the result is useful sinrce it allows the differential equation
associated with the optimization problem to be expressed either in terms of stochastic sales
or stochastic price. Empirically, it turns out to be convenient to treat price as stochastic
since there is an explicit first-order condition from which the price-expectation error can be
estimated.
Returning to the copper-inventory problem, when sales are treated as stochastic then
z= z,_ l y[p(s)] - s. Utilizing Merton's result, the stochastic differential equation can be
written as:
dz = E(y-s)dt + (yPpP-l)QS(t)u8(t)dt112 (A1.17)
When the demand equation is inverted and price is treated as stochastic,
Zt = *y(p) + s(p) and the associated stochastic differential equation can be written as:
Either equation A1. 17 or AI. 18 can be normalized to yield a version similar to the constaint
73
dZ = E(y-s)dt + (yp _SP)aO(,)uP(,)d,112 (Al.18)
in A1.5:
dz = E(y-s)dt + oa(t)u(t)dt112. (A1.19)
In the next section, estimation results are presented based on monthly data for US
copper refiners. After calculating the money return to storage,
m(t) a p(t)-p(t-l)-r(t-l)p(t-l), the following two equations are estimated jointly:
I"(t) - C,(t) -. P2 2(t) + U,(t)
2 (A1.20)
p(t) = C(t) + u(t)
The first equation in A1.20 is a discrete version of the price-arbitrage condition
given in Al.11 plus an error term u. The second equatiou is a combination of i and ii
from the first-order conditions given in Al.7 plus an error term up. Merton's result is
especially handy in this application, since the second of the two equations provides a formal
model for u, the stochastic component of price. By using instruments, consistent (in the
econometric sense) estimated values of up can then be used to form a logically self-
consistent estimate of a2. As noted earlier, expressing the constraint in terms of the
unanticipated component of price rather than sales or inventories is essentially a simple re-
scaling of the constraint. The symbol P. is therefore used to reflect the re-scaled value of
V.. The term P. is treated as a constant and estimated directly as a parameter. This
treatment is equivalent to assuning that the underlying marginal cost function can be
reasonably approximated by a quadratic.'
'Recall equation 3.15.
74
These results are employed in Section 4 of this paper to test for convexity in the
shadow price for copper and provide an estimate of the dispersion premium associated with
copper inventories. In addition, the model can also provide a test of jointness between
production and inventory holding, which forms the ecomonic rationale for a Kaldor-
convenience benefit.
75
q a
I _ 0 E ot
S~~~~~~~. .9 .a *@ 3
4t -I '4 U
Annex 3: Tho QuadratI-PIn Modd
In order to test whether the results rqorted above are dependent on the functional
form chosen, a more general form, quadratic with third-order terms for z and y, was
estimated. Specifically, the following marginal cost functions were substituted for 4.2:
C,bo+ XiS b X, +S SEbVXj +b,
I b I (A3.1)
C, . co + E £Xx + Ec79
1 1 1
where Xi, for i = 1,2,...,5 represents inventories (z), production (y), the price of electricity
(x), the price of scrap copper (w), and refining capacity (f). The estimation procedure was
repeated on:
S 5 5
i(t) m bo + E biyxl + E E b"XX, + bk(t) + b7* - Xt + d (t)
1 1 (A3.2)
P(t) "Co + ClX + Z CUX,X + C.k(t) + C7 4 dt(t)
1 110t
using the three estimation techniques and the six specifications of 0;(t). Paameters from
one of the eighteen estimations, where a six-month moving average was used to calculate
6 (t), are given in Table A3. 1. The parameters were estimated using three-stage least-
squares.
77
Tabb A3.1: Paameter estimdons for tbe quadratic-plus model.
Parameter Estimate t-score Parmmeter Estinate t-score
CZ O 0.72 0.16 CY O -0.82 -0.17
CZ Z -I.OOE-02 -1.45 CY Z 0.93E-02 1.36
CZ Y *0.03 -2.06' CY_Y -0.02 -1.05
CZ X 51.69 0.33 CY X 13.09 0.08
CZ W -7.05 -2.44' CY W 7.89 2.97'
CZF 0.24E-02 3.10' CY_F -0.34E-03 -0.40
CZ ZZ 4.39E-04 -1.92* CY ZZ 0.401-05 0.82
CZ ZY -0.58E405 -0.39 CY7ZY -L.OOE-05 40.67
CZ ZX 0.27 2.50* CY_ZX *0.22 -2.24*
CZ ZW 1.001-03 0.47 CY ZW -O.IIE-02 *0.43
CZ77P O.JOE-05 1.44 CY ZF 0.12E-05 1.47
CZ YY 0.57E-05 0.48 CY_YY 0.48E-04 0.34
CZ YX 0.48 2.03* CY-YX 0.17 0.54
cZ YW 0.02 2.83* CY YW -0.54E402 -0.85
CZ YF -0.48E46 -0.20 CYYF 0.48145 1.72*
CZ XX 4120.33 -0.69 CY7XX 66.20 0.06
CZ XW 93.13 1.96* CY_XW -98.85 -2.32'
CZ XF 40.05 -3.36* CYXF 0.01 0.86
CZ WW 1.12 1.62 CYWW -0.21 -0.33
CZ7WF -0.50E43 -1.68* CY WF -0.43E-03 -1.37
CZ FF 0.52E07 0.51 CY_FF -0.16E-06 -2.12*
CZZZZ 0.84-07 1.77* CYYYY -0.16E-06 -0.41
C5ZK 0 10 3.00* CY K 0.01 0.31
v-ZZZ 6.57 2.43*
Note: T-scorms markod with an * ar sinifiant at a 90% + level of confidence. The parameters are
named using the folowing convention: CZ_W is the parameter in the marginal cost of storale function
(C.) on w, CY_WF is the parmmeter in the marginal cost of production function (C,) on the product of
w and f, CZ ZZZ is the coefficie.nt in C, on z', where z is inventories, y is producdon, x is the price of
elctricity, w is the price of scrap copper, f is refining capacity, and k is the strike dummy. V ZZZ is
the coefficient on the six-month moving varince, -2(t).
78
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