It should be obvious that holders of commodities incur a storage cost for financing and storing inventories, including warehousing, handling charges and insurance. The conundrum is that not infrequently prices of the deferred futures are below that of the nearby future and/or spot price. When such situation occurs the market is said to reflect “inverse carrying charge” or “backwardation.” Much academic study has been dedicated to this seemingly strange phenomenon, and in providing a satisfactory answer as to why such conditions exist.
Working, whose 1949 paper The Theory of Price of Storage evolved from Keynes’ writings on commodity pricing, investigates the “theoretical problem… that it is existing supply rather than expected change in the supply which is involved in determining inter-temporal price relations.” This statement is more interesting if one goes back to Working’s 1948 article, Theory of the Inverse Carrying Charge in Futures Markets, in which he discusses “four different lines of attempt to explain inverse carrying charges.”
There seems to be substantial agreement among writers on futures markets that positive carrying charges tend to reflect marginal net costs of storage, and that when carrying charges are positive, prices of near and distant futures must respond about equally to any causes of price change. With regard to inverse carrying charges there is more difference of opinion and much reason for the student of futures markets to be dissatisfied with the present state of theory. How shall one explain a large inverse carrying charge between December and May in a United States wheat market? Do inverse carrying charges reliably forecast price declines? When prices of deferred futures fall below the spot price, does the futures market tend to lose effective connection with the cash market? These are some specific questions that call for answer and to which no satisfactory reply seems to be offered by prevailing theory.Theory of the Inverse Carrying Charge - Working
The first explanation that Working (1948) explores is the idea that “cash and futures prices, though related, are not equivalents aside from the time element.” In the narrow sense the basis difference may be due to quality differentials or delivery locations. However, as Working noted even back in the 1940s, the cash market is “clearly subsidiary from the standpoint of price formation” and that cash buyers and sellers ordinarily “bargain in terms of cents ‘over’ and cents ‘under’.” More broadly, it can be argued that “cash and futures prices may differ because they reflect the opinions of substantially different groups of traders.” However, this implies that cash-futures arbitrage is ineffective. In this regard Working points out that hedging is essentially a form of arbitrage, and notes that hedging persists even when markets are backwardated.
Next, Working addresses the view that futures prices tend to have a downward bias due to risk aversion. Keynes’ (1930) posits that producers are willing remunerate speculators in order “to avoid the risk of price fluctuations… during the period of production” causing the spot price to exceed the forward price. Vance (1946) adds “that future events always bear some degree of uncertainty is perhaps sufficient to justify a discounting of expectations.” Working admits that such ideas probably have some validity, but points out that risk avoidance can be applied to “possible future events” which are “price-elevating” as well as “price-depressing.” Rather, risk aversion is “pertinent as partial explanation of the ‘supply curve for storage’.”
The third point that Working disparages is “the belief that price differences between futures commonly reflect expectations regarding future developments.” Working argues that there is a continuity between nearby futures and deferred futures to the extent that “expectations arising from an existing supply situation” have a bearing on “expectations regarding future supply or demand developments” and vice-versa. This hypothesis is substantiated by empirical research in which evidence of “large changes in supply prospect on price relations” between nearby and distant futures contracts “lasted for not more than a week or two”, and was “followed quickly by a return toward the previous price relation” prior to the disturbance.
Working’s conclusion is that inverse carrying charges are best “explained in terms of the concept of price of storage.” First, “spot and futures prices for a commodity are intimately connected at all times” and “do not, in general, measure expected consequences of future developments.” Second, “inverse carrying charges are reliable indications of current shortage; the forecast of price decline which [the term structure implies] is no more reliable than a forecast… of the current supply situation itself.” Third, inverse carry is explained within a commercial context whereby hedgers “are willing to risk loss on a fraction of the stocks for the sake of assurance against having their… activities handicapped by shortage of supplies.”
From these thoughts Working (1949) developed his price-of-storage theory which “exposes clearly the fact that in the presence of hedging much storage does occur in response to a recorded, and competitively determined, assurance of return specifically for the storage itself.” In other words, the futures market “coupled with the practice of hedging, gives potential holders of [a commodity] a precise or at least a good approximate index of the return to be expected from storing [such commodity]. A known return for storage is, in essentials, a price of storage… determined in a free market through the competition of those who seek to supply storage service.”
With respect to inverse carrying charges, Working noted that “the problem [with the theory] tends to emerge clearly only when the price for deferred delivery is below the ‘nearer’ price.” Nevertheless, Working argues that such conditions can be rationalized by Kaldor’s (1939) ‘convenience yield’ whereby “stocks of all goods posses a yield… which is a compensation to the holder of stocks, [and] must be deducted from carrying costs proper in calculating net carrying cost.” In other words, to remain long in business a merchant “must carry stocks beyond known immediate needs and take his return in general customer satisfaction.” Thus “convenience yield may offset what appears as a fairly large loss from exercise of the storage function itself.”
In effect, the price-of-storage theory, in which “positive carrying charges tend to reflect marginal net costs of storage,” can be extended to include negative carrying charges. According to Working (1948, 1949), “the supply-curve relationship between amount of storage and price of storage does not break down when the ‘price’ becomes negative… [which] occur when supplies are relatively scarce.” Rather, “a negative price of storage makes available for consumption in a year of shortage, supplies which would otherwise remain tied up in ‘convenience stocks’.” On the other hand, “if stocks to be stored are exceptionally large, the return for carrying [commodities] may exceed the ‘cost’ of storage… If stocks are quite moderate, competition among firms with storage facilities tends to result in the storage being provided for a rather small return.”
Michael Brennan in his 1958 paper, The Supply of Storage, both clarifies and expands on Working’s (1949) theory through description, formulae and empirical analysis. First he introduces the ‘demand for storage’ equation, Pt+1 – Pt = ft+1(St + Xt+1 – St+1) – ft(St-1 + Xt+1 – St), wherein “the demand curve for storage of a commodity from period t to period t+1 will shift upward (e.g., to D΄D΄ in Figure 1) as result of an increase in production in t,… [and] opposite movements of [this variable] will produce a shift downward.” Brennan then defines the ‘net marginal cost of storage’ in period t as “marginal outlay on physical storage plus a marginal risk-aversion factor minus the marginal convenience yield on stocks.” He also assumes that “marginal outlay is approximately constant until total warehouse capacity is almost fully utilized”, and beyond this, outlay rises at an increasing rate. The net marginal cost of storage equation is mt΄(St) = ot΄(St) + rt΄(St) – ct΄(St).
From such equations Brennan (1958) derives two graphics which elegantly reveal the key dynamics influencing the determination of positive versus negative carry prices. In the first graphic below, DD, D΄D΄ and D΄΄D΄΄ are demand-for-storage curves, and CC is the supply curve for storage. It is noted that the horizontal axis represents supply of storage at end of t, and the vertical axis represents the basis between deferred and nearby futures.
The second graphic reveals the interplay between convenience yield, storage outlay and risk aversion upon the net marginal cost of storage. Importantly, Brennan notes that risk aversion should be an increasing function of inventories held. “If a comparatively small quantity of stocks is held, the risk involved in undertaking the investment in stocks is also small.” However, “there is probably some critical level of stocks at which the loss would seriously endanger the firm’s credit position, and as stocks increase up to this point the risk incurred in holding them will steadily increase also—the risk of loss will constitute a part of the cost of storage.”
Brennan also adds to Kaldor’s and Working’s discussion on the convenience yield by noting it is “attributed to the advantage (in terms of less delay and lower costs) of being able to keep regular customers satisfied or of being able to take advantage of a rise in demand and price without resorting to a revision of the production schedule. Similarly, for a processing firm the availability of stocks as raw materials permits variations in production without incurring the trouble, cost and perhaps delays of frequent spot purchases and deliveries. A wholesaler can vary his sales in response to an increased flow of orders only if he has sufficient stocks on hand. The smaller the level of stocks on hand the greater will be the convenience yield of an additional unit.” Alternatively, “it is assumed that there is some quantity of stocks so large that the marginal convenience yield is zero.” Further, distinction is sometimes made between ‘surplus’ stocks, which Brennan notes are distinguished by a speculative motive, versus ‘pipeline’ or ‘working’ stocks.
To wrap up this article, we leave the reader with Lester Telser’s insights. Telser (1958) analyzed storage in relation to firms’ stockholding schedule, and then related the results of his study to conventionally-held ideas about commodity pricing which emphasizes the role of the speculator and the role of commodity futures contracts as insurance. In certain respects Telser’s findings support Working’s (1949) conclusion that “only some direct explanation of the price relation in terms of an existing condition can account for the fact that expectations regarding future events, which are directly pertinent to a distant forward price, have approximately the same effect on spot and near forward prices as on a distant forward price.”
A widely accepted theory advanced by Keynes and Hicks which relates the futures price and the expected spot price regards hedgers as buyers of insurance and speculators as sellers of insurance who must be induced to bear the risk of price changes. When statistical evidence was examined to see whether futures prices display an upward trend as they approach maturity predicted by this theory, it was found instead that futures prices display no trend. Although hedgers may be willing to pay speculators to bear the risks of price changes, they need not do so if speculators are eager to speculate. Firms that hedge can reduce their price risks at little or no cost to themselves. I accept the hypothesis that the futures price equals the expected spot price.The forgoing was presented to help educate readers on the “theory of storage”, specifically key concepts in the ever-evolving debate of what factors most influence commodity prices. The conclusion we arrive at as a result of this exercise is the same conclusion we have deduced in other studies—pricing models are first and foremost informed by perspective, and perspective is informed by assumptions. As a result, the legacy of research will always remain inconclusive with respect to modeling the sources of returns in the commodity futures markets largely because these models have inherent shortcomings in being able to pinpoint an authoritative source of structural risk premium within the complexity of such markets.
 “Dwarfs standing on the shoulders of giants” (Latin: nanos gigantium humeris insidentes) is a Western metaphor meaning, one who develops intellectual works by understanding the research created by notable thinkers of the past. The saying is attributed to Bernard of Chartres due to John of Salisbury’s reference in Metalogicon (1159).
 Hilary Till (2007). “Part I of A Long-Term Perspective on Commodity Futures Returns: Review of the Historical Literature” Intelligent Commodity Investing, Till and Eagleeye, Ed., London: Risk Books, p 39.
 In the paper, The Supply of Storage, Brennan (1958) clarifies that “supply of storage refers not to the supply of storage space but to the supply of commodities as inventories. In general, a supplier of storage is anyone who holds title to stocks with a view to their future sale, either in their present or in a modified form.”
 In the context of financial futures, “basis” is defined as spot price minus the futures price. There is a different basis for each delivery month for each contract.
 “One condition which makes [inverse carrying charges] possible is the fact that storage of [commodities] is an enterprise in which most of the costs are fixed costs, from a short-run standpoint. Another important condition is that for 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 facilities largely as a necessary adjunct to their merchandising or processing 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.” (Working, 1949)
 Let Pt be the price in period t and let Ct be consumption during t. Consumption in any period equals stocks carried into the period plus current production minus stocks carried out of the period. Consequently consumption, ft(Ct), can be rewritten as Pt = ft(St-1 + Xt – St), where St-1 is stocks at the end of period t-1, Xt is production during t and St is stocks at the end of t. In general, price in the next period minus price in the current period may be expressed as a decreasing function of stocks carried out of the current period. Symbolically the demand for storage from period t to period t+1 can be represented as Pt+1 – Pt = ft+1(Ct+1) – ft(Ct), which is equivalent to formula presented in article.
 mt(St) is the net total cost of storage and mt΄(St) is the net marginal cost of storage. Again let St denote the stocks carried out of period t. Let ot(St) be the total outlay on physical storage where ot΄>0, rt(St) the total risk-aversion factor where rt΄>0, and ct(St) the total convenience yield where ct΄≥0. The net marginal cost of storage need not be positive.
Blas, Javier, “Growing demand for bullion hands banks gold opportunity in storage” Financial Times, June 12, 2010.
Brennan, M. (1958). “The Supply of Storage” American Economic Review, 47(1), pp 50-72.
Fama, Eugene F. and French, Kenneth R. (1988). “Business Cycles and the Behavior of Metals Prices” Journal of Finance, 43(5), pp 1075-1093.
Keynes, John Maynard (1930). A Treatise on Money, Volume II: The Applied Theory of Money, London: Macmillan, 1930, pp 142-147.
Kaldor, Nicholas (1939). “Speculation and Economic Stability” Review of Economic Studies, 7(1), pp 1-27.
Telser, L.G. (1958). “Futures Trading and the Storage of Cotton and Wheat” Journal of Political Economy, 66, pp 223-255.
Working, Holbrook (1948). “Theory of the Inverse Carrying Charge in Futures Markets” Journal of Farm Economics, 30(1), pp 1-28.
Working, Holbrook (1949). “The Theory of Price of Storage” American Economic Review, 39(6), December 1949, pp 1254-1262.