Futures Term Structure in Depth: Theory of Storage, Convenience Yield, and Normal Backwardation

The term structure of a commodity futures market is the set of prices for every listed delivery month, and for storable commodities its shape is set mainly by how much of the commodity is sitting in inventory. When stocks are plentiful, later months trade above nearer ones by roughly the cost of carrying the physical good; when stocks are scarce, owning the commodity now is worth more than owning it later, and the curve inverts.

In short: two classic theories explain futures curves, and they answer different questions. The theory of storage (Kaldor 1939, Working 1949, Brennan 1958) explains the spread between months: futures price equals spot plus interest plus storage cost minus a convenience yield, a benefit of holding inventory that rises sharply as inventories fall. The theory of normal backwardation (Keynes 1930, Hicks 1939) explains the risk premium: the gap between today’s futures price and the spot price people expect at expiry, paid by hedgers to speculators. Gorton, Hayashi and Rouwenhorst (Review of Finance, 2013) tied the two together with 40 years of inventory data and found that low inventories predict both a steeper backwardation and a higher risk premium.

This is the professional-depth companion to contango and backwardation and how futures prices are set. Those articles cover the definitions and the cost-of-carry arithmetic. Here we go through the theory those ideas come from, the evidence for it, and where it breaks.

The misconception: “the curve is the market’s forecast”

The most common error in reading a futures curve is to treat each price as a forecast of the future spot price. A crude oil curve that slopes upward is read as “the market expects oil to rise.” For a storable commodity, that reading is mostly wrong. A contango curve is first a statement about the cost of storage, and a backwardated curve is first a statement about scarcity today. Expectations matter, but they enter through inventories: if a shortage is expected next year, the rational response is to store more now, which raises the spot price today and pulls the curve back toward full carry.

The forecasting interpretation is closer to right for commodities that cannot be stored at all, such as electricity, where nothing links today’s price to next quarter’s except expectations and risk premiums. Most traded commodities sit somewhere between: grains are storable within a crop year but not across many of them, natural gas is storable but with limited capacity, and live cattle are storable only in a loose sense.

The theory of storage

Kaldor and Working: the spread is a price

Nicholas Kaldor’s “Speculation and Economic Stability” (Review of Economic Studies, 1939) introduced the idea that holding stocks yields a return in kind, which he called a convenience yield: a processor with grain in its own elevator can keep its mill running without waiting for deliveries. Holbrook Working’s “The Theory of Price of Storage” (American Economic Review, December 1949) made the key empirical move. Working argued that the difference between two futures prices is itself a market price, the price of storage, set by the competition of those who supply storage. He used wheat data to trace out a supply of storage curve: when stocks are large, the spread is positive and close to the full cost of storage; as stocks shrink, the spread falls and can become negative, and yet some stocks are still carried.

That last observation was the puzzle. Why would a merchant hold wheat when the futures market pays less for it in three months than it costs today? Working’s answer, building on Kaldor, was that the merchant is not really being paid by the spread. Holding a working stock has value of its own.

Brennan: the supply of storage

Michael Brennan’s “The Supply of Storage” (American Economic Review, March 1958) formalised the curve. Brennan treated the amount stored as the outcome of a marginal decision: an inventory holder keeps one more unit if the expected price gain covers the marginal net cost of storage, which he built from three parts: the physical outlay (warehousing, insurance, interest), plus a risk-aversion premium for carrying an unhedged or partly hedged stock, minus the marginal convenience yield. Because convenience yield is high when stocks are low and falls toward zero when stocks are ample, the net cost of storage can be negative at low inventories. Brennan estimated the relationship empirically on historical stocks and price spreads for several storable farm commodities.

The equation every desk uses

Gorton, Hayashi and Rouwenhorst write the theory of storage as a single no-arbitrage condition (their Equation 1):

F(t,T) = S(t) + S(t) x r + w - c

where F is the futures price for delivery at T, S the spot price, r the interest cost from t to T, w the storage cost per unit, and c the convenience yield. Interest and storage are observable. Convenience yield is not, so in practice it is backed out of the curve: it is whatever makes the equation hold. That is why desks talk about “implied” convenience yield, much as options desks talk about implied volatility.

The equation gives the curve an upper bound and no lower bound. If the futures price rose above spot plus interest plus storage, anyone with storage capacity could buy spot, store it, sell the future and lock in a profit. So the contango spread is capped at full carry. Nothing equivalent caps backwardation, because you cannot borrow a barrel from the future to sell today. A backwardated spread can be very wide when inventories are very low.

Worked example: implied convenience yield

Take an illustrative crude oil market with spot at $70.00 a barrel, an interest rate of 4% a year (simple), storage of $0.40 a barrel a month, and a six-month futures contract. These are round numbers chosen for the arithmetic, not current prices.

Full carry for six months:

Interest: $70.00 x 4% x 0.5 = $1.40
Storage: $0.40 x 6 = $2.40
Full-carry futures price: $70.00 + $1.40 + $2.40 = $73.80

Now compare two curves, one in contango and one in backwardation:

	Contango market	Backwardated market
Six-month futures price	$73.20	$67.50
Spread over spot	+$3.20 (+4.57%)	-$2.50 (-3.57%)
Implied convenience yield (73.80 - F)	$0.60	$6.30
Annualized, as % of spot	1.71%	18.00%
Spread as % of full carry ($3.80)	84%	negative

The contango market is close to full carry: inventories are comfortable and holding a barrel earns almost nothing beyond what the futures pay. The backwardated market implies a convenience yield of 18% a year. That is not a rate anyone receives in cash. It is the value refiners and traders place on having barrels on hand when they are scarce. Grain traders express the same idea when they quote calendar spreads as a percentage of full carry: spreads near 100% signal ample stocks, and spreads well below it signal tightness.

What happens to a long futures holder if spot stays at $70.00 for six months? In the contango market the contract bought at $73.20 converges down to $70.00, a loss of 4.37%. In the backwardated market the contract bought at $67.50 converges up to $70.00, a gain of 3.70%. The spot price did not move in either case. This convergence is what practitioners call roll yield, and it is entirely a consequence of the curve’s shape. As we will see, whether it is a return or just a mechanical identity is a subtle question.

Normal backwardation: the risk premium

Keynes and Hicks

John Maynard Keynes, in A Treatise on Money (1930), and John Hicks, in Value and Capital (1939), offered a different explanation for futures prices. Their argument starts with who is in the market. Producers and inventory holders are naturally long the commodity and want to hedge by selling futures. If hedgers who want to sell outnumber those who want to buy, speculators must be induced to take the other side, and the inducement is a futures price set below the spot price expected at maturity. Keynes called this normal backwardation. The risk premium is the size of the discount.

The word “backwardation” causes endless confusion here, because it means something different from the curve shape. Normal backwardation compares the futures price with the expected future spot price, which nobody observes. Curve backwardation compares the futures price with the current spot price, which everyone observes. A market can be in contango on the screen and still pay a positive risk premium to long holders, if the futures price, though above today’s spot, is below where spot is expected to be at expiry.

Separating premium from surprise

Gary Gorton and Geert Rouwenhorst (“Facts and Fantasies about Commodity Futures,” Financial Analysts Journal, 2006) set out the arithmetic cleanly. The return to a futures position held to expiry is (S at expiry minus F today) divided by F. Split it in two:

Risk premium: expected spot at expiry minus F, divided by F. This is what the long is paid, on average, for providing price insurance.
Surprise: actual spot at expiry minus expected spot, divided by F. This averages to zero if expectations are unbiased.

Continuing the backwardated example: if the market’s expectation of spot in six months is $70.00 and the futures price is $67.50, the expected return to a long is 3.70%. If spot actually ends at $66.00, the realized return is -2.22%: a premium of +3.70% and a surprise of -5.93%. Gorton and Rouwenhorst stress the corollary: expected trends in spot prices are not a source of return to a futures investor, because they are already in the futures price. Only the premium and the surprise are.

Does the premium exist?

The empirical case for a positive average premium is strong at the portfolio level. Gorton and Rouwenhorst built an equally weighted, fully collateralized index of commodity futures from July 1959 to December 2004 and found an average annualized excess return over Treasury bills of 5.23%, with a t-statistic of 2.92, compared with 5.65% for the S&P 500 over the same period. Their footnote cites earlier estimates: Bodie and Rosansky (1980) found 9.5% a year for 1950-1976, and Fama and French (1987) a continuously compounded premium of 0.45% a month for 1966-1984.

The case at the level of individual commodities is much weaker, and the gap between the two is the subject of commodity index investing.

Gorton, Hayashi and Rouwenhorst: inventories tie it together

Until 2013, the two theories largely ran on separate tracks: one about spreads, the other about premiums. “The Fundamentals of Commodity Futures Returns” by Gary Gorton, Fumio Hayashi and Geert Rouwenhorst (Review of Finance, volume 17, 2013) built a simple two-period model in which both the basis and the risk premium depend on inventories, then tested it on 31 commodity futures with matched physical inventory data from 1971 to 2010.

The model

In their model, when initial inventories are high enough to carry the commodity forward, the convenience yield is zero and the spread is pinned at the cost of storage. When a stock-out is possible, the convenience yield turns positive, reflecting the spot price spike a shortage would cause. So the convenience yield, and with it the basis, is a decreasing, convex function of inventories: flat when stocks are ample, rising steeply as they approach zero. The risk premium also falls with inventories, provided demand is not too inelastic, because low inventories mean volatile future spot prices and more risk for whoever is long.

This draws on a modern line of theory: Deaton and Laroque (1992), who showed that the non-negativity constraint on inventories makes prices more volatile when stocks are low, and Routledge, Seppi and Spatt (2000), who showed that the convenience yield can arise as an option value on a possible stock-out even if nobody gets a direct benefit from holding the good.

The evidence

The main findings, from the paper’s own tables and text:

The basis-inventory curve is nonlinear, as theory predicts. For many commodities the basis rises more steeply as normalized inventories fall below normal, and the slope at normal inventory levels is steeper for commodities that are hard to store. The authors call this “a strong confirmation of the basic assumption of the Theory of Storage.”
Average contango, positive average premium. Across the 31 commodities, the equally weighted index earned an excess return of 5.75% a year, while the average basis was -1.1% a year: on average, futures traded above spot. A positive premium coexisted with negative average roll yield. The authors say the practitioner view that positive roll yield is required for a positive premium is, in theory, unwarranted.
Low inventories, high premiums. Sorting commodities each month by lagged normalized inventory, the low-inventory half outperformed the high-inventory half, and the long-short portfolio’s t-statistic was -2.78.
Price signals proxy for inventories. A portfolio of high-basis (most backwardated) commodities beat a low-basis portfolio by 10.64% a year (t = 3.92) over the full sample. A high-momentum portfolio beat a low-momentum one by 11.94% a year (t = 4.40). Both selected commodities with below-normal inventories.
Positions do not predict returns. Using CFTC data on the positions of commercials and non-commercials, they found that positions move with inventories and prices at the same time, but found “no evidence that the positions of participants in futures markets predict risk premiums.” That is a direct challenge to the hedging-pressure version of Keynes.

The practical meaning: the shape of the curve is informative about expected returns, not because the curve is a forecast, but because it is a readout of inventories, and inventories determine how much risk the long is bearing.

How professionals read a curve

With the theory in place, desk practice makes more sense:

Calendar spreads as inventory gauges. The spread between the first and second contract, or the spread as a percentage of full carry, is watched as a real-time proxy for stocks, because official inventory data arrive with a lag. A strengthening nearby spread is read as tightening supply.
Separating the front from the back. Near-dated spreads are dominated by current inventories and convenience yield. Long-dated contracts reflect the cost of new supply and long-run expectations. A curve can be backwardated in the first year and in contango thereafter.
Seasonality. For grains and natural gas, inventories have strong seasonal cycles, so the “normal” spread changes through the year. Gorton, Hayashi and Rouwenhorst net out seasonal effects before relating the basis to inventories. (See seasonality in commodity futures.)
Implied convenience yield as a time series. Because the convenience yield is the residual in the carry equation, its history is a cleaner signal than the raw spread, which also moves with interest rates and storage costs.

Where the theory breaks or bends

The theory of storage is an equilibrium story, and real markets add frictions:

Storage capacity is finite. Full carry assumes anyone can store at a known cost. When tanks fill, the marginal cost of storage rises without limit, and contango can briefly exceed any normal “full carry.” The extreme case was the May 2020 WTI contract, which settled at -$37.63 a barrel on April 20, 2020, when storage at Cushing, Oklahoma was close to full; see negative oil prices in April 2020.
Delivery mechanics matter. The carry equation compares a future with a specific spot price, at a specific location and grade. Where the deliverable is narrow, local logistics, not global inventories, set the front spread.
Non-storable or hard-to-store commodities (electricity, livestock, to some degree natural gas) have weak links between months, so the curve carries more expectation and risk premium and less carry.
Inventory data are noisy. Gorton, Hayashi and Rouwenhorst acknowledge measurement error in inventories; part of their contribution is showing that prices carry the same information, which is reassuring and also a reminder that the evidence is statistical, not exact.
Risk premiums vary and can be negative. The premium is an average over long periods and many commodities. For any single commodity over a few years, the surprise term dominates.

The short version

A futures curve for a storable commodity is not a price forecast. Its slope is governed by the theory of storage: futures equal spot plus interest plus storage minus convenience yield, and convenience yield rises steeply as inventories run down. That caps contango near full carry and leaves backwardation unbounded. Separately, Keynes’s theory of normal backwardation says futures prices sit below expected spot prices to pay speculators for insurance, a risk premium that long-run portfolio studies estimate at roughly 5% a year. Gorton, Hayashi and Rouwenhorst showed the two theories are linked through inventories: low stocks mean steep backwardation, high volatility and higher expected returns to long positions. None of this makes the curve a trading signal on its own. It makes the curve a measurement of how scarce a commodity is today, which is the single most useful thing a futures market tells you.