# Prices and cycles: Storage, convenience yield and the forward curve

Source page: https://www.materialinsights.org/learn/commodities/prices-and-cycles/p3/

[Understanding Commodities](https://www.materialinsights.org/learn/commodities/) / [Module 5: Prices and cycles](https://www.materialinsights.org/learn/commodities/prices-and-cycles/) Part 3 of 5

## Storage, convenience yield and the forward curve

A commodity has many prices at once: one for delivery today and one for each future month. The line joining them is the forward curve. The theory of storage explains its shape through what it costs, and what it is worth, to hold physical stocks. With it you can read tightness, glut and even negative prices off the curve.

### 3.1 Holding stocks earns a convenience yield, and its size depends on how scarce stocks are

The theory of storage, developed by Kaldor and Working in the 1930s and 1940s, asks why anyone would hold a physical commodity rather than a contract for later delivery. Holding stocks costs storage, insurance, deterioration and interest. It also has a benefit. Stocks let a firm meet an unexpected order, avoid shutting a production line, or keep running through a supply disruption in a producing country. Kaldor and Working called this benefit the convenience yield: a return that accrues to the owner of the physical good but not to the holder of a futures contract, rather as a dividend goes to a shareholder but not to someone holding an option on the share.

Geman defines the convenience yield as a rate, and net of storage cost, so it can be positive or negative. Its key property is that it depends on inventory. When stocks are ample, one more tonne in the warehouse adds little security, so the convenience yield is small. When stocks are scarce, the last tonnes are precious and the yield is high.

Geman draws out the testable predictions. Volatility rises as inventory falls, because thin stocks cannot buffer shocks; Geman and Nguyen found soybean volatility an almost exact inverse function of world inventory. Price and volatility therefore rise together, the opposite of equity markets, where volatility jumps when prices crash. Geman calls this the inverse leverage effect. And near-dated contracts are more volatile than distant ones, the Samuelson effect, because news hits immediate supply while production can adjust before a distant contract matures.

- Geman (2005), Commodities and Commodity Derivatives · pdf pp. 36-40

### 3.2 Contango: when stocks are ample, the forward price is capped by the cost of carry

Geman derives the link between spot and forward prices from a no-arbitrage argument. In its simple linear form, the forward price for delivery after a time T equals the spot price multiplied by one plus T times (interest rate plus storage cost minus the benefit of holding the physical good). Interest and storage push the forward above spot; the convenience benefit pulls it below.

When stocks are plentiful the convenience benefit is close to zero, the forward price sits above spot, and the curve slopes upward. This is contango. Arbitrage puts a ceiling on it. If the forward price exceeds spot by more than the full cost of carry, a trader can borrow, buy spot, store it, and sell forward, locking in a riskless profit. The buying lifts spot and the selling depresses the forward until the gap shrinks to the cost of carry. Radetzki and Wårell's numbers: with storage at 3% a year and interest at 4%, the 12-month price cannot exceed spot by much more than 7%.

Contango lets banks and traders finance surplus stocks with little risk, which is why gluts fill warehouses rather than crashing prices further. It also transmits financial buying: in strong contango, anything that pushes futures up pulls spot up through the same arbitrage. Tilton and Guzmán distinguish strong contango, where the spread covers carrying costs, from weak contango, where it does not and the arbitrage link is looser.

Two cautions. The ceiling is set by the marginal storer, and Geman shows storage costs rise steeply in the short run as space runs out. And the theory needs storability: for electricity the spot-forward link collapses.
 Worked example

#### Worked example: the contango ceiling

Take an illustrative spot copper price of $10,000 a tonne, interest at 4% and storage at 3% a year (Radetzki and Wårell's figures), and no convenience benefit. Geman's formula gives a 12-month forward of 10,000 x (1 + 0.04 + 0.03) = $10,700. If the market quotes $11,000, borrow, buy spot, pay $400 interest and $300 storage, and deliver against the forward: $300 riskless profit. Now let holding metal be worth 10% a year because stocks are scarce. The forward becomes 10,000 x (1 + 0.07 - 0.10) = $9,700, below spot. Same formula, opposite shape.

- Geman (2005), Commodities and Commodity Derivatives · pdf pp. 47-51, 184, 195-198
- Radetzki & Wårell (2021), A Handbook of Primary Commodities · pdf pp. 156-157
- Tilton & Guzmán (2016), Mineral Economics and Policy · pdf pp. 123-124

### 3.3 Backwardation: when stocks are scarce, spot can rise without limit

When immediate supply is short, the convenience benefit exceeds interest plus storage and the curve slopes down: spot above forward. This is backwardation. Unlike contango it has no arbitrage ceiling, because the trade that would close the gap requires borrowing physical metal that nobody wants to lend. A shortage today can send spot soaring even when everyone expects new mines to end it next year. Radetzki and Wårell identify the only real limit: at some level of backwardation, the reward for selling stock now and buying it back cheaply forward outweighs the convenience of keeping it, and inventory is released.

That makes the shape of the curve a direct reading of physical tightness. Humphreys notes that metal markets switched from contango into backwardation in the second half of 2003, an early sign that the long slump was ending. Geman observes that oil curves tend to be backwardated when spot prices are high and in contango when they are low, because both reflect the level of commercial stocks.

Backwardation also matters to investors. A fund that holds near-dated futures and rolls them forward sells the expiring contract and buys a cheaper later one. In a backwardated market that roll earns a return; in contango it costs one. This is the roll yield, and it means a commodity fund's return can diverge sharply from the spot price it seems to track. Radetzki and Wårell, discussing Iwarson's analysis, put the average roll return at about 2% a year for the Goldman Sachs index between 1970 and 2005.
 Worked example

#### Worked example: copper's implied convenience yield, May 2006

Humphreys reports copper averaging $8,044 a tonne in May 2006, with spot more than $2,000 above the 27-month price, so that price was about $6,044 at most. Put these into Geman's linear formula with an assumed 7% a year for interest plus storage: 6,044 = 8,044 x (1 + (0.07 - y) x 2.25). Solving gives y of about 0.18: the market valued having metal now at roughly 18% a year, well above the 7% cost of carrying it. Geman's continuous version gives about 20%. That is a market with almost no spare stock.

- Radetzki & Wårell (2021), A Handbook of Primary Commodities · pdf pp. 157, 164-165
- Humphreys (2015), The Remaking of the Mining Industry · pdf pp. 103, 105
- Geman (2005), Commodities and Commodity Derivatives · pdf pp. 49-51, 225

### 3.4 A futures price is not a forecast, and in 2020 storage ran out

It is natural to read a futures price as the market's prediction of the future spot price. Geman shows it is a mix of two things that cannot be separated in advance: the expected price, and a risk premium that hedgers will pay to fix a price today. Keynes argued that producers, as net sellers of futures, would accept a price below the expected spot price, paying speculators to carry risk: normal backwardation. Telser found no such bias in wheat, cotton and corn, and Geman regards the theory as largely obsolete, since the premium's sign depends on the commodity and its stocks.

The forward curve is better read as a statement about storage economics than about the future. April 2020 showed that literally. As lockdowns spread, Blas and Farchy report, oil demand fell by as much as 30%, against perhaps 4% in a severe recession, while Russia and Saudi Arabia fought a price war. Prices fell from about $60 a barrel to below $30, then below $20. Tanks filled. The US benchmark, WTI, is delivered physically at Cushing, Oklahoma, and a holder of an expiring contract must take oil there. With nowhere to put it, for a few hours in April the price fell below zero: sellers paid buyers to take the oil.

The contango ceiling had not failed. The cost of storage, the c in Geman's formula, had become enormous because storage itself was scarce. The arbitrage still worked for anyone who had tanks, and the traders did.
 Case

#### Storing the glut at sea, 2020

Glencore, warned by its China staff, began hiring tankers early in 2020, including the Europe, able to hold 3.2 million barrels. It bought US shale oil for as little as $10 a barrel while futures for delivery three months later traded at three times that. Buy, store, sell forward: after tanker hire and financing, returns ran at 50-100%. About a billion barrels were stored worldwide in a few months, and Glencore made $1.3 billion from energy trading in the first half of 2020 (Blas and Farchy).

- Geman (2005), Commodities and Commodity Derivatives · pdf pp. 43-46, 224
- Blas & Farchy (2021), The World for Sale · pdf pp. 20, 309-311

### See it in the table
[Cu Copper](https://www.materialinsights.org/commodities/copper/)[Wh Wheat](https://www.materialinsights.org/commodities/wheat/)[Ct Cotton](https://www.materialinsights.org/commodities/cotton/)

### Check your understanding

Why does arbitrage cap contango but not backwardation?
 Cash-and-carry needs only money and storage. The reverse trade needs someone to give up physical stock in a shortage, which they will do only once the backwardation exceeds the convenience of holding it.

What does Geman's 'inverse leverage effect' describe?
 Low stocks raise prices and also remove the buffer against shocks, so high prices and high volatility coincide. In equities the opposite pattern, the leverage effect, holds.

In April 2020 three-month oil futures traded at about three times the price traders paid for spot crude. Within the theory of storage, what allowed such a steep contango?
 The contango ceiling equals interest plus storage cost for the marginal storer. With storage nearly exhausted that cost exploded, so a vast contango persisted, and those with tanks could still profit.

[← Part 2: Why commodity prices swing so hard](https://www.materialinsights.org/learn/commodities/prices-and-cycles/p2/)[Mark read · Part 4: Real prices and the long-run trend →](https://www.materialinsights.org/learn/commodities/prices-and-cycles/p4/)

## Other sections

- [Module overview](https://www.materialinsights.org/learn/commodities/prices-and-cycles/index.md)
- [How a commodity price is made](https://www.materialinsights.org/learn/commodities/prices-and-cycles/p1/index.md)
- [Why commodity prices swing so hard](https://www.materialinsights.org/learn/commodities/prices-and-cycles/p2/index.md)
- [Real prices and the long-run trend](https://www.materialinsights.org/learn/commodities/prices-and-cycles/p4/index.md)
- [Booms, super-cycles, money and the dollar](https://www.materialinsights.org/learn/commodities/prices-and-cycles/p5/index.md)
- [Key terms, quiz and sources](https://www.materialinsights.org/learn/commodities/prices-and-cycles/review/index.md)