Convenience yield, contango and backwardation
1. The idea
Oil, copper and wheat are held mainly to be used in production. A refinery that owns oil can keep running when supplies are short; a futures contract cannot do that for it. This benefit of holding the physical commodity is its convenience yield, written as a yearly share of its value, like a dividend yield. It lowers the futures price, which the cost of carry sets just as it sets a forward price, below what the cost of carry alone would give:
\[ F_0 = S_0\, e^{(r + u - y)T}, \]
where \(F_0\) is the futures price for delivery in \(T\) years, \(S_0\) the spot price, \(r\) the risk-free rate, \(u\) the storage cost and \(y\) the convenience yield, all as yearly shares with continuous compounding, as decimals, and \(T\) the time to delivery in years. The convenience yield cannot be seen directly, but a futures price reveals it:
\[ y = r + u - \frac{\ln(F_0/S_0)}{T}. \]
Here \(\ln\) is the natural logarithm. When the convenience yield is below the cost of carry, \(r + u\), futures prices are above the spot price and rise with the time to delivery: the market is in contango. When it is above, as when inventories are low and oil is scarce, futures prices are below the spot price: the market is in backwardation.
Arbitrage does not stop this. If gold futures were too cheap, gold owners would sell gold and buy futures. A refinery will not sell the oil it needs to keep running, so for commodities held to be used, the formula from the cost of carry gives only an upper limit for the futures price.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The futures curve on a chart
In the question, the spot price of oil is $80 a barrel, and the futures price for delivery in a year is $76. The futures price is 5% below the spot price: a yearly growth, with continuous compounding, of \(\ln(76/80)/1 = -0.0513\), or −5.13% a year. The cost of carry is 4% + 2% = 6% a year, so the convenience yield is \(0.04 + 0.02 + 0.0513 = 0.1113\), or 11.13% a year. It is above the cost of carry: the market is in backwardation.
Now move the convenience yield. Below 6%, the futures price for delivery in a year is above the spot price: contango. Above 6%: backwardation.
And here is the futures curve itself, at a convenience yield of 11%: the later the delivery, the lower the futures price.
4. What the shape of the curve means for investors
A futures contract ends at the spot price on its delivery date. An investor who wants to own oil without storing it buys a futures contract and, before it expires, sells it and buys the next one. This is called rolling the position. In contango, a futures price is above the spot price and falls toward it as delivery comes closer. So if the spot price and the shape of the curve stay as they are, each contract loses value while it is held: a negative roll yield. Rolling into the next contract does not itself cost anything; that contract simply starts the same fall from a higher price. In backwardation, futures prices rise toward the spot price, and the roll yield is positive. A change in the spot price or in the shape of the curve can outweigh either effect.
The extreme case came in April 2020. Demand for oil had collapsed, and the storage tanks at Cushing, Oklahoma, where the main US oil futures contract is delivered, were nearly full. Holders of the May contract who could not take delivery had to sell it, and on 20 April it settled at −$37.63 a barrel: they paid others to take the oil.
Investors trade the shape of the curve in two main ways:
- Carry, or term structure: buy futures in backwardation and sell futures in contango, to earn the roll yield if the curve keeps its shape.
- Momentum: buy futures whose prices have risen and sell those that have fallen. Many hedge funds and commodity trading advisers follow such trends across futures markets.
The two often agree. When inventories are low, the convenience yield is high and the market is in backwardation, so carry says buy. The shortage that runs inventories down also tends to push prices up, so momentum often says buy as well.
5. One worked example, then your turn
The worked example is a market in contango.
Now one with hints, one step at a time.
And one on your own.
6. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a futures price above the spot price and one for a futures price below it. You can always open the solution, but then that question does not count.
7. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: