Cost of Carry and Storage

Buy a barrel of oil today and you own a barrel. Agree today to buy one for delivery in six months and you own a promise. Those two are not worth the same, and the gap between them is usually not anybody's forecast. It is a bill. Somebody has to fund that barrel, rent a tank for it, insure it and guard it for six months, and nobody does that for free. Cost of carry is the total of that bill.

Here is the everyday version. You need a tonne of rice next March. Buy it in March at whatever it costs then, or buy it today, pay a warehouse to hold it, tie up your cash, and walk into your own store in March. The second route has a price you can work out right now: today's price, plus the interest you gave up, plus the rent. If March futures ever traded far above that number, every warehouse operator alive would buy rice, sell March futures against it and pocket the difference risk-free. The futures price is tethered to the cash price by the cost of doing it yourself.

The three items on the bill

Carry has exactly three components.

• Financing. Money spent on rice today is money not earning interest. Call that rate $r$, per year. It is almost always the biggest item.
• Storage. Tank rent, silo fees, insurance, security, and losses to spoilage or rust. Call the total $u$, expressed as a percentage of value per year so it sits on the same footing as the interest rate.
• Convenience yield. The benefit of having the actual goods in your shed rather than a contract in your drawer. A refinery that runs out of crude stops running; owning real barrels means it never has to. Call that $y$, again per year. It is a benefit, so it subtracts from the bill.

Put them together and the fair futures price for delivery in $T$ years is

$$F = S\,e^{(r + u - y)T}.$$

In words: take today's spot price $S$ and grow it at the net carry rate, the interest you forgo plus what storage costs you minus what possession is worth to you. For mental arithmetic the simple version $F \approx S\,[1 + (r + u - y)T]$ is close enough at short tenors.

Worked example: gold, the clean case

Gold is the textbook case: a bar in a vault does almost nothing for you that a futures contract doesn't, so the convenience yield is near zero. Take spot at $2,400 an ounce, financing at $4.5\%$, vaulting and insurance at $0.4\%$ a year, and a six-month contract, so $T = 0.5$.

Net carry is $4.5\% + 0.4\% - 0\% = 4.9\%$. Over half a year that is $0.049 \times 0.5 = 0.0245$, and $e^{0.0245} = 1.0248$. So

$$F = 2400 \times 1.0248 = 2459.5 .$$

The fair six-month future is about $2,459.50, roughly $59.50 above spot. Not one word of that calculation was about whether gold is going up: it is pure rent and interest, which is why gold curves are almost always gently upward-sloping.

Worked example: crude, backing out the convenience yield

Now run it backwards. Spot crude is $78, the three-month future trades at $76.20, financing is $4.5\%$ and storage runs $3\%$ a year. The future is below spot, so the equation can only balance if $y$ is large:

$$\frac{76.20}{78} = e^{(0.045 + 0.03 - y)\,0.25}.$$

The left side is $0.9769$, and $\ln(0.9769) = -0.02335$. Divide by $T = 0.25$ to get $-0.0934$, so $0.075 - y = -0.0934$ and $y = 0.168$, about $16.8\%$ a year. That is the market saying barrels are scarce: refiners will pay the equivalent of a $16.8\%$ annual yield to have oil in the tank now rather than next quarter. See Convenience Yield.

Try the mechanics below. Set the principal to a spot price and the rate to your net carry, then watch the compounded value trace out the fair futures price at each tenor. Short tenors barely move; the gap only grows when carry is high or delivery is far away.

What this means in practice

The arbitrage only works cleanly in one direction. If futures are too high, anyone can buy the physical, store it and sell the future: the classic cash-and-carry trade. If futures are too low, the reverse needs you to borrow physical barrels and sell them, and there is no deep lending market for oil sitting in a tank. So carry gives a firm ceiling on the futures price and only a soft floor, which is why deep backwardation can persist for months while extreme contango is arbitraged away in days.

The other limit is that storage is finite. When tanks fill, the marginal cost of storage stops being a modest rent and explodes. That is how West Texas crude printed a negative price in April 2020: nobody would take delivery because there was nowhere to put it. See Cushing Storage Limits and Negative Oil Prices.

Key terms

• Cost of carry — financing plus storage minus convenience yield.
• Convenience yield ($y$) — the value of having the actual commodity rather than a claim on it.
• Cash-and-carry — buy spot, store it, sell the future, lock in the difference.
• Full carry — a curve priced at exactly financing plus storage, meaning zero convenience yield.