Storage, Carry & Convenience Yield: What Prices a Commodity Forward

Last lesson you met the basis — the gap between the spot price (the price for the physical commodity right now) and the futures price — and you saw curves that slope up (contango, futures above spot) or down (backwardation, futures below spot). You learned that the curve has a shape. What you did not get was a reason. The curve isn’t decorating itself for fun.

This lesson hands you the reason. Almost every commodity forward price falls out of one tidy idea: the cost of carry. A forward isn’t a fortune-teller’s guess about next year’s price — it’s today’s price plus the bill for keeping the physical thing alive until delivery, minus the perk of actually owning it now. Financing costs money. Storage costs money. Holding the real barrel buys you peace of mind, which is worth something. Add those up with the right signs and you’ve priced the future.

By the end you’ll be able to derive a fair forward from spot, explain why bulky perishable commodities behave nothing like gold, run the cash-and-carry arbitrage that pins the upper bound, and — the headline act — explain why a shortage today drives the curve into backwardation, not because anyone expects cheaper prices later, but because owning the physical right now is suddenly worth a fortune.

The cost-of-carry formula

Analogy. Think of “next year’s barrel of oil” as a product you can manufacture today: buy a barrel now, then babysit it for a year. The price you’d fairly charge for delivering it next year is what you paid plus every bill you racked up babysitting it (interest on the cash you tied up, the rent on the tank) minus any perk you enjoyed from having the real thing on hand. That total is the forward price. No crystal ball required.

Definition. The cost-of-carry model says the fair forward price $F$ equals the spot price $S$ scaled up by the net cost of carrying the physical to delivery over time $T$ (in years). In simple (annualized, non-compounded) form:

$F = S\,(1 + (r + u - y)\,T)$

and in the cleaner continuously-compounded form practitioners actually use:

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

where

• $r$ = the financing rate — interest on the cash locked up in the inventory,
• $u$ = the storage cost as a fraction of the commodity’s value per year (tanks, warehouses, insurance, spoilage),
• $y$ = the convenience yield — the non-cash benefit of physically holding the commodity (more on this villain later).

The quantity $r + u - y$ is the net cost of carry. Positive net carry pushes $F$ above $S$ (contango); negative net carry pulls $F$ below $S$ (backwardation).

Worked example — gold (pure financing carry). Gold is the textbook clean case: it’s cheap to store ($u \approx 0$) and nobody runs a factory on gold bars, so its convenience yield is roughly zero ($y \approx 0$). Net carry collapses to just the interest rate. Take $S$ = $2,000/oz, $r = 5\%$, $u = 0$, $y = 0$, $T = 1$:

So the formula $S(1 + (r + u - y)T)$ here is $2{,}000 \times (1 + 0.05) = 2{,}100$ in dollars, i.e. $2,100. Continuous compounding gives $2{,}000 \times e^{0.05} \approx 2{,}102.54$, i.e. about $2,102.54 — same idea, a couple of dollars apart because of compounding frequency. Gold sitting in mild contango is the formula doing its job, not a bullish prophecy.

When to use it

Reach for cost-of-carry whenever the commodity is genuinely storable and shortable enough to support arbitrage — precious metals, most base metals, crude, refined products, grains. It’s your default lens for reading a futures curve. It gets shaky when storage is impossible (electricity) or when you literally cannot borrow the physical to short it — both of which we’ll dissect.

Storage and financing — the cost side

Analogy. Owning a physical commodity is like owning a horse. The horse cost you money up front (that cash could’ve been earning interest — that’s financing), and now it needs a stable, hay, a vet, and insurance against it kicking a wall (that’s storage). To sell “a horse delivered next year” you’d have to recoup all of that. The bigger and hungrier the animal, the bigger the carry. Gold is a pet rock — basically free to keep. Crude oil is a temperamental, flammable horse.

Definition. Break the cost side of carry into two pieces:

• Financing ($r$): the interest on the cash tied up in the inventory. Buy $80 of crude, and that $80 isn’t earning the risk-free rate anymore — the foregone interest is a real carrying cost.
• Storage ($u$): the physical bill — tank or warehouse rent, insurance, security, and spoilage/shrinkage for anything that rots, evaporates, or leaks. Expressed as a fraction of value per year so it scales like the other terms.

Together they form the positive part of net carry. With $y$ set aside, $F = S(1 + (r + u)T) > S$ — the forward sits above spot, which is exactly the contango you met last lesson.

Worked example — crude oil with real storage. Take $S$ = $80/bbl, $r = 5\%$, $u = 3\%$ (oil storage is not free), $y = 0$ for now, $T = 1$:

The forward lands at $86.40, a $6.40 contango over spot — $4.00 of it financing, $2.40 of it the cost of keeping oil in a tank for a year. The chunkier the storage bill, the steeper the contango. Play with the sliders below to watch financing and storage stack up onto spot, and convenience yield eat into them from the top.

When to use it

Lead with the cost side whenever inventories are comfortable and the commodity is expensive or annoying to store. Plentiful supply means the convenience yield (next section) is low, so $r + u$ dominates and the curve tends to sit in contango. The fatter the storage bill, the steeper that contango can get — right up until the warehouses are full, when storage stops being a smooth percentage and becomes a hard physical wall.

Convenience yield — the benefit of holding the physical

Analogy. Why keep cash in your wallet when a one-year CD pays interest? Because cash in hand lets you grab the taxi, the sandwich, the parking meter — the convenience of having it now is worth giving up a little interest. A commodity has the same perk. A refinery that owns crude in its tanks can keep the plant humming, dodge a ruinous shutdown, and honor delivery contracts the moment demand spikes. That operational peace of mind is a real yield — it just doesn’t show up as a cash payment. It’s the convenience yield, and it works against the cost side.

Definition. The convenience yield $y$ is the non-cash benefit of physically holding the commodity rather than holding a paper claim (a future) on it. Owning the real thing lets you keep production running, avoid a stockout, and meet obligations on a moment’s notice. Because it’s a benefit, it enters carry with a minus sign:

$F = S\,(1 + (r + u - y)\,T)$

Here’s the punchline. When inventories are tight — refineries scrambling, warehouses near empty — the convenience yield spikes, because having the physical right now is suddenly priceless. If $y$ climbs high enough to exceed financing plus storage ($y > r + u$), net carry goes negative, and the forward drops below spot. That’s backwardation, derived from first principles.

Worked example — the same crude, now in a tight market. Keep $S$ = $80, $r = 5\%$, $u = 3\%$, $T = 1$, but a supply scare drives the convenience yield to $y = 12\%$:

Net carry is $r + u - y = 0.05 + 0.03 - 0.12 = -0.04$, a negative 4%. The forward falls to $76.80, sitting $3.20 below spot. Nothing about anyone’s price forecast changed — the only thing that moved was $y$, the value of owning the physical today. Same commodity, same rates, same storage; flip convenience yield from low to high and the curve flips from contango to backwardation.

When to use it

Convenience yield is the term to watch whenever inventories swing — energy and agricultural commodities especially. In gluts, $y$ is near zero and the curve relaxes into contango; in shortages, $y$ explodes and yanks the curve into backwardation. It’s the single knob that explains why the same commodity flips between curve shapes across the seasons and the business cycle, with no change in long-run price expectations at all.

Storable vs non-storable

Analogy. Cost-of-carry assumes you can buy it now and keep it until delivery — like stockpiling canned beans for the winter. That only works if the thing stores. You can hoard beans; you cannot hoard a lightning bolt. The instant a commodity stops being storable, the bridge between today’s price and the forward collapses, and the forward has to be built from something else entirely.

Definition. Cost-of-carry requires storability, because the whole argument rests on physically carrying the commodity from now until delivery. Commodities live on a spectrum:

• Electricity (essentially non-storable). At grid scale you can’t warehouse power, so the carry link breaks. A power forward for “August 2pm” reflects the expected supply and demand in that specific delivery window — generation capacity, weather, demand peaks — not today’s spot scaled by carry. Different delivery hours can have wildly different forwards with no carry relationship at all.
• Gold (one extreme — pure financing carry). Cheap to store ($u \approx 0$), near-zero convenience yield ($y \approx 0$), so net carry is basically just $r$. Gold therefore sits in mild, persistent contango almost always — boring and predictable, exactly as the formula demands.
• Agricultural & energy commodities (the messy middle). Real, sometimes large storage costs and real, volatile convenience yields. Their curves swing between contango and backwardation as harvests come in, tanks fill or drain, and seasons turn. This is where cost-of-carry is most alive — both $u$ and $y$ are doing real work.

The cleaner the storage and the lower the convenience yield, the more tightly the forward tracks pure financing carry. The messier the storage and the more volatile the convenience yield, the more the curve dances.

When to use it

Before applying cost-of-carry to any commodity, ask: can I actually store this? If yes (gold, metals, crude, grains), carry is your model and the curve reads as $r + u - y$. If no (electricity, and partly fresh produce or perishables with brutal spoilage), throw carry out and price the forward off expected fundamentals in the delivery window. Same futures plumbing, completely different pricing logic.

Cash-and-carry arbitrage enforces the bound

Analogy. Imagine you could buy a sandwich today for $5, keep it perfectly fresh for $1, and someone has already promised to buy that sandwich from you next week for $10. You’d do it every single time — lock in $4 of riskless profit, no opinion about sandwich prices required. So would everyone else, and all that buying-today / selling-the-promise would drag the two prices back into line. That’s cash-and-carry arbitrage, and it’s the enforcement officer behind the cost-of-carry formula.

Definition. Cash-and-carry arbitrage: when the future is priced too high relative to carry ($F > S(1 + (r + u - y)T)$), you:

1. Borrow cash and buy the commodity at spot $S$ today,
2. Store it (paying $u$) and carry the financing cost ($r$),
3. Sell the overpriced future at $F$ today, and
4. Deliver the stored commodity into the future at expiry, collecting $F$.

You pocket $F - S(1 + (r + u - y)T)$ with no price risk — you owned the physical and had a locked-in sale the whole time. Crowds of arbitrageurs doing this sell the future down until the gap vanishes, which caps the forward at $S$ plus carry. Watch the cash and the physical flow through the trade below.

Worked example — an arbitrage when the future is too rich. Crude: $S$ = $80, $r = 5\%$, $u = 3\%$, $y = 0$, $T = 1$, so the fair forward $F$ = $86.40. Suppose the market future trades at $95:

You walk away with $95 − $84.00 − $2.40 = $8.60 of riskless profit per barrel — exactly the $95 market future minus the $86.40 fair forward. Do it at scale and your selling drags the future back toward $86.40, which is why the forward can’t durably exceed $S$ plus carry.

The asymmetry that lets backwardation survive. The reverse trade — reverse cash-and-carry, used when $F$ is too low — requires you to short the physical: borrow the commodity, sell it spot, buy the cheap future, and return the borrowed commodity at delivery. But you generally cannot borrow a tanker of crude the way you borrow a stock. Physical shorting is hard or outright impossible for most commodities. So the lower bound is not well enforced — which is exactly why backwardation can persist without arbitrage erasing it. The upper bound (cash-and-carry) is sturdy; the lower bound (reverse cash-and-carry) is leaky. The carry model gives a clean ceiling and a much mushier floor.

When to use it

Run the cash-and-carry calculation whenever a future looks rich to carry and you can actually source, store, and deliver the physical — it’s a genuine locked-in profit and the mechanism that caps the curve. But never assume the symmetric floor holds: when a curve is in steep backwardation, don’t expect arbitrage to “fix” it, because the reverse trade needs a physical short that usually isn’t available. Cash-and-carry is a ceiling-enforcer, not a two-sided clamp.

Putting it together

Strip away the jargon and a commodity forward is one sentence: today’s price, plus the bill to babysit the physical until delivery, minus the perk of owning it now. Financing ($r$) and storage ($u$) are the babysitting bill — positive carry that lifts the forward above spot into contango. Convenience yield ($y$) is the perk — negative carry that pulls the forward down, and when inventories are tight enough that $y$ overwhelms $r + u$, it drags the whole curve into backwardation. That backwardation isn’t a forecast of cheaper prices; it’s a scarcity alarm, ringing because barrels-in-hand are precious today.

The model only works where you can actually store the thing: gold is the clean pure-financing case (perpetual mild contango), energy and grains are the swinging middle (storage and convenience yield both alive), and non-storables like electricity fall outside carry entirely, priced off expected supply and demand per delivery window. And the whole structure is policed by cash-and-carry arbitrage — buy spot, store, sell the rich future, deliver — which firmly caps the forward at $S$ plus carry. Its mirror image needs a physical short you usually can’t get, which is precisely why the floor leaks and backwardation can persist for months. Carry is a sturdy ceiling and a soft floor.

Key Takeaways