11.3 Commodity Forwards and Futures

Commodity Forwards and Futures

FMP–11 extends forward pricing from paper assets to physical commodities. The formulas look similar—spot, rates, and “yields”—but the economics change: you cannot freely short a barrel someone needs to refine, storage is costly and capacity-constrained, and holding inventory can be valuable when markets are tight.

Commodities Versus Financial Assets

Gold sits in between: it is a commodity with large investment demand, active lease markets, and lower convenience yield than crude in a squeeze—so gold forwards often behave more “financial.”

Storage, Carry, Lease Rate, and Convenience Yield

Storage cost U (as PV of storage over the life) or continuous storage rate u raises the forward: you must be compensated to hold physical.

With PV of storage U:

F0 = (S0 + U) × e^(r × T)

With continuous storage u:

F0 = S0 × e^((r + u) × T)

Lease rate (especially precious metals): owners can lend metal and earn a lease rate δ. Economically this acts like a dividend yield, lowering F:

F0 = S0 × e^((r − δ) × T)

Convenience yield y: the non-monetary benefit of holding the physical commodity (ability to keep a production process running, meet unexpected demand, avoid stock-outs). It behaves like a yield on the spot, lowering the forward relative to pure storage+finance:

F0 = S0 × e^((r + u − y) × T)

When inventories are low, convenience yield spikes and forward curves often backwardate (near contracts above deferred, or F < S depending on quote/tenor presentation).

Net cost of carry b ≈ r + u − y (continuous). Then F0 = S0 × e^(b × T). Positive net carry → upward-sloping tendency; negative net carry (high y) → downward pressure on forwards versus spot.

Sector Factors

Commodity curves are not generic:

• Energy (oil, gas, power): strong seasonality (heating/cooling), refinery constraints, geopolitics; WTI/Brent inventory reports move the front of the curve.
• Agriculture: plantings, harvests, weather, crop year transitions; optionality around weather markets.
• Industrial metals: global manufacturing demand, Chinese credit cycles, warehouse stocks.
• Precious metals: monetary demand, ETF flows, central-bank buying, lease rates.
• Livestock: feed costs, biology (you cannot store live cattle like copper).

Exam vignettes often hint at inventory levels: high stocks → low convenience yield → more contango; tight stocks → high y → backwardation pressure.

Commodity Forward Pricing and Arbitrage

Cash-and-carry (upper bound): If F is too high versus spot + storage + finance, buy spot, store, sell forward. This requires storage capacity and financing. It enforces:

F0 ≤ (S0 + U) e^(rT) (or the continuous analogue)

Reverse cash-and-carry (lower bound): Short commodity, invest proceeds, buy forward. This requires the ability to borrow the commodity. If end users will not lend scarce inventory, the lower bound can fail and F can trade below the pure financial-storage parity—explained by convenience yield.

So for commodities you often get:

(S0) e^((r+u−y)T) ≈ F0 with y ≥ 0, and y endogenous to scarcity.

Worked storage example

Spot copper S0 = $8,000/ton. Continuous r = 5%, continuous storage u = 1%, T = 1 year, assume y = 0 for a well-supplied market.

F0 = 8,000 × e^((0.05+0.01)×1) = 8,000 × e^0.06 ≈ 8,000 × 1.06184 = $8,494.7

If convenience yield rises to y = 4% because of a supply scare:

F0 = 8,000 × e^((0.05+0.01−0.04)×1) = 8,000 × e^0.02 ≈ $8,161

Same spot and storage, much lower forward—the market pays for access to physical now.

Worked mispricing sketch

If quoted F = $8,700 with y ≈ 0 and fair carry ≈ $8,495, cash-and-carry: buy copper, pay storage, sell forward, finance the purchase. At T, deliver on the forward and repay the loan—profit if costs were estimated correctly. If quoted F = $7,900 while theory (with low y) says $8,495, reverse cash-and-carry only works if you can borrow copper; otherwise the cheap forward can persist and signal high convenience yield rather than “free arb.”

Cost of Carry and Synthetic Commodity

Long synthetic commodity via forwards/futures: long forward/futures + invest PV of F (or post margin and earn interest on collateral—details vary). At maturity you effectively acquire exposure as if you paid F.

Synthetic spot from carry: long forward, short the financing/storage package—rearrangements of cash-and-carry identities. Risk managers use synthetics when physical logistics are impractical but price exposure is needed (e.g., airline jet-fuel hedges via heating-oil/crack proxies—with basis risk).

Remember: futures on commodities introduce basis risk between the hedged item and the contract grade/location/timing. Pricing parity assumes the deliverable grade; real hedges often cannot.

Contango, Backwardation, and Normal Backwardation

Language is a classic FRM trap. Separate curve shape from Keynes’s hedging theory.

Critical point: a market can be in contango (F > S) and still be in normal backwardation if F is nonetheless below E[S_T]. Conversely, backwardation of the curve (F < S) is not the same phrase as normal backwardation. Questions that say “normal backwardation” want the risk-premium / hedging-pressure story, not merely F < S.

Roll yield intuition

In sustained contango, a long who rolls short-dated futures may earn negative roll (sell cheap near expiry relative to richer deferred—sign depends on curve). In backwardation, rolls can be positive for longs. This is about the curve, not a free lunch; expected spot moves and risk premia still matter.

Putting Commodity Forwards Together

Price commodities with net carry b = r + u − y. Use cash-and-carry to cap rich forwards when storage exists. Treat seemingly “cheap” forwards as possible convenience yield rather than automatic reverse arb. When a vignette mentions producer hedging pressure and futures under expected spot, name normal backwardation—and do not confuse it with a backwardated curve.