Commodity Forwards and Futures: formula sheet

Key formulas

Payoff of long forward or futures at maturity

Payoff (long) = S_T − K

S_T is the spot price at maturity and K is the delivery price (or futures price when the position was opened). Multiply by contract size and number of contracts.

Payoff of short position

Payoff (short) = K − S_T

The short's gain is the long's loss. The two sides sum to zero.

Futures daily gain or loss

Daily P&L (long) = (F_t − F_t−1) × contract size × number of contracts

Short position has the opposite sign. Cash moves through the margin account each day.

Hedged price for a short hedger

Effective price ≈ S_T + (F_0 − F_T) = F_0 + Basis_T

Basis_T = S_T − F_T. If basis is zero at close-out, the hedger locks in F_0.

Basis

Basis = Spot price − Futures price

Use the question's convention. Some texts define basis the other way, so read carefully.

Forward price, storage costs as PV of cash amounts

F₀ = (S₀ + U) × e^(rT), where U = PV of all storage costs

Use when storage costs are given as fixed cash amounts at known dates. Discount each at r (continuous) before adding to S₀.

Forward price, storage as continuous proportional rate

F₀ = S₀ × e^((r + u)T)

u is storage cost as an annual rate of the spot price. r and u use the same compounding basis.

Forward price with convenience yield

F₀ = S₀ × e^((r + u − y)T)

y is the convenience yield. It is usually inferred from market prices, not observed.

Consumption commodity bounds

F₀ ≤ (S₀ + U) × e^(rT) or F₀ ≤ S₀ × e^((r + u)T)

For consumption commodities the upper bound is enforced by cash-and-carry arbitrage. There is no firm lower bound.

Implied convenience yield

y = r + u − ln(F₀ ÷ S₀) ÷ T

Rearranged from the continuous formula.

Annual compounding version

F₀ = S₀ × (1 + r + u)^T (approximate form) or (S₀ + U) × (1 + r)^T

Use only if the question states annual compounding. Check which form the question implies.

Forward price with continuous compounding

F = S × e^((r + u − y)T)

r = risk-free rate, u = storage cost as a proportion of spot per year, y = convenience yield, T in years. All rates continuously compounded.

Implied convenience yield

y = r + u − ln(F ÷ S) ÷ T

Use the market F and S. Gives an annualised, continuously compounded yield.

Net cost of carry

c = r + u − y

F = S × e^(cT). If c < 0, the market is in backwardation.

Storage cost as a dollar amount

F = (S + U) × e^(rT) × e^(−yT)

Here U is the present value of storage costs. Use when storage is given in currency, not as a rate.

Bounds without convenience yield

F ≤ S × e^((r + u)T)

With convenience yield, F can fall well below S × e^(rT). Arbitrage gives only this upper bound.

Contango

F(T₂) > F(T₁) > S (upward-sloping curve)

Futures prices rise with maturity. Typical when carry costs dominate and inventories are high.

Backwardation

F(T₂) < F(T₁) < S (downward-sloping curve)

Futures prices fall with maturity. Typical when convenience yield is high and inventories are low.

Cost of carry with convenience yield

F = S × e^((r + u − y) × T)

r = risk-free rate, u = storage cost rate, y = convenience yield, all continuously compounded. Contango if r + u > y; backwardation if r + u < y.

Roll yield (per roll, long position)

Roll yield = (Price of expiring contract − Price of next contract) ÷ Price of expiring contract

Positive in backwardation, negative in contango. This is a common simple definition; some texts use the spot-futures difference, so follow the question's wording.

Futures return decomposition

Total return ≈ Spot return + Roll yield + Collateral return

For a fully collateralised long futures position. Roll yield is the part due to curve shape.

Forward price with lease rate (continuous compounding)

F(0,T) = S(0) × e^((r − δ_l) × T)

r is the continuously compounded risk-free rate and δ_l the continuously compounded lease rate, both annual. T is in years.

Implied lease rate from spot and forward

δ_l = r − (1 ÷ T) × ln(F ÷ S)

The term (1 ÷ T) × ln(F ÷ S) is the annualised continuously compounded forward premium.

Annualised forward premium

(1 ÷ T) × ln(F ÷ S) = r − δ_l

Positive means contango. Negative means backwardation. The forward premium is the net cost of carry.

Forward ratio between two maturities

F(0,T2) ÷ F(0,T1) = e^((r − δ_l) × (T2 − T1))

Use it to find the implied lease rate between two forward dates. Rates are those for that period.

Discrete compounding version

F = S × (1 + r)^T ÷ (1 + δ_l)^T

Use only if the question states annual compounding. Do not mix with the continuous formula.

Basis

b = S − F

Spot minus futures at the same date. b1 is the basis when you open the hedge, b2 when you close it. Check the question's sign convention.

Effective price, short hedge

Price received = S2 + (F1 − F2) = F1 + b2

You sell the physical at S2 and gain F1 − F2 on the short futures. Only b2 is uncertain.

Effective cost, long hedge

Price paid = S2 − (F2 − F1) = F1 + b2

You buy the physical at S2 and gain F2 − F1 on the long futures. Same expression as the short hedge.

Minimum variance hedge ratio

h* = ρ × σS ÷ σF

ρ is the correlation between changes in spot and futures prices. σS and σF are the standard deviations of those changes over the hedge life.

Number of contracts

N* = h* × QA ÷ QF

QA is the exposure size, QF is the size of one futures contract. Round to the nearest whole contract.

Hedge effectiveness

R² = ρ²

The proportion of the variance of the spot price change removed by the optimal hedge.

Basis effect on a hedge

Strengthening basis helps shorts; weakening basis helps longs

Holds when basis is defined as spot minus futures.

Quick revision

• Cost of carry (continuous): F = S × e^((r + u − y) × T), with u as storage cost rate and y as convenience yield.
• Higher interest rates or storage costs raise the forward price; higher convenience yield lowers it.
• Contango: futures prices rise with maturity, so F is above S.
• Backwardation: futures prices fall with maturity, so F is below S.
• Convenience yield is the benefit of holding the physical commodity, not the futures contract.
• Implied convenience yield: y = r + u − ln(F ÷ S) ÷ T.
• Futures are marked to market daily; forwards usually settle at maturity.
• Basis = spot price of the asset being hedged − futures price.
• Basis risk arises when the hedged asset and the futures contract do not move together perfectly.
• A short hedge gains if the basis strengthens; a long hedge gains if it weakens.
• Minimum-variance hedge ratio h* = ρ × σS ÷ σF.
• Match contracts to exposure: number of contracts = h* × exposure ÷ contract size.

Common mistakes

• Saying forwards and futures have identical cash flows. Fix: Futures settle daily through margin. Forwards typically settle once at maturity. This also creates counterparty risk for forwards.
• Reversing the hedge direction. Fix: A producer fears a price fall and sells (shorts). A consumer fears a price rise and buys (goes long).
• Adding undiscounted storage costs to the spot price. Fix: Discount each storage payment to today at r for its payment date, then add to S₀. Equivalent: compound each payment forward to T and add to S₀e^(rT).
• Subtracting storage costs as if they were income. Fix: Storage is a cost, so it raises F. Income lowers F. Convenience yield lowers F like income.
• Adding convenience yield to the cost of carry Fix: It is a benefit. It enters with a minus sign: r + u − y.
• Treating convenience yield as an observable cash flow Fix: Convenience yield is implied from prices. It is a plug that makes the cost-of-carry formula fit.
• Saying contango always means futures prices are expected to rise Fix: The curve reflects carry costs and convenience yield. It is not a pure forecast of spot prices. Futures converge to spot at expiry.
• Treating roll yield as a loss in any contango market regardless of position Fix: It is bad for long investors who roll. Short positions gain a positive roll yield in contango.
• Adding the lease rate to r instead of subtracting it. Fix: Treat δ_l like a yield. The exponent is (r − δ_l), so a higher lease rate lowers the forward price.
• Dropping the time factor when solving for the lease rate. Fix: Divide ln(F ÷ S) by T before subtracting from r. This gives an annual rate.

Exam tips

• Expect conceptual comparisons: forward versus futures on credit risk, liquidity, customization and settlement. Know the table by heart.
• Always attach a sign to the payoff. Check whether the question asks about the long or the short.
• In hedging questions, compute the effective price as spot plus futures gain (short) or spot minus futures gain (long). Verify with the basis shortcut.
• Watch the units: barrels, bushels, ounces, metric tons. Contract size changes the answer by orders of magnitude.
• Remember that commodity-specific features (storage, convenience yield, delivery grade and location) separate commodity contracts from financial ones.
• Read how storage is quoted. A cash amount means PV and add to S₀. A percentage means put it in the exponent.
• Check whether the commodity is for investment or consumption before writing an equality.
• Keep five decimals on e^x values, because answer options can be close together.