Walk through an arbitrage trade using commodity forwards and compute the profit that could be earned.
The idea
A cash-and-carry arbitrage works when the forward price is above the full cost of carry, which is the spot price compounded at the risk-free rate plus the storage costs carried to expiry. If storage is a continuous yearly rate u, the same quantity can be written as one exponential: Carry value = S0 × e^((r + u)T). Borrow the spot price, buy and store the commodity, and sell it forward at no cost. At expiry, deliver the commodity, collect F0, repay the loan with interest and pay for storage. The riskless profit on each unit is what the forward pays less the loan and the storage: Profit = F0 − S0 × e^(rT) − FV of storage. Multiply by the contract size for the profit on the whole trade. Its present value is that profit discounted at r. If the forward is too low and the commodity can be borrowed, do the reverse: short it, lend the proceeds and buy forward. The usual errors are leaving out storage or lease fees and mixing up present value with value at expiry, so check which one the question asks for.
Formulas
Cash-and-carry arbitrage profit per unit at expiry
profit = F0 − S0 × e^(r × T) − FV of storage
- F0
- the forward price the commodity is sold at
- S0
- the spot price today
- r
- the continuously compounded risk-free rate, per year
- T
- the time to delivery, in years
- FV of storage
- the storage costs carried to expiry
Positive only when the forward exceeds the full cost of carry. This is the profit at expiry; its present value is the same amount discounted at r, so check which one is asked.
A practice question
The forward price of a storable commodity exceeds its full cost of carry, S x exp((r + storage) x T). Which trade locks in a riskless profit?
- Borrow money, buy and store the commodity, and sell it forward. - correct
- Short the commodity, invest the proceeds and buy it forward.Why not: This is the reverse cash-and-carry, profitable when the forward is too low.
- Buy the commodity forward and sell it in the spot market.Why not: Sells spot without owning it and gives no financing or storage offset; this is the wrong direction.
- Lend money, buy the commodity and buy it forward.Why not: Takes the long position twice; it does not hedge price risk and does not use the overpricing.
An overpriced forward is sold, and the spot commodity is bought and stored against it (cash-and-carry). The loan finances the purchase.
A harder one
A commodity trades at USD 75, the one-year forward is USD 82, the continuously compounded interest rate is 5% and storage costs are 2% per year (continuous). An arbitrageur trades 10,000 units in a cash-and-carry. The profit at expiry is closest to:
- USD 31,547Why not: Ignores storage costs in the carry value.
- USD 15,619 - correct
- USD 70,000Why not: Ignores both financing and storage costs, taking the gap to spot as profit.
- USD 14,857Why not: Gives the present value of the profit rather than its value at expiry.
Carry value = 75 x exp(0.07) = 80.44. Profit per unit = 82 - 80.44 = 1.56; for 10,000 units = USD 15,619 at expiry.
The hardest tier
Gold trades at USD 2,000 per ounce and the one-year forward is USD 2,020. The continuously compounded interest rate is 4%. Gold can be borrowed for one year with a lease fee of USD 30 per ounce paid at expiry. Using one ounce, the arbitrage profit at expiry from borrowing, selling and investing the proceeds and buying forward is closest to:
- USD 61.62Why not: Ignores the lease fee that the borrower must pay at expiry.
- USD 91.62Why not: Adds the lease fee as income instead of deducting it as a cost.
- USD -91.62Why not: Reverses the trade direction (forward minus carry minus fee), giving a loss instead of the profit.
- USD 31.62 - correct
Invest 2,000 to get 2081.62; pay 2,020 for forward gold and 30 lease fee: profit = 2081.62 - 2,020 - 30 = 31.62.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Commodity Forwards and Futures
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