The futures price is the cost of buying today and holding until delivery. The contango that follows is not a cost of hedging, and a quote below it tells you something.
A futures price is fair when it equals the cost of buying the commodity today and holding it to delivery: spot, plus the interest on the money tied up, plus storage and insurance, less any convenience yield from holding the physical. On an illustrative metal at 2,450 a tonne, with 5 per cent financing for 180 days and warehouse rent of 0.45 a tonne a day, the six-month fair value is 2,592.25, a contango of 142.25 that is not a cost of hedging.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A buyer who needs metal in six months has two routes. Buy the future at F, or buy the metal now at spot, borrow to pay for it, and rent a warehouse until the date. Both deliver the same tonne on the same day, so they must cost the same; if they did not, a trader would buy the cheaper route and sell the dearer one. The same no-arbitrage identity sets the FX forward rate, where the interest differential plays the part that storage plays here.
| Input | Value |
|---|---|
| Spot price | 2,450.00 |
| Financing rate, a year | 5.0% |
| Days to delivery | 180 |
| Warehouse rent and insurance, per tonne per day | 0.45 |
| Quantity to hedge, tonnes | 10,000 |
| Contract size, tonnes per lot | 25 |
F = S × (1 + r × days ÷ 360) + storage − convenience
Financing: 2,450.00 × 0.05 × 180 ÷ 360 = 61.25
Storage: 0.45 × 180 = 81.00
F = 2,450.00 + 61.25 + 81.00 = 2,592.25
Excel, with spot, rate, days and daily rent in B1:B4: =B1*(1+B2*B3/360)+B4*B3
The contango, futures over spot, is 142.25 a tonne, 5.81 per cent of spot for six months. In continuous form the same carry is written F = S × e(r + u − y)T, with storage u expressed as a yield on the spot price (6.6122 per cent here) and y the convenience yield. That version gives 2,596.46: the gap is compounding, not a different answer. Use the convention of the contract and of the rate you are quoting.
The market quotes 2,560. That is 32.25 below full carry. Buying spot and selling the future would lose money, and the reverse trade, selling metal short and buying the future, needs metal that someone will lend, so the gap can persist: holders of metal are willing to forgo some of the carry because having the physical is worth something to them. Express it as an implied convenience yield:
y = (full carry − market) ÷ (S × T) = 32.25 ÷ 1,225.00 = 2.63% a year
The market quotes 2,620. That is 27.75 above full carry. Buy spot at 2,450, finance and store it, and sell the future: 27.75 a tonne locked in, 693.75 a lot. Traders with warehouse space and cheap funding do exactly this, which is why a future rarely stays above full carry for long. For a hedger it is a signal to check the inputs, not a gift.
For a hedger the comparison matters in a practical way. A buyer who sees the six-month future trade well below full carry is being offered metal for later at less than it would cost to buy and hold it now, and has little reason to buy physical early. A buyer who sees it trading at full carry is indifferent between the two routes and should choose on storage capacity, credit lines and the cash it would tie up.
| Days to delivery | 3% financing | 4% | 5% | 6% |
|---|---|---|---|---|
| 90 | 2,508.88 | 2,515.00 | 2,521.12 | 2,527.25 |
| 180 | 2,567.75 | 2,580.00 | 2,592.25 | 2,604.50 |
| 360 | 2,685.50 | 2,710.00 | 2,734.50 | 2,759.00 |
At 5 per cent, the contango is 71.12 for 90 days and 284.50 for a year. Storage is the larger part of the carry for a heavy, cheap commodity; for gold, where storage is negligible, the contango is almost all financing, and for an equity index the dividends run the other way.
A buyer hedging 10,000 tonnes six months out buys 400 lots at 2,592.25. Against today's spot of 2,450, the report shows 1,422,500 of "hedging cost". It is not one. The alternative to the future was never metal at 2,450 delivered in six months; it was metal at 2,450 today, paid for now and kept in a warehouse, which costs the same 61.25 of financing and 81.00 of rent per tonne. The forward price is the spot price carried forward.
The real costs of the hedge are elsewhere: the bid and offer on each lot, broker and exchange fees, the funding of initial and variation margin, and basis risk if the future does not track the metal actually bought. Those are the lines to report and to manage. How large the margin calls can grow is worked in how much more margin a five-year hedge calls.
Backwardation, a future below spot, is the case full carry cannot reach. If spot were squeezed to 2,700 with the six-month future at 2,560, the market would be 140.00 below spot and 288.50 below the full-carry price of 2,848.50: a shortage priced into the curve.
F = spot × (1 + r × t) + storage − convenience: 2,450.00 + 61.25 + 81.00 = 2,592.25. A quote below full carry measures the convenience yield, 2.63 per cent a year at 2,560; a quote above it is an arbitrage that rarely lasts. Report the contango as carry, not cost. The same identity applied to a currency hedge, where the reported "cost" turns out to be the interest differential, and the three real cost lines beside it, are in the free workbook for this case. All figures are illustrative.
Spot plus the interest on holding the index, less the dividends received over the period. With the index at 5,000, rates at 5 per cent and a 1.5 per cent dividend yield, a three-month future is 5,000 x (1 + 0.05 x 0.25) minus 18.75 of dividends, which is 5,043.75: a basis of 43.75 points. Dividends play the part that storage plays for a metal, with the opposite sign.
Contango is a futures price above spot, the normal state for a storable commodity because buying later saves the financing and the rent. Backwardation is a futures price below spot, which full carry cannot explain and which signals a shortage: holders of the physical value having it now. If spot jumped to 2,700 with the six-month future at 2,560, the market would be 140.00 backwardated against a full-carry price of 2,848.50.
No. Buying 10,000 tonnes six months forward at 2,592.25 rather than spot at 2,450 looks like 1,422,500 of extra cost, but buying spot now means paying 61.25 a tonne of financing and 81.00 of rent to hold it. The forward price is the same purchase with the carrying cost included. The real costs of a hedge are the bid and offer, the broker's fees and the margin, not the contango.
This article is one calculation from Derivatives. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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