Basis risk and the effective price
1. The idea
A company that will buy or sell a commodity later can fix most of its price today with futures. A buyer fears a rise in the price, so it buys futures: a long hedge. A seller fears a fall, so it sells futures: a short hedge. When the company finally buys or sells the commodity at the spot price of that day, \(S_2\), it closes out the futures at \(F_2\), and their gain or loss makes up for the change in the spot price.
The basis is the spot price less the futures price, \(b = S - F\). With every unit hedged, the price paid or received once the futures’ gain or loss is counted, the effective price, is
\[ S_2 - (F_2 - F_1) = F_1 + b_2, \]
where \(F_1\) is the futures price when the hedge starts and \(b_2 = S_2 - F_2\) the basis when it is closed out. The formula is the same for both sides: a buyer pays \(S_2\) less the gain on its long futures, \(F_2 - F_1\); a seller receives \(S_2\) plus the gain on its short futures, \(F_1 - F_2\). So the hedge locks in the futures price \(F_1\), except for the basis at the end, which no one knows when the hedge starts. That uncertainty is basis risk. On a futures contract’s delivery date the basis is zero, because the futures price converges to the spot price, but a hedge usually ends before then: the company chooses a contract that expires after its date.
An example. In March a bakery buys futures at $6.40 a bushel to fix the price of wheat it will buy in June. In June the spot price is $7.10 and the futures price $7.25. The futures gained \(7.25 - 6.40 = 0.85\) a bushel, so the effective price of the wheat is \(7.10 - 0.85 = \$6.25\) a bushel: the futures price of March, $6.40, plus the basis of June, \(7.10 - 7.25 = -0.15\).
A partial hedge covers only a share \(h\) of the amount. Each unit then gets the share \(h\) of the futures’ gain or loss: the effective price is \(S_2 - h\,(F_2 - F_1)\), an average of \(F_1 + b_2\) on the hedged part and \(S_2\) on the rest. Questions on hedges usually leave out the interest on the futures’ daily gains and losses; the minimum variance hedge ratio shows how tailing the hedge takes it into account.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The hedge on a chart
First, the basis when the hedge is closed out.
Now move the spot price in June, with the basis kept as it is. Without a hedge the mill pays whatever the spot price is; with the hedge it pays the same effective price whatever happens, because only the basis could change it.
4. One worked example, then your turn
The worked example is a seller’s hedge, for the whole amount.
Now a partial hedge, with hints, one step at a time.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a buyer’s hedge, a seller’s hedge and a partial hedge. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: