The minimum variance hedge ratio
1. The idea
Futures often do not follow the price being hedged one for one. A trucking company that buys diesel may hedge with heating oil futures, whose price moves with diesel’s but not exactly; hedging with futures on a related asset is a cross hedge. Then one unit of futures for each unit of exposure is not the best hedge.
Let \(\Delta S\) and \(\Delta F\) be the changes in the spot price and the futures price over the life of the hedge. A hedge with \(h\) units of futures for each unit of exposure changes in value by \(\Delta S - h\,\Delta F\), and its variance, \(\sigma_S^2 + h^2 \sigma_F^2 - 2h\rho\,\sigma_S \sigma_F\), is smallest at the minimum variance hedge ratio
\[ h^* = \rho\,\frac{\sigma_S}{\sigma_F}, \]
where \(\sigma_S\) and \(\sigma_F\) are the standard deviations of \(\Delta S\) and \(\Delta F\), and \(\rho\) their correlation. The optimal number of contracts is then
\[ N^* = h^*\,\frac{Q_A}{Q_F}, \]
where \(Q_A\) is the amount hedged and \(Q_F\) the amount in one contract. A buyer buys these futures and a seller sells them, as in the lesson on the basis.
An example. A trucking company will buy 1,050,000 gallons of diesel and hedges with heating oil futures, each on 42,000 gallons. Over the hedge’s life, \(\sigma_S = 0.040\), \(\sigma_F = 0.045\) and \(\rho = 0.90\), so \(h^* = 0.90 \times 0.040/0.045 = 0.80\), and \(N^* = 0.80 \times 1{,}050{,}000/42{,}000 = 20\): the company buys 20 contracts. The hedged position’s standard deviation is then \(\sigma_S\sqrt{1 - \rho^2}\), the part of the spot price’s changes that the futures do not follow.
Tailing the hedge. Futures are settled every day, so their gains and losses arrive before the hedge ends, and the hedge works day by day. Tailing takes this into account by using the values of the position and of one contract at today’s prices, \(V_A\) and \(V_F\): \(N^* = \hat h\, V_A / V_F\), where \(\hat h\) is the hedge ratio estimated from one-day percentage changes in the two prices. When a question gives only \(h^*\), as here, \(h^*\) is used for \(\hat h\). With diesel at $3.10 and heating oil futures at $2.95, that is \(0.80 \times 3{,}255{,}000/123{,}900 = 21.02\), or 21 contracts.
From data. In practice \(\sigma_S\), \(\sigma_F\) and \(\rho\) are estimated from past changes in the two prices, with sample standard deviations that divide by \(n - 1\), where \(n\) is the number of observed pairs of price changes. The estimate of \(h^*\) is also the slope of a straight line fitted through the pairs \((\Delta F, \Delta S)\).
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The hedge ratio on a chart
First, a check of the formula in the simplest case.
Now move the hedge ratio. The hedged position’s standard deviation falls as futures are added, is lowest at \(h^*\), and rises again with more futures than that.
4. One worked example, then your turn
In the worked example, we find the number of contracts for a buyer.
Now a tailed hedge for a seller, with hints, one step at a time.
And one on your own, from data.
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 the hedge ratio, the number of contracts, a tailed hedge and a ratio from data. 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: