Case 091Hedging a bookHard

An airline can hedge its jet fuel only with crude oil futures. With monthly volatilities of 2.6% and 3.0% and a correlation of 0.9, compute the minimum-variance hedge ratio, the share of risk removed and the basis risk that remains.

The situation

Skyvara Airways buys about 1,00,000 tonnes of jet fuel a year. At an assumed Rs 80,000 a tonne, that is Rs 800 crore of fuel. There is no liquid jet fuel futures market it can use, so the treasury hedges with crude oil futures.

Over the last three years, monthly percentage changes in the jet fuel price have had a volatility of 2.6% and changes in the crude futures price 3.0%, with a correlation of 0.9. The treasurer's draft policy says to hedge one rupee of crude for every rupee of fuel, because the airline wants to be fully hedged.

Your task

What hedge ratio minimises risk, how much of the risk does it remove, and what basis risk is left? Is the one-for-one policy right?

Quick check

What hedge ratio minimises the variance of the hedged fuel bill?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The minimum-variance hedge ratio is 0.78, it removes 81% of the variance, and it leaves a monthly basis risk of about 1.13%, Rs 9.1 crore on Rs 800 crore of fuel. The one-for-one policy leaves more risk, 1.31% a month, because crude is more volatile than jet fuel and hedging it fully adds crude risk the airline does not have. What no crude hedge can remove is the gap between jet fuel and crude, the crack spread.

Step 1Why is a cross hedge not one for one?

Because the hedge and the exposure are cousins, not twins. If you insure your bicycle against theft by betting on the city's overall theft rate, you would size the bet to how closely bicycle thefts track the city figure and how much more the city figure swings. The minimum-variance hedge ratioThe size of hedge, per unit of exposure, that makes the hedged position as steady as possible; it equals the correlation times the ratio of the two volatilities. is the correlation times the volatility of the exposure over the volatility of the hedge: 0.9 times 2.6 over 3.0, which is 0.78. It is also the slope of a line fitted through jet fuel changes against crude changes.

The relationship

h^* = \rho\,\frac{\sigma_{jet}}{\sigma_{crude}} = 0.9 \times \frac{2.6}{3.0} = 0.78 \qquad \sigma_{left} = \sigma_{jet}\sqrt{1-\rho^2} = 2.6 \times \sqrt{0.19} = 1.13\%

\rho	correlation of monthly jet fuel and crude changes, 0.9
\sigma_{jet}, \sigma_{crude}	monthly volatilities, 2.6% and 3.0%
\sigma_{left}	monthly volatility of the hedged position

What it says in wordsHold 0.78 of crude per rupee of fuel; what is left over moves about 1.13% a month, whatever the crude market does.

Monthly changes in jet fuel plotted against crude futures cluster around a line with slope 0.78, the minimum-variance hedge ratio, and the band of about 1.13% either side of it is the basis risk a crude hedge cannot remove.

Step 2How much risk does the hedge remove, and what is left?

The share of variance a hedge can remove is the correlation squared. A correlation of 0.9 removes 81% of the variance, and the volatility left is 2.6% times the square root of 0.19, about 1.13% a month. On Rs 800 crore of fuel that is Rs 9.1 crore of monthly standard deviation, down from Rs 20.8 crore, and roughly Rs 31 crore over a year if months are independent. That remainder is basis riskThe risk that the hedge and the exposure move by different amounts, left over after hedging with a related but different instrument.: jet fuel and crude drifting apart, for example when refining margins on jet fuel widen.

Step 3Is the one-for-one policy wrong?

It leaves more risk, not less. Hedging Rs 800 crore of fuel with Rs 800 crore of crude leaves a monthly volatility of 1.31%, Rs 10.5 crore, against Rs 9.1 crore at the 0.78 ratio, because crude swings more than jet fuel and the extra 22 paise of crude per rupee of fuel is an open bet on crude. Fully hedged is a feeling, not a measurement. The policy should hedge about Rs 624 crore of crude value and recheck the ratio each quarter, because correlations estimated from three years of data move.

On Rs 800 crore of annual fuel, monthly risk falls from Rs 20.8 crore unhedged to Rs 9.1 crore at a 0.78 hedge ratio, while a one-for-one crude hedge leaves Rs 10.5 crore, because it over-hedges.

Close with the limit. The ratio is only as good as the correlation behind it, and correlations tend to fall in the months that matter, when refinery outages or jet demand shocks move jet fuel on its own. A risk manager reports the hedge's effectiveness monthly and watches the jet-crude spread directly, rather than assuming the 0.9 holds.

Where candidates lose it

The common slip is to answer 1.0 because the goal is to be fully hedged, or 0.9 because the correlation is 0.9. The ratio needs both the correlation and the volatility ratio; either alone gives the wrong size.

The second is saying the hedge removes 90% of the risk. The share of variance removed is the correlation squared, 81%, and the share of volatility removed is only about 56%, from 2.6% to 1.13%.

What the interviewer asks next

The correlation falls to 0.7 in a refinery outage. What ratio, and how much risk is left?
Would a heating oil or diesel futures contract be a better hedge? What would you check?
How would hedge accounting rules treat a hedge with this much basis risk?

Company names and figures are illustrative.