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How do you calculate the optimal hedge ratio for a futures hedge?

The minimum variance hedge ratio from twelve months of price changes, the number of futures contracts, and why a one-for-one cross hedge leaves almost twice the residual risk.

Julian R. Sterling · from Derivatives

The optimal (minimum variance) hedge ratio is the correlation between changes in the price you pay and changes in the futures price, times the ratio of their volatilities: h* = ρ × σS / σF. On twelve months of illustrative data, a correlation of 0.9233 and a volatility ratio of 0.6426 give a hedge ratio of 0.5933: 237 contracts for 10,000 tonnes, not 400. Hedging one for one would leave almost twice the residual risk.

Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →

The case

A manufacturer buys 10,000 tonnes a month of a grade of metal that has no futures contract of its own. Its physical price is a regional price that follows the exchange contract but moves less, and not always in step. The exchange contract is 25 tonnes. Before setting the hedge, the treasurer takes the last twelve monthly changes in both prices, per tonne. The data are illustrative.

Monthly price changes, per tonne

Month	Physical price change (ΔS)	Futures price change (ΔF)
1	30	70
2	−13	−60
3	112	135
4	−75	−55
5	54	40
6	−105	−160
7	28	80
8	30	55
9	−28	−95
10	43	95
11	−16	−50
12	66	120

Step 1: volatility and correlation of the changes

Work on changes, never on price levels. The sample standard deviation of the physical changes is 60.89 a tonne and of the futures changes 94.76. Their covariance is 5,327.50, so the correlation is 5,327.50 / (60.89 × 94.76) = 0.9233. The physical price moves about two thirds as much as the futures (a volatility ratio of 0.6426), and in the same direction most of the time.

Step 2: the hedge ratio and the number of contracts

The hedge ratio is the slope of a regression of the physical change on the futures change, which is why SLOPE gives the same number. It answers one question: for each tonne of exposure, how many tonnes of futures minimise the variance of the combined position? Because the physical price is less volatile than the futures, the answer is well under one.

The result: what the hedge leaves behind

The variance of the hedged position is σS² + h²σF² − 2h × covariance. At the optimum this reduces to σS × √(1 − ρ²) = 23.39 a tonne, and the share of variance removed is ρ², 85.2 per cent.

Residual monthly risk on 10,000 tonnes

Hedge	Ratio	Contracts	Residual sd per tonne	Monthly sd on position	95% one-month loss	Variance removed
Unhedged	0.0000	0	60.89	608,911	1,001,598	0.0%
Minimum variance	0.5933	237	23.39	233,857	384,672	85.2%
Policy ratio of 80%	0.8000	320	30.50	305,043	501,766	74.9%
One for one	1.0000	400	45.08	450,786	741,498	45.2%

The one-for-one hedge, 400 contracts, feels safer and is not. It leaves a residual standard deviation of 45.08 a tonne against 23.39 at the optimum: 216,929 a month more risk on the position, because 163 of the 400 contracts are no longer hedging the metal but speculating on the futures. The 95 per cent one-month loss uses a normal multiplier of 1.6449.

Over-hedging a cross hedge does not add protection. Beyond the optimal ratio, each extra contract adds more futures risk than it removes physical risk, and at a ratio of one the position has given back almost half of the variance reduction available.

What if the correlation is lower?

The correlation does two jobs: it scales the hedge ratio and it caps how much risk any hedge can remove. Holding the volatility ratio at 0.6426:

Hedge ratio and effectiveness by correlation

Correlation	Hedge ratio	Contracts	Variance removed	Residual sd per tonne
0.70	0.4498	180	49.0%	43.48
0.80	0.5141	206	64.0%	36.53
0.90	0.5783	231	81.0%	26.54
0.9233	0.5933	237	85.2%	23.39
0.95	0.6105	244	90.2%	19.01

At a correlation of 0.70 the best hedge removes less than half the variance. That is the point at which a treasurer should ask whether the futures contract is the right instrument at all, or whether a bilateral swap on the regional index, with its own credit and collateral cost, is worth paying for.

The common mistakes

• Regressing price levels. Two trending prices show a high correlation in levels whatever their month-to-month relationship. The hedge is rebalanced on changes, so it must be estimated on changes.
• Mixing returns and money. If the inputs are percentage returns, the ratio is in value terms and must be multiplied by the spot-to-futures price ratio before converting to contracts.
• Treating the ratio as permanent. Twelve observations give an estimate, not a constant. Re-estimate on a rolling window and expect the ratio to move; the correlation table above is the range to plan for.
• Confusing variance with accounting. A hedge that minimises variance can still fail an effectiveness test under the accounting rules, which compare cumulative changes period by period.

Takeaway

Estimate ρ and both volatilities on price changes, multiply, and convert to contracts: 0.5933 and 237 contracts here, removing 85.2 per cent of the variance. The accounting test of the same hedge is a different calculation, worked in how to test hedge effectiveness with the dollar offset method, and what a hedge costs in cash and collateral rather than in variance is the subject of the free workbooks that go with the book.

Questions readers ask

Is the minimum variance hedge ratio the same as the regression beta?

Yes. Regressing the change in the hedged price on the change in the futures price gives a slope equal to correlation times the volatility ratio. In Excel, SLOPE on the two columns of changes returns 0.5933 in this case, and RSQ returns 0.8525, the share of variance the optimal hedge removes.

How do you convert a hedge ratio into a number of futures contracts?

Multiply the hedge ratio by the exposure and divide by the contract size, in the same units. Here 0.5933 times 10,000 tonnes divided by 25 tonnes a contract is 237.3, rounded to 237 contracts. If the ratio was estimated on percentage returns, first multiply by the value of the exposure over the value of one contract.

What hedge effectiveness can a cross hedge achieve?

At most the correlation squared. With a correlation of 0.9233 the best hedge removes 85.2 per cent of the variance; at 0.80 it removes 64.0 per cent and at 0.70 only 49.0 per cent. Whatever the number of contracts, the remaining basis risk cannot be hedged with that futures contract.

Also on this site

• How do you test hedge effectiveness with the dollar offset method?Hedge change over hedged-item change, within 80 to 125 per cent: a USD forward reads 100.3 per cent cumulative but 167.8 per cent in a 12-pip quarter.
• Is a zero-cost collar really zero cost?Zero premium is not zero value. Priced with Black-76, a cap at 2,680 is worth 182.84 a tonne and the floor sold at 2,300 only 122.18: 1,577,235 changes hands.
• Did my currency hedge fail if it lost money?A twelve-month forward showed a loss of 8,130,844.04 and did exactly what it was for. The break-even of a hedge is the forward, not the spot it was dealt at.
• How much more margin does a five-year hedge call than a rolling one?A fall of 10 per cent in sterling calls 52,378,487 from a locked five-year hedge and 11,696,910 from a rolling one: 4.48 times, same exposure.
• How do you calculate an FX forward rate?Forward = spot x (1 + quote-currency rate) / (1 + base-currency rate). At 1.2700 spot, 4.5% USD and 4.0% GBP, one-year GBP/USD is 1.2761.
• How do you calculate CVA on an interest rate swap?Sum expected exposure x default probability x LGD, discounted: 143,510 on a 120M five-year swap, about 2.7bp a year against an 11bp credit charge.

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