FMP 8: Using Futures for Hedging

Futures serve two opposite purposes, and which one applies depends on what the trader already owns. A firm exposed to an exchange rate, an interest rate, an equity index or a commodity price can take a futures position that shrinks that exposure. A trader with no such exposure who takes the identical position is speculating. Only the first use is hedging.

Wiping out every trace of risk with futures is normally impossible, so the practical question is how to cut risk as far as it can be cut. Early sections treat futures as if they were forwards, with the position set once and left alone until the exposure matures.

When a short futures position is the right hedge

Two situations call for one. A company already holds a quantity of an asset and knows the date on which it will sell, or it expects to receive an asset later and plans to sell on arrival. The danger in both cases is a falling price, and a short futures position pays off precisely when prices fall.

Consider a producer due to take delivery of 2 million barrels of crude oil three months from now, with the whole cargo earmarked for immediate sale. A one-cent move in the oil price shifts the proceeds by USD 20,000, being USD 0.01 applied to 2,000,000 barrels, and management treats a swing of that size as unacceptable. Oil futures traded by the CME Group cover 1,000 barrels apiece, so the exposure is offset by shorting 2,000 three-month contracts, that count being 2,000,000 barrels spread over 1,000 barrels per lot. A hedge assembled from a short futures position is called a short hedge.

Example 1 · Worked

Today oil trades at a spot price of USD 59.50 per barrel, while the futures price for delivery in three months is USD 60.00 per barrel. The producer shorts 2,000 contracts, each on 1,000 barrels. The oil arrives during the delivery period of those contracts, so the futures price then equals the spot price or comes very close.

1. Oil is delivered on a day when the spot price stands at USD 50. What does the producer end up with?

Solution. The physical sale of 2,000,000 barrels at 50 brings in 100,000,000. Each of the 2,000 short contracts gains 10 per barrel across 1,000 barrels, worth 10,000 apiece and 20,000,000 across the book. Adding 100,000,000 and 20,000,000 gives net proceeds of USD 120 million, and spreading 120,000,000 over 2,000,000 barrels leaves USD 60 per barrel.

2. Now let the spot price on the delivery day be USD 68 instead.

Solution. Selling 2,000,000 barrels at 68 raises 136,000,000. The short futures position now loses 8 per barrel, which is 8,000 on each of the 2,000 contracts of 1,000 barrels, or 16,000,000. Take 16,000,000 from 136,000,000 and net proceeds come to USD 120 million once more, again USD 60 per barrel.

3. Which price has been locked in, and what risk remains?

Solution. The locked-in figure is the initial futures price of USD 60, not the initial spot price of USD 59.50. Oil carries no ceiling price three months out, so the loss on the short futures contracts has no limit either, and that is tolerable only because a matching gain on the physical oil arrives alongside it.

Two mechanical routes give that result: hand the oil over on receipt as the futures contract provides, or close out the position at the prevailing spot price and sell the oil through the usual channels. Real hedges rarely run this cleanly, for reasons taken up below.

Figure 1: How a short hedge flattens realised proceeds

The unhedged producer receives whatever the market offers; the hedged producer receives the initial futures price whatever the market offers.

Check yourself

A short hedge locks in USD 60 per barrel even though the spot price on the day the hedge is placed is USD 59.50. Why is the futures price and not the spot price the relevant benchmark?

The producer sells nothing on the day the hedge is placed, so today’s spot price is not one it can actually obtain. What it can obtain is the price at which the futures market takes delivery in three months, USD 60.00.

A long hedge is the mirror image. It fits a company that already knows it must buy some quantity of an asset at a future date and wants protection against the price rising first. Buying futures now means a rise in the asset price shows up as a futures gain offsetting the extra amount paid in the market, and the effective cost settles back to the futures price prevailing when the hedge was placed.

Example 2 · Worked

A power generator will need to buy 600,000 MMBtu of natural gas in three months. The spot price is USD 3.10 per MMBtu and the three-month futures price is USD 3.20 per MMBtu. The exchange contract is on 10,000 MMBtu, so the generator goes long 60 (= 600,000/10,000) contracts. Assume the futures price equals the spot price on the purchase date.

1. Gas costs USD 3.90 per MMBtu on the purchase date. What is the effective cost of the fuel?

Solution. Buying in the market costs 600,000 x 3.90 = USD 2,340,000. The long futures position gains 60 x 10,000 x (3.90 – 3.20) = USD 420,000. Net outlay is 2,340,000 – 420,000 = USD 1,920,000, which is USD 3.20 per MMBtu.

2. Gas costs USD 2.60 per MMBtu instead. Repeat the calculation.

Solution. The purchase costs 600,000 x 2.60 = USD 1,560,000, and the futures position loses 60 x 10,000 x (3.20 – 2.60) = USD 360,000. Net outlay is 1,560,000 + 360,000 = USD 1,920,000, again USD 3.20 per MMBtu.

Why not simply buy the asset today

An obvious alternative is to buy the physical asset now and hold it. Two things make that less attractive. The asset may be dearer in the spot market than in the futures market, and the buyer takes on the cost of financing the purchase and storing the goods for the whole waiting period.

Suppose instead the futures price sat above the spot price. Buying today might then look appealing, but financing and storage over the period would cost at least as much as the gap between the two prices. Were the costs smaller, a trader could buy spot, store, and sell in the futures market for a riskless arbitrage profit, and that opportunity would be competed away.

Check yourself

A manufacturer needs an industrial metal in two months and notices that the two-month futures price is below the current spot price. Does that discount by itself make the long hedge worthwhile?

It reinforces the choice but is not the reason for it. The reason is that the purchase price is uncertain and the firm wants it fixed. The discount only says the futures route is also cheaper than buying and storing the metal today.

Hedging damps the swings in earnings that come from moving asset prices, and steadier earnings can make a company more appealing to investors. Plenty of companies still choose not to hedge.

Shareholders may prefer no hedging

One line of argument says the company should leave the decision alone, because shareholders can hedge on their own account. Not hedging hands the owners the choice of whether to carry the risk.

Shareholders normally hold many companies at once, and spreading a portfolio across industries and geographies removes much of the risk that hedging would otherwise address: a fall in oil prices hurts one holding and helps another. The same logic applies to corporate diversification. Firms are tempted to buy other companies or move into new lines of business, but a shareholder can assemble that diversification more cheaply, so a company should extend beyond its own expertise only where genuine synergy exists, meaning the combination is worth more than the parts added up.

Two counterarguments push back. Shareholders in a public company know less than management about the risks being run, which leaves managers better placed to identify and deal with them. And a shareholder who does understand the exposure may still be unable to act, because the position needed might amount to a small fraction of one futures contract.

Whichever way the decision goes, the board should set the hedging policy and communicate it plainly to shareholders, so that the owners know what risks they are carrying. Gold mining illustrates the point. Developing a mine takes years, and the gold price can move against the operator throughout, so some miners hedge future production and others do not. Investors wanting exposure to gold buy the unhedged producers; those who do not buy the hedgers.

Check yourself

Why does the argument that shareholders can hedge for themselves weaken when the exposure is small relative to a futures contract?

Futures contracts come in fixed sizes and cannot be traded in fractions. A shareholder whose slice of the exposure amounts to a fraction of one contract cannot put the position on at all, whereas the company hedges its whole exposure in whole contracts.

Before sizing a hedge, a company has to work out how large its exposure genuinely is, which means looking at the complete risk profile rather than one input in isolation.

There may be little or no exposure

Take a manufacturer of 24 carat gold jewelry whose production consumes 100 ounces of gold, bought every two months. The gold price feeds straight into its costs, so it locks in purchases for the next two years using a series of long futures contracts. Now suppose the data show that competitive pressure makes the wholesale price of jewelry track the price of the gold inside it, so gold and jewelry prices move together. If gold falls over the two years, the manufacturer loses on the hedges and the improvement in gross margin it was expecting never appears. If gold rises, the gains on the hedges are cancelled by the deterioration in gross margin that also never appears. The strategy called hedging has increased risk rather than reduced it.

Demand for jewelry may itself respond to the gold price, which could justify some long gold position. But a hedge built for that purpose would be sized quite differently from one built to cover gold purchases. Suppose the firm overstates the exposure and buys ten gold futures contracts where one contract would cover it. The other nine are speculation that nobody intended, and a fall in the gold price turns them into a loss.

Hedging may lose money

People often assume hedging exists to raise profits. It does not. It exists to reduce the variability of profits. Sometimes the hedged outcome is worse than the unhedged one and sometimes better, but it should always be more certain.

The way results are received makes some treasurers reluctant to act. A treasurer at an oil producer who sells oil futures to fix the price of future production looks clever when the oil price falls and careless when it rises, and the criticism can be severe even though the policy was correct. Some treasurers respond by hedging with options instead. Buying put options on oil protects against a fall while leaving the upside of a sharp rise intact. That protection is not free: options require a premium paid upfront by the purchaser.

Check yourself

The jewelry manufacturer’s hedge failed a test that the oil producer’s hedge passed. What was the test?

Whether a real exposure exists once the whole business is considered. The oil producer sells at a price it does not control, so a falling oil price cuts revenue with nothing to offset it. The jewelry manufacturer sells a product whose price already moves with gold, so cost and price rise and fall together. Hedging an exposure that is not there creates one.

The examples so far rested on two simplifications. The asset underlying the futures contract was the same asset the hedger was exposed to, and the futures price equalled the spot price when the hedge was closed out. The first assumption is often reasonable. The second is almost never true.

Most futures positions are closed before the delivery period named in the contract, partly because the market can behave unreliably during the delivery month. A serviceable rule of thumb follows: pick the contract whose maturity is the earliest month after the date the hedge ends. With contract months of March, May, July, September and December, a December, January or February exposure is hedged with the March contract, a March or April exposure with the May contract, and so on up the calendar.

Closing a futures contract before maturity is exactly what exposes the hedger to basis risk.

Definition of the basis at any point in the life of a futures contract.

Where the hedged asset differs from the asset underlying the contract, the definition stretches to cover the mismatch.

The general form, which reduces to the first version when the two assets are identical.

Basis risk is the risk attached to the level of the basis at the moment the hedge is closed. One warning: some writers, particularly when the contract is on a financial asset, define the basis the other way round as futures price minus spot price. The convention used here is spot price minus futures price.

Check yourself

A hedger needs cover through the end of April and the available contract months are March, May, July, September and December. Which contract should be used, and why not the March one?

The May contract, because its maturity is the earliest month falling after the end of the hedge. The March contract expires before the exposure has run its course, leaving the last stretch uncovered.

A little algebra shows where the residual uncertainty in a futures hedge lives. Write F_a for the futures price at initiation, F_t for the futures price when the hedge is closed, S_t for the spot price of the hedged asset on that closing date, and b_t for the basis at time t, which is S_t minus F_t.

Start with a hedger due to sell an asset and therefore running a short hedge. At time t the asset fetches S_t and the short futures position has gained F_a minus F_t.

The short hedge outcome, rearranged so that the basis appears on its own.

Now take a hedger due to buy an asset and therefore running a long hedge. At time t the asset costs S_t and the long futures position has gained F_t minus F_a.

Net cost when a long hedge is used, which lands on the same expression.

The two results coincide: selling under a short hedge and buying under a long hedge both settle at F_a plus b_t. Since F_a is known on the day the hedge goes on, every remaining doubt about the hedge is doubt about b_t, the basis at the close. That doubt tends to be larger for commodity futures than for futures on financial instruments.

In the crude oil example the hedge was closed while the futures price and the spot price were equal, which set b_t at zero and delivered the USD 60.00 futures price at initiation in every scenario. Close the hedge before the delivery period, or hedge one grade of crude with a contract written on another, and b_t stops being zero and stops being known.

Figure 2: The basis narrowing as the delivery date approaches

The gap between the two prices is the basis. It converges to zero at delivery, but a hedge closed earlier settles at whatever gap prevails on the day.

Check yourself

A company is hedging a future purchase and the basis turns out positive when the hedge is closed. Is that good or bad for the company?

Bad. Net cost equals the futures price at initiation plus whatever the basis turns out to be, so a positive basis lifts the effective purchase price. A company hedging a sale gains from the same positive basis, because its net price received is built the same way.

Basis risk arrives through two doors. One is timing: the hedge is unwound before delivery, so the futures price has not yet been pulled into line with the spot price. The other is asset mismatch, where the hedged asset is not the one the contract is written on and the two prices need never converge.

A currency hedge closed out before maturity

A U.S. company plans to buy 250,000 British pounds (GBP) in February and hedges with the CME Group’s March futures contract. Each contract covers GBP 62,500, so four contracts on the long side are required. The futures price when the hedge is initiated is 1.25 USD per GBP, the futures price when the position is closed in February is 1.30 USD per GBP, and the spot price in February is 1.31 USD per GBP. Counting the gain from hedging, the cost in USD of the pounds is

Cost of the currency purchase after the futures gain is applied.

That works out at USD 1.26 (= 315,000/250,000) per GBP. Read that in two pieces. The exchange rate rose, so hedging improved the price paid by 0.05 USD per GBP against the spot rate. But the basis is USD 0.01 (= 1.31 – 1.30), so the company cashes out its futures at 1.30 while paying 1.31 for the pounds. With no basis it would have paid USD 1.25 (= 1.30 – 0.05); the basis pushes that to USD 1.26 (= 1.31 – 0.05).

A commodity hedge with a grade mismatch

A grower will sell 50,000 bushels of corn during June, hedged with the July futures contract. Each contract is on 5,000 bushels, so ten contracts are shorted. The futures price at initiation is 300 cents per bushel and stands at 320 cents per bushel when the hedge is closed in June. The corn actually being sold, possibly a different variety from the one the contract is written on, fetches a spot price of 325 cents per bushel. Taking the hedging loss into account, the USD proceeds are

Proceeds from the corn sale net of the loss on the short futures position.

Dividing 152,500 by 50,000 leaves a net price of 305 cents per bushel. Corn rose, so the grower gives up 20 (= 320 – 300) cents per bushel on the futures. The basis of 5 (= 325 – 320) cents works the other way, because the corn itself sells at 325 cents. Netting the two leaves 305 cents per bushel, above the 300 cents at which the hedge started.

Check yourself

Twenty futures contracts, each on 5,000 bushels, hedge an exposure to soybeans. When the hedge is closed the basis is 20 cents per bushel. What does that basis do to the hedger?

Twenty contracts of 5,000 bushels at USD 0.20 a bushel move the net price by USD 20,000. If the hedge covers a purchase of soybeans, that USD 20,000 is an extra cost. If the hedge covers a sale of soybeans, it is an extra USD 20,000 received.

Every hedge so far matched the futures position to the exposure one for one, as when the grower shorted contracts covering exactly 50,000 bushels of corn. That is only defensible when the contract is written on the asset being hedged. Once the two differ, the right size must be worked out.

The hedge ratio compares the size of the futures position with the size of the position in the underlying asset. Hedging exposure to one asset with a futures position in a different asset is called cross hedging.

Write the change in the spot price over an interval matching the life of the hedge as the change in S, and the change in the futures price over that same interval as the change in F. Linear regression on historical data gives the best fit relationship.

Spot price changes regressed on futures price changes, with a and b constants and epsilon the error term.

With hedge ratio h, the change in value per unit of the asset hedged is the spot change less h times the futures change.

The residual exposure of the hedged position, one unit at a time.

Only the middle term depends on the futures price. Setting h equal to b kills that term, which is exactly what minimises the variance left on the right hand side. So the minimum variance hedge ratio is the regression slope, and that slope decomposes: the covariance of the two changes divided by the variance of the futures change equals the correlation times the ratio of the standard deviations.

Equation 8.1. Rho is the coefficient of correlation between the spot change and the futures change; the sigmas are their standard deviations.

Two sanity checks follow. Under perfect correlation with equal standard deviations, the ratio is 1 and the hedge is one for one. Under perfect correlation with the spot standard deviation 20% higher, the ratio is 1.2, which is what a hedger wants when spot price moves always run 20% above futures price moves. A correlation of zero drives the ratio to zero, telling the hedger not to trade.

Figure 3: Variance of the hedged position as a function of the hedge ratio

Both an undersized and an oversized futures position leave more variance than the optimum.

Check yourself

Why does setting the hedge ratio equal to the regression slope minimise variance rather than eliminate it?

Choosing h equal to b removes the term in the futures price change, but the regression error term survives untouched. That error is the part of the spot price movement the futures price does not explain, and no size of futures position can offset it.

A hedge ratio is a proportion, not an order ticket. Converting it into contracts needs two more quantities: Q_A, the size of the position being hedged in units, and Q_F, the units of the asset underlying one futures contract. N* is the optimal number of futures contracts.

Equation 8.2. The hedged quantity, scaled by the hedge ratio, divided by the contract size.

Airlines supply the standard case. Jet fuel futures exist but trade far less actively than heating oil futures, so airlines routinely hedge jet fuel with heating oil contracts and accept the mismatch.

Example 3 · Worked

An airline puts the correlation between monthly moves in the heating oil futures price and monthly moves in the jet fuel price at 0.9. Monthly changes in the heating oil futures price carry a standard deviation of USD 0.03 per gallon, against USD 0.025 per gallon for jet fuel. One heating oil futures contract is on 42,000 gallons.

1. Calculate the optimal hedge ratio.

Solution. Equation 8.1 gives h* = 0.9 x 0.025/0.03 = 0.75, an optimal hedge ratio of 75%. Jet fuel is the less volatile of the two series, which pulls the ratio below one, and the imperfect correlation of 0.9 pulls it down further.

2. The airline needs to cover a purchase of 1 million gallons of jet fuel one month out. How many contracts should it take?

Solution. Equation 8.2 gives N* = 0.75 x 1,000,000/42,000 = 17.86, and that becomes 18 contracts after rounding. They must be long contracts, since the airline buys fuel and needs a position that gains when energy prices rise.

3. How effective is this hedge expected to be?

Solution. Effectiveness is the square of the correlation, 0.9 x 0.9 = 0.81. Around 81% of the variance of the jet fuel price change should disappear, and roughly 19% survives as basis risk from the difference between the two fuels.

Check yourself

Using the optimal hedge ratio of 0.72 from the previous section, with 200,000 units of the commodity being hedged and one futures contract covering 5,000 units, how many contracts are needed?

Applying Equation 8.2 gives 0.72 x 200,000/5,000 = 28.8, and rounding to the nearest whole number leaves 29 contracts.

Everything above is exactly right for a hedge built out of forward contracts. Futures settle daily, so the hedger is really running a sequence of one-day hedges, and a small correction called tailing the hedge follows. The name reflects the idea that the position should in theory be trimmed as time passes.

For any single day the hedge ratio still comes from Equation 8.1, with the correlation and the standard deviations now measured on one-day changes. Analysts prefer one-day returns, that is percentage changes, because those tend to be more stable than price changes. So take the standard deviation of each one-day spot return and each one-day futures return, plus the correlation between them. Write V_A for the value of the hedged position, the spot price multiplied by Q_A, and V_F for the value of one futures contract, the futures price multiplied by Q_F.

Equation 8.3. The hedge ratio built from daily returns rather than from absolute price changes.

Equation 8.4. The optimal number of contracts in value terms.

Keep the two versions apart. In Equation 8.1 the standard deviations describe price changes over the whole life of the hedge; in Equation 8.3 they describe daily returns, and the correlation shifts with them.

In principle the position needs constant maintenance, since V_A and V_F drift as prices move and the right number of contracts drifts with them. A discount factor also belongs in each day’s ratio, because a gain settled today is worth more than the same gain settled at maturity. In practice V_A divided by V_F is frozen at its opening value and the discounting is left out.

Work through a case. Daily returns on the spot price have a standard deviation of 1.2% and daily returns on the futures price have a standard deviation of 1%, with a correlation of 0.88 running between them. The hedged position is worth 1 million USD and one futures contract is worth USD 20,000, so V_A = 1,000,000 and V_F = 20,000. Equation 8.3 gives a hedge ratio of 0.88 x 0.012/0.01 = 1.056, and Equation 8.4 turns that into 1.056 x 1,000,000/20,000 = 52.8, which rounds up to 53 whole contracts.

Check yourself

Why is a futures hedge described as a series of one-day hedges, and would the same description fit a hedge built with forwards?

Because futures are settled daily, so gains and losses are realised in cash every day rather than accumulating to a single settlement. A forward contract has no daily settlement, so the whole gain or loss lands once at maturity and the tailing adjustment does not arise.

Stock index futures let an investor dial equity market exposure up or down without disturbing the holdings and without the cost of selling and repurchasing a portfolio. Someone expecting a turbulent three months can shed exposure temporarily; someone bullish can add to it. The most interesting user is the stock picker with no view on the market, since hedging away the market return leaves only the excess return of the chosen stocks.

Suppose an investor holds a well diversified portfolio of USD 1 million that closely tracks the S&P 500 and wants no market exposure for the next two months. The mini CME futures contract on the S&P 500, written on USD 50 multiplied by the index, is the natural instrument. A regular contract covers 250 times the index; the mini gets chosen because it trades more actively and lets a position be sized more finely.

For a portfolio tracking the index, the daily return standard deviations for the portfolio and for the futures price can be taken as equal, and their correlation as 1.0, so Equations 8.3 and 8.4 collapse to the ratio of the two values. One contract controls 125,000 (= 50 x 2,500) of index value when the futures price is USD 2,500, so a portfolio of 1,000,000 calls for 8 contracts short.

A portfolio that does not track the index needs one more ingredient. Beta scales the portfolio’s daily return standard deviation against that of the futures, and the correlation between the two series stays close to 1.0.

Number of index futures contracts to short in order to remove market exposure from a portfolio of value V sub A.

Example 4 · Worked

A portfolio is worth USD 1 million and has a beta of 1.25. The index futures price for a six-month contract is USD 2,500 and the contract is on USD 50 times the index. The risk-free rate runs at 4% per year, and the index pays a dividend yield of 2% per year.

1. How many contracts are required, and what is the spot level of the index?

Solution. With beta at 1.25 the count becomes 1.25 multiplied by 8, which is 10 contracts shorted. As for the index level, the futures price of an index equals its spot price compounded forward at the amount by which the risk-free rate exceeds the dividend yield, here 2% (= 4% – 2%) per year, so across six months the futures price sits roughly 1% (= 0.5 x 2%) above the spot price. Working back from a futures price of USD 2,500, the spot index sits at about USD 2,475, since 2,500 divided by 1.01 gives that figure.

2. The index stands at 2,300 after six months. Value the hedged position.

Solution. The index has dropped 175 points from 2,475, a capital loss of -7.07%, and a dividend gain of 1% arrives on top, being half the annual dividend yield, so the total index return is -6.07%. The capital asset pricing model, using a six-month risk-free rate of half the annual figure, gives an expected portfolio return of 0.02 plus 1.25 multiplied by (-0.0607 – 0.02), that is -0.0809 or -8.09%. Applying 1 – 0.0809 to USD 1,000,000 leaves USD 919,100. Meanwhile each of the 10 contracts gains 50 for every one of the 200 index points, so the futures leg adds 100,000 and the total reaches USD 1,019,100.

3. What does the pattern across scenarios show?

Solution. The hedged total lands close to 1,020,000 wherever the index ends up, and 1,020,000 is what USD 1 million placed at the risk-free rate is worth after six months. Removing market risk leaves the risk-free return.

Hedged outcomes across five index scenarios, portfolio beta 1.25

Index in six months (level)	Capital gain	Dividend	Index return	Portfolio return	Portfolio (USD)	Futures gain (USD)	Final value (USD)	Gap from 1,020,000
2,100	-15.15%	1.00%	-14.15%	-18.19%	818,100	200,000	1,018,100	-1,900
2,300	-7.07%	1.00%	-6.07%	-8.09%	919,100	100,000	1,019,100	-900
2,500	1.01%	1.00%	2.01%	2.01%	1,020,100	0	1,020,100	100
2,700	9.09%	1.00%	10.09%	12.11%	1,121,100	-100,000	1,021,100	1,100
2,900	17.17%	1.00%	18.17%	22.21%	1,222,100	-200,000	1,022,100	2,100

Source: chapter figures, transposed so each row is one scenario, with a final column of our own. Daily settlement is ignored.

Check yourself

An investor good at stock selection but with no market view holds a portfolio and shorts index futures against it. What return is left?

The excess return of the selected stocks over the market, plus the risk-free return on the hedged position. The futures leg strips out whatever the market does, so the outcome depends on stock picking rather than market direction.

Beta measures the sensitivity of a portfolio’s return to the return on the market. A portfolio with a beta of 1.0 mirrors the market, one with a beta of 0.5 moves half as far, and one with a beta of 2.0 moves twice as far. The capital asset pricing model links beta to the expected return.

Equation 8.5. On the right are the risk-free rate, the return on the market portfolio and the beta of the portfolio.

Beta magnifies whatever the market earns above the risk-free rate, so a portfolio with a beta of 1.5 picks up a 50% larger excess return than the market itself. Take a risk-free rate of 3% and a market return of 7%. The market earns 4% above the risk-free rate; a beta of 1.5 turns that into an expected 6% above the risk-free rate for the portfolio, giving 9% in total. Let the market instead return -3%. Measured against the same 3%, the shortfall is -6%, beta of 1.5 magnifies it to -9% relative to the risk-free rate, and adding the 3% back leaves an expected portfolio return of -6%.

Hedging out all market risk drives beta to zero, but the same instrument can move beta to any chosen level. Where the current beta exceeds the target, contracts are shorted.

Reducing beta to a lower target.

Return to the USD 1 million portfolio with a beta of 1.25, hedged with contracts on 50 times an index priced at 2,500. Cutting beta from 1.25 to 0.5 strips out 0.75 of beta, and since one contract carries 125,000 of index value the portfolio needs 0.75 multiplied by 8, which is 6 contracts short. Where the target beta sits above the current beta, the position reverses.

Raising beta to a higher target.

Six contracts bought rather than sold lift beta from 1.25 to 2.0, since the required increase of 0.75 applied to the same eight-contract base again gives six. The same six contracts cut beta to 0.5 when shorted and raise it to 2.0 when bought.

Check yourself

Beta is 1.5 on a stock portfolio worth 1 million dollars. Each contract on the index future covers 50 times an index quoted at 3,000. How many contracts hedge the market risk, and how would beta instead be moved to 0.9 or to 1.8?

One contract carries 150,000 of index value, so the portfolio spans about 6.67 contracts. A full hedge multiplies that by 1.5 and shorts 10 contracts. Moving beta down to 0.9 removes 0.6 of beta and calls for 4 contracts short, while moving it up to 1.8 adds 0.3 of beta and calls for 2 contracts long.

Liquidity concentrates in the nearby maturities, so a hedger needing cover several years out often finds nothing tradable at that horizon. The workaround is a stack and roll strategy: put on a short-maturity hedge, close it just before its delivery period, replace it with another, and repeat for as long as the exposure lasts. Companies usually face exposure in every month rather than one, so enough contracts are entered to cover each month and the stack is rolled forward together.

Example 5 · Worked

It is January of Year 1 and a company knows it must sell 5,000 ounces of gold in 18 months. Only contracts maturing in seven months or fewer carry sufficient liquidity. CME Group gold futures mature in February, April, June, August, October and December, and one contract is on 100 ounces, so 50 contracts are needed. The company sells 50 August of Year 1 futures at USD 1,500 and closes them at USD 1,450 in July of Year 1, replacing them with 50 February of Year 2 futures sold for USD 1,470. Those come off at USD 1,410 in January of Year 2, giving way to 50 August of Year 2 futures sold for USD 1,430 and unwound in July of Year 2 at USD 1,440, with the spot price then at USD 1,435.

1. Calculate the total gain on the short futures positions and the price realised for the gold.

Solution. Leg by leg the company gains 50 per ounce, then 60 per ounce, then loses 10 per ounce, which on 100 ounces a contract comes to 5,000, plus 6,000, minus 1,000, or 10,000 for each contract. Across 50 contracts the total gain is USD 500,000. Dividing 10,000 by 100 ounces gives USD 100 per ounce, so the gold realises USD 1,535 per ounce once that is added to the USD 1,435 spot price. The net outcome is close to what a single August of Year 2 contract would have delivered had one been available.

2. What risk did the company carry that a single long-dated contract would have avoided?

Solution. Roll risk. Each time an old contract is closed and a new one opened, the difference between the two futures prices is unknown in advance, and this schedule contains two such junctures. The gold price is covered throughout; the shape of the futures curve on each roll date is not.

Figure 4: Rolling a stack of short-dated contracts across an 18-month exposure

Three short-dated hedges laid end to end cover an exposure no single liquid contract could reach. The vertical lines mark the roll dates.

Cash flow risk in a rolled hedge

Daily settlement creates a timing mismatch between the cash flows on the futures contracts and those on the exposure being hedged, and it bites hardest on long-dated hedges. A company running one should satisfy itself that it can fund the futures losses until the matching gains arrive.

The German company Metallgesellschaft learned this in the early 1990s. It had sold a huge volume of 5-to-10-year fixed-price heating oil and gasoline contracts to customers and hedged by rolling long positions in short-maturity futures. Heating oil and gasoline prices then fell, so the fixed-price contracts were expected to become valuable in time. The futures losses arrived immediately, and the cash outflows could not be financed. The contracts were abandoned at a cost of over USD 1 billion.

Check yourself

Metallgesellschaft’s hedge was expected to be profitable in the end. How did a hedge with a sound economic case still destroy over USD 1 billion?

The gains sat in customer contracts running out five to ten years, while the futures losses settled in cash every day. Being right about the eventual outcome does not help a company that cannot fund the interim margin calls.

What a short hedge is and how to calculate its profit and loss

When a short futures position is the right hedge

What a long hedge is and when to use one instead of buying spot

Why not simply buy the asset today

Arguments for and against hedging a corporate exposure

Shareholders may prefer no hedging

When hedging increases risk: overstated exposure and hedging losses

There may be little or no exposure

Hedging may lose money

How to calculate the basis and choose a futures contract maturity

How the basis determines the net price achieved by a hedge

Sources of basis risk in a currency hedge and a commodity hedge

A currency hedge closed out before maturity

A commodity hedge with a grade mismatch

What cross hedging is and how the minimum variance hedge ratio is derived

Hedge effectiveness and how the hedge ratio is estimated from historical data

How to calculate the optimal number of futures contracts to hedge an exposure

Tailing the hedge: adjusting the hedge ratio for daily settlement

How to hedge a stock portfolio with stock index futures

How to use stock index futures to change a stock portfolio's beta

Creating a long-term hedge with a stack-and-roll strategy and its risks

Cash flow risk in a rolled hedge