22.5Hedging Long-Dated Commitments with Short-Maturing FuturesorForward Contracts

Afundamental hedging problem faced by many corporations arises because most finan-

cial instruments, particularly futures, have relatively short maturities. Corporate commit-

ments, in contrast, are often long term. Even when longer maturities exist for the desired

hedging instruments, it is difficult to use them for sizable hedging tasks because very

little trading takes place in the longer maturing financial instruments. The sheer market

impact of a hedge would make hedging prohibitively costly. Corporations with long-dated

obligations thus tend to hedge them by using the shorter maturing futures or forwards.

As we show below, for the special case where convenience yields are constant, a

perfect hedge can be implemented with short-term futures and forward contracts. This

perfect hedging can be extended to the case where changes in the convenience yield

are perfectly correlated with changes in the price of the commodity. However, because

this assumption of perfect correlation is unrealistic, it is generally not possible to per-

fectly hedge a long-term obligation by rolling over a series of shorter term forward or

futures contracts.

Grinblatt1589Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1589Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

789

Maturity, Risk, and Hedging in the Presence of a Constant Convenience Yield

To understand the hedging of long-dated commitments with shorter-maturing futures

and forwards, it is necessary to link the risk of each commitment and hedging instru-

ment to the risk of holding the commodity. For example, assuming that oil has a

convenience yield of 2 percent per year, the forward commitment to buy a barrel of oil

10years from nowis equivalent in risk to a position in 11.02¹⁰barrels of oil pur-

chased today. Similarly, a commitment to buy oil one year from nowis equivalent in

risk to a position in 11.02 barrels of oil purchased today. Hence, to perfectly hedge

the obligation to buy a barrel of oil 10 years from now by selling a forward contract

to purchase oil one year from now, sell

111.02¹⁰

.837

1.02⁹

11.02

forward contracts maturing one year from now.

Rollovers at the Forward Maturity Date.As each day passes, it is unnecessary to

alter the position in the one-year forward contract. For example, one-half year from

now, the obligation to buy oil long term would be 9.5 years away, while the short-term

forward would be 0.5 years from maturity. Hence, the proper number of short-term for-

ward contracts being sold still would be

111.02^9.5

1.02⁹11.02^.5

However, as the year elapses and the short-term forward contract matures, it is

important to roll over the old contract and enter into a new one-year forward con-

tract. For this new contract, the obligation would be nine years out. At this point,

selling

111.02⁹

.853

1.02⁸11.02

of the new one-year forward contracts perfectly hedges the risk of the (now) nine-

year-out obligation. Altering the number of offsetting forward contracts at the maturity

date of the short-term hedging instrument is a form of tailing the hedge, similar to the

tailing observed earlier with long-dated futures contracts.

Exhibit 22.5 illustrates the use of this tailed rollover strategy for hedging. The

exhibit indicates that at each annual rollover date of the series of short-term forward

contracts, the number of forward contracts sold that would perfectly hedge the obliga-

tion increases until in the final year—because of the matched maturity between the

commitment and the hedging instrument—the perfect hedge involves selling exactly

one forward contract.

Futures Hedges.Earlier, we noted that a one-year futures contract is more risky than

a one-year forward contract because of the mark-to-market feature of the futures. With

a 10 percent risk-free rate, each one-year futures contract is equivalent in risk to 11.1

one-year forward contracts. Hence, for the illustration above, a perfect futures hedge

would involve selling

1

.761

9

1.02(1.1)

Grinblatt1591Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1591Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

790	Part VIRisk Management

EXHIBIT22.5

Hedging a Long-Term Obligation with Short-Maturing ForwardsRolled Overat Maturity (Positive Convenience Yield)

Hedge

ratio

1

				Time
1st	2d	3d	Obligation
maturity	maturity	maturity	date
(rollover)	(rollover)	(rollover)
date	date	date

one-year futures contracts at the outset. However, as each day elapses in that first year,

it is necessary to tail the futures hedge because it is getting closer to the forward con-

tract in terms of its risk. For example, one-half year from now, the obligation to buy

oil long term would be 9.5 years away, implying that the number of short-term futures

contracts being sold would have to equal

111.02^9.5

1.1^.5

.798

9^.5)11.02^.5

1.02(1.1

Thus, as Result 22.2 noted, the hedge ratio for a hedge involving a futures contract is

always changing, both when there is and when there is not a convenience yield to hold-

ing the underlying commodity. Exhibit 22.6 illustrates the process of hedging with the

futures rollover strategy. Note the difference between the hedge ratios from this exhibit

and those in Exhibit 22.5; in particular, note the difference in the way the hedge ratios

evolve between rollover dates.

Example 22.7 applies these insights to a hypothetical hedging problem that is sim-

ilar to the problem faced by Metallgesellschaft.

Example 22.7:Hedging Oil Price Risk with Short-Dated Futures and Forwards

Assume that Metallgesellschaft has an obligation to deliver 1.25 million barrels of oil one

year from now at a fixed price of $25 per barrel.

a.	How can it hedge this obligation in the forward or futures markets, using forwardsand futures maturing one month from now and then rolling over into new one-monthforwards and futures as these mature? Assume that the convenience yield is

Grinblatt1593Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1593Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

791

EXHIBIT22.6

Hedging a Long-Term Obligation with Short-Maturing Futures Rolled Overat Maturity (Positive Convenience Yield)

Hedge

ratio

1

				Time
1st	2d	3d	Obligation
maturity	maturity	maturity	date
(rollover)	(rollover)	(rollover)
date	date	date

Forward contract hedge ratio

Futures contract hedge ratio

5percent per year and the risk-free interest rate is 10 percent per year, both

compounded annually.

b.	How would your answer change if you want to hedge against the change in valuefrom owning 1.25 million barrels of oil?

Answer:

a.	The sum of 1 plus the one-month convenience yield per dollar invested is 1.051/12. Hence, the forward hedge would buy 1.25 million/1.0511/12barrels of oil one-month forward.In the futures market, one would buy futures to acquire 1.25million/[(1.0511/12)(1.11/12 )] barrels of oil.

b.

The convenience yield makes the risk from holding 1.25 million barrels of oil greaterthan the risk associated with the present value of the obligation to receive 1.25million barrels in one year.The risk-minimizing hedge would be to sell 1.051/121.25 million barrels of oil for one-month forward delivery.A futures position to sell (1.05/1.1)1/12 1.25 million barrels of oil also eliminates the oil price risk.

Quantitative Estimates of the Oil Futures Stack Hedge Error

Mello and Parsons (1995) constructed a simulation model for Metallgesellschaft’s hedg-

ing problem. Their model assumes that Metallgesellschaft has an obligation to deliver

1.25 million barrels of oil at a fixed price on a monthly basis for the next 10 years.

Grinblatt1595Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1595Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

792Part VIRisk Management

This amounts to delivery obligations of 150 million barrels of oil at a fixed price. Mello

and Parsons reported that Metallgesellschaft used what is known as a rolling stack of

short-term futures contracts to undertake this hedge. With a rolling stack, the obliga-

tion to sell (buy) a commodity is offset with a series of short-term futures or forward

contracts to buy (sell) the same amount of the commodity. Metallgesellschaft’s

commitment to sell 150 million barrels of oil at a preset price is hedged (albeit imper-

fectly) with a rolling stack that takes a position in short-term futures contracts to buy

150 million barrels of oil. As the futures contracts expire, they are replaced with futures

contracts in amounts that maintain a one-to-one relation between the futures contracts

to buy oil and the forward commitment to sell oil. Mello and Parsons assumed that the

convenience yield of oil is 0.565 percent per month and the risk-free rate is 0.565 per-

cent per month. Based on the analysis above, the obligation to sell 1.25 million barrels

of oil at month tcould be perfectly hedged by purchasing one-month futures contracts

that are rolled over. If the convenience yield is constant, the number of futures con-

tracts should be

1.25 million1.25 million

1.00565^{tt 1})(1.00565)

(1.00565

For the obligation at each of the 120 months, the annuity formula gives a futures posi-

tion (in barrels of oil) of

1.25 million1.25 million1.25 million1.25 million

108.7 million. . .. . .

1.005651.00565²1.00565^t120

1.00565

Hence, a rolling stack of 150 million barrels in futures positions overhedges the risky

obligation of Metallgesellschaft by a considerable degree.