22.3Hedging Short-Term Commitments with Maturity-Matched

Futures Contracts

This section investigates how to hedge obligations that generate risk exposure with

futures contracts that mature on the same date as the obligation. As we will see, there

is an important difference between hedging with futures and hedging with forwards.

Review of Futures Contracts, Marking to Market, and Futures Prices

In contrast to forward contracts, which can be tailored to the individual needs of the

corporation, futures contracts are standardized. For example, the contracts on Globex2,

an electronic currency futures market affiliated with the Chicago Mercantile Exchange,

is limited to standard lot sizes, which differ between currencies, have standard matu-

rity dates (quarterly), and involve only a selected number of major currencies.⁶

Recall that the essential distinction between a forward and a futures contract lies

in the timing of their cash flows. With a futures contract, profit (or loss) is received

(paid) on a daily basis, instead of being paid in one large sum at the maturity date as

is the case with a forward contract.

Because each party to a futures contract keeps a small amount of cash on deposit

(that is, margin) with a broker to cover potential losses, brokers automatically execute

the daily cash transfer, requiring only occasional notification to the two parties when

margin funds are running low. If the futures price increases from the previous day’s

price, cash is taken from the accounts of investors who have short positions in the con-

tract and placed in the accounts of those with long positions in the contract. If the

futures price goes down, the reverse happens.

This procedure, known as marking to market(see Chapter 7), has a negligible effect

on the fair market price of the futures relative to the forwards (with the notable excep-

tion of long-term interest rate contracts). This means that forwards and futures con-

tracts can be treated the same, for the most part, for valuation purposes. Despite this

valuation similarity, the next subsection points out that futures and forwards cannot be

treated as if they are the same for hedging purposes.

Tailing the Futures Hedge

It is easy to become confused about how to hedge with futures because, as we shall

see, futures hedges require tailing(defined shortly). The futures position in a tailed

futures hedge is smaller than it is in a hedge that uses forward contracts because it

needs to account for the interest earned on the marked-to-market cash. The proper way

to perform tailing on a hedge is a source of confusion for many practitioners, and it

has caused grief for a number of corporations. Therefore, we need to go through the

logic of futures hedge tailing carefully.

No Arbitrage Futures and Forward Prices.Assume that gold trades at $400 an

ounce. To compute the futures price for gold, recall from Chapter 7 that for an invest-

ment that pays no dividends, the no-arbitrage T-year forward price—and, because their

values are approximately the same, the T-year futures prices—is given by the equation

FS(1 r)^T

00f

⁶See www.cme.comfor a description of these standard features.

Grinblatt1579Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1579Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

784Part VIRisk Management

where

Ffutures price

0

S today’s spot price of the underlying investment

0

rannually compounded yield on a T-year zero coupon bond

f

The futures price for gold delivered one year from now, with a risk-free interest rate of

10 percent per year, would then be $440 [$400(1.1)] per ounce of gold.

Creating a Perfect Futures Hedge.Suppose you own an ounce of gold that you wish

to sell in one year. To fix the selling price today by selling futures contracts, it is nec-

essary to tailyour hedge. That is, you should sell less than one ounce in futures for

each ounce that you plan to sell in one year.

Why is selling a futures contract on one ounce of gold overhedging in this case?

Well, picture what would happen if the spot price of gold instantly changed today from

$400 per ounce to $401 per ounce. According to the latest equation, the gold futures

price would then change from $440 to $441.10 per ounce. Hence, as line aof Exhibit

22.3 illustrates, selling one futures contract to hedge the change in the price of gold

would overhedge the gold price risk. As the gold price jumps from $400 to $401 per

ounce, we gain $1 from holding one ounce of gold, but lose $1.10 from having sold a

futures contract on one ounce of gold.

Selling less than one futures contract remedies this overhedging problem. Specifi-

cally, for a sale of 11.1 futures contracts, the loss on the futures contracts associated

with the $1 gold price increase would be (11.1) $1.10 or $1.00, which would exactly

offset the $1.00 gain from holding one ounce of gold. This is shown in line bof Exhibit

22.3. The practice of selling less than one financial contract to hedge one unit of the

spot asset is known as tailing the hedge.

Contrasting the Futures Hedge with the Forward Hedge.In our gold example,

the no-arbitrage forward price, like the future price, is initially $440. This makes the

EXHIBIT22.3Hedging a Decline in the Price of Gold with Futures and Forwards

	(1)	(2)	(3)	(4)
	Position Value at	Position Value if
	Initial Gold Price of	Gold Price Rises to	Mark-
	$400oz. (⇒Zero PV	$401oz (⇒Zero PV	to-	Gain from
	forward and futures	forward and futures	Market	Position
Position	price$440)	price$441.10)	Cash	(2)(3) (1)

Hold 1 oz gold	$400	$401	$0	$1
Sell 1 futures contract	0	0	1.1	1.1
Sell 1/1.1 futures contracts	0	0	1	1
Sell 1 forward contract	0	1 440/1.1 401	0	1

a.Hold 1 oz of gold and
sell 1 futures contract	400	401	1.1	.1

b.Hold 1 oz of gold and
sell 1/1.1 futures contracts	400	401	10

c.Hold 1 oz of gold and
sell 1 forward contract	400	400 401 10	0

Grinblatt1581Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1581Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

785

forward contract, like the futures contract, a zero-PVinvestment. It seems curious

that the minimum-risk hedge with the forward contract, where the hedge ratio is one-

to-one, should alwaysdiffer from the hedge ratio with the futures contract. Note,

however, that the forward contract, in contrast with the futures contract, need not

have a zero present value after the contract terms are set. This difference explains

why futures hedges require tailing, but (maturity-matched) forward hedges do not.

Consider what happens to the present values of the two sides of the forward

contract when the price of gold instantly jumps from $400 to $401 on the first day of

the contract. The present value of the forward contract’s risk-free payment of $440 at

a 10 percent discount rate remains the same (that is, $400), but this payment is

exchanged for gold that has a present value of $401 after the $1 price increase. Thus,

the forward contract’s present value jumps from zero to $1. In other words, instanta-

neous changes in the price of gold do not affect the present value of the cash payout

of the forward contract, but they do affect the present value of the gold received and,

hence, the forward contract’s value, on a one-for-one basis.

In other words, if the price of gold increases from $400 to $401 per ounce, the for-

ward contract, formerly a zero-PVinvestment, becomes an investment with a positive

PVof $1. As line cof Exhibit 22.3 illustrates, immediately after the increase the clos-

ing out of one short positionin a forward contract, which loses $1 in value, exactly

offsets the $1 gain from holding one ounce of gold.

Result 22.1 summarizes the distinction between hedging with futures and hedging

with forwards.

Result	22.1	Futures hedges must be tailed to account for the interest earned on the cash that is exchanged
		as a consequence of the futures mark-to-market feature. Such tailed hedges require holding
		less of the futures contract the further one is from the maturity date of the contract. The
		magnitude of the tail relative to an otherwise identical forward contract hedge depends on
		the amount of interest earned (on a dollar paid at the date of the hedge) to the maturity date
		of the futures contract.