22.4Hedging and Convenience Yields

The forward price of gold is generally close to its spot price times one plus the risk-

free return to the forward maturity date. In this respect, gold is very similar to a stock

that pays no dividend.⁷Since the entire return from holding a stock that pays no div-

idend comes from capital appreciation, the present value of receiving a non-dividend-

paying stock in the future must be the current price of the stock. As Chapter 8 noted,

this is not true for stocks that pay a dividend. The discounted value of the forward price

of a dividend-paying stock is less than its current price by an amount equal to the pres-

ent value of the dividends that will be paid between the current date and the maturity

date of the forward contract.

From a valuation perspective, most commodities are like dividend-paying stock

because there is a benefit to owning them aside from their potential for price appreci-

ation. The direct benefit from owning such commodities is called a convenience yield.

The convenience yieldis the benefit from holding an inventory of the commodity net

⁷There is some dispute about this among academics and practitioners. Central banks are willing to

pay money to lease gold. This may imply that gold is more like a stock that pays dividends. Other

researchers have argued that this lease value is tied to default risk.

Grinblatt1583Titman: Financial	VI. Risk Management	22. The Practice of Hedging	© The McGraw1583Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

786Part VIRisk Management

of its direct storage costs that arises because it is more convenient to have the inven-

tory on hand than to have to purchase the commodity every time it is needed. For

commodities with convenience yields, the number of the futures or forward contracts

used to hedge the commodity would vary depending on the date of the future obligation.

When Convenience Yields Do Not Affect Hedge Ratios

Convenience yields do not affect forward or futures hedge ratios when the maturity of

the future obligation one is trying to hedge matches the maturity of the futures or for-

ward contract used as the hedging instrument. This point is an obvious one with for-

ward hedges. Example 22.3, for instance, shows that a forward obligation is offset

exactly with an opposite position in a maturity-matched forward contract with the same

terms as the forward obligation. Example 22.3 is based on oil, a commodity that has

long been known to have a convenience yield; Example 22.6 illustrates this point by

showing how to hedge a forward obligation to deliver with a maturity-matched futures

contract for oil.

Example 22.6:Hedging Oil Price Risk with a Futures Contract

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

year out at a fixed price of $25 per barrel.How can it hedge this obligation in the futures

market if the risk-free rate is 10 percent per year?

Answer:Since forward contracts to buy 1.25 million barrels hedged this same obligation

in Example 22.3, tailed positions in futures contracts to buy 1.25/1.1 million barrels also

would hedge this obligation.The number of futures contracts would increase every day to

reflect the shortening maturity of the contract.

Whenever the obligation is a forward contract and risk therefore is eliminated with

an offsetting opposite forward contract, the futures contract can generate the same per-

fect hedge provided that it is tailed for interest earned on the marked-to-market cash,

just as in the gold illustration in Exhibit 22.3. When there is a mismatch in the matu-

rity of the hedging instrument and the obligation to be hedged, the convenience yield

affects the hedge ratio whether the hedge is executed with forward contracts or futures

contracts. Before analyzing this issue, it is important first to understand what deter-

mines convenience yields.

How Supply and Demand for Convenience Determine Convenience Yields

Consider the gasoline that is used to fill up the tank in your automobile. When you go

to the gas station, you fill up your tank instead of pumping a single gallon of gas into

the tank because it is convenient not to stop at a gas station every 25 miles or risk run-

ning out of gas. There is a small cost to this convenience: The gasoline in the tank, on

average, is not appreciating in value, whereas the money used to pay for the gas might

have earned interest (or you might have owed less interest on a credit card balance) if

it had been in the bank instead of in the tank.

It also would be convenient to have even greater inventories of gasoline. You would

rarely have to stop at a gas station if you could dig holes in your backyard and keep

gasoline storage tanks there, have extra storage tanks in your car, or have a gasoline

tanker truck follow you wherever you drive. You don’t do this because it would be pro-

hibitively costly.

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Strategy, Second Edition

Chapter 22

The Practice of Hedging

787

If storage of large amounts of gasoline were free, you would probably choose to

store the gasoline. Note that freemeans not only free of the direct costs of storage, but

also free in the sense that interest earned from holding gasoline (in terms of its expected

price appreciation, adjusted for systematic risk) would be comparable to that earned

from cash deposited in the bank. Of course, this situation is not possible even if the

direct costs of storage were zero. If everyone stored gasoline to an unlimited degree,

the price of gasoline would be bid upward, making its return smaller than the interest

on cash deposited in the bank.

In essence, the demand for convenience and the supply of convenience, which

depend on the cost of supplying convenience, determine the inventory of any com-

modity. For supply to equal demand, the difference between the expected price

appreciation of the commodity and that of any other investment of identical risk

must be the difference in their net convenience yields (the latter being the value of

convenience less direct storage costs as a proportion of the commodity’s price). For

simplicity of exposition, we will refer to the net convenience yield as the conve-

nience yield.

Hedging the Risk from Holding Spot Positions in Commodities with Convenience Yields

Convenience yields tend to reduce the ratio of forward prices to spot prices. Consider,

for example, the forward prices of crude oil. Refineries with oil inventories earn a con-

venience yield (that exceeds the cost of storage) because they avoid the risk of having

to shut down the refinery if supplies are interrupted. Assume that crude oil has a con-

venience yield of 2 percent per year. If oil is currently selling at $25 a barrel and the

risk-free rate is 10 percent per year, then the no-arbitrage futures and forward price for

oil delivered one year from now is

$25(1.1)

$26.96 per barrel

1.02

More generally, if the commodity’s convenience yield to the forward commitment’s

maturity date is y,and Fis the no-arbitrage forward price that would apply in the

0

absence of a convenience yield, the forward price for the commodity should be

F

0

1y

The division of Fby one plus the convenience yieldaffects hedge ratios when

0

there is a mismatch between the maturity of the futures and the date of the position

one is trying to hedge. For example, when trying to use forwards or futures to perfectly

hedge the oil price risk from holding an inventory of oil, the convenience yield would

affect the hedge ratio used. As Exhibit 22.4 illustrates, an increase in the current price

of oil from $25 per barrel in column (1) to $30 per barrel in column (2)—which results

in a zero-PVfutures and forward price of $32.35—would be offset by a short position

in 1.021.1 futures contracts (row a) or 1.02 forward contracts (row b).

Hence, the 2 percent convenience yield makes the hedge ratios for oil differ from

those for gold, which we believe has very little or no convenience yield. In contrast

with gold, the present value of the obligation to buy a barrel of oil one year from now

is less volatile than the value of the purchase of a barrel of oil today. Apurchase of oil

one year from now is, in essence, equivalent to a purchase of 11.02 barrels of oil today,

in terms of risk.

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

788	Part VIRisk Management

EXHIBIT22.4

Hedging a Decline in the Price of Oil with a 2 Percent Convenience Yield=CurrentOil Position

(1)(2)(3)(4)

Position Value at Initial Position Value if Oil Mark-

Oil Price of $25BarrelPrice Rises to $30barrel to-Gain from

(⇒Zero PVForward and(⇒Zero PVForward andMarket Position

PositionFutures Price $26.96)Futures Price $32.35)Cash(2)(3) (1)

Hold 1 barrel oil	$25	$30	$0	$5
Sell 1 futures contract	0	0	5.4	5.4
Sell 1.021.1 futures contracts	0	0	5	5

		26.96 32.35
Sell 1 forward contract	0	4.9	0	4.9
		1.1

Sell 1.02 forward contracts

0

5 1.02( 4.9)

0

5

Hold 1 barrel of oil and sell
1 futures contract	25	$30	5.4	.4

Hold 1 barrel of oil and sell
1 forward contract	25	25.1 30 4.9	0	.1

a.Hold 1 barrel of oil and sell
1.02/1.1 futures contracts	25	$30	50

b.Hold 1 barrel of oil and
sell 1.02 forward contracts	25	25 30 1.02(4.9)	0	0

It is important to recognize that this risk comparison has been oversimplified by

our assumption that the convenience yield of a commodity does not fluctuate with the

commodity’s price. For most commodities, convenience yields tend to increase as the

price of the commodity increases and decrease as the price of the commodity decreases.

When this is the case, the forward price is even less volatile relative to the volatility

of the spot price. This is discussed in detail in the next section.