Intuition for Hedging with a Maturity Mismatch in the Presence of a Constant Convenience Yield

We summarize the results of this section as follows:

Result 22.2

Long-dated obligations hedged with short-term forward agreements need to be tailed if theunderlying commitment has a convenience yield. The degree of the tail depends on the con-venience yield earned between the maturity date of the forward instrument used to hedgeand the date of the long-term obligation. When hedging with futures, a greater degree oftailing is needed (see Result 22.1).

Result 22.2 derives from the fact that a convenience yield makes the receipt of a

commodity at a future date less risky than receiving the same commodity now. Just as

50 percent of an investor’s risk disappears when he sells 50 percent of his shares of

stock in a company or if the stock has a dividend equal to 50 percent of its value, so

does a 50 percent convenience yield reduce the risk of a commodity received in the

future by 50 percent. Because ypercent of risk disappears with a ypercent convenience

yield, eliminating the risk of a forward commitment by taking on a position in a (more

risky) underlying (spot) commodity requires less than a one-to-one hedge ratio when

the commodity has a convenience yield.

Perfect hedging of long-dated obligations with short-term forwards (or futures) has

a less than one-to-one hedge ratio because perfect hedging is a matter of matching up

risks. The short-dated forward (or futures) contract is more like the underlying com-

modity, which has more risk than the long-term obligation. The closer the maturity date

of the forward, the closer the hedge ratio (in a perfect hedge) is to the “less-than-one”

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Chapter 22

The Practice of Hedging

793

ratio for hedging the obligation by holding the underlying commodity. The longer the

maturity of the forward contract, the more the forward contract looks like the forward

commitment and the closer the hedge ratio is to one.

Convenience Yield Risk Generated by Correlation between Spot Prices and Convenience Yields

The analysis up to this point has assumed that the convenience yield is constant. Gen-

erally, however, the convenience yield is uncertain and correlated with the price of the

commodity. This subsection examines the effect of the correlation on the hedge ratios.

It is not difficult to see that convenience yields are generally positively related to

the price of the underlying commodity. In the late spring of 2000, for example, gaso-

line prices in the United States rose over a matter of weeks by close to 50 percent.

Industry forecasters suggested that this was caused by a temporary shortage of gaso-

line and higher than expected demand, but that gasoline prices would be coming down

by the end of the summer.

Earlier, we argued that gasoline consumers fill up their tanks for the convenience

of not having to stop again. However, when gasoline prices rose in the summer of 2000,

some motorists did not fill up their tanks when they were empty. Instead, they put in

five gallons at a time, hoping that in a few days the price of gasoline would comedown.

Because gasoline prices were expectedto depreciate, the cost of convenience went up.

After all, storing an inventory is not very profitable when the inventory value is declin-

ing. Note, however, that the convenience yield of gasoline also went up at this time

because the convenience benefit of that last gallon in a 5-gallon fill-up is a little higher

than the convenience of the last gallon in a 15-gallon fill-up.⁸

Generally, the size of a commodity’s convenience yield fluctuates inversely with

the aggregate inventory of a commodity, which, in turn, is driven by supply and demand

shocks. Anecessary ingredient for a convenience yield, that is, for there to be a bene-

fit from holding inventory, is that there must be some transaction cost, above and

beyond the ordinary cost of the commodity, to acquire the commodity at certain times.

Agasoline station may pay the ordinary price for gasoline when the supplier’s tanker

truck shows up on its weekly route. However, if between its regular deliveries the gaso-

line station requires a special delivery of gasoline because demand is exceptionally

high, the supplier may impose a surcharge. This surcharge reflects the tanker truck dri-

ver’s inability to deliver gasoline using the most efficient route possible. At times of

low aggregate inventory nationwide, for example, the summer driving season, the

benefit of a large inventory of gasoline—gasoline’s convenience yield—is high.

Hedge ratios are further reduced by convenience yields that tend to increase when-

ever the price of the commodity increases. The intuition for this insight, as with Result

22.2, is based on a comparison of the risk of the commodity’s present value at differ-

ent dates. In particular, the following result suggests that a convenience yield that

increases a lot as spot prices increase tends to dampen the change in the long-term for-

ward prices more than when the convenience yield exhibits only a mild increase in

response to a spot price increase.

⁸Some motorists of course continued to put as much gasoline in their tanks as they had before the

price increase. These motorists always place a high value on the convenience of gasoline. However, these

motorists were never the marginal investor in determining the convenience yield in the marketplace.

Rather, it was the motorists who were willing to cut their inventory of gasoline who determined the price

of convenience in the marketplace.

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794	Part VIRisk Management

Result 22.3

The greater the sensitivity of the convenience yield to the commodity’s spot price, the lessrisky is the long-dated obligation is to buy or sell a commodity.

The positive correlation between the convenience yield and the commodity’s spot price

means that the convenience yield acts as a partial hedge against spot price movements.

Usually, the convenience yield is high when the commodity price is high and vice versa.

Thus, a more realistic picture of the convenience yield suggests that in hedging long-

dated commitments with short-term forwards or futures, the hedge ratio should be

smaller than the ratio computed for a convenience yield that is assumed to be certain.

The exact computation of the hedge ratio in cases where the convenience yield

changes depends on the process that generates the underlying commodity’s spot price,

which is tied to the fluctuating convenience yield.⁹

We summarize the results of this subsection as follows:

Result 22.4

The greater the sensitivity of the convenience yield to the price of the underlying commod-ity, the lower is the hedge ratio when hedging long-dated obligations with short-term for-ward agreements. When hedging with futures, a further tail is needed (see Result 22.1).

In simple models, sensitivity as used in Results 22.3 and 22.4 can be thought of

as the covariance between changes in the convenience yield and changes in the com-

modity’s price. Alternatively, one can view sensitivity as the slope coefficient from

regressing changes in the convenience yield on changes in the commodity’s price.

Basis Risk

When hedging an obligation with a rollover position, there is an additional considera-

tion known as basis risk. The basisat date tof a futures contract, B, is the difference

t

between the futures (or forward) price and the spot price,

BF S

ttt

Basis riskis the degree to which fluctuations in the basis are unpredictable given per-

fect foresight about the path the price takes in the future. Since the basis is simply the

difference between the futures (or forward) price and the spot price, basis risk is also

the degree to which the futures (or forward) price is unpredictable, given perfect fore-

sight about the path the spot price will take.

Sources of Basis Risk.Basis risk may arise because investors are irrational or face

market frictions that prevent them from arbitraging a mispriced futures or forward con-

tract. We are somewhat skeptical about this as a cause of basis risk, especially as it

applies to financial markets from about the late 1980s on. Basis risk may also arise

because changes in interest rates are unpredictable. While this eliminates the ability to

arbitrage any deviation from the futures-spot pricing relation, the effect on the pricing

relation and on hedge ratios has to be negligible. [See Grinblatt and Jegadeesh (1996),

for example.] Finally, basis risk may arise because of variability in convenience yields

that is not determined by changes in the spot price of the commodity. Convenience yield

risk of this type is unhedgeable and eliminates not only the ability to perfectly hedge

long-term obligations with short term forwards, but also the ability to arbitrage devia-

tions from the forward spot pricing relation. It is largely this unhedgeable convenience

yield risk that the analyst needs to be concerned about when estimating hedge ratios.

⁹The interested reader is referred to Gibson and Schwartz (1990) and Ross (1997).

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Chapter 22

The Practice of Hedging

795

How Unhedgeable Convenience Yield Risk Affects Hedge Ratios.Thinking about

convenience yields in the same way one thinks about dividend yields aids in

understanding the effect of unhedgeable convenience yield risk on the variance min-

imizing hedge ratio. Recall from Chapter 11 that the stock price is the present value

of future dividends. Hence, the stock price 10 years in the future is the present value

(PV) of thedividends from year 10 onward, while the stock price 1 year in the future

is the PVof the dividends from year 1 onward. The difference is the PVof the div-

idends paid from years 1 through 10. Now, consider the hedge of a forward obliga-

tion to pay the year 10 stock price offset with a 1-year forward contract on the stock

and examine how variable the PVs of the two hedge components are over time. The

variability of the PVof the 1-year forward contract, which is entirely due to changes

in PVof the stock price 1 year in the future, exceeds the variability in the PVof the

10-year forward obligation, which stems entirely from the PVof the year 10 stock

price. The difference in variability is the variability in the PVof the dividends paid

from years 1 through 10.There is a portion of this that cannot be hedged because it

is unrelated to the stock price. As this chapter’s prior and subsequent analysis shows

(for example, Section 22.9), when the hedging instrument is more volatile than the

obligation, the hedge ratio is generally lower, implying:

Result 22.5

The greater the unhedgeable convenience yield risk, the lower is the hedge ratio for hedg-ing a long-term obligation with a short-term forward or futures contract.

When hedging with a rollover strategy, the largest risk arises at the rollover dates

of the short-term contracts. At these dates, the benefit of convenience embedded in the

value of the hedging instrument drops abruptly as a result of the change in the forward

maturity date. By contrast, the risk from changes in the convenience yield over small

intervals of time between rollover dates is orders of magnitude smaller.

Situations Where Basis Risk Does Not Affect Hedge Ratios.Basis risk does not

affect the size of the minimum variance hedge ratio when hedging a long-dated

commitment with a comparably long-dated forward contract on the same underlying

commodity or asset. In this case, the forward contract and the forward obligation are

essentially the same investment, implying a hedge ratio of one. As long as there is no

arbitrage at the maturity date, FS. Hence, any basis risk before the maturity date

TT

is irrelevant.