804
P A R T V I
Options, Futures, and Other Derivatives
depreciates by 10% relative to the dollar, the firm would need to increase the pound price
of its goods by 10% in order to maintain the dollar-equivalent price. However, the firm
might not be able to raise the price by 10% if it faces competition from British produc-
ers, or if it believes the higher pound-denominated price would reduce demand for its
product.
To offset its foreign exchange exposure, the firm might engage in transactions that bring
it profits when the pound depreciates. The lost profits from business operations resulting
from a depreciation will then be offset by gains on its financial transactions. For example,
if the firm enters a futures contract to deliver pounds for dollars at an exchange rate agreed
to today, then if the pound depreciates, the futures position will yield a profit.
To illustrate, suppose that the futures price is currently $2 per pound for delivery in
3 months. If the firm enters a futures contract with a futures price of $2 per pound, and
the exchange rate in 3 months is $1.90 per pound, then the profit to the short position is
F
0
2 F
T
5 $2.00 2 $1.90 5 $.10 per pound.
How many pounds should be sold in the futures market to most fully offset the exposure
to exchange rate fluctuations? Suppose the dollar value of profits in the next quarter will
fall by $200,000 for every $.10 depreciation of the pound. To hedge, we need a futures
position that provides $200,000 extra profit for every $.10 that the pound depreciates.
Therefore, we need a futures position to deliver £2,000,000. As we have just seen, the
profit per pound on the futures contract equals the difference in the current futures price
and the ultimate exchange rate; therefore, the foreign exchange profits resulting from a
$.10 depreciation
1
will equal $.10 3 2,000,000 5 $200,000.
The proper hedge position in pound futures is independent of the actual depreciation in
the pound as long as the relationship between profits and exchange rates is approximately
linear. For example, if the pound depreciates by only half as much, $.05, the firm would
lose only $100,000 in operating profits. The futures position would also return half
the profits: $.05 3 2,000,000 5 $100,000, again just offsetting the operating exposure.
If the pound appreciates, the hedge position still (unfortunately in this case) offsets the
operating exposure. If the pound appreciates by $.05, the firm might gain $100,000 from
the enhanced value of the pound; however, it will lose that amount on its obligation to
deliver the pounds for the original futures price.
The hedge ratio is the number of futures positions necessary to hedge the risk of the
unprotected portfolio, in this case the firm’s export business. In general, we can think of
the hedge ratio as the number of hedging vehicles (e.g., futures contracts) one would
establish to offset the risk of a particular unprotected position. The hedge ratio, H, in this
case is
H
5
Change in value of unprotected position for a given change in exchange rate
Profit derived from one futures position for the same change in exchange rate
5
$200,000 per $.10 change in $/£ exchange rate
$.10 profit per pound delivered per $.10 change in $/£ exchange rate
5 2,000,000 pounds to be delivered
Because each pound-futures contract on the Chicago Mercantile Exchange calls for
delivery of 62,500 pounds, you would sell 2,000,000/62,500 per contract 5 32 contracts.
1
Actually, the profit on the contract depends on the changes in the futures price, not the spot exchange rate.
For simplicity, we call the decline in the futures price the depreciation in the pound.
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C H A P T E R
2 3
Futures, Swaps, and Risk Management
805
One interpretation of the hedge ratio is as a ratio of sensitivities to the underlying source
of uncertainty. The sensitivity of operating profits is $200,000 per swing of $.10 in the
exchange rate. The sensitivity of futures profits is $.10 per pound to be delivered per swing
of $.10 in the exchange rate. Therefore, the hedge ratio is 200,000/.10 5 2,000,000 pounds.
We could just as easily have defined the hedge ratio in terms of futures contracts.
Because each contract calls for delivery of 62,500 pounds, the profit on each contract per
swing of $.10 in the exchange rate is $6,250. Therefore, the hedge ratio defined in units of
futures contracts is $200,000/$6,250 5 32 contracts, as we found above.
Suppose a U.S. investor is harmed when the dollar depreciates. Specifically, suppose that its profits decrease
by $200,000 for every $.05 rise in the dollar/pound exchange rate. How many contracts should the firm
enter? Should it take the long side or the short side of the contracts?
CONCEPT CHECK
23.2
Given the sensitivity of the unhedged position to changes in the exchange rate, calculat-
ing the risk-minimizing hedge position is easy. Estimating that sensitivity is much harder.
For the exporting firm, for example, a naive view might focus only on the expected pound-
denominated revenue, and then contract to deliver that number of pounds in the futures or
forward market. This approach, however, fails to recognize that pound revenue is itself a
function of the exchange rate because the U.S. firm’s competitive position in the U.K. is
determined in part by the exchange rate.
One approach relies, in part, on historical relationships. Suppose, for example, that the
firm prepares a scatter diagram as in Figure 23.3 that relates its business profits (mea-
sured in dollars) in each of the last 40 quarters to the dollar/pound exchange rate in that
Profits per
Quarter
$2.2 million
$2.0 million
$2.00/£
$1.90/£
Exchange Rate
Slope
= 2 million
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