Certain Selected Problems
Chapter 8
• 1. On Monday morning, an investor takes a
long position in a pound futures contract that
matures on Wednesday afternoon. The
agreed-upon price is $1.78 for £62,500. At the
close of trading on Monday, the futures price has
risen to $1.79. At Tuesday close, the price rises
further to $1.80. At Wednesday close, the price
falls to $1.785, and the contract matures. The
investor takes delivery of the pounds at the
prevailing price of $1.785. Detail the daily
settlement process (see Exhibit 8.3). What will
be the investor's profit (loss)?
ANSWER
Time
Action
Cash Flow
------------------------------------------------------------------------------------------------------------------------------------- ----Monday
Investor buys pound futures
None
morning
contract that matures in two
days. Price is $1.78.
Monday
close
Futures price rises to $1.79.
Contract is marked-to-market.
Investor receives
62,500 x (1.79 - 1.78) = $625.
Tuesday
close
Futures price rises to $1.80.
Contract is marked-to-market.
Investor receives
62,500 x (1.80 - 1.79) = $625.
Wednesday
close
Futures price falls to $1.785.
(1)
(1) Contract is marked-to-market.
(2) Investor takes delivery of £62,500. (2)
Net profit is $1,250 - 937.50 = $312.50.
Investor pays
62,500 x (1.80 -1.785) = $937.50
Investor pays 62,500 x 1.785 = $111,562.50.
• 2. Suppose that the forward ask price for
March 20 on euros is $0.9127 at the same
time that the price of IMM euro futures for
delivery on March 20 is $0.9145. How
could an arbitrageur profit from this
situation? What will be the arbitrageur's
profit per futures contract (size is
€125,000)?
• Answer. Since the futures price exceeds
the forward rate, the arbitrageur should
sell futures contracts at $0.9145 and buy
euro forward in the same amount at
$0.9127. The arbitrageur will earn
125,000(0.9145 - 0.9127) = $225 per euro
futures contract arbitraged.
• 3. Suppose that DEC buys a Swiss franc
futures contract (contract size is SFr
125,000) at a price of $0.83. If the spot
rate for the Swiss franc at the date of
settlement is SFr 1 = $0.8250, what is
DEC's gain or loss on this contract?
• Answer. DEC has bought Swiss francs
worth $0.8250 at a price of $0.83. Thus, it
has lost $0.005 per franc for a total loss of
125,000 x .005 = $625.
• 4. On January 10, Volkswagen agrees to
import auto parts worth $7 million from the
United States. The parts will be delivered
on March 4 and are payable immediately
in dollars. VW decides to hedge its dollar
position by entering into IMM futures
contracts. The spot rate is $0.8947/€ and
the March futures price is $0.9002/€.
• a. Calculate the number of futures
contracts that VW must buy to offset its
dollar exchange risk on the parts contract.
• Answer. Volkswagen can lock in a euro price for
its imported parts by buying dollars in the futures
market at the current March futures price of
€1.1109/$1 (1/0.9002). This is equivalent to
selling euro futures contracts. At that futures
price, VW will sell €7,776,050 for $7 million. At
€125,000 per futures contract, this would entail
selling 62 contracts (7,776,050/125,000 = 62.21)
at a total cost of €7,750,000.
• b. On March 4, the spot rate turns out to
be $0.8952/€, while the March futures
price is $0.8968/€. Calculate VW's net
euro gain or loss on its futures position.
Compare this figure with VW's gain or loss
on its unhedged position.
• Answer. Under its futures contract, Volkswagen has
agreed to sell €7,750,000 and receive $6,976,550
(7,750,000 x 0.9002). On March 4, VW can close out its
futures position by buying back 62 March euro futures
contracts (worth €7,750,000). At the current futures rate
of $0.8968/€, VW must pay out $6,950,200 (7,750,000 x
0.8968). Hence, VW has a net gain of $26,350
($6,976,550 - $6,950,200) on its futures contract. At the
current spot rate of $0.8952/€, this translates into a gain
of €29,434.76 (26,350/0.8952). Upon closing out the 62
futures contracts, VW will then buy $7 million in the spot
market at a spot rate of $0.8952/€. Its net cost is
€7,790,046.92 (7,000,000/0.8952 - 29,434.76).
• If VW had not hedged its import contract, it could have
bought the $7 million on March 10 at a cost of €
7,819,481.68 (7,000,000/0.8952). This contrasts with a
projected cost based on the spot rate on January 10th of
€7,823,851.57 (7,000,000/0.8947). However, the latter
“cost” is irrelevant since VW had no opportunity to buy
March dollars at the January 10th spot rate of $0.8947/€.
By not hedging, VW would have paid an extra
€29,434.76 for the $7,000,000 to satisfy its dollar liability,
the difference between the cost of $7 million with
hedging (€ 7,790,046.92) and the cost without hedging
(€7,819,481.68).
• 5. Citigroup sells a call option on euros
(contract size is €500,000) at a premium of
$0.04 per euro. If the exercise price is
$0.91 and the spot price of the euro at
date of expiration is $0.93, what is
Citigroup's profit (loss) on the call option?
• Answer. Since the spot price of $0.93 exceeds
the exercise price of $0.91, Citigroup's
counterparty will exercise its call option, causing
Citigroup to lose 2¢ per euro. Adding in the 4¢
call premium it received gives Citigroup a net
profit of 2¢ per euro on the call option for a total
gain of .02 x 500,000 = $10,000.
•
• 6. Suppose you buy three June PHLX call
options with a 90 strike price at a price of
2.3 (¢/€).
• a. What would be your total dollar cost for
these calls, ignoring broker fees?
• Answer. With each call option being for
€62,500, the three contracts combined are
for €187,500. At a price of 2.3¢/€, the total
cost is therefore 187,500 x $0.023 =
$4,312.50.
• b. After holding these calls for 60 days,
you sell them for 3.8 (¢/€). What is your
net profit on the contracts assuming that
brokerage fees on both entry and exit
were $5 per contract and that your
opportunity cost was 8% per annum on the
money tied up in the premium?
• Answer. The net profit would be 1.5¢/€
(3.8 - 2.3) for a total profit before expenses
of $2,812.50 (0.015 x 187,500). Brokerage
fees totaled $10 per contract or $30
overall. The opportunity cost would be
$4,312.50 x 0.08 x 60/365 = $56.71. After
deducting these expenses (which total
$86.71), the net profit is $2,725.79.
• 7. Apex Corporation must pay its Japanese supplier
¥125 million in three months. It is thinking of buying 20
yen call options (contract size is ¥6.25 million) at a strike
price of $0.00800 in order to protect against the risk of a
rising yen. The premium is 0.015 cents per yen.
Alternatively, Apex could buy 10 three-month yen futures
contracts (contract size is ¥12.5 million) at a price of
$0.007940 per yen. The current spot rate is ¥1 =
$0.007823. Suppose Apex's treasurer believes that the
most likely value for the yen in 90 days is $0.007900, but
the yen could go as high as $0.008400 or as low as
$0.007500.
• a. Diagram Apex's gains and losses on
the call option position and the futures
position within its range of expected prices
(see Exhibit 8.4). Ignore transaction costs
and margins.
• Answer. In all the following calculations, note
that the current spot rate is irrelevant. When a
spot rate is referred to, it is the spot rate in 90
days. If Apex buys the call options, it must pay a
call premium of 0.00015 x 125,000,000 =
$18,750. If the yen settles at its minimum value,
Apex will not exercise the option and it loses the
call premium. But if the yen settles at its
maximum value of $0.008400, Apex will exercise
at $0.008000 and earn $0.0004/¥1 for a total
gain of .0004 x 125,000,000 = $50,000. Apex's
net gain will be $50,000 - $18,750 = $31,250.
OPTION
Inflow
Outflow
Call premium
Exercise cost
Profit
FUTURES
Inflow
Outflow
Profit
75
79.4
81.5
84
--
--
$1,018,750
$1,050,000
-$18,750
-_______
-$18,750
-$18,750
-________
-$18,750
-18,750
-1,000,000
__________
$0
-1,000,000
-18,750
_________
$31,250
$937,500
-992,500
_______
-$55,000
$992,500
-992,500
________
$0
$1,000,000
-992,500
_________
$7,500
$1,050,000
-992,500
__________
$57,500
• As the diagram shows, Apex can use a futures contract
to lock in a price of $0.007940/¥ at a total cost
of .007940 x 125,000,000 = $992,500. If the yen settles
at its minimum value, Apex will lose
$0.007940 - $0.007500 = $0.000440/¥ (remember it is
buying yen at 0.007940, when the spot price is only
0.007500), for a total loss on the futures contract of
0.00044 x 125,000,000 = $55,000. On the other hand, if
the yen appreciates to $0.008400, Apex will earn
$0.008400 - $0.007940 = $0.000460/¥ for a total gain on
the futures contracts of 0.000460 x 125,000,000 =
$57,500.
• b. Calculate what Apex would gain or lose
on the option and futures positions if the
yen settled at its most likely value.
• Answer. If the yen settles at its most likely
price of $0.007900, Apex will not exercise
its call option and will lose the call
premium of $18,750. If Apex hedges with
futures, it will have to buy yen at a price of
$0.007940 when the spot rate is $0.0079.
This will cost Apex $0.000040/¥, for a total
futures contract cost of 0.000040 x
125,000,000 = $5,000.
• c. What is Apex's break-even future spot
price on the option contract? On the
futures contract?
• Answer. On the option contract, the spot
rate will have to rise to the exercise price
plus the call premium for Apex to break
even on the contract, or $0.008000 +
$0.000150 = $0.008150. In the case of the
futures contract, break-even occurs when
the spot rate equals the futures rate, or
$0.007940.
• d. Calculate and diagram the
corresponding profit and loss and
break-even positions on the futures and
options contracts for the sellers of these
contracts.
• Answer. The sellers' profit and loss and break-even
positions on the futures and options contracts will be the
mirror image of Apex's position on these contracts. For
example, the sellers of the futures contract will
breakeven at a future spot price of ¥1 = $0.007940,
while the options sellers will breakeven at a future spot
rate of ¥1 = $0.008150. Similarly, if the yen settles at its
minimum value, the options sellers will earn the call
premium of $18,750 and the futures sellers will earn
$55,000. But if the yen settles at its maximum value of
$0.008400, the options sellers will lose $31,250 and the
futures sellers will lose $57,500.