CHAPTER 14
1.
The open interest includes:
Longs
Bob
Lois
15 contracts
5 contracts
Shorts
Bill
10 contracts
Helen 10 contracts
Determine the new open interest under the following assumptions:
(a) Bill goes long 1 contract and Gene shorts 1 contract.
(b) Bill goes long 1 contract and Bob shorts 1 contract.
a.
Bob
Lois
b.
Bob
Lois
4.
Longs
15
5
Bill
Helen
Gene
20
Shorts
9
10
1
20
Longs
14
5
19
Bill
Helen
Shorts
9
10
19
Suppose you go long in gold futures at $300 per ounce. Your broker requires you to put
up 5% of this price as collateral. The net day gold futures settle at $320. Compute the
gain as a percent of your equity in the position. What is the general relationship between
the percent change in the investor’s equity and the percent change in the futures price?
% change underlying
% put down
 320 - 300 
 300 
= 1.33 = 133.33%
= 
0.05
% change equity =
5.
Assume the following information about the futures price for gold:
Delivery dates (number of years into future)
1
2
3
$300
$350
$400
Current price of futures
contract per ounce of gold
If the current spot price of gold is $260, determine spot interest rates for periods 1, 2, and
3, and forward interest rates for periods 2 and 3, assuming no marking-to-market and no
storage or transactions costs.
a. Delivery date = 1
300 = 260(1 + R0,1); R0,1 = 0.1538 = 15.38%
b. Delivery date = 2
350 = 260(1 + R0,2)2; R0,2 = 0.1602
f0,2 = 0.1666 = 16.67%
c. Delivery date = 3
400 = 260(1 + R0,3)3; R0,3 = 0.1544
f0,3 = 0.1429 = 14.29%
6.
Assume no marking-to-market or storage costs. The spot price of gold is $300 and the
futures price for delivery in 1 year is $360. The annual interest rate is 10 percent. Is the
preceding information mutually consistent? If not, how can investors exploit the
situation for their own profit?
The spot price and interest rate imply a future price of 330 = 300(1.10). Investors can
engage in the following arbitrage:
+300
-300
Points in Time
1
Deliver gold and receive
+360
Repay
-330
Deliver into the short futures position
0
+30
0
Short futures
Borrow
Buy gold
Net
Arbitragers will purchase gold in the spot market and short gold futures, forcing the
prices to converge to an equilibrium level.
7.
Suppose that oil in the spot market is selling for $30 per barrel. Oil futures for delivery
in two years are quoted at $20 per barrel. The two-year spot interest rate is 8%. Explain
the perfect market arbitrage to profit from this situation. In practice, what would prevent
this arbitrage from occurring?
The theoretical futures price equals the spot price times (1 + R0,2)2. If the actual futures
price equals $20, the following arbitrage could occur.
Points in Time
1
0
Long futures
Short oil in spot market
Lend short sale proceeds
2
-20 and receive oil
Deliver oil to close short
+30(1.08)2 = 34.992
+30
-30
Net
Cumulative Net
0
0
0
0
+14.992
+14.992
In practice, short selling oil is impossible; the arbitrage does not work.
9.
A farmer plants enough wheat to harvest 1000 bushels. The cost of planting the wheat is
$2.50 per bushel. On the harvest date, the wheat price will be one of three prices with
equal probability: $2.00, $4.00, or $6.00. Compute the farmer’s profit for each of these
possibilities. The futures price for wheat is $3.80. Compute the farmer’s profit for this
price. When is hedging with futures the best choice for the farmer?
The total cost of planting is $2,500.
Sales Price
Total Revenue
Profit of Loss
2.00
2,000
-500
4.00
4,000
1,500
6.00
6,000
3,500
3.80
3,800
1,300
The expected profit if the wheat is sold at the spot price on the harvest date is:
(-500)(1/3) + (1,500)(1/3) + (3,500)(1/3) = 1,500.
Sales at the futures price results in a sale with a profit of 1,300. For a risk averse farmer,
the certainty of a profit of $1,300 may be better than the gamble with an expected value
of $1,500 but with the possibility of a loss of $500.
10.
You buy gold in the spot market at $280 per ounce. You decide to hedge the gold
position with a cross hedge in platinum futures. Suppose platinum futures are quoted at
$400 per ounce. For every dollar that gold advances (or declines), platinum futures are
likely to change by $1.25. Draw a profit profile for a one-to-one hedge—one ounce of
short platinum futures for each ounce long spot gold. Then, derive the hedge ratio for a
perfect hedge and draw the profit profile.
Since platinum is more volatile than gold, a one-to-one hedge results in a net loss on the
hedge when gold increases in price and a net gain when gold decrease in price. The
optimal hedge would short .8 platinum contracts.
Profit Profile
Profit
Long
Gold
1 to 1 Hedge
Loss
Short Platinum
Futures
11.
The open interest includes:
Longs
Helen 9 contracts
Ellen 12 contracts
Shorts
Bill
Phil
Will
7 contracts
7 contracts
7 contracts
Determine the new open interest under the following assumptions:
Helen shorts 2 contracts and Ellen goes long 3 contracts.
Bill goes long 1 contract, Phil goes long 2 contracts, and Will goes short 4 contracts.
Helen
Ellen
Longs
7
15
Bill
Phil
Will
22
12.
Shorts
6
5
11
22
Suppose you go long in gold futures at $280 per ounce. Your broker requires you to put up
5% of this price as collateral. What would the futures price have to become for you to earn
a rate of return of 200% on your investment?
Put down (280)(0.05) = $14
200% Return is a profit of $28
New Price = 280 + 28 = $308
13.
Gold is selling for $280 in the spot market. Gold futures for delivery in one year are quoted
at $315 per ounce. You observe that one-year Treasury strips are selling at $91.74 per $100
of par and two-year strips are selling at $85.73 per $100 of par value. Are there are any
arbitrage opportunities?
F(theoretical) = 280/0.9174 = 305.21.
0
|
Short Future
Buy Gold
Borrow
Net
0
-280
+280
0
Del
1
|
+315 + delivery
-305.21 (Repay)
+9.79
2
|
14. You buy gold in the spot market at $280 per ounce. You decide to hedge the gold position
with a cross hedge in platinum futures. Suppose platinum futures are quoted at $400 per
ounce. For every eighty cents that gold advances (or declines), platinum futures are likely to
change by $1.20. What hedge ratio (number of futures contracts per spot contract) gives an
optimal hedge. The slope is 1.20/.80. The hedge ratio is .80/1.20.
Hedge Ratio = .80/1.20 = 2/3