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CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
PROBLEM SETS
1. In formulating a hedge position, a stock’s beta and a bond’s duration are used similarly
to determine the expected percentage gain or loss in the value of the underlying asset
for a given change in market conditions. Then, in each of these markets, the expected
The major difference in the calculations necessary to formulate a hedge position in
each market lies in the manner in which the first step identified above is computed.
For a hedge in the equity market, the product of the equity portfolio’s beta with respect
to the given market index and the expected percentage change in the index for the
A secondary difference in the calculations necessary to formulate a hedge position
in each market arises in the calculation of the hedge ratio. In the equity market, the
hedge ratio is typically calculated by dividing the total expected dollar change in the
2. One of the considerations that would enter into the hedging strategy for a U.S.
exporting firm with outstanding bills to its customers denominated in foreign
currency is whether the U.S. firm also has outstanding payables denominated in the
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CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
receivables. Equivalently, if the U.S. firm expects to incur ongoing obligations
denominated in the same foreign currency in order to meet expenses required to
deliver additional products to its customers, then the firm would reduce its short
position in the foreign currency futures. In general, if the U.S. firm incurs expenses
in the same foreign currency, then the firm would take a short position in the
3. The hedge will be much more effective for the gold-producing firm. Prices for distant
maturity oil futures contracts have surprisingly low correlation with current prices
because convenience yields and storage costs for oil can change dramatically over
time. When near-term oil prices fall, there may be little or no change in longer-term
In contrast, both convenience yields and storage costs for gold are substantially
smaller and more stable; the result is that the correlation between short-term and
more distant gold futures prices is substantially greater. In other words, the basis
4. Municipal bond yields, which are below T-bond yields (tax-exempt status), are
expected to close in on Treasury yields. Because yields and prices are inversely
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CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
c. The futures price is too low. Buy futures, short the index, and invest the
proceeds of the short sale in T-bills:
CF Now CF in 1 Year
Buy futures 0 ST 2,012
6. a. The value of the underlying stock is:
The transaction cost in the stock market is 10.00 times the transaction cost in
the futures market.
7. a. You should be short the index futures contracts. If the stock value falls, you
need futures profits to offset the loss.
b. Each contract is for $50 times the index, currently valued at 1,950. Therefore,
9. You would short $0.50 of the market index contract and $0.75 of the computer
industry contract for each dollar held in IBM.
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CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
10. The dollar is depreciating relative to the euro. To induce investors to invest in the
U.S., the U.S. interest rate must be higher.
11. a. From parity:
US
0 0
UK
11.01
1.50 1.49
1 1.02
r
F E r
+
= ´ = ´ =
+
b. Let F0 = $1.52/£. Dollars are relatively too cheap in the forward market ↔
pounds are too expensive. Borrow the present value of £1, use the proceeds to
buy pound-denominated bills in the spot market, and sell £1 forward:
Action Now CF in $ Action at period-end CF in $
Sell £1 forward for $1.52 0 Collect $1.52,
deliver £1 $1.52 – $E1
Buy £1/1.02 in spot market;
invest at the British risk-free rate –1.50/1.02 = –$1.47 Exchange £1 for $E1$E1
Borrow $1.47 $1.47 Repay loan;
U.S. interest rate = 2% –$1.49
Total 0 Total $0.03
12. a. Lend in the U.K.
b. Borrow in the U.S.
c. Lending in the U.S. offers a 4% rate of return. Lending in the U.K. and
covering interest rate risk with futures or forwards offers a rate of return of
0
US UK
0
1.98
(1 ) 1 1.07 1 0.0593 5.93%
2.00
F
r r E
é ù é ù
= + ´ - = ´ - = =
ê ú ê ú
ë û
ë û
An arbitrage strategy involves simultaneous lending (UK) and borrowing
(US) with the covering of interest rate risk:
Action Now CF in $ Action at period-end CF in $
Borrow $2.00 in U.S. $2.00 Repay loan –$2.00 × 1.04
Convert borrowed dollars to
pounds; lend £1 pound in U.K.–$2.00 Collect repayment; exchange
proceeds for dollars 1.07 × E1
Sell forward £1.07 at F0 = $1.98 0 Unwind forward
1.07 × ($1.98 – E1)
Total 0 Total $0.0386
13. The farmer must sell forward
This requires selling 111,111/5,000 = 22.2 contracts
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CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
14. The closing futures price will be: 100 0.60 = 99.40
The initial futures price was 99.1175, so the gain to the long side is 28.25 basis
points or
15. Suppose the yield on your portfolio increases by 1.5 basis points. Then the yield
on the T-bond contract is likely to increase by 1 basis point. The loss on your
portfolio will be
16. She must sell:
8.0$
10
8
million 1$
million of T-bonds
17. If yield changes on the bond and the contracts are each 1 basis point, then the bond
value will change by
The contract will result in a cash flow of
If F0 = 1545, you could earn arbitrage profits as follows:
CF Now CF in 1 year
Buy gold 1500 ST
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CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
b. F0 = S0(l + rf )T = 1500 1.02 = 1530
If F0 = 1550, you could earn arbitrage profits as follows:
CF Now CF in 1 year
19. If a poor harvest today indicates a worse than average harvest in future years, then the
futures prices will rise in response to today’s harvest, although presumably the two-year
Suppose first that corn is never stored across a harvest, and second that the quality of a
harvest is not related to the quality of past harvests. Under these circumstances, there is
20. The required rate of return on an asset with the same risk as corn is
Thus, in the absence of storage costs, three months from now corn would sell for
The future value of three month’s storage costs is
where FA stands for the future value factor for a level annuity with a given interest rate
and number of payments. Thus, in order to induce storage, the expected price would
have to be
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CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
21. If the exchange of currencies were structured as three separate forward contracts,
the forward prices would be determined as follows:
Forward exchange rate $1 million euros = Dollars to be delivered
Year 1: 1.20 (1.04/1.03) 1 million euros = $1.2117 million
Instead, we deliver the same number of dollars (F*) each year. The value of F* is
determined by first computing the present value of this obligation:
1 2 3 1 2 3
* * * 1.2116 1.2234 1.2535 3.3943
1.04 1.04 1.04 1.04 1.04 1.04
F F F
+ + = + + =
F* equals $1.2231 million per year (PV = 3.3943; N = 3; I = 4; FV = $0; Solve for
PMT = 1.2231).
22. The firm’s overall cost of the fund will equal the spread between the LIBOR rate
23. a. The swap rate moved in favor of firm ABC. ABC should have received 1%
more per year than it could receive in the current swap market. Based on
notional principal of $10 million, the loss is
b. The market value of the fixed annual loss is obtained by discounting at the
c. If ABC had become insolvent, XYZ would not be harmed. XYZ would be
24. The firm receives a fixed rate that is 2% higher than the market rate. The extra
payment of (0.02 × $10 million) has present value equal to
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CHAPTER 23: FUTURES, SWAPS, AND RISK MANAGEMENT
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