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Problem Set 1
Financial Derivatives , Spring 2016
Equilibrium in Futures Markets
Note: The first two problems are warm-ups.
1.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Euro per Dollar exchange rate is 0.90 spot.
• Pound per Dollar exchange rate is .70 spot.
• Euro per Pound exchange rate is 1.35 spot.
2.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Swiss Franc per Dollar exchange rate is 1.02 spot.
• Swiss Franc per Pound exchange rate is 1.53 spot.
• Pound per Dollar exchange rate is 0.60 spot.
Note: Next, we’ll consider some basic commodity problems.
3.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Wheat is $2.00 per bushel spot and $2.30 per bushel for 180-day futures.
• U.S. interest rate is 10.00% APR compounded daily.
• Storage cost in a bonded, insured warehouse is $0.10 per bushel (prepaid) for a
180-day period, and you already have an inventory of one million bushels in
storage.
4. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Wheat is $2.00 per bushel spot and $2.03 per bushel for 180-day futures.
• U.S. interest rate is 10.00% APR compounded daily.
• Storage cost in a bonded, insured warehouse is $0.10 per bushel (prepaid) for a
180-day period, and you already have an inventory of one million bushels in
storage.
5. Problems 3 and 4 illustrate the role of storage cost and interest in establishing the
basis. Some have argued that “convenience yield” is also a factor in establishing the
basis. Convenience value may be real for specific individuals or companies, but
some conditions must be met before such factors translate into reductions in the basis
for a commodity. Discuss the necessary conditions for convenience value to
influence market value (so that the futures price is at “less than full carry”).
6. Be prepared to describe “backwardation” and discuss the factors that cause it. Ditto
for “contango” and the factors that cause it. Can backwardation and contango both
be present for a particular commodity, for contracts with different expirations? (If
so, why?)
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Problem Set 1
Financial Derivatives , Spring 2016
7. Suppose that in March you observe the price for soybeans is $1.57 for May delivery,
$1.60 for June, and $1.55 for September. Is there anything apparently out of
balance? How might it be explained?
8. If the risk of holding a commodity is diversifiable, what should the risk premium be?
Suppose the holder’s risk can be fully hedged? (See p. 311 of the text for a
discussion of the risk premium in futures pricing.)
9. Why is the value of a futures contract at the time it is purchased equal to zero?
10. Comment on the following statement made by a futures trader: “Futures prices are
determined by either expectations or the cost of carry.”
11. The cost of carry futures pricing equation may appear slightly flawed to some
people. To derive it, they argue, we equate a payoff at expiration with the initial
outlay for the asset. Since these cash flows occur at different times, the time value
of money seems to have been omitted. Explain why this interpretation is not
correct.
12. What factors influence the effectiveness of cross hedging? (See the discussion of
cross hedging in the text for reference.)
Note: Now, we’ll consider some international situations.
13. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Euro per Dollar exchange rate is 0.77 spot and 0.80 for 180-day forward
($1 = € 0.77 spot and $1 = € 0.80 forward)
• German interest rate is 3.00% APR compounded daily.
• U.S. interest rate is 1.00% APR compounded daily.
14. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• London gold price per ounce is 1020 spot and 1050.60 for 180-day forward.
• London silver price per ounce is 20.40 spot and 21.012 for 180-day forward.
• London tin price per pound is 6.5806 spot and 6.7781 for 180-day forward.
• London Dollar exchange rate is .60 spot and .58 for 180-day forward
($1 = £ 0.60 spot and $1 = £ 0.58 forward).
• U.S. interest rate is 1.00% APR compounded daily.
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Problem Set 1
Financial Derivatives , Spring 2016
15. You are an expatriate working for CommerzBank in Frankfurt, Germany, and
observe the following prices. Formulate an arbitrage strategy to profit from the
situation.
• Euro per Dollar exchange rate is 0.77 spot and 0.80 for 180-day forward.
• German interest rate is 5.00% compounded daily.
• U.S. stock market index is 1324 today.
• At today's level of the index, the average annual dividend yield on the stocks in
the index is 3% (for simplicity, assume the dividends for your six-month holding
period will all be paid at the end of 180 days).
• The U.S. stock market index 180-day futures price is 1363.72
16. You are an expatriate working for Bank America in Hong Kong, and observe the
following prices. Formulate an arbitrage strategy to profit from the situation.
• Swiss Franc per Dollar exchange rate is 0.91 spot and 0.90 for 180-day forward.
• Swiss interest rate is 3.00% compounded daily.
• U.S. stock market index is 1324 today.
• At today's level of the index, the average annual dividend yield on the stocks in
the index is 3% (for simplicity, assume the dividends for your six-month holding
period will all be paid at the end of 180 days).
• The U.S. stock market index 180-day futures price is 1363.72
17. Be prepared to discuss the role of index futures in supporting and stabilizing the
stock market.
18. Round Rock National Bank lent $1,000,000 to Block Watne Homes, with very
substantial collateral, at a floating rate pegged at 2% above the T-Bill rate. The
Bank borrowed $1,000,000 in Eurodollars from HSB Bank in England, at 1% over
LIBOR (London Interbank Order Rate). Round Rock National's correspondent,
Citicorp, offered to arrange a swap with $1,000,000 principal that would allow
Round Rock to receive interest at 1% over LIBOR and pay at 1% over T-bill. Is the
swap attractive to Round Rock National?
19. Be prepared to discuss the role of the swap market in the supply of funds for
domestic loans.
Note: Next, we’ll consider some situations involving interest rate futures.
20. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Interest rate is 1.53% APR compounded daily, for 180-day T-bills in the spot
market.
• Interest rate is 1.50% APR compounded daily, for 90-day T-bills in the spot
market.
• The futures rate is 1.52% APR for T-bills with 90 days to maturity, to be
delivered 90 days from now.
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Problem Set 1
Financial Derivatives , Spring 2016
21. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Interest rate is 7.25% APR compounded daily, for 270-day T-bills in the spot
market.
• Interest rate is 7.00% APR compounded daily, for 90-day T-bills in the spot
market.
• The futures rate is 7.50% APR for T-bills with 180 days to maturity, to be
delivered 90 days from now.
22. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Interest rate is 7.25% APR compounded daily, for 270-day T-bills in the spot
market.
• Interest rate is 7.10% APR compounded daily, for 180-day T-bills in the spot
market.
• The futures rate is 7.50% APR for T-bills with 90 days to maturity, to be
delivered 180 days from now.
23. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Interest rate is 7.12% APR compounded daily, for 180-day T-bills in the spot
market.
• Interest rate is 7.00% APR compounded daily, for 90-day T-bills in the spot
market.
• The futures rate is 7.15% APR for T-bills with 90 days to maturity, to be
delivered 90 days from now.
Note: Now, we’ll do some practice with multiple choice problems.
24. Suppose the futures price of Plantonium (a mineral which your firm uses heavily) is
$55 per unit for delivery in six months. At the same time the spot price is $60.
Assuming that the futures market is reasonably efficient, which of the following is
the best choice?
a. The market expects a significant increase in available supplies of plantonium
between now and the delivery date.
b. The market expects a significant decrease in available supplies of plantonium
between now and the delivery date.
c. There is nothing out of the ordinary in this situation, as the quoted prices reflect
the normal relationship between spot and future.
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Problem Set 1
Financial Derivatives , Spring 2016
25. After the Chernobyl nuclear disaster in Russia, the prices of agricultural
commodities were quickly bid much higher in active trading on the U.S. spot and
futures markets.
a. This was a perverse response, leading to further reductions in the potential future
supplies of basic foodstuffs.
b. Automatically, without government intervention, this put in place incentives to
reduce consumption and increase production of basic foodstuffs; in turn leading
to an increase in potential future supplies.
c. This action was motivated by the desire for profits.
d. Both b and c.
e. None of the above.
Note: Now, we’ll do some more practice with essay questions.
26. Why does the text say it is naïve to construct a hedge for 10,000 bushels of corn by
selling 10,000 bushels of futures contracts?
27. Explain a minimum variance hedge.
28. Explain a price sensitivity hedge.
29. Discuss special factors that might arise in stock index futures hedging.
30. Explain “tailing the hedge”
31. Explain why a strengthening basis benefits a short hedge and hurts a long hedge.
32. What factors must one consider in deciding the appropriate futures commodity for a
partial hedge?
Financial Derivatives
1.
2.
3.
Solutions: Problem Set 1
Relative to the Euro, the Pound is worth
twice as much as the Dollar; but the
Pound/Dollar exchange rate does not
reflect the same relationship (the Dollar
buys too many Pence). An arbitrage to
take advantage of this involves the
following steps:
a. Borrow $1,000,000 and buy
700,000 Pounds.
b. Exchange the Pounds for Euro
945,000.
c. Exchange the Euro for $1,050,000.
d. Pay off the loan and keep $50,000
profit.
Relative to the Swiss Franc, the Pound
is worth 1.5 times as much as the
Dollar; but the Pound/Dollar exchange
rate does not reflect the same
relationship (the Dollar buys too few
Pence). An arbitrage to take advantage
of this involves the following steps:
a. Borrow $1,000,000 and buy CHF
1,020,000.
b. Exchange the Francs for £666,667
c. Exchange the Pounds for
$1,111,111
d. Pay off the loan and keep $111,111
profit.
An arbitrage to take advantage of this
involves the following steps:
a. Buy wheat and store it, for an
investment of $2.10 per bushel.
b. Sell futures at $2.30 per bushel.
Instead of putting money into the bank,
you have put money an inventory of
wheat that will provide an annualized
risk-free rate of return of 18.45%
(compounded daily over the 180-day
period).
4.
An arbitrage to take advantage of this
involves the following steps:
a. Sell wheat from inventory,
releasing $2.00 per bushel.
b. Buy futures at $2.03 per bushel.
Spring 2016
Instead of storing wheat, you should
move your money into the bank. By
selling wheat you can “borrow” money
at an annualized interest rate of 3.02%.
5-12. For class discussion.
13. An arbitrage to take advantage of this
involves the following steps:
a. Borrow € 770,000 in Germany.
b. Convert this to Dollars and buy
$1,000,000 worth of U.S. bonds.
c. Contract to exchange the future
value of your Dollars for Euros at
$1 = € 0.80 in 180 days.
The riskfree profit at expiration will be
€ 803,954.90 – € 781,475.99 =
€22,478.90.
14. An arbitrage to take advantage of this
involves the following steps:
a. Borrow $1,700,000 in the U.S.
b. Buy 1,000 ounces of London gold.
c. Sell London gold futures.
d. Contract to exchange Pounds for
Dollars at the rate of .58 Dollars
per Pound.
The riskfree profit at expiration will be
$1,811,379.31 – $1,708,404.15 =
$102,975.16. A similar arbitrage can be
done with any of the commodities, with
the same profit.
15. An arbitrage to take advantage of this
involves the following steps:
a. Borrow € 770,000 in Frankfurt, at
5% compounded daily.
b. Convert it to $1,000,000 and invest
in the stocks in the U.S. stock
market index.
c. Sell stock index futures to cover
the position.
d. Contract to exchange Dollars back
to Euro at the rate of $1 = €0.80
At the termination of the hedge, collect
dividends of $15,000 then sell the stock
and settle the index futures contract to
Financial Derivatives
Solutions: Problem Set 1
net $1,030,000. You will have a total of
$1,045,000. Convert this into Euro to
yield € 836,000. You will have to pay
only € 789,220.98 to settle your debt,
leaving a profit of € 46,779.02. The
only uncovered source of risk arises
from the dividends. Although
individual company dividends are
somewhat unpredictable, however, the
average dividend yield for a hundred
companies is reasonably stable, so the
risk is minimal.
16. An arbitrage to take advantage of this
involves the following steps:
a. Borrow CHF 910,000, at 3%
compounded daily.
b. Convert it to $1,000,000 and invest
in the stocks in the U.S. stock
market index.
c. Sell stock index futures to cover
the position.
d. Contract to exchange Dollars back
to SF at the rate of 0.90 CHF per
dollar.
At the termination of the hedge, collect
dividends of $15,000 and sell the stock
to net $1,030,000. Convert your
$1,045,000 into CHF at 0.90, to yield
CHF 940,500. You will have to pay
only CHF 923,562.53 to settle your
debt, leaving a profit of CHF 16,937.47.
The only uncovered source of risk
arises from the dividends. Although
individual company dividends are
somewhat unpredictable, however, the
average dividend yield for five hundred
companies is reasonably stable, so the
risk is minimal.
17. For class discussion.
18. Yes. Round Rock Bank will have the
following annual cash flow stream for
Spring 2016
the life of the arrangement:
⎡ TBill + 2%
⎤
⎢ − ( LIBOR + 1%)⎥
$1,000, 000 × ⎢
+ ( LIBOR + 1%)⎥
⎢
⎥
⎣ − (TBill + 1%) ⎦
= $1,000, 000 × 1% = $10, 000
19. For class discussion.
20. In this problem, there are two ways to
invest for a 180-day period. One way is
to buy 180-day bills, which yields
1.53%. The other is to buy 90-day bills
and then contract to roll them over in 90
days, which yields only 1.51%.
Clearly, the 180-day bills are more
attractive. An arbitrage to take
advantage of this involves the following
steps:
a. Borrow $1,000,000 for 90 days at
1.50% compounded daily
b. Invest it in 180-day bills.
c. Contract to sell 90-day bills in 90
days (by which time the bills you
bought in the previous step will
have 90 days left to maturity).
The risk-free profit from this arbitrage
will be $1,003,804.40 – $1,003,705.40
= $99.00. Repeat this a thousand times,
and then celebrate.
21. In this problem, there are two ways to
invest for a 270-day period. One way is
to buy 270-day bills, which yields
7.25%. The other is to buy 180-day
bills and then contract to roll them over,
which yields 7.33%. Clearly, the rollover strategy is more attractive. An
arbitrage to take advantage of this
involves the following steps:
a. Borrow $1,000,000 for 270 days at
7.25% compounded daily
b. Invest it in 90-day bills.
c. Contract to buy 180-day bills in 90
days to yield 7.50%.
Financial Derivatives
Solutions: Problem Set 1
The risk-free profit from this arbitrage
will be $1,055,739.13 – $1,055,088.67=
$650.46. Repeat this ten thousand
times and then retire.
22. In this problem, there are two ways to
invest for a 270-day period. One way is
to buy 270-day bills, which yields
7.25%. The other is to buy 180-day
bills and then contract to roll them over,
which yields only 7.23%. Clearly, the
270-day bills are more attractive. An
arbitrage to take advantage of this
involves the following steps:
a. Borrow $1,000,000 for 180 days at
7.10% compounded daily
b. Invest it in 270-day bills.
c. Contract to sell 90-day bills in 180
days (by which time the bills you
bought in the previous step will
have 90 days left to maturity).
The risk-free profit from this arbitrage
will be $1,035,758.04 – $1,035,630.37
= $127.67. Repeat this 10,000 times
and then retire.
23. In this problem, there are two ways to
invest for a 180-day period. One way is
to buy 180-day bills, which yields
7.12%. The other is to buy 90-day bills
and then contract to roll them over in 90
days, which yields only 7.075%.
Clearly, the 180-day bills are more
attractive. An arbitrage to take
advantage of this involves the following
steps:
a. Borrow $1,000,000 for 90 days at
7% compounded daily
b. Invest it in 180-day bills.
c. Contract to sell 90-day bills in 90
days (by which time the bills you
bought in the previous step will
have 90 days left to maturity).
The risk-free profit from this arbitrage
will be $1,017,634.17 – $1,017,408.41
= $225.76. Repeat this 1,000 times and
then retire.
24. A
25. D
26-32. For class discussion.
Spring 2016