Fina 5210
Problem Set 4
Fall 2013
Arbitrage
Problems 1 & 2 are “warm-up” for the subsequent problems. The fundamental “clue” in arbitrage is that
when there are two ways to do the same thing, the cost and the outcome should be the same either way.
1.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Euro per Dollar exchange rate is 0.70 spot ($1=€0.70).
• Pound per Dollar exchange rate is 0.60 spot ($1=£0.60).
• Euro per Pound exchange rate is 1.40 spot (£1=€1.40).
2.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Swiss Franc per Dollar exchange rate is 0.90 spot ($1=CHF0.90).
• Pound per Dollar exchange rate is 0.48 spot ($1=£0.48).
• Swiss Franc per Pound exchange rate is 1.50 spot (£1=CHF1.50).
For preparation prior to problem 3, see lecture notes concerning interest rate parity from Topic 7, slides
28 through 30.
3.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Dollar per Euro exchange rate is 1.40 spot and 1.42 for 180-day forward.
• German interest rate is 5.00% compounded daily.
• U.S. interest rate is 3.00% compounded daily.
For preparation prior to problem 4, see lecture notes concerning purchasing power parity from Topic 7,
slides 31through 35.
4.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• London gold price per ounce is 150 spot and 154.50 for 180-days forward.
• London silver price per ounce is 3.00 spot and 3.09 for 180-days forward.
• London tin price per ton is 100 spot and 103 for 180-days forward.
• London Dollar exchange rate is $1 = £0.50 spot and $1 = £0.48 for 180-days
forward.
• U.S. interest rate is 3.00% compounded daily.
Prof. Kensinger
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Fina 5210
Problem Set 4
Fall 2013
For preparation prior to problem 5, review lecture notes on basis arbitrage from Topic 7, slides 16
through 27; and purchasing power parity in the previous problem. Then treat equity as a commodity that
works the same way as any physical commodity, except that “storage cost” is changed into a benefit
because the holder of the stock collects dividends.
5.
You are an expatriate working for CommerzBank in Frankfurt, Germany, and
observe the following prices. Formulate an arbitrage strategy to profit from the
situation.
• Dollar per Euro exchange rate is 1.40 spot and 1.42 for 180-day forward.
• German interest rate is 5.00% compounded daily.
• U.S. stock market index is 1500 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 1.5%, paid at the end of 180 days).
• The U.S. stock market index 180-day futures price is 1560
For preparation prior to problems 6 & 7, see lecture notes concerning basis arbitrage from Topic 7, slides
16 through 27.
6.
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 3.00% 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.
7. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation.
• Wheat is $2.00 per bushel spot and $2.02 per bushel for 180-day futures.
• U.S. interest rate is 3.00% 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.
For preparation prior to problem 8, review lecture notes on basis arbitrage from Topic 7, slides 16
through 27; and the basis arbitrage in problems 6 & 7 above. Then treat bonds as a commodity that
works the same way as any physical commodity, except that “storage cost” may be a benefit if the holder of
the bond collects coupon payments. Also, the price is quoted as a discount rate in the case of financial
futures.
8.
The following conditions are observed. Formulate an arbitrage strategy to profit
from the situation.
• Interest rate is 3.20% compounded daily, for 180-day T-bills in the spot market.
• Interest rate is 3.00% compounded daily, for 90-day T-bills in the spot market.
• The futures rate is 3.15% for T-bills with 90 days to maturity, to be delivered 90
days from now.
Prof. Kensinger
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Fina 5210
Problem Set 4
Fall 2013
For preparation prior to problem 9, see lecture notes in Swaps.pdf, slide 3. This problem is an application
of a standard floating rate to floating rate swap. Using the swap, an underwriter can arrange an arbitrage
that benefits all parties. The source of the arbitrage is the party’s inability to connect fully into the global
financial marketplace. The underwriter bridges this gap.
9.
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 Midland Bank PLC in England, at
1% over LIBOR (London Interbank 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?
Preparation prior to problem 10 is just basic addition and subtraction. Value Additivity should hold (the
Value Additivity Principal is introduced in Topic 1).
10. Suppose that analysts in the mergers and acquisitions department analyzed the
breakup value of BigWorld Corp, which operates BigWorld Airlines, Shug's
Restaurants, and Betty's Boutiques. What is the implied market value of the airline?
Is there an arbitrage opportunity here?
• There are 10 million shares of BigWorld outstanding, and the market price is
$100 per share.
• Shug's Restaurant chain could be sold for $750 million.
• Betty's Boutiques could be sold for $600 million.
• The airline owns $200 million worth of aircraft, net of all debt.
For preparation prior to problem 11, see lecture notes from Topic8, slides 6 through 8. This problem is an
application of put-call parity. The math involved is just basic addition and subtraction.
11. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation (ignoring transactions costs).
• Digital Datawhack stock is selling for $95 per share.
• Call options on Digital Datawhack at 90, with April expiration, are selling for $9
per share.
• Put options on Digital Datawhack at 90, with April expiration, are selling for
$2.50 per share.
• At the current T-Bill rate, $89 invested today would grow to $90 at the expiration
date of the options.
Prof. Kensinger
page 3 of 7
Fina 5210
Problem Set 4
Fall 2013
For preparation prior to problem 12, review the lecture notes concerning the box spread (from Topic8,
slides 35 through 47). This problem is another application of put-call parity. The math involved is just
basic addition and subtraction.
12. The following prices are observed. Formulate an arbitrage strategy to profit from the
situation (ignoring transactions costs).
• Digital Datawhack stock is selling for $95 per share.
• Call options on Digital Datawhack at 85, with April expiration, are selling for
$12 per share.
• Put options on Digital Datawhack at 85, with April expiration, are selling for
$1.25 per share.
• Call options on Digital Datawhack at 90, with April expiration, are selling for
$10 per share.
• Put options on Digital Datawhack at 90, with April expiration, are selling for
$1.75 per share.
• At the current T-Bill rate, $89 invested today would grow to $90 at the expiration
date of the options (and $84.06 invested today would grow to $85).
Problems 13 through 16 involve the linear relationship between risk and return (CAPM), covered in Topic
5.
13. Suppose Phillips Petroleum stock has expected return of 10% with standard
deviation of 8%. At the same time the riskfree rate is 3% and the market portfolio
has expected return of 8% with standard deviation of 5%. According to "textbook"
finance theory,
a. Phillips is overvalued.
b. Phillips is undervalued.
c. Phillips is priced in equilibrium.
d. There is not enough information to tell.
14. Suppose ARCO stock has expected return of 12% with standard deviation of 8%. At
the same time the riskfree rate is 3% and the market portfolio has expected return of
8% with standard deviation of 5%. Assuming the above data is correct, and the
world conforms to the assumptions underlying mean-variance portfolio theory, then:
a. ARCO is overvalued.
b. ARCO is undervalued.
c. ARCO is priced in equilibrium.
d. There is not enough information to tell.
Prof. Kensinger
page 4 of 7
Fina 5210
Problem Set 4
Fall 2013
15. Suppose there were four stocks with the following attributes.
• Stock A has expected return of 15% with beta of 1.
• Stock B has expected return of 11% with beta of 0.5.
• Stock C has expected return of 20% with beta of 1.5.
• Stock D has expected return of 25% with beta of 2.
Assume that the world conforms to the CAPM assumptions, and that the above data
is correct.
a. Stock A is undervalued.
b. Stock B is undervalued.
c. Stock C is overvalued
d. Stock D is overvalued.
e. There is not enough information to tell.
16. Suppose that Stock A has beta of 0.8, and expected return of 13%. At the same time
the market portfolio has expected return of 15%, and the risk-free rate is 5%. Which
of the following is most accurate?
a. Stock A is temporarily overvalued.
b. Stock A is temporarily undervalued.
c. Stock A is priced at equilibrium.
d. There is not enough information to tell.
For preparation prior to problems 17 through 20, review lecture notes from Topic8, slide 47.
17. Digital Datawhack stock rose by $2, from $60 to $62. “Crunch” Murdoch owns call
options on Datawhack, which he bought two months ago for $5 per share. They
expire in seven months, and have an exercise price of $55. Which of the following
is a theoretically possible response of the option value (on a per-share basis), to the
recent move of the stock?
a. rise by $2.10
b. rise by $1.80
c. fall by $2.05
d. fall by $1.75
e. None of the above
18. Option A is a call option written on Digital Datawhack stock, with expiration in one
month. Option B is a call on Datawhack with expiration in six months. Both
options have an exercise price of $50 per share. Which of the following is a
theoretically correct relationship?
a. Option A is worth more than Option B.
b. Option B is worth more than Option A.
c. Option B is worth the same as Option A.
Prof. Kensinger
page 5 of 7
Fina 5210
Problem Set 4
Fall 2013
19. Option A is a call option written on Digital Datawhack, with exercise price of $50.
Option B is also a call option on Datawhack, with exercise price of $55. Both
options expire six months from now. Which of the following is a theoretically
correct relationship?
a. Option A is worth more than Option B
b. Option B is worth more than Option A.
c. Option B is worth the same as Option A.
20. Suppose treasury bill rates went up, but nothing else changed. What would be the
theoretical response of call option values?
a. rise
b. fall
c. remain unchanged
Preparation prior to problem 10 is just basic addition and subtraction. Value additity should hold (the
Value Additivity Principal is introduced in Topic 1).
21. Megalithic Ironworks, Inc. has market value of $100 million. Newton Brickyards,
Inc. has market value of $50 million. Megalithic management decided to acquire
Newton, and paid $100 million for it. Assuming the Value Additivity Principle
holds in this case, what prediction would it lead to?
a. Megalithic stockholders lost $50 million.
b. Newton stockholders lost $50 million.
c. Megalithic stockholders made $50 million.
d. Stockholders of both firms made $50 million.
For preparation prior to problems 22 & 23, see lecture notes conerning basis arbitrage from Topic 7,
slides 16 through 27.
22. 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.
Prof. Kensinger
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Fina 5210
Problem Set 4
Fall 2013
23. 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.
Prof. Kensinger
page 7 of 7
Fina 4200
1.
Solutions: Problem Set 4
Relative to the Euro, the Pound is worth
twice as much as the Dollar. This is not
reflected in the exchange between
Dollar and Pound. An arbitrage to take
advantage of this imbalance involves
the following steps:
a. Borrow $1,000,000 and buy
600,000 Pounds.
b. Exchange the Pounds for Euro
840,000.
c. Exchange the Euros for
$1,200,000.
d. Pay off the loan and keep $200,000
profit.
2.
An arbitrage to take advantage of this
involves the following steps:
a. Borrow $1,000,000 and buy
900,000 Swiss Francs.
b. Exchange the Francs for 600,000
Pounds.
c. Exchange the Pounds for
$1,250,000.
d. Pay off the loan and keep $250,000
profit.
3.
An arbitrage to take advantage of this
involves the following steps:
a. Borrow $1,000,000 in the U.S.
(Repayment will be 1,014,903.88)
b. Buy Euro 714,285.71 worth of Euro
bonds. (This will grow to FV of
732,115.94)
c. Contract to exchange your Euro for
dollars at 1.42 in 180 days.
The riskfree profit at expiration is
$1,039,604.63 - $1,014,903.88 =
$24,700.75
4.
An arbitrage to take advantage of this
involves the following steps:
a. Borrow $1,000,000 in the U.S.
(Repayment will be 1,014,903.88)
b. Buy London gold, 3,333.33 ounces.
c. Sell London gold futures.
Prof. Kensinger
d.
Fall 2013
Contract to exchange Pounds for
Dollars at the rate of .48 Dollars
per Pound.
When you sell the gold in London at
£154.50 per ounce the inflow will be
£515,000, which would convert to
$1,072,916.67. The riskfree profit at
expiration is $1,072,916.67 $1,014,903.88 = $58,012.78 A similar
arbitrage could be done with silver, or
even tin.
5.
An arbitrage to take advantage of this
involves the following steps:
a. Borrow Euro 1,071,428.57 in
Frankfurt, at 5% compounded
daily. (Repayment will be
€1,098,173.91)
b. Convert it to $1,500,000 and invest
in the stocks in the U.S. stock
market index (a million dollars
worth of stock is 1,000 times the
index).
c. Sell stock index futures to cover
the position.
d. Contract to exchange Dollars back
to Euro at the rate of 1.42 Euro per
dollar.
At the termination of the hedge, collect
dividends of $22,500 and sell the stock
to net $1,560,000 (1,000 times the
index futures price) for a total of
$1,582,500. Convert your Dollars into
Euro at $1.42 = €1, to yield Euro
1,114,436.62. You will have to pay only
€1,098,173.91 to settle your debt,
leaving a profit of €16,262.71. The only
uncovered source of risk arises from the
dividends. Although individual
company dividends are somewhat
unpredictable, the average dividend
yield for five hundred companies is
reasonably stable; so the risk is
minimal.
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Fina 4200
6.
Solutions: Problem Set 4
7.
The basis is not enough to cover storage
costs. An arbitrage to take advantage of
this situation involves the following
steps:
a. Sell wheat from inventory,
releasing $2.00 per bushel.
b. Buy futures at $2.02 per bushel.
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 2.02%.
8.
the futures contract with discount
rate 3.15%).
An arbitrage to take advantage of this
involves the following steps:
a. Buy more 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 into an inventory
of wheat that will provide an annualized
risk-free rate of return of 18.45%
(compounded daily over the 180-day
period).
In this problem, there are two ways to
invest for a 180-day period. One way is
to buy 180-day bills, which yields
3.20%. The other is to buy 90-day bills
and then contract to roll them over in 90
days, which yields only 3.075%.
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
3% compounded daily.
(Repayment will be $1,007, 424.38
on day ninety.)
b. Invest it in 180-day bills. (FV will
be $1,015, 905.29 on day 180)
c. Contract to sell 90-day bills in 90
days. (By then the bills you bought
in the previous step will have 90
days left to maturity, with value at
day 90 of 1,008,045.53 because of
Prof. Kensinger
Fall 2013
The risk-free profit from this arbitrage
will be 1,008,045.53 – $1,007,424.38 =
$621.15 to be received 90 days from
now. Repeat this 10,000 times, and
then retire.
9.
Yes. Round Rock Bank will have the
following annual cash flow stream for
the life of the arrangement:
&
# TBill + 2%
% − ( LIBOR + 1%)(
$1,000, 000 × %
+ ( LIBOR + 1%)(
(
%
$ − (TBill + 1%) '
= $1,000, 000 × 1% = $10, 000
10. The airline's implied value as a going
concern is $1 billion – $750 million –
$600 million – $200 million = negative
$550 million. Thus there is an arbitrage
opportunity with potential profit of
$550 million. To collect, buy BigWorld
for $1 billion and sell the two viable
businesses for $1.35 billion. Then
liquidate the airline and collect another
$200 million from its planes. (Oh! How
dastardly!)
11. This is a violation of put-call parity,
which offers a nice opportunity for
arbitrage. To take advantage of it, buy
stock and puts for $97.50 per share.
Sell calls and bonds (that is, write a call
and borrow the present value of the
exercise price) and receive $98.00 per
share. Having bought a put and written
a call with exercise price of $90 per
share, you are committed to sell your
stock for $90, which will be just enough
money to pay off the loan. Your future
obligations are covered, and you can
still pocket a profit of 50¢ per share. If
you can do this in a large volume, say
100 contracts at 100 shares per contract,
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Fina 4200
Solutions: Problem Set 4
you'll make an arbitrage profit of
$5,000.
12. This is a violation of put-call parity
which offers an opportunity for
arbitrage via a “box” spread. You could
take advantage of it without leaving the
options market: First, sell calls at 90
and puts at 85. Then buy calls at 85 and
puts at 90. The net outflow would be
$2.50 per share of volume. Regardless
of what happens to the market price of
the stock, you are committed to buy
stock for $85 and sell it for $90 on the
expiration date (placing $5 in your
pocket at expiration). Thus, you will
double your money in a few months
with no risk. (For comparison, if you
bought Treasury securities, it would
cost $4.94 today in order to receive $5 at
the expiration date of the options.)
13. Phillips is not earning as much reward
in relation to total risk as the market
portfolio, but not all of the risk involved
is relevant. We need to know Phillip’s
β. Therefore, D is the right choice.
14. Unlike Phillips, ARCO is earning more
reward in relation to risk than the
market, even in terms of total risk.
Therefore, ARCO's price should
increase to reflect its high performance.
The correct choice is B.
Fall 2013
15. As we move from one stock to another,
we exchange 10 units of reward for
each unit of risk, except in the case of
stock B. To be in line with the others, it
should have expected return of 10%. It
has abnormally high reward, and
therefore is undervalued.
16. C. The opportunity cost of capital for
the stock is 5% + .8(15%–5%) = 13%.
Since this is in balance with the
expected return, the stock is priced at
equilibrium.
17. B. Option response is in the same
direction, but less than 1-for-1
18. B. Option B has longer to expiration.
19. The lower the exercise price, the higher
the value of the call option. Prior to
expiration, A is worth more than B.
After expiration, both are worthless.
20. A
21. A. The combination would be worth
$150 million. After paying $100
million for Newton, that leaves only
$50 million for Megalithic
shareholders. This is $50 million less
than they had before.
22. A. Market is inverted, thus
discouraging storage and lowering the
price today (in anticipation of the
increased supply in the future).
23. D
Prof. Kensinger
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