In-class questions
1. Assume the following information:
-
Spot rate of Canadian dollar: $.80
90-day forward rate of Canadian dollar: $.79
90-day Canadian interest rate: 4%
90-day US interest rate: 2.5%
Assume the investor has $1m or C$ 1,25m.
a.
b.
c.
d.
Graph the IRP.
What is the arbitrage opportunity?
How much is the profit?
What market forces would occur to eliminate any further possibilities of covered
interest arbitrage?
ANSWER:
a. Graph the IRP => b. arbitrage opportunity
c. $1,000,000/$.80 = C$1,250,000 × (1.04) = C$1,300,000 × $.79 = $1,027,000
Yield = ($1,027,000 – $1,000,000)/$1,000,000 = 2.7%, which exceeds the yield in
the U.S. over the 90-day period.
d. The Canadian dollar's spot rate should rise, and its forward rate should fall; in
addition, the Canadian interest rate may fall and the U.S. interest rate may rise.
2. Assume the following information:
- Spot rate of Mexican Peso: $0.100
- 180-day forward rate of Mexican Peso: $0.098
- 180-day Mexican interest rate: 6%
- 180-day U.S. interest rate: 5%
Assume the investor has $1m or MXP 10m.
a.
b.
c.
d.
Graph the IRP.
What is the arbitrage opportunity?
How much is the profit?
What market forces would occur to eliminate any further possibilities of covered
interest arbitrage?
ANSWER:
- You are supposed to do that yourselves.
Chapter 4: Problems
A. Speculation Based on Expected Exchange Rate
Movements
1. You are working for Bank of America as a financial analyst. You expect that the MXP/$
spot rate will change in 15 days, from $.16 to $.13. You have the following information:
Lending Rate
Borrowing Rate
U.S. dollar
8.0%
8.3%
Mexican peso
8.5%
8.7%
Assume that you can borrow either $10 million or 70 million MXP in the interbank
market, depending on which currency it wants to borrow.
a. Assume that you have no deposited funds, what is your strategy and how much is
the profit that you can get from the strategy?
b. Assume all the preceding information with this exception: the spot rate will change
from $.15 to $.17 in 30 days, what is your strategy and how much is the profit that
you can get from the strategy?
2. You are CFO of Bank of America. Your staff tell you that the S$/$ spot rate is likely to
change in 60 days, from $.43 to $.42. The lending and borrow rates in the interbank
market are as below:
Lending Rate
U.S. dollar
Singapore dollar
Borrowing Rate
7.0%
22.0%
7.2%
24.0%
Your staff suggest that you should borrow S$ 10 million in the interbank market and
invest the funds in $ for 60 days. Will you follow their suggestion and why?
3. Your bank expects that the pound will depreciate from $1.70 to $1.68 in one year. You
have no money to invest, but you could borrow money to invest. You can borrow either 1
million dollars or 1 million pounds for one year. You can borrow dollars at 6% or Bitish
pounds at 5% for one year. You can invest in a risk-free dollar deposit at 5% for one year
or a risk-free British deposit at 4% for one year. How much is the profit that you can get
from the strategy?
B. International Arbitrage and Interest Rate Parity
4. Locational Arbitrage. Assume the following information:
Beal Bank Yardley Bank
Bid price of New Zealand dollar
$.401
$.398
Ask price of New Zealand dollar
$.404
$.400
Given this information, is locational arbitrage possible? If so, explain the steps involved
in locational arbitrage, and compute the profit from this arbitrage if you had $1,000,000
to use. What market forces would occur to eliminate any further possibilities of
locational arbitrage?
5. Triangular Arbitrage. Assume the following information:
Quoted Price
Value of Canadian dollar in U.S. dollars
$.90
Value of New Zealand dollar in U.S. dollars
$.30
Value of Canadian dollar in New Zealand dollars
NZ$3.02
Given this information, is triangular arbitrage possible? If so, explain the steps that
would reflect triangular arbitrage, and compute the profit from this strategy if you had
$1,000,000 to use. What market forces would occur to eliminate any further possibilities
of triangular arbitrage?
6 Covered Interest Arbitrage. Assume the following information:
Quoted Price
Spot rate of Canadian dollar
$.80
90-day forward rate of Canadian dollar
$.79
90-day Canadian interest rate
90-day U.S. interest rate
4%
2.5%
Given this information, what would be the yield (percentage return) to a U.S. investor
who used covered interest arbitrage? (Assume the investor invests $1,000,000.) What
market forces would occur to eliminate any further possibilities of covered interest
arbitrage?
7. Covered Interest Arbitrage. Assume the following information:
Spot rate of Mexican peso
= $.100
180-day forward rate of Mexican peso
= $.098
180-day Mexican interest rate
= 6%
180-day U.S. interest rate
= 5%
Given this information, is covered interest arbitrage worthwhile for Mexican investors
who have pesos to invest? Explain your answer.
8. Covered Interest Arbitrage in Both Directions. The one-year interest rate in New
Zealand is 6 percent. The one-year U.S. interest rate is 10 percent. The spot rate of the
New Zealand dollar (NZ$) is $.50. The forward rate of the New Zealand dollar is $.54. Is
covered interest arbitrage feasible for U.S. investors? Is it feasible for New Zealand
investors? In each case, explain why covered interest arbitrage is or is not feasible.
9. Case Problem: Blades, Inc.
1. The first arbitrage opportunity relates to locational arbitrage. Holt has obtained spot
rate quotations from two banks in Thailand, Minzu Bank and Sobat Bank, both located
in Bangkok. The bid and ask prices of Thai baht for each bank are displayed in the table
below:
Minzu
Sobat
Bank
Bank
Bid
$0.0224
$0.0228
Ask
$0.0227
$0.0229
Determine whether the foreign exchange quotations are appropriate. If they are not
appropriate, determine the profit you could generate by withdrawing $100,000 from
Blades’ checking account and engaging in arbitrage before the rates are adjusted.
2. Besides the bid and ask quotes for the Thai baht provided in the previous question,
Minzu Bank has provided the following quotations for the U.S. dollar and the Japanese
yen:
Quoted Bid Price
Quoted Ask Price
Value of a Japanese yen in U.S.
dollars
$0.0085
$0.0086
Value of a Thai baht in Japanese yen
¥2.69
¥2.70
Determine whether the cross exchange rate between the Thai baht and Japanese yen is
appropriate. If it is not appropriate, determine the profit you could generate for Blades
Inc, by withdrawing $100,000 from Blades’ checking account and engaging in triangular
arbitrage before the rates are adjusted.
3. Ben Holt has obtained several forward contract quotations for the Thai baht to
determine whether covered interest arbitrage may be possible. He was quoted a forward
rate of $0.0225 per Thai baht for a 90-day forward contract. The current spot rate is
$0.0227. Ninety-day interest rates available to Blades in the U.S. are 2 percent, while
90-day interest rates in Thailand are 3.75 percent (these rates are not annualized). Holt
is aware that covered interest arbitrage, unlike locational and triangular arbitrage,
requires an investment of funds. Thus, he would like to be able to estimate the dollar
profit resulting from arbitrage over and above the dollar amount available on a 90-day
U.S. deposit.
Determine whether the forward rate is priced appropriately. If it is not priced
appropriately, determine the profit you could generate for Blades by withdrawing
$100,000 from Blades’ checking account and engaging in covered interest arbitrage.
Measure the profit as the excess amount above what you could generate by investing in
the U.S. money market.
4. Why are arbitrage opportunities likely to disappear soon after they have been
discovered? To illustrate your answer, assume that covered interest arbitrage involving
the immediate purchase and forward sale of baht is possible. Discuss how the baht’s
spot and forward rates would adjust until covered interest arbitrage is no longer
possible. What is the resulting equilibrium state called?
C. Relationships Among Inflation, Interest Rates, and
Exchange Rates
10. Estimating Depreciation Due to PPP. Assume that the spot exchange rate of the
British pound is $1.73. How will this spot rate adjust according to PPP if the United
Kingdom experiences an inflation rate of 7 percent while the United States experiences an
inflation rate of 2 percent?
11. Deriving Forecasts of the Future Spot Rate. As of today, assume the following
information is available:
U.S.
Mexico
2%
2%
11%
15%
Spot rate
—
$.20
One-year forward rate
—
$.19
Real rate of interest required
by investors
Nominal interest rate
a. Use the forward rate to forecast the percentage change in the Mexican peso over the
next year.
b. Use the differential in expected inflation to forecast the percentage change in the
Mexican peso over the next year.
c. Use the spot rate to forecast the percentage change in the Mexican peso over the
next year.
12. Integrating IRP and IFE. Assume the following information is available for the U.S.
and Europe:
U.S.
Europe
Nominal interest
rate
4%
6%
Expected inflation
2%
5%
Spot rate
-----
$1.13
One-year forward
rate
a.
b.
c.
d.
-----
$1.10
Does IRP hold? Graph the IRP.
According to PPP, what is the expected spot rate of the euro in one year?
According to the IFE, what is the expected spot rate of the euro in one year?
Reconcile your answers to parts (a). and (c).
13. IRP, PPP, and Speculating in Currency Derivatives. The U.S. three-month
interest rate (unannualized) is 1%. The Canadian three-month interest rate (unannualized)
is 4%. Interest rate parity exists. The expected inflation over this period is 5% in the U.S.
and 2% in Canada. A call option with a three-month expiration date on Canadian dollars is
available for a premium of $.02 and a strike price of $.64. The spot rate of the Canadian
dollar is $.65. Assume that you believe in purchasing power parity.
a. Determine the dollar amount of your profit or loss from buying a call option contract
specifying C$100,000.
b. Determine the dollar amount of your profit or loss from buying a futures contract
specifying C$100,000.
14. PPP and Speculating with Currency Futures. Assume that you believe purchasing
power parity (PPP) exists. You expect that inflation in Canada during the next year will be
3%, and inflation in the U.S. will be 8%. Today the spot rate of the Canadian dollar is $.90
and the one-year futures contract of the Canadian dollar is priced at $.88. Estimate the
expected profit or loss if an investor sold a one-year futures contract today on one million
Canadian dollars and settled this contract on the settlement date.
15. IRP. The one-year risk-free interest rate in Mexico is 10%. The one-year risk-free rate
in the U.S. is 2%. Assume that interest rate parity exists. The spot rate of the Mexican peso
is $.14.
a. What is the forward rate premium?
b. What is the one-year forward rate of the peso?
c. Based on the international Fisher effect, what is the expected change in the spot
rate over the next year?
d. If the spot rate changes as expected according to the IFE, what will be the spot rate
in one year?
e. Compare your answers to (b) and (d) and explain the relationship.
ANSWERS
1a.
1. Borrow MXP70 million
2. Convert the MXP70 million to dollars:
MXP70,000,000 × $.16 = $11,200,000
3. Lend the dollars through the interbank market at 8.0% annualized over a 15-day
period. The amount accumulated in 15 days is:
$11,200,000 × [1 + (8% × 15/360)] = $11,237,333
4. Repay the peso loan. The repayment amount on the peso loan is:
MXP70,000,000 × [1 + (8.7% × 15/360)] = MXP70,253,750
5. Based on the expected spot rate of $.13, the amount of dollars needed to repay the peso
loan is:
MXP70,253,750 × $.13 = $9,132,988
6. After repaying the loan, you will have a speculative profit (if its forecasted exchange
rate is accurate) of:
$11,237,333 – $9,132,988 = $2,104,345
b. You can capitalize as follows:
1. Borrow $10 million
2. Convert the $10 million to pesos (MXP):
$10,000,000/$.15 = MXP66,666,667
3. Lend the pesos through the interbank market at 8.5% annualized over a 30-day
period. The amount accumulated in 30 days is:
MXP66,666,667 × [1 + (8.5% × 30/360)] = 66,666,667 × [1.007083] =
MXP67,138,889
4. Repay the dollar loan. The repayment amount on =the dollar loan is:
$10,000,000 × [1 + (8.3% × 30/360)] = $10,000,000 × [1.006917] = $10,069,167
5. 5. Convert the pesos to dollars to repay the loan. The amount of dollars to be
received in 30 days (based on the expected spot rate of $.17) is:
MXP67,138,889 × $.17 = $11,413,611
6. The profits are determined by estimating the dollars available after repaying the
loan:
$11,413,611 – $10,069,167 = $1,344,444
2.
1. Borrow S$10 million
2. Convert the S$10 million to $:
S$10,000,000 × $.43 = $4,300,000
3. Lend the $ through the interbank market at 7% annualized over a 60-day
period. The amount accumulated in 60 days is:
$4,300,000 × [1 + (7% × 60/360)] = $4,350,167
4. Repay the S$ loan. The repayment amount on the S$ loan is:
S$10,000,000 × [1 + (24% × 60/360)] = S$10,400,000
5. Convert the S$ to $ to repay the loan. The amount of $ to be received in 60 days
(based on the expected spot rate of $.42) is:
$4,350,167: $.42 = S$10,357,540
6. The profits are determined by estimating the dollars available after repaying the
loan:
S$10,357,540 - S$10,400,000 = - S$42,460
ð You shouldn’t follow their sugesstion.
3.
1. Borrow £1 million
2. Convert the £1 million to $:
£1,000,000 × $1.70 = $1,700,000
3. Lend the $ through the interbank market at 5% annualized over a year. The amount
accumulated in a year is:
$1,700,000 × (1 + 5%) = $1,785,000
4. Repay the £ loan. The repayment amount on the £ loan is:
£1,000,000 × (1 + 5%) = £1,050,000 = $1,764,000
5. The profits are determined by estimating the dollars available after repaying the
loan:
$1,785,000- $1,764,000 = $21,000 or £12,500
4. Yes. One could purchase New Zealand dollars at Yardley Bank for $.40 and sell them to
Beal Bank for $.401. With $1 million available, 2.5 million New Zealand dollars could be
purchased at Yardley Bank. These New Zealand dollars could then be sold to Beal Bank
for $1,002,500, thereby generating a profit of $2,500.
The large demand for New Zealand dollars at Yardley Bank will force this bank's ask
price on New Zealand dollars to increase. The large sales of New Zealand dollars to Beal
Bank will force its bid price down. Once the ask price of Yardley Bank is no longer less
than the bid price of Beal Bank, locational arbitrage will no longer be beneficial.
5. Yes. The appropriate cross exchange rate should be 1 Canadian dollar = 3 New Zealand
dollars. Thus, the actual value of the Canadian dollars in terms of New Zealand dollars is
more than what it should be. One could obtain Canadian dollars with U.S. dollars, sell the
Canadian dollars for New Zealand dollars and then exchange New Zealand dollars for U.S.
dollars. With $1,000,000, this strategy would generate $1,006,667 thereby representing a
profit of $6,667.
[$1,000,000/$.90 = C$1,111,111 × 3.02 = NZ$3,355,556 × $.30 = $1,006,667]
The value of the Canadian dollar with respect to the U.S. dollar would rise. The value
of the Canadian dollar with respect to the New Zealand dollar would decline. The value
of the New Zealand dollar with respect to the U.S. dollar would fall.
6. $1,000,000/$.80 = C$1,250,000 × (1.04)
= C$1,300,000 × $.79
= $1,027,000
Yield = ($1,027,000 – $1,000,000)/$1,000,000 = 2.7%, which exceeds the yield in the U.S.
over the 90-day period.
The Canadian dollar's spot rate should rise, and its forward rate should fall; in addition,
the Canadian interest rate may fall and the U.S. interest rate may rise.
7. To answer this question, begin with an assumed amount of pesos and determine the yield
to Mexican investors who attempt covered interest arbitrage. Using MXP1,000,000 as the
initial investment:
MXP1,000,000 × $.100 = $100,000 × (1.05) = $105,000/$.098 = MXP1,071,429
Mexican investors would generate a yield of about 7.1% ([MXP1,071,429 –
MXP1,000,000]/MXP1,000,000), which exceeds their domestic yield. Thus, it is
worthwhile for them.
8. To determine the yield from covered interest arbitrage by U.S. investors, start with an
assumed initial investment, such as $1,000,000.
$1,000,000/$.50 = NZ$2,000,000 × (1.06)
= NZ$2,120,000 × $.54 = $1,144,800
Yield = ($1,144,800 – $1,000,000)/$1,000,000 = 14.48%
Thus, U.S. investors can benefit from covered interest arbitrage because this yield
exceeds the U.S. interest rate of 10 percent.
To determine the yield from covered interest arbitrage by New Zealand investors, start
with an assumed initial investment, such as NZ$1,000,000:
NZ$1,000,000 × $.50 = $500,000 × (1.10)
= $550,000/$.54 = NZ$1,018,519
Yield = (NZ$1,018,519 – NZ$1,000,000)/NZ$1,000,000 = 1.85%
Thus, New Zealand investors would not benefit from covered interest arbitrage since the
yield of 1.85% is less than the 6% that they could receive from investing their funds in
New Zealand.
9. Solution to Continuing Case Problem: Blades, Inc.
Locational Arbitrage
1. Buy Thai baht from Minzu Bank ($100,000/$0.0227)
4,405,286.34
2. Sell Thai baht to Sobat Bank (THB4,405,286.34 × $0.0228)
100,440.53
3. Dollar profit ($100,440.53 – $100,000)
440.53
ANSWER: Triangular arbitrage is possible.
Triangular Arbitrage
1. Exchange dollars for Thai baht ($100,000/$0.0227)
4,405,286.34
2. Convert the Thai baht into Japanese yen (THB4,405,286.34 ×
¥2.69)
11,850,220.25
3. Convert the Japanese yen into dollars (¥11,850,220.26 × $0.0085)
100,726.87
4. Dollar profit ($100,726.87 – $100,000)
726.87
ANSWER: Covered interest arbitrage is possible.
Covered Interest Arbitrage
1. On Day 1, convert U.S. dollars to Thai baht and set up a 90-day
deposit
account at a Thai bank ($100,000/$0.0227)
4,405,286.34
2. In 90 days, the Thai deposit will mature to THB4,405,286.34 ×
1.0375,
which is the amount to be sold forward
4,570,484.58
3. In 90 days, convert the Thai baht into U.S. dollars at the agreedupon rate
(THB4,570,484.58 × $0.0225)
4. Dollar amount available on a 90-day U.S. deposit ($100,000 × 1.02)
102,835.90
102,000.00
5. Dollar profit over and above the dollar amount available on a 90-day 2,835.90
U.S. deposit ($102,835.90 – $100,000)
ANSWER: Arbitrage opportunities are likely to disappear soon after they have been
discovered because of market forces. Due to the actions taken by arbitrageurs, supply
and demand for the foreign currency adjust until the mispricing disappears. For
example, covered interest arbitrage involving the immediate purchase and subsequent
sale of Thai baht would place upward pressure on the spot rate of the Thai baht and
downward pressure on the Thai baht forward rate until covered interest arbitrage is no
longer possible. At that point, interest rate parity exists, and the interest rate
differential between the two countries is exactly offset by the forward premium or
discount.
10. According to PPP, the exchange rate of the pound will depreciate by 4.7 percent.
Therefore, the spot rate would adjust to $1.73 × [1 + (–.0467)] = $1.649.
11. a. ($.19 – $.20)/$.20 = –.05, or –5%
b. (1.09)/(1.13) – 1 = -.0353 or -3.53%; the negative sign represents depreciation of the
peso.
c. zero percent change
12. a.
p=
(1 + ih )
−1
(1 + i f )
(1.04)
−1
(1.06)
= −1.89%
=
Therefore, the forward rate of the euro should be $1.13 × (1 – .0189) = $1.109. IRP does
not hold in this case.
b.
ef =
(1 + I h )
−1
(1 + I f )
(1.02)
−1
(1.05)
= −2.86%
=
According to PPP, the expected spot rate of the euro in one year is $1.13 × (1 – 2.86%) =
$1.098.
c.
ef =
(1 + ih )
−1
(1 + i f )
(1.04)
−1
(1.06)
= −1.89%
=
According to the IFE, the expected spot rate of the euro in one year is $1.13 × (1 –
1.89%) =
$1.1086.
Parts a and c combined say that the forward rate premium or discount is exactly equal
to the expected percentage appreciation or depreciation of the euro.
13. a. The expected change in the Canadian dollar’s spot rate is:
(1.05)/(1.02) – 1 = 2.94%.
Therefore, the expected spot rate in 3 months is $.65 × (1.0294) = $.66911.
The net profit per unit on a call option is $.66911 – $.64 – $.02 = $.00911.
For the contract, the net profit is $.00911 × 100,000 = $911.
b. According to IRP, the futures rate premium should be (1.01)/(1.04) – 1 = –2.88%
Therefore, the futures rate should be $.65 × (1 – .0288) = $.6313.
Recall that the expected spot rate in 3 months is $.65 × (1.0294) = $.66911. The expected
net profit per unit from buying a futures contract is $.66911 – $.6313 = $.03781. For the
contract, the net profit is $.03781 × 100,000 = $4,341.
14.
a. According to interest rate parity, the forward premium is
(1 + .02)
− 1 = −.07273
(1 + .10)
b. The forward rate is $.14 × (1 – .07273) = $.1298.
c. According to the IFE, the expected change in the peso is:
(1 + .02)
− 1 = −.07273
(1 + .10)
or –7.273%
d. $.14 × (1 – .07273) = $.1298
e. The answers are the same. When IRP holds, the forward rate premium and the
expected percentage change in the spot rate are derived in the same manner. Thus, the
forward premium serves as the forecasted percentage change in the spot rate according
to IFE.