Chapter
2
The Foreign Exchange Market
QUESTIONS
1. What is an exchange rate?
Answer: An exchange rate is the relative price of two currencies, like the U.S. dollar price of
the euro, the Thai baht price of the Malaysian ringgit, or the Mexican peso price of the
Canadian dollar.
2. What is the structure of the foreign exchange market? Is it like the New York Stock
Exchange?
Answer: The interbank foreign exchange market is a very large, diverse, over-the-counter
market, not a physical trading place where buyers and sellers gather to agree on a price to
exchange currencies. Traders, who are employees of financial institutions in the major
financial cities around the world, deal with each other primarily over the phone or via
computer, with written or formal electronic confirmations of transactions occurring only
later.
3. What is a spot exchange rate contract? When does delivery occur on a spot contract?
Answer: When currencies in the interbank spot market are traded, certain business
conventions are followed. For example, when the trade involves the U.S. dollar, business
convention dictates that spot contracts are settled in 2 business days—that is, the payment of
one currency and receipt of the other currency occurs in 2 business days. One business day is
necessary because of the back-office paperwork involved in any financial transaction. The
second day is needed because of the time zone differences around the world.
Several exceptions to the 2-business-day rule are noteworthy. First, for exchanges
between the U.S. dollar and the Canadian dollar or the Mexican peso, the rule is 1 business
day. Second, if the transaction involves the dollar and the first of the 2 days is a holiday in
the United States but not in the other settlement center, the first day is counted as a business
day for settlement purposes. Third, Fridays are not part of the business week in most Middle
Eastern countries, although Saturdays and Sundays are. Hence, non–Middle Eastern
currencies settle on Fridays, and Middle Eastern currencies settle on Saturdays.
©2017 Cambridge University Press
2 Chapter 2 The Foreign Exchange Market
4. What was the Japanese yen spot price of the U.S. dollar on July 21, 2015?
Answer: Examining Exhibit 2.5 for July 21, 2015 we find that the Japanese yen spot price of
the U.S. dollar was ¥123.84/$.
5. What was the U.S. dollar spot price of the Swiss franc on July 21, 2015?
Answer: Examining Exhibit 2.5 for July 21, 2015, we find that the U.S. dollar spot price of
the Swiss franc was $1.0425/CHF.
6. How large are the bid–ask spreads in the interbank spot market? What is their
purpose?
Answer: The purpose of the bid-ask spread is to allow traders to profit by buying a currency
at a low bid price and selling that currency at a higher ask price. Bid–ask spreads in the spot
foreign exchange market are quite small, often only two or three basis points. For example, a
yen–dollar trader might quote a bid price of yen per dollar at which she is willing to buy
dollars in exchange for yen of, say, ¥123.83/$. The trader would then quote a higher ask price
at which she is willing to sell dollars for yen, say, at an exchange rate of ¥123.85/$. This
percentage bid-ask spread is
¥123.85/$ - ¥123.83/$
100 = 0.02%
( ¥123.85/$ + ¥123.83/$ ) / 2
7. What was the euro price of the British pound on July 21, 2015? Why?
Answer: We can find this information two ways. The cross-rate quote from Exhibit 2.6 is
€1.42725/£. The exchange rates of euros per dollar and dollars per pound from Exhibit 2.5
are €0.9153/$ and $1.5593/£. We know that there would be a triangular arbitrage possibility
if the cross-rate differs from the indirect rate using the dollar as an intermediary. Thus, we
find the same value if we calculate
€0.9153/$  $1.5593/£ = €1.42723/£
8. If the direct euro price of the British pound is higher than the indirect euro price of the
British pound using the dollar as a vehicle currency, how could you make a profit by
trading these currencies?
Answer: If the direct euro price of the British pound is higher than the indirect euro price of
the British pound using the dollar as a vehicle currency, there would be a triangular arbitrage.
We would want to buy pounds at the indirect low price and sell pounds at the direct high
price. Suppose in Question 7 that the direct price euros per pound were €1.4650/£ and the
exchange rates versus the dollar are €0.9153/$ and $1.5593/£. The indirect euro price of the
pound is therefore
€0.9153/$  $1.5593/£ = €1.4272/£
©2017 Cambridge University Press
Chapter 2: The Foreign Exchange Market 3
If we start with €10,000,000, we can convert euros to dollars and get
€10,000,000 / €0.9153/$ = $10,925,380.
Converting these dollars into pounds gives
$10,925,380 / $1.5593/£ = £7,006,593
Converting these pounds into euros gives
€1.4650/£  £7,006,593 = €10,264,659.
Thus, we make a profit of €264,659 or 2.65%.
9. What is an appreciation of the dollar relative to the pound? What happens to the dollar
price of the pound in this situation?
Answer: An appreciation of the dollar relative to the pound means that it takes fewer dollars
to buy a pound, so the dollar price of the pound falls. This situation is also described as the
dollar is stronger in the foreign exchange market, the pound has depreciated versus the dollar,
and the pound is weaker in the foreign exchange market.
10. What is a depreciation of the Thai baht relative to the Malaysian ringgit? What
happens to the baht price of the ringgit in this situation?
Answer: A depreciation of the Thai baht relative to the Malaysian ringgit means that it will
take more baht to buy one ringgit. Thus, the baht price of the ringgit is now higher after the
depreciation of the baht.
PROBLEMS
1. Mississippi Mud Pies, Inc. needs to buy 1,000,000 Swiss francs (CHF) to pay its Swiss
chocolate supplier. Its banker quotes bid–ask rates of CHF1.3990–1.4000/USD. What
will be the dollar cost of the CHF1,000,000?
Answer: The bank’s bid rate is CHF1.3990/$. That is the price at which the bank is willing to
buy $1 in return for CHF1.3990. The bank sells dollars at its ask price CHF1.4000/$.
Mississippi Mud Pies must sell dollars to the bank to buy CHF. Therefore, Mississippi Mud
Pies will receive the bank’s bid rate of CHF1.3990/$. The dollar cost of CHF1,000,000 is
consequently
CHF 1,000,000 / CHF1.399/$ = $714,796
2. If the Japanese yen–U.S. dollar exchange rate is ¥104.30/$, and it takes 25.15 Thai bahts
to purchase 1 dollar, what is the yen price of the baht?
©2017 Cambridge University Press
4 Chapter 2 The Foreign Exchange Market
Answer: To prevent triangular arbitrage, the direct quote of the yen price of the baht (¥/THB)
must equal the yen price of the dollar times the dollar price of the baht (which is the
reciprocal of the baht price of the dollar):
¥104.30/$  1/(THB25.15/$) = ¥104.30/$  $0.03976/THB = ¥ 4.1471/THB
3. As a foreign exchange trader, you see the following quotes for Canadian dollars (CAD),
U.S. dollars (USD), and Mexican pesos (MXN):
USD0.7047/CAD
MXN6.4390/CAD MXN8.7535/USD
Is there an arbitrage opportunity, and if so, how would you exploit it?
Answer: The direct quote for the cross-rate of MXN6.4390/CAD should equal the implied
cross-rate using the dollar as an intermediary currency; otherwise there exists a triangular
arbitrage opportunity. The indirect cross rate is
MXN8.7535/USD  USD0.7047/CAD = MXN6.1686/CAD
This indirect cross rate is less than the direct quote so there is an arbitrage opportunity to
exploit between the three currencies. In this situation, buying the CAD with MXN by first
buying USD with MXN and then buying the CAD with the USD and finally selling that
amount of CAD directly for MXN would make a profit because we would be buying the
CAD at a low MXN price and selling the CAD at a high MXN price.
4. The Mexican peso has weakened considerably relative to the dollar, and you are trying
to decide whether this is a good time to invest in Mexico. Suppose the current exchange
rate of the Mexican peso relative to the U.S. dollar is MXN9.5/USD. Your investment
advisor at Goldman Sachs argues that the peso will lose 15% of its value relative to the
dollar over the next year. What is Goldman Sachs’s forecast of the exchange rate in 1
year?
Answer: One way to think of this is to say that the investment advisor is referring to the fact
that the Mexican peso price of the dollar will be 15% higher next year. In this case, the
forecast of the MXN/USD exchange rate in year 1
MXN9.5/USD  1.15 = MXN 10.925/USD
A 15% loss of value of the Mexican peso versus the U.S. dollar technically means that dollar
price of the peso is 15% lower. We know that the current USD price of the peso is
1 / (MXN9.5/USD) = USD0.105263/MXN
If this exchange rate falls by 15%, the new exchange rate will be
0.85  USD0.105263/MXN = USD0.089474/MNX
In this case the forecast for the future exchange rate measured in pesos per dollar is
1 / (USD0.089474/MXN) = MXN11.1765/USD
©2017 Cambridge University Press
Chapter 2: The Foreign Exchange Market 5
The difference arises because the simple percentage change in the exchange rate depends on
how the exchange rate is quoted.
5. Deutsche Bank quotes bid–ask rates of $1.3005/€ - $1.3007/€ and ¥104.30 - 104.40/$.
What would be Deutsche Bank’s direct asking price of yen per euro?
Answer: The direct asking price of yen per euro (¥/€) is the amount of yen that the bank
charges someone who is buying euros with yen. The bank would want this to be the same as
the price at which it sells dollars for yen (the bank’s ask price) times the price at which it
sells euros for dollars (also the bank’s ask price). Thus, the asking price of yen per euro
should be
(¥104.40/$)  ($1.3007/€) = ¥135.79/€
6. Alumina Limited of Australia has called Mitsubishi UFJ Financial Group to get its
opinion about the Japanese yen–Australian dollar exchange rate. The current rate is
¥67.72/A$, and Mitsubishi thinks the Australian dollar will weaken by 5% over the next
year. What is Mitsubishi UFJ’s forecast of the future exchange rate?
Answer: If the Australian dollar weakens by 5% over the next year, it will take 5% fewer
Japanese yen to purchase the Australian dollar. Thus, the forecast is
¥67.72/A$  (1 – 0.05) = ¥64.334/A$
7. Go to www.fxstreet.com, find the “Live Charts Window,” and plot the exchange rate of
the dollar vs. the euro with a “candle stick” high-low chart at 5 minute intervals for one
day, daily intervals for one month, and weekly intervals for one year. Now, cover the
units and ask a classmate to identify the different graphs. Are you surprised?
Answer: The Web site, fxstreet.com, provides some interesting ways to look at the data. The
point of this exercise is that the dynamics of exchange rates look quite similar at the different
intervals. The eye easily sees “trends” and other “reversals.” These are often quite difficult
to detect with statistical analysis, and in real time one never knows when the “trend” will
stop.
8. Pick 3 currencies, and go to www.oanda.com to get their current bilateral exchange
rates. Is there an arbitrage opportunity?
Answer: Oanda is an excellent source of high quality data. When we checked prices, they all
satisfied the no arbitrage requirement.
9. Go to the CLS Bank web site, www.cls-group.com, and read about In/Out Swaps. How
do they help participants manage their risks?
©2017 Cambridge University Press
6 Chapter 2 The Foreign Exchange Market
Answer: Here is the quote from the Web site:
The In/Out Swap service is an additional liquidity management tool which helps financial
institutions manage intraday liquidity effectively.
How It Works
The In/Out Swap is designed to reduce the payment obligations to CLS and to mitigate liquidity
pressures. An In/Out Swap is an intraday swap consisting of two equal and opposite FX
transactions that are agreed as an intraday swap.
One of the “legs” is settled inside CLS in order to reduce each Settlement Member’s net position
in the two relevant currencies. The other “leg” is settled outside CLS.
Benefits
The combined effect of these two transactions is a reduction in the intraday funding requirements
of the two Settlement Members, while leaving the institutions’ overall FX positions unchanged.
On average, In/Out Swaps reduce payment obligations in CLS by 75%, which results in a
funding requirement in CLS of less than 1% of the total gross settlement value.
As In/Out Swaps reduce these "in-CLS" cash positions as well as the corresponding liquidity
positions outside of CLS, Members can more easily manage liquidity flows for their non-CLS
needs, as well as in the CLS system.
©2017 Cambridge University Press
Chapter
3
Forward Markets and Transaction
Exchange Risk
QUESTIONS
1. What is a forward exchange rate? When does delivery occur on a 90-day forward
contract?
Answer: The forward exchange rate is a price quoted today for the exchange of currencies
at the maturity of the forward contract. To find the delivery date for a 90-day forward
contract, one first finds the spot value date, which is typically two business days in the
future relative to the day that the contract is made. Then, to find the forward value date, one
goes to the calendar date in three months corresponding to the calendar date of the spot
value date. If that calendar date in three months is a legitimate business day in both
countries, that date is the forward value date. If the banks in one of the countries are closed
on that date, because it is a weekend or holiday, the forward value date is the next available
business day without going out of the month. If going forward in time would take you out
of the month, you go backward in time. This rule is followed except when the spot value
day is the last business day of the current month, in which case the forward value day is the
last business day in both countries in three months (this is referred to as the end-end rule).
2. If the yen is selling at a premium relative to the euro in the forward market, is the
forward price of EUR per JPY larger or smaller than the spot price of EUR per JPY?
Answer: When the yen is selling at a premium in the forward market, the euro price of the
yen in the forward market, EUR per JPY, would be larger than the spot price of EUR per
JPY.
3. What do we mean by the expected future spot rate?
Answer: The expected future spot rate is the conditional mean of the probability
distribution of future spot rates. The probability distribution describes all of the possible
realizations (or ranges of realizations) of the future spot rate and assigns probabilities to
those values (or ranges of values). The conditional mean of the probability distribution of
future spot rates takes a probability weighted average of those possible realizations (or
©2017 Cambridge University Press
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Chapter 3: Forward Markets and Transaction Exchange Risk
ranges of realizations). We say that the expectation is conditional because we use all
available information at the time when we are describing the probability distribution.
4. How much of the probability distribution of future spot rates is between plus or
minus 2 standard deviations?
Answer: For a normal distribution, when we go from 2 standard deviations below the
conditional mean to 2 standard deviations above the conditional mean, we encompass
95.44% of the probability distribution.
5. If you are a U.S. firm and owe someone ¥10,000,000 in 180 days, what is your
transaction exchange risk?
Answer: Because you owe ¥10,000,000 in 180 days, you have a transaction exchange risk
because if you do nothing to hedge and the yen strengthens, it will take more dollars to
eliminate your yen liability.
6. What is a spot–forward swap?
Answer: A spot-forward swap involves either the purchase of foreign currency spot against
the sale of the same amount of foreign currency forward, or the sale of foreign currency
spot against the purchase of the same amount of foreign currency forward.
7. What is a forward–forward swap?
Answer: A forward-forward swap involves either the purchase of foreign currency at a
short maturity forward against the sale of the same amount of foreign currency at a longer
maturity forward, or the sale of foreign currency at the short maturity forward against the
purchase of the same amount of foreign currency at a longer maturity forward.
PROBLEMS
1. If the spot exchange rate of the yen relative to the dollar is ¥105.75, and the 90-day
forward rate is ¥103.25/$, is the dollar at a forward premium or discount? Express
the premium or discount as a percentage per annum for a 360-day year?
Answer: When the forward rate of yen per dollar is less than the spot rate of yen per dollar,
the dollar is said to be at a discount in the forward market. The magnitude of the discount is
expressed in percentage per annum by dividing the difference between the forward rate and
the spot rate by the spot rate and multiplying by reciprocal of the fraction of the year
corresponding to the maturity of the forward contract (360/N days) and by 100. Thus, the
©2017 Cambridge University Press
Chapter 3: Forward Markets and Transaction Exchange Risk 3
annualized forward discount is 9.46% because
¥103.25/$ - ¥105.75/$ 360

100 = −9.46%
¥105.75/$
90
Notice that the word “discount” implies that the forward rate is less than the spot rate.
2. Suppose today is Tuesday, August 4, 2015. If you enter into a 30-day forward contract
to purchase euros, when will you pay your dollars and receive your euros? (Hints:
September 4, 2015, is a Friday, and the following Monday is a holiday.)
Answer: To determine the value date of the forward contract, which is the day on which the
exchange of currencies happens, one must first find the spot value date. For dollar-euro
contracts, the spot value date is two business days in the future. Thus, for a spot contract on
Tuesday, August 4, 2015, the exchange of currencies would take place on Thursday,
August 6, 2015. The 30-day forward contract settles on the calendar day in the next month
corresponding to the date of spot settlement if that is a legitimate business day. The
forward contract would therefore settle on September, 6, 2015 if that is a legitimate
business day, but that date is a Sunday. Furthermore, Monday, September 7, 2015, is a
holiday, so the settlement of the forward contract would be on Tuesday, September 8,
2015.
3. As a foreign exchange trader for JPMorgan Chase, you have just called a trader at
UBS to get quotes for the British pound for the spot, 30-day, 60-day, and 90-day
forward rates. Your UBS counterpart stated, “We trade sterling at $1.7745-50, 47/44,
88/81, 125/115.” What cash flows would you pay and receive if you do a forward
foreign exchange swap in which you swap into £5,000,000 at the 30-day rate and out
of £5,000,000 at the 90-day rate? What must be the relationship between dollar
interest rates and pound sterling interest rates?
Answer: The fact that you are swapping into £5,000,000 at the 30-day rate forward rate
means that you are paying dollars and buying pounds. You would do this transaction at the
bank’s 30-day forward ask rate. To find the forward ask rate, you must realize that the 30day forward points of 47/44 indicate the amounts that must be subtracted from the spot bid
and ask quotes to get the forward rates. We know to subtract the points because the first
forward point is greater than the second. Hence, the first part of the swap would be done at
$1.7750/£ - $0.0044/£ = $1.7706/£. Therefore, to buy £5,000,000 you would pay
$1.7706/£  £5,000,000 = $8,853,000
In the second leg of the swap, you would sell £5,000,000 for dollars in the 90-day forward
market. Because you are selling pound for dollars, you transact at the 90-day forward bid
rate of $1.7745/£ - $0.0125/£ = $1.7620/£. Therefore, you would receive
$1.7620/£  £5,000,000 = $8,810,000
Notice that you get back fewer dollars than you paid, but you had use of £5,000,000 for 60
days. Thus, the pound must be the higher interest rate currency.
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4
Chapter 3: Forward Markets and Transaction Exchange Risk
4. Consider the following spot and forward rates for the yen–euro exchange rates:
Spot
30 days
60 days
90 days
180 days
360 days
146.30
145.75
145.15
144.75
143.37
137.85
Is the euro at a forward premium or discount? What are the magnitudes of the
forward premiums or discounts when quoted in percentage per annum for a 360-day
year?
Answer: The forward rates of yen per euro are lower than the spot rates. Therefore, the euro
is at a discount in the forward market. The annualized forward premium or discount for the
N day forward contract is
F-S
360

= 100
S
N days
If the value of this calculation is negative, say -2%, we say there is a 2% discount.
Therefore, the discounts are 4.51% for 30 days, 4.72% for 60 days, 4.24% for 90 days,
4.01% for 180 days, and 5.78% for 360 days.
5. As a currency trader, you see the following quotes on your computer screen:
Exch. Rate
Spot
1-month
2-month
3-month
6-month
USD/EUR
1.0435/45
20/25
52/62
75/90
97/115
JPY/USD
98.75/85
12/10
20/16
25/19
45/35
USD/GBP
1.6623/33
30/35
62/75
95/110
120/130
a. What are the outright forward bid and ask quotes for the USD/EUR at the 3-month
maturity?
Answer: The spot bid and ask quotes for USD/EUR are 1.0435/45. These quotes mean that the
bank buys euros with dollars spot at $1.0435/€, and the bank sells euros for dollars at
$1.0445/€. Because the forward points at the 3-month maturity are 75/90, we know that we
must add the points to get the outright forward bid and ask rates. Adding the points makes the
bid-ask spread in the forward market larger than the bid-ask spread in the spot market.
Consequently, the forward bid rate is $1.0435/€ + $0.0075/€ = $1.0510/€, and the forward ask
quote is $1.0445/€ + $0.0090/€ = $1.0535/€.
©2017 Cambridge University Press
Chapter 3: Forward Markets and Transaction Exchange Risk 5
b. Suppose you want to swap out of $10,000,000 and into yen for 2 months. What are the
cash flows associated with the swap?
Answer: When you swap out of $10,000,000 into yen in the spot market, you are selling
dollars to the bank. The bank buys dollars at its low bid rate of ¥98.75/$, so you get
¥98.75/$ $10,000,000 = ¥987,500,000
When you contract to buy the $10,000,000 back from the bank in the 2-month forward
market, you must pay the bank’s ask rate of
¥98.85/$ - ¥00.16/$ = ¥98.69/$
You subtract the points because the 2-month forward quote is 20/16. Subtracting the
points makes the bid-ask spread in the forward market larger than the bid-ask spread in
the spot market. Hence, the amount of yen you pay is
¥98.69/$ $10,000,000 = ¥986,900,000
c. If one of your corporate customers calls you and wants to buy pounds with
dollars in 6 months, what price would you quote?
Answer: If the customer wants to buy pounds with dollars, the customer must pay the
bank’s 6-month ask rate. The spot quotes are 1.6623/33 which means the spot ask rate is
$1.6633/£. The 6-month forward points are 120/130. We add the points because the first
one, 120, is less than the second, 130. Hence, the outright forward quote would be
$1.6633/£ + $0.0130/£ = $1.6763/£
6. Intel is scheduled to receive a payment of ¥100,000,000 in 90 days from Sony in
connection with a shipment of computer chips that Sony is purchasing from Intel.
Suppose that the current exchange rate is ¥103/$, that analysts are forecasting that the
dollar will weaken by 1% over the next 90 days, and that the standard deviation of 90day forecasts of the percentage rate of depreciation of the dollar relative to the yen is
4%.
a. Provide a qualitative description of Intel’s transaction exchange risk.
Answer: Intel is a U.S. company, and it is scheduled to receive yen in the future. A
weakening of the yen versus the dollar causes a given amount of yen to convert to fewer
dollars in the future. This loss of value could be severe if the yen depreciates by a
significant amount.
b. If Intel chooses not to hedge its transaction exchange risk, what is Intel’s expected
dollar revenue?
Answer: If Intel chooses not to hedge, the expected dollar revenue is the expected dollar
value of the ¥100,000,000. The expected spot rate incorporates a 1% weakening of the
dollar. This means that the expected yen price of the dollar is 1% less than the current
spot rate of ¥103/$ or
Et[S(t+90, ¥/$)] = 0.99 ¥103/$ = ¥101.97/$
Hence, Intel expects to receive ¥100,000,000 / ¥101.97/$ = $980,681
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Chapter 3: Forward Markets and Transaction Exchange Risk
c. If Intel does not hedge, what is the range of possible dollar revenues that
incorporates 95.45% of the possibilities?
Answer: We are told that the standard deviation of the rate of depreciation of the dollar is 4%.
The standard deviation of the future spot rate is therefore 4% of the current spot rate or 0.04
¥103/$ = ¥4.12/$. Thus, plus or minus 2 standard deviations around the conditional expected
future spot rate is
¥101.97/$ + ¥8.24/$ = ¥110.21/$
¥101.97/$ - ¥8.24/$ = ¥93.73/$
The range that encompasses 95.45% of possible future values for Intel’s receivable is therefore
¥100,000,000 / ¥110.21/$ = $907,359
¥100,000,000 / ¥93.73/$ = $1,066,894
7. Go to the Wall Street Journal’s Market Data Center and find New York closing prices for
currencies. Calculate the 180-day forward premium or discount on the dollar in terms of the
yen.
F (t ,180) − S (t )
 200
S (t )
Answer: The correct calculation is
.
8. Go to the St. Louis Federal Reserve Bank’s data base, FRED, at
http://research.stlouisfed.org/fred2/ and download data for the exchange rate of the Brazilian
real vs. the U.S. dollar. Calculate the percentage changes over a one month interval. What loss
would you take if you owed BRL 1 million in one month and the dollar depreciated by two
standard deviations.
Answer: Data on FRED for the Brazilian real per dollar start in January 1995 until the present. Our
data ended with June 2017. Using the full sample available to us, the monthly standard deviation of
the rate of appreciation of the dollar relative to the real was 4.28%. Thus, a 2 standard deviation
move would be 8.57% (we use higher accuracy than the numbers indicated here). The spot rate was
BRL2.3437/USD.
Thus, the exchange rate could change to BRL2.3437/USD x (1 - 0.0857)= BRL2.1429/USD [if the
dollar weakened by 2 standard deviations.
BRL1, 000, 000
= USD426,676
The dollar cost of the BRL1 million could go from (BRL2.3437 / USD)
BRL1, 000, 000
= USD466,657
to (BRL2.1429 / USD)
.
This is a loss of USD39,981 . Of course, this calculation assumes that the spot rate is not expected
to change so that the expected spot rate is the current spot rate.
©2017 Cambridge University Press
Chapter
6
Interest Rate Parity
QUESTIONS
1. Explain the concepts of present value and future value.
Answer: These concepts relate to the time value of money. Because interest rates are positive,
a given amount of money in the future is not worth as much today. If you want to know how
much a future amount of money is worth, you take the present value. This is the amount that
you could borrow against the future amount while using the future amount to pay the interest
plus principal on the loan. Analogously, the future value of an amount of money available
today is the value that would be available in the future if you invested today and received the
principal plus interest on the investment in the future.
2. If the dollar interest rate is positive, explain why the value of $1,000,000 received every
year for 10 years is not $10,000,000 today.
Answer: If you were to borrow against each of the annual $1,000,000 payments, the bank
would lend you progressively smaller amounts. The present value of the 10 payments could
be found by using the spot interest rates, i(t,k), at time t for year t+k in the future. The present
value would be
10
$1,000,000
PV = 
k
k=1 (1 + i(t,k) )
3. Describe how you would calculate a 5-year forward exchange rate of yen per dollar if
you knew the current spot exchange rate and the prices of 5-year pure discount bonds
denominated in yen and dollars. Explain why this has to be the market price.
Answer: The 5-year forward rate would be equal to the spot rate of yen per dollar times the
ratio of the future value in 5 years of one yen to the future value in 5 years of one dollar. The
logic is the following. If you can invest directly in yen for 5 years, you can convert one yen
into the future value of one yen in 5 years. Alternatively, you can convert the one yen into
dollars in the spot foreign exchange market, invest that dollar principal for 5 years to get the
future value of dollars, and contract today to sell those dollars in the forward market to get
back to future yen in 5 years. If the two amounts of future yen differ, there would be an
©2017 Cambridge University Press
2
Chapter 6: Interest Rate Parity
arbitrage available in which you would borrow future yen where they are cheap and invest in
future yen where they are expensive. Hence, the 5 year forward rate should satisfy
5
1 + i(t,5,¥) )
(
F(t,5,¥/$) = S(t,¥/$) ×
5
(1 + i(t,5,$) )
4. If interest rate parity is satisfied, there are no opportunities for covered interest
arbitrage. What does this imply about the relationship between spot and forward
exchange rates when the foreign currency money market investment offers a higher
return than the domestic money market investment?
Answer: If the foreign currency money market investment offers a higher return (in the
foreign currency) than the domestic money market investment, the foreign currency must be
at a discount in terms of the domestic currency in the forward market. The forward discount
locks in a capital loss when the transaction exchange risk is offset, which reduces the higher
return of the foreign currency back to the lower return offered in the domestic money market.
5. It is often said that interest rate parity is satisfied when the differential between the
interest rates denominated in two currencies equals the forward premium or discount
between the two currencies. Explain why this is an imprecise statement when the
interest rates are not continuously compounded.
Answer: Interest rate parity requires the equality of returns from investing directly in the
domestic money market versus converting domestic currency into foreign currency, investing
the foreign currency, and selling the foreign currency forward. Symbolically, we have
1
× (1 + i(t,FC) ) × F(t,DC/FC)
(1 + i(t,DC) ) =
S(t,DC/FC)
If we divide by (1 + i(t,FC) ) on both sides and subtract one from both sides, we get
i(t,DC) - i(t,FC)
F(t,DC/FC) - S(t,DC/FC)
=
S(t,DC/FC)
(1 + i(t,FC) )
The left-hand side is the interest differential between the domestic and foreign rates adjusted
for the denominator term and the right-hand side is the forward premium or discount on the
foreign currency in terms of the domestic currency.
6. What do economists mean by the external currency market?
Answer: The external currency market is an interbank market for deposits and loans that are
denominated in currencies that are not the currency of the country in which the bank is
operating. Its settlement procedures are identical to those of the foreign exchange market.
The first currency for which these deposits and loans began to trade was the dollar, and the
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Chapter 6: Interest Rate Parity
3
deposits were called Eurodollars because they were dollar-denominated deposits at European
banks. The market for other currencies came to be called the Eurocurrency market, even
though the trading might be done in Asia or the Americas. With the advent of the euro as a
currency, the term external currency market seems less confusing.
7. What determines the bid–ask spread in the external currency market? Why is it usually
so small?
Answer: The bid-ask spread in the external currency market is the difference between the bid
rate, which is the interest rate that the bank pays on its deposits and the ask rate, which is the
interest rate that the bank charges on its loans. The market is very competitive, and the bidask spreads are small. The reason is because the banks accepting the deposits and making the
loans are subject only to the regulations of the government of the country in which the bank
is operating, not the government of the country that issues the money in which the deposits
and loans are denominated. These regulations include how much banks must keep on reserve
with their nation’s central bank. Because reserve requirements are often lower for foreign
currency deposits than for domestic currency deposits, banks can lend out a larger part of
these deposits. Thus, the foreign currency deposits are potentially more profitable.
8. Explain why the absence of covered interest arbitrage possibilities can be characterized
by two inequalities in the presence of bid–ask spreads in the foreign exchange and
external currency markets.
Answer: Because there are bid-ask spreads in the foreign exchange market and in the external
currency market, we do not convert from one currency to another at the same spot or forward
exchange rates, and we do not borrow at the same rate at which we lend. The absence of
covered interest arbitrage therefore is characterized by two inequalities. We cannot profit by
borrowing the domestic currency (at the ask domestic interest rate), converting to the foreign
currency (at the ask spot rate of domestic currency per unit of foreign currency), lending the
foreign currency (at the foreign bid interest rate), and converting to the domestic currency in
the forward market (at the bid forward rate of domestic currency per unit of foreign
currency). Similarly, we cannot profit by borrowing the foreign currency (at the ask foreign
interest rate), converting to domestic currency (at the bid spot rate of domestic currency per
unit of foreign currency), lending the domestic currency (at the domestic bid interest rate),
and converting to the foreign currency forward (at the ask forward rate of domestic currency
per unit of foreign currency).
9. Describe the sequence of transactions required to do a covered interest arbitrage out of
Japanese yen and into U.S. dollars.
Answer: To do a covered interest arbitrage out of Japanese yen and into U.S. dollars, one
would borrow yen from the bank at the bank’s ask interest rate. You would owe interest on
the yen and would have to return the yen principal at the end of the investment horizon. You
©2017 Cambridge University Press
4
Chapter 6: Interest Rate Parity
would then convert the yen principal into dollars at the ask spot exchange rate of yen per
dollar. You would pay the ask rate because you are buying dollars from the bank with yen.
You would then invest the dollar principal at the bank’s bid dollar interest rate. Because you
would know how much the dollar interest plus principal would be at the end of the
investment horizon, you would contract to sell that amount of dollars forward for yen. This
forward contract would be made at the bank’s forward bid rate of yen per dollar. If the
amount of yen that you get from the forward contract exceeds the amount of yen that you
owe the bank from the initial borrowing, you have successfully done a covered interest
arbitrage.
10. Suppose you saw a set of quoted prices from a U.S. bank and a French bank such that
you could borrow dollars, sell the dollars in the spot foreign exchange market for euros,
deposit the euros for 90 days, and make a forward contract to sell euros for dollars and
make a guaranteed profit. Would this be an arbitrage opportunity? Why or why not?
Answer: It could be an arbitrage opportunity, but it could also reflect the fact that
counterparty risk differs across banks. It may be that the market knows that the default risk of
the French bank is higher than other banks, which has induced the French bank to increase its
promised deposit rates above rates charged by other banks with lower default risk. The
perceived arbitrage opportunity would be illusory in this case.
11. The interest rates on U.S. dollar–denominated bank accounts in Mexican banks are
often higher than the interest rates on bank accounts in the United States. Can you
explain this phenomenon?
Answer: Mexico has periodically gotten into balance of payments difficulties and suffered
severe depreciations of the peso. During these periods of crisis, the Mexican government
converted dollar-denominated bank deposits into peso-denominated accounts at exchange
rates that were unfavorable to the depositor, effectively expropriating some of an investor’s
principal. If there is a possibility of this type of risk or just higher default risk by the Mexican
banks than at U.S. banks, the higher dollar-denominated Mexican deposit rates would be
required to induce depositors to invest in Mexican banks.
12. What is a money market hedge? How is it constructed?
Answer: In a money market hedge you offset the underlying transaction exchange risk with
borrowing or lending in the foreign money market rather than with a forward market
transaction. For example, if the underlying business transaction gives you a liability in
foreign currency, you can borrow domestic currency, convert the principal from the
borrowing into foreign currency, and invest the foreign currency thereby acquiring a foreign
currency asset that is equivalent in value to the underlying foreign currency liability. You
would want to borrow an amount of domestic currency equal to the present value of the
foreign currency liability when converted at the spot exchange rate.
©2017 Cambridge University Press
Chapter 6: Interest Rate Parity
5
13. Suppose you are the French representative of a company selling soap in Canada.
Describe your foreign exchange risk and how you might hedge it with a money market
hedge.
Answer: As a French company, you are interested in euro profits. Selling soap in Canada will
give you Canadian dollar revenues. The euro value of these Canadian dollar revenues will
fall in value if the Canadian dollar weakens relative to the euro. To offset this loss in value,
your company should borrow in Canadian dollars.
14. What is a pure discount bond?
Answer: A pure discount bond only has one cash flow at the maturity of the bond. Hence, the
bond’s price is less than its face value, and this discount of the price from the face value
provides the return to holding the bond.
15. What is the term structure of interest rates? How are spot interest rates determined
from coupon bond prices?
Answer: The term structure of interest rates is the relation between the maturities of bonds
and the pure discount bond yields, which are the spot interest rates for the various maturities.
At the shortest maturities, one typically has spot interest rates from pure discount bond
prices. One can then begin to use coupon bonds at longer maturities, discounting the early
coupons at the known spot interest rates for those maturities and determining the final spot
interest rate by finding the discount rate such that discounting the final coupon and principal
payment provides a present value equal to the bond price minus the present value of the
intervening coupon payments.
16. How does a coupon bond’s yield to maturity differ from the spot interest rate that
applies to cash flows occurring at the maturity of the bond? When are the two the
same?
Answer: A coupon bond’s yield to maturity is the internal rate of return that sets the present
value of the promised coupon payments and principal payment equal to the bond price. The
yield to maturity is therefore a kind of average of the spot interest rates at various maturities.
The yield to maturity equals the spot interest rates at various maturities only when the term
structure of interest rates is flat, that is, when the spot interest rates at various maturities are
all identical.
©2017 Cambridge University Press
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Chapter 6: Interest Rate Parity
PROBLEMS
1. In the entry forms for its contests, Publisher’s Clearing House states, “You may have
already won $10,000,000.” If the Prize Patrol visits your house to inform you that you
have won, it offers you $333,333.33 each and every year for 30 years. If the interest rate
is 8% p.a., what is the actual present value of the $10,000,000 prize?
Answer: The present value of 30 annual payments of $333,333.33 when discounted at 8% is
30
$333,333.33
= $3, 752,594.41

1.08k
k =1
This value can be found in Excel by using the function NPV(rate, cashflows), where rate =
8% and cashflows refers to a sequence of 30 cells that all have the value $333,333.33.
2. Suppose the 5-year interest rate on a dollar-denominated pure discount bond is 4.5%
p.a., whereas in France, the euro interest rate is 7.5% p.a. on a similar pure discount
bond denominated in euros. If the current spot rate is $1.08/€, what is the value of the
forward exchange rate that prevents covered interest arbitrage?
Answer: We know that the 5-year forward rate must satisfy
5
1+i(t,5,$) )
(
1.0455
F(t,5,$/€) = S(t,$/€)×
=
$1.08/€
×
= $0.9375/€
5
1.0755
(1+i(t,5,€) )
3. Carla Heinz is a portfolio manager for Deutsche Bank. She is considering two
alternative investments of EUR10,000,000: 180-day euro deposits or 180-day Swiss
francs (CHF) deposits. She has decided not to bear transaction foreign exchange risk.
Suppose she has the following data: 180-day CHF interest rate, 8% p.a., 180-day EUR
interest rate, 10% p.a., spot rate EUR1.1960/CHF, 180-day forward rate,
EUR1.2024/CHF. Which of these deposits provides the higher euro return in 180 days?
If these were actually market prices, what would you expect to happen?


180 
Answer: The euro return to investing directly in euros is 5% = 10% 
 , so the euros
360

available in 180 days is EUR10,000,000  1.05 = EUR10,500,000. Alternatively, the
EUR10,000,000 can be converted into Swiss francs at the spot rate of EUR1.1960/CHF. The
Swiss francs purchased would equal EUR10,000,000 / EUR1.1960/CHF = CHF8,361,204.


180 
This amount of Swiss francs can be invested to provide a 4% =  8% 
 return over the
360

next 180 days. Hence, interest plus principal on the Swiss francs is CHF8,361,204  1.04 =
CHF8,695,652. If we sell this amount of Swiss francs forward for euros at the 180-day
forward rate of EUR1.2024/CHF, we get a euro return of CHF8,695,652  EUR1.2024/CHF
= EUR10,455,652. This is less than the return from investing directly in euros.
©2017 Cambridge University Press
Chapter 6: Interest Rate Parity
7
If these were the actual market prices, you should expect investors to do covered interest
arbitrages. Investors would borrow Swiss francs, which would tend to drive the CHF interest
rate up; they would sell the Swiss francs for euros in the spot foreign exchange market,
which would tend to lower the spot rate of EUR/CHF; they would deposit euros, which
would tend to drive the EUR interest rate down; and they would contract to buy CHF with
EUR in the 180-day forward market, which would put upward pressure on the forward rate of
EUR/CHF. Each of these actions would help bring the market back to equilibrium.
4. If the 30-day yen interest rate is 3% p.a., and the 30-day euro interest rate is 5% p.a., is
there a forward premium or discount on the euro in terms of the yen? What is the
magnitude of the forward premium or discount?
Answer: We know that the high interest rate currency must sell at a forward discount when
priced in the low interest rate currency to prevent a covered interest arbitrage. Therefore the
euro is at a discount in the forward market. To determine the magnitude of the discount,
recognize that interest rate parity requires equality of the return to investing in yen versus
converting the yen principal into euros, investing the euros, and selling the euro principal
plus interest in the forward market for yen:
1
× (1 + i(€) ) × F(¥/€)
(1 + i(¥) ) =
S(¥/€)
Solving this expression for the forward premium, we find
F(¥/€) - S(¥/€)
i(¥) - i(€)
=
S(¥/€)
(1 + i(€) )
The de-annualized interest rates are 0.0025 = (3/100)  (30/360) for the yen and
0.004167 = (5/100)  (30/360) for the euro. The right-hand side of the above expression is
therefore -0.00166. The annualized value is -0.00166  (100)  (360/30) = -1.99%. We
therefore say that the euro sells at an annualized discount of 1.99%.
5. Suppose the spot rate is CHF1.4706/$ in the spot market, and the 180-day forward rate
is CHF1.4295/$. If the 180-day dollar interest rate is 7% p.a., what is the annualized
180-day interest rate on Swiss francs that would prevent arbitrage?
Answer: Interest rate parity requires equality of the return to investing in CHF versus
converting the CHF principal into dollars, investing the dollars, and selling the dollar
principal plus interest in the forward market for CHF:
1
× (1 + i($) ) × F(CHF/$)
(1 + i(CHF) ) =
S(CHF/$)
If we de-annualize the dollar interest rate, we find that the 180 day interest rate is 0.035.
Hence, the Swiss franc interest rate that prevents arbitrage is
1
i(CHF) =
× 1.035 × CHF1.4295/$ - 1 = 0.0061
CHF1.4706/$
©2017 Cambridge University Press
8
Chapter 6: Interest Rate Parity
If we annualize this value, we find 0.0061  (100)  (360/180) = 1.21%.
6. As a trader for Goldman Sachs you see the following prices from two different banks:
1-year euro deposits/loans:
6.0% – 6.125% p.a.
1-year Malaysian ringgit deposits/loans: 10.5% – 10.625% p.a.
Spot exchange rates:
MYR 4.6602 / EUR – MYR 4.6622 / EUR
1-year forward exchange rates:
MYR 4.9500 / EUR – MYR 4.9650 / EUR
The interest rates are quoted on a 360-day year. Can you do a covered interest
arbitrage?
Answer: We need to check the two inequalities that characterize the absence of covered
interest arbitrage. In the first, we will borrow euros at 6.125%, convert to ringgits in the spot
market at MYR4.6602 / EUR, invest the ringgits at 10.5%, and sell the ringgit principal plus
interest forward for euros at MYR4.9650 / EUR. We find that
MYR4.6602
1
1.06125 >
× 1.105 ×
= 1.0372
EUR
MYR4.9650/EUR
Thus, it is not profitable to try to arbitrage in this direction as the amount that we would
owe is greater than the amount that we would gain.
Let’s try the other direction, arbitraging out of ringgits into euros and covering the
foreign exchange risk. We will borrow ringgits at 10.625%, convert to euros in the spot
market at MYR4.6622 / EUR, invest the euros at 6.0%, and sell the euro principal plus
interest forward for ringgits at MYR4.9500 / EUR. We find that
1
MYR4.9500
1.10625 <
× 1.06 ×
= 1.1254
MYR4.6622/EUR
EUR
Thus, there is a possible arbitrage opportunity because the amount that we owe from
borrowing ringgits is less than the amount that we gain by converting from ringgits to euros,
investing the euros, and covering the transaction exchange risk with a forward sale of euros
for ringgits.
7. As an importer of grain into Japan from the United States, you have agreed to pay
$377,287 in 90 days after you receive your grain. You face the following exchange rates
and interest rates: spot rate, ¥106.35/$, 90-day forward rate ¥106.02/$, 90-day USD
interest rate, 3.25% p.a., 90-day JPY interest rate, 1.9375% p.a.
a. Describe the nature and extent of your transaction foreign exchange risk.
Answer: As a Japanese grain importer, you are contractually obligated to pay $377,287 in
90 days. Any weakening of the yen versus the dollar will increase the yen cost of your
grain. The possible loss is unbounded.
©2017 Cambridge University Press
Chapter 6: Interest Rate Parity
9
b. Explain two ways to hedge the risk.
Answer: You could hedge your risk by buying dollars forward at ¥106.02/$.
Alternatively, you could determine the present value of the dollars that you owe and buy
that amount of dollars today in the spot market. You could borrow that amount of yen to
avoid having to pay today.
c. Which of the alternatives in part b is superior?
Answer: If you do the forward hedge, you will have to pay
¥106.02/$  $377,287 = ¥39,999,967.74
in 90 days. If you do the money market hedge, you first need to find the present value of
$377,287 at 3.25%. The de-annualized interest rate is (3.25/100)  (90/360) = 0.008125.
Thus, the present value is
$377,287 / 1.008125 = $374,246.25
Purchasing this amount of dollars in the spot market costs
¥106.35/$  $374,246.25 = ¥39,801,088.69
To compare this value to the forward hedge, we must take its future value at 1.9375% p.a.
The de-annualized interest rate is (1.9375/100)  (90/360) = 0.00484375, and the future
value is
¥39,801,088.69  (1.00484375) = ¥39,993,875.21
The cost of the money market hedge is essentially the same as the cost of the forward
hedge because interest rate parity is satisfied.
8. You are a sales manager for Google Nexus and export cellular phones from the United
States to other countries. You have just signed a deal to ship phones to a British
distributor. The deal is denominated in pounds, and you will receive £700,000 when the
phones arrive in London in 180 days. Assume that you can borrow and lend at 7% p.a.
in U.S. dollars and at 10% p.a. in British pounds. Both interest rate quotes are for a
360-day year. The spot exchange rate is $1.4945/£, and the 180-day forward exchange
rate is $1.4802/£.
a. Describe the nature and extent of your transaction foreign exchange risk.
Answer: As a U.S. exporter, you have a contract to receive £700,000 in 180 days. Any
weakening of the pound versus the dollar will decrease the dollar value of your pounddenominated receivable. Large losses are possible as the dollar value could go to zero,
although that is highly unlikely.
©2017 Cambridge University Press
10 Chapter 6: Interest Rate Parity
b. Describe two ways of eliminating the transaction foreign exchange risk.
Answer: You could hedge by selling pounds forward for dollars. Alternatively, you could
do a money market hedge in which you borrow the present value of the pounds, and
convert the loan principal to dollars in the spot market, and then use the pound receivable
to pay off the interest plus principal on the loan at maturity.
c. Which of the alternatives in part b is superior?
Answer: The forward hedge gives
$1.4802/£  £700,000 = $1,036,140
in 180 days. The money market hedge requires the present value of the £700,000. The
interest rate is (10/100)  (180/365) = 0.0493. Thus, the present value is
£700,000 / 1.0493 = £667,111.41
The dollar value of this is
$1.4945/£  £667,111.41 = $996,998
To compare this to the forward hedge we must take its future value at 7% p.a. The
interest rate is (7/100)  (180/360) = 0.035. Therefore the future value is
$996,998  1.035 = $1,031,892.93
The forward hedge provides slightly more dollar revenue.
d. Assume that the dollar interest rate and the exchange rates are correct. Determine
what sterling interest rate would make your firm indifferent between the two
alternative hedges.
Answer: We know that if interest rate parity is satisfied, the money market hedge and the
forward hedge will provide the same revenue. The pound interest rate that satisfies
interest rate parity is
1
(1 + i(£) ) = S($/£) × (1 + i($) ) ×
F($/£)
The value of the right-hand side is $1.4945/£  1.035 / $1.4802/£ = 1.0450. Thus the
annualized pound interest rate that would make the firm indifferent between the forward
hedge and the money market hedge is 0.0450  100  (365/180) = 9.12%.
9. Suppose that there is a 0.5% probability that the government of Argentina will
nationalize its banking system and freeze all foreign deposits indefinitely during the
©2017 Cambridge University Press
Chapter 6: Interest Rate Parity 11
next year. If the dollar deposit interest rate in the United States is 5%, what dollar
interest would Argentine banks have to offer in order to attract deposits from foreign
investors?
Answer: If the freezing of deposits is an idiosyncratic event, then the expected value of the
return should equal the risk free return of 5%. If investors effectively get a return of zero with
0.5% probability, they must get a return of (1 + X%) with 99.5% probability, such that
[(1 + X%)  0.995] + [0  0.005] = 1.05
When we solve this equation for X%, we find X% = 5.53%. Of course, the more that you
eventually recover in the event of a freeze of deposits, the smaller the interest rate can be.
10. Suppose the market price of a 20-year pure discount bond with a face value of $1,000 is
$214.55. What is the spot interest rate for the 20-year maturity expressed in percentage
per annum?
Answer: We know that the relationship between the price of a pure discount bond and the
spot interest rate at the 20 year maturity satisfies
$1,000
P(t) =
20
(1 + i(t,20) )
Substituting the price of $214.55 and solving for i(t,20), we find
1/20
 $1,000 
i(t,20) = 
- 1 = 0.08
 $214.55 
Therefore, the spot interest rate for the 20-year maturity expressed in percentage per annum
is 8%.
11. Consider a 2-year euro-denominated bond that has a current market price of €970, a
face value of €1,000, and an annual coupon of 5%. Suppose the 1-year eurodenominated spot interest rate is 5.5%. What is the 2-year euro-denominated spot
interest rate?
Answer: The present value of a coupon paying bond is found by discounting each annual
coupon and the final principal payment at the appropriate spot interest rates for those
maturities. Thus, to find the 2-year euro-denominated spot interest rate we must solve for the
two-period spot interest rate in the following equation:
€50
€1050
€970 =
+
1.055 (1+i(t,2) )2
The answer is i(t,2) = 6.68%.
©2017 Cambridge University Press
12 Chapter 6: Interest Rate Parity
12. Consider some data drawn from Exhibit 6.5. The 1-year rates can be viewed as spot
interest rates, and the 2-year rates are yields to maturity in annualized percent. The
spot exchange rate is ¥132.192/£.
U.K.
Japan
1 year
1.105
0.370
2 year
1.770
0.430
What should be the 2-year forward rate to prevent arbitrage?
Answer: We know that if the coupon on a bond is equal to the yield to maturity on the bond, then
the bond is selling for face value. Therefore,without loss of generality, we assume that a twoyear coupon bond has a coupon of 1.77% in the U.K. and 0.43% in Japan. Thus, in the U.K. the
two-year spot interest rate satisfies
1=
0.0177
1.0177
+
(1 + 0.01105) (1 + i (2)) 2
Solving for i(2) gives 0.01776. Doing the same for the yen, we have
1=
0.0043
1.0043
+
(1 + 0.0037) (1 + i (2)) 2
Solving for i(2) gives 0.004301. Hence, the 2-year forward rate that prevents arbitrage would
satisfy
F (t , 2) =
¥132.192 1.0043012 ¥128.719

=
£
1.017762
£
13. Internet question: Go to the ICE website and find out which banks are on the LIBOR
panel for all five currencies for which the organization determines LIBOR rates.
Answer: The information can be found at https://www.theice.com/iba/libor. There are 20 banks
on the panel, but only 8 provide interest rate data on all five currencies. On June 5, 2017 these
banks were the following:
Lloyds Bank plc,
Bank of Tokyo-Mitsubishi UFJ Ltd,
Barclays Bank plc,
HSBC Bank plc,
Deutsche Bank AG (London Branch),
JPMorgan Chase Bank, N.A. London Branch,
The Royal Bank of Scotland plc,
UBS AG.
©2017 Cambridge University Press
Chapter
7
Speculation and Risk in the Foreign
Exchange Market
QUESTIONS
1. What are two ways to speculate in the currency markets without investing any money
up front?
Answer: To be long in the foreign currency, one can borrow domestic currency, convert to
foreign currency in the spot foreign exchange market, and invest in the foreign money market
while leaving the transaction exchange risk unhedged. The alternative way is to enter into a
forward contract to buy the foreign currency forward. To be short in the foreign currency,
one can borrow foreign currency, convert to domestic currency in the spot foreign exchange
market, and invest in the domestic money market while leaving the transaction exchange risk
unhedged. The alternative way is to enter into a forward contract to sell the foreign currency
forward.
2. What do financial economists mean when they discuss the conditional expectation of the
future spot exchange rate?
Answer: The conditional expectation of the future spot exchange rate is the probability
weighted average of the future possible exchange rates. It is the mean of the conditional
probability distribution of future spot rates.
3. What is the main determinant of the variability of forward market returns?
Answer: The main and only determinant of the variability of forward market returns is the
variance of the future exchange rate.
4. Describe how you construct the uncertain yen-denominated return from investing 1 yen
in the Swiss franc money market.
Answer: If you invest yen in the Swiss money market, you first must convert from yen into
Swiss francs in the spot foreign exchange market. With the Swiss francs that you get, you
invest in the Swiss money market, leaving the investment unhedged. At the end of the
©2017 Cambridge University Press
2
Chapter 7: Speculation and Risk in the Foreign Exchange Market
investment horizon, you convert from Swiss francs back into yen at the future spot exchange
rate.
5. What is a hedged foreign currency investment? What happens if you hedge your return
in Question 4?
Answer: A hedged foreign currency investment sells the known foreign currency return in the
forward market at the time of the investment. This eliminates exposure to foreign exchange
risk, but it also elements possible gains from appreciation of the foreign currency. By interest
rate parity, we know that the domestic currency return from the hedged foreign currency
investment is just the domestic currency money market return.
6. What does it mean for the 90-day forward exchange rate to be an unbiased predictor of
the future spot exchange rate?
Answer: If the forward exchange rate for 90 days is an unbiased predictor of the future spot
rate, the forward rate is equal to the expected future spot rate. While there will be forecast
errors that may be large, there will not be systematic errors on one side or the other.
Therefore, the expected forward market return is zero.
7. Why is it true that the hypothesis that the forward exchange rate is an unbiased
predictor of the future spot exchange rate is equivalent to the hypothesis that the
forward premium (or discount) on a foreign currency is an unbiased predictor of the
rate of its appreciation (or depreciation)?
Answer: When the forward exchange rate is an unbiased predictor of the future spot
exchange rate, we know that the forward rate equals the conditional expectation of the future
spot rate. For example, at the 90 day maturity, we have
F(t,90) = E t S(t+90)
Because the current spot rate, S(t), is in the information set that is used to take the conditional
expectation, we can divide by it on both sides of the above equation. Subtracting one from
both sides then gives
 S(t+90) - S(t) 
F(t,90) - S(t)
= Et 

S(t)
S(t)


This equation states that the forward premium on the foreign currency equals the expected
rate of appreciation of the foreign currency.
8. It is often claimed that the forward exchange rate is set by arbitrage to satisfy (covered)
interest rate parity. Explain how interest rate parity can be satisfied and how the
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Chapter 7: Speculation and Risk in the Foreign Exchange Market
3
forward exchange rate can be set by speculators in reference to the expected future spot
exchange rate.
Answer: Interest rate parity is a no arbitrage relation between 4 variables, the spot and
forward exchange rates and the interest rates on the two currencies. If the forward exchange
rate is set by speculators in reference to the expected future spot exchange rate, the current
spot rate or the two interest rates can adjust to satisfy interest rate parity. The speculative
dimension of trading must also be satisfied in equilibrium.
9. It is sometimes asserted that investors who hedge their foreign currency bond or stock
returns remove the foreign exchange risk associated with the investment, reduce the
volatility of their domestic currency returns, and thus get a “free lunch” because the
mean return in domestic currency remains the same as the mean return in the foreign
currency. Is this true or false? Why?
Answer: If forward rates are unbiased predictors of future spot rates, hedging foreign
currency investments does not change their expected returns and effectively removes a
source of volatility. Some would say that this provides a “free lunch” because volatility is
reduced without a reduction in mean. By hedging foreign investments, you can increase your
Sharpe ratio for this particular asset class, which allows you to lever the return to the same
volatility and get more return. But, there is no “free lunch” because in this case the foreign
exchange risk is not a priced risk. If there is a risk premium in the foreign exchange market,
the statement is wrong. Hedged foreign bond and equity investments would have different
expected returns than unhedged investments.
10. It is often argued that forward exchange rates should be unbiased predictors of future
spot exchange rates if the foreign exchange market is efficient. Is this true or false?
Why?
Answer: Market efficiency means that asset prices accurately incorporate all available
information and expected returns on assets correspond to true sources of risk. If forward rates
are biased predictors of future spot exchange rates, the source of the bias could be an
equilibrium risk premium. Thus, the claim that forward exchange rates should be unbiased
predictors of future spot exchange rates if the foreign exchange market is efficient is wrong.
11. What is the prediction of the CAPM for the relationship between the forward exchange
rate and the expected future spot exchange rate?
Answer: The CAPM predicts that the expected excess return on an asset is the beta of the
asset times the expected excess return on the market portfolio. The beta is the covariance of
the return on the asset with the return on the market portfolio divided by the variance of the
market portfolio. By applying the CAPM to uncovered foreign money market investments
and using interest rate parity, one can demonstrate that the expected return on a forward
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Chapter 7: Speculation and Risk in the Foreign Exchange Market
contract equals the beta of the return on the forward contract times the expected excess return
on the market portfolio. The expected return on a forward contract to purchase foreign
currency is the difference between the expected future spot exchange rate and the forward
exchange rate scaled by the current spot rate. If the change in the future spot rate is positively
correlated with the return on the market portfolio, forward rates would be less than expected
future spot rates, and if the change in the future spot rate is negatively correlated with the
return on the market portfolio, forward rates would be greater than expected future spot rates.
12. If the CAPM explains deviations of the forward exchange rate from the expected future
spot exchange rate, explain why one party involved in a forward contract would be
willing to enter into a contract with an expected loss.
Answer: If the party that is long in the forward market has an expected profit, the party that is
short in the forward market has an expected loss. The CAPM explains this seeming dilemma
because the party that is long would have a risky asset. The covariance of the profit on the
long position with the return on the market portfolio would be positive, and this positive
covariance is what makes the contract risky. The person who is on the short side of the
forward contract would have an asset whose covariance with the world market portfolio was
negative. They would receive profit when the market portfolio was doing poorly, and they
would be willing to take an expected loss because of the desirable property of the negative
covariance, which is like having portfolio insurance.
13. Why is it only the covariance of an asset’s return with the return on the world market
portfolio that determines whether there is a risk premium associated with the asset’s
expected return?
Answer: The uncertain part of an asset’s return can be broken into two components. The nondiversifiable part is determined by the covariance of the return on the asset with the return on
the market portfolio. This part is the source of risk and the part that gives rise to a risk
premium on the asset. The other component of the uncertain part of an asset’s return is the
part that is diversifiable and therefore does not contribute to the variance of a large, welldiversified portfolio. This latter part is therefore not a source of risk to an investor who holds
a large, well-diversified portfolio.
14. What is the rational expectations hypothesis, and how is it applied to tests of hypotheses
about expected returns in financial markets?
Answer: The rational expectations hypothesis states that we can break the realization of a
return into an expected return that depends on the current information set and an unexpected
component that depends only on new information and consequently does not depend in any
way on the current information set. Theories of risk premiums generate hypotheses that relate
unobservable expected rates of return to variables that are in the current information set. By
using the rational expectations hypothesis, we can replace the expected return with the
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Chapter 7: Speculation and Risk in the Foreign Exchange Market
5
realized return minus an error term and recognize that the error term has a mean of zero and
does not covary with anything in the current information set. These statistical properties
allow us to test hypotheses about expected returns in financial markets.
15. Suppose that the forward premium equals the conditional expectation of the future rate
of appreciation of the foreign currency relative to the domestic currency. If we form the
average realized rate of appreciation from a large sample of data and compare it to the
average forward premium, what should be true?
Answer: If we impose the assumption of rational expectations, we know that we can
decompose the realized rate of appreciation into the conditional expectation of the future rate
of appreciation plus an error term that has a mean of zero. Therefore, the average of the
realized rate of appreciation from a large sample of data should equal the average of the
conditional expectations of the rate of appreciation. If the forward premium on the foreign
currency at each moment in time equals the conditional expectation of the future rate of
appreciation of the foreign currency relative to the domestic currency at that moment in time,
the average or mean of the forward premium should be equal to the average of the realized
rate of appreciation from a large sample of data.
16. Explain how you would use a regression to test the unbiasedness hypothesis.
Answer: The unbiasedness hypothesis states that the forward premium on the foreign
currency at each moment in time equals the conditional expectation of the future rate of
appreciation of the foreign currency relative to the domestic currency at that moment in time.
If we impose the assumption of rational expectations, we know that we can decompose the
realized rate of appreciation into the conditional expectation of the future rate of appreciation
plus an error term that has a mean of zero. Thus, at the 30-day horizon, the unbiasedness
hypothesis predicts
s(t+30) = fp(t,30) + ε(t+30)
where s(t+30) is the realized rate of appreciation between time t and time t+30 and fp(t,30) is
the 30-day forward premium at time t. This representation of the theory looks exactly like a
regression equation
s(t+30) = α + βfp(t,30) + ε(t+30)
in which α = 0 and β = 1.
17. Suppose you regress the realized rate of appreciation of a foreign currency on a
constant and the forward premium on the foreign currency. What interpretation can
you give to the estimated slope coefficient? If the slope coefficient is negative, is it true
that the forward premium is predicting the wrong sign for the rate of appreciation?
Answer: The slope coefficient is the covariance of the rate of appreciation with the forward
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Chapter 7: Speculation and Risk in the Foreign Exchange Market
premium divided by the variance of the forward premium. The slope coefficient indicates
how much the expected rate of appreciation moves with the forward premium. If the slope
coefficient is negative, some people interpret the regression as saying that the forward
premium is predicting the wrong direction for the rate of appreciation. To get a correct
interpretation of the regression, one must remember to include the constant term in the
forecast. The correct interpretation is that the forecast of the expected rate of appreciation is
the constant plus the slope coefficient multiplied by the forward premium. A negative
coefficient certainly does predict that the expected rate of appreciation predicted by an
increase in the forward premium is smaller than before the increase in the forward premium.
18. What does a negative slope coefficient in an unbiasedness regression imply about the
variability of risk premiums relative to variability of expected rates of appreciation?
Answer: The volatilities of forward premiums on the major currencies are about 3% (on an
annualized basis). It turns out that the regression evidence presented in Exhibit 7.5 implies
that both the volatilities of expected exchange rate changes and risk premiums are often
(much) larger than the volatilities of forward premiums. The regression states that
Et ( st +1 ) = a + bfpt
The variance of expected exchange rate changes is therefore
VAR  Et  s ( t + 1)   = VAR  a + b fp ( t )  = b 2VAR  fp ( t ) 
Hence, if b 2  1 , which is the case for all pairs involving the yen, and the $/£ pair, expected
exchange rate changes are more variable than forward premiums. To find the variance of the
risk premium, recall that the risk premium is simply the expected forward market return.
Therefore,
rp(t ) = Et fmr ( t + 1)  = Et  s ( t + 1)  − fp ( t ) = a + ( b − 1) fp ( t )
Hence,
VAR  rp(t)  = ( b − 1) VAR  fp ( t ) 
2
Consequently, as long as b is negative, which is the case for all currencies, the implied
variance of the risk premium is not only larger than the variance of the forward premium, it is
also larger than the implied variance of the expected exchange rate changes.
19. What is a carry trade?
Answer: The carry trade involves borrowing low interest rate currencies and lending high
interest rate currencies. If the high interest rate currency fails to depreciate as much as the
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Chapter 7: Speculation and Risk in the Foreign Exchange Market
7
interest differential is predicting, the carry trade has a positive return. The carry trade can be
done by going long in currencies that are selling at a forward discount.
20. What is a Sharpe ratio?
Answer: The Sharpe ratio is a measure of the risk-return tradeoff on an asset or a portfolio of
assets. It is measured by taking the ratio of the average excess return on the asset divided by
the standard deviation of the return on the asset.
21. Do carry trades contain risks that may not be reflected in their Sharpe ratios?
Answer: Carry trades appear to have negative skewness. That is, the distribution of the
returns to the carry trade is not symmetrically distributed like the normal distribution.
Instead, there are more realizations of large negative returns than would be predicted by a
normal distribution. The distribution of returns on the carry trade is sometimes characterized
as picking up nickels in front of a steam roller.
22. What is a peso problem? Explain the term within the context of its original derivation.
Now, explain how peso problems can generally plague the study of financial market
returns.
Answer: A phenomenon called the peso problem arises when rational investors anticipate
events that do not occur during the sample or at least do not occur with the frequency the
investors expect. This invalidates statistical inferences conducted under the rational
expectations assumption based on data drawn from the period. The peso problem got its
name from the experience of Mexico during 1955–1976. During this period, the Mexican
authorities were attempting to peg the peso–dollar exchange rate at MXP12/USD, and they
did so successfully between 1955 and 1975. Suppose we assume that the market sets the
forward rate in such a way that it is an unbiased predictor of the future spot rate—that is, we
assume that the unbiasedness hypothesis holds. Now, let’s see if a statistician would
conclude that the forward rate is an unbiased or a biased predictor using the Mexican data.
Let Speg be the peso–dollar exchange rate at which the Mexican authorities are currently
pegging. Let Sdev be the rate that the Mexican authorities will choose if they devalue the
peso, so Sdev > Speg. Suppose that the Mexican authorities successfully peg the peso to the
dollar between time T0 and time T2, when they eventually devalue the peso. Suppose also that
the market knows during the time period between T0 and T2 that the Mexican authorities
might devalue the peso at any time. It is assumed that everyone knows the value of S dev. Let
probt be the probability that the market assigns to the event that the peso will be devalued
during the next month. Then, the 1-month forward rate is an unbiased predictor of the future
spot rate when it is the probability-weighted average of the two possible events:
F ( t ) = E t S ( t+1)  = (1 - prob t ) Speg + prob t Sdev
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Chapter 7: Speculation and Risk in the Foreign Exchange Market
The forward rate is the probability of no devaluation multiplied by the current exchange rate
plus the probability of a devaluation multiplied by the new exchange rate. To be concrete,
let’s assume a value for Sdev = MXP18/$; which represents a 50% appreciation of the dollar
relative to the peso. As the above equation makes clear, when the market’s assessment of the
strength of the government’s commitment to the peg changes over time, so will prob t, and
thus, so will the forward rate. As long as the devaluation does not materialize, the dollar will
trade at a forward premium relative to the peso (in pesos per dollar, F > Speg), and peso
money market investments will carry higher interest rates than dollar investments.
If the statistician takes data from an interval of time during which no devaluation occurs, say,
between T0 and T1, where T1 < T2, and compares forward rates with realized future spot
rates, the person will conclude that the forward rate is a biased predictor of the future spot
rate. During the statistician’s sample, the realized future spot rate is always below the
forward rate. Hence, the statistician rejects the null hypothesis that the forward rate is an
unbiased predictor of the future spot rate. The statistician has rejected the null hypothesis, but
the null hypothesis is true.
How did the statistician go wrong? In other words, what led to the peso problem in this case?
When we do statistical analysis on a financial time series using the rational expectations
assumption, we assume that a reasonably long sample of returns is representative of the true
distribution of returns that investors thought they faced when they made their investments.
For the forward market example, we would assume that the ex post spot rates reflect all the
possible events that investors thought might happen when they entered into their forward
contracts. If there are important events that investors thought might happen but that did not
happen, or if relatively rare events happened too frequently, the historical sample means,
variances, and correlations in the data may tell us very little about the means, variances, and
correlations of returns that investors thought they faced. The historical means, variances, and
correlations may also be relatively uninformative about the moments that investors will face
in the future. It is in this sense that the past performance of foreign investments, whether they
are hedged or unhedged, may be poor indicators of the returns that investors can expect in the
future.
In the case of the Mexican peso, even though the forward rate seemed to be a biased
predictor of the future spot rate over 20 years, the devaluation eventually occurred in 1976,
thereby validating the prediction embedded in the forward rate.
23. How can you use interest rate differentials to understand the probability of devaluation
and the potential magnitude of the devaluation?
Answer: In a fixed exchange rate regime, the interest differential provides information about
the probability of the devaluation multiplied by the magnitude of the devaluation. To see this,
consider the situation of the domestic currency. There are two possible events:
1. A devaluation with probability of occurrence equal to prob
2. No devaluation with probability of occurrence equal to (1 – prob).
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Chapter 7: Speculation and Risk in the Foreign Exchange Market
9
When the domestic central bank successfully holds the peg, the exchange rate remains equal
to the current spot rate, which we will measure in domestic currency per unit of foreign
currency as S(t, DC/FC). Let Z% denote the magnitude of the percentage change in the
exchange rate, which formally refers to the percentage revaluation of the foreign currency
versus the domestic currency if the pegged exchange rate does not hold, but which we will
call the magnitude of the devaluation. Then, the interest rate differential tells us something
about the probability of devaluation, prob, and the percentage magnitude of the devaluation,
Z%. Consider the expected returns in the domestic currency on two investments for a period
of n days, with interest rates measured at annual rates and with exchange rates measured in
domestic currency per foreign currency:
domestic money market investment: 1 +i(DC)
foreign money market investment:
1 + i(FC) × E t S(t+n,DC/FC)
S(t,DC/FC)
According to uncovered interest rate parity, these two investments yield the same expected
return. Because there are two possible events—devaluation or no devaluation—the expected
spot rate is simply
E t S(t+n,DC/FC)  = (1 - prob)  S(t,DC/FC) + prob  S(t,DC/FC)  (1 + Z%)
Therefore, by equating the two rates of return, substituting for the expected spot rate, and
solving for the intensity of the devaluation (which is the probability of the devaluation
multiplied by the size of the devaluation), we find
1 + i(DC)
prob  Z% =
-1
1 + i(FC)
By placing the right-hand side over a common denominator, we find
i(DC) - i(FC)
<DM> prob  Z% =
1 + i(FC)
Consequently, if domestic interest rates are higher than foreign interest rates, there is a
chance of a devaluation of some magnitude. The higher the interest differential, the higher
the market assesses the chance and/or the magnitude of a devaluation of the domestic
currency.
PROBLEMS
1. Over the next 30 days, economists forecast that the pound may weaken relative to the
dollar by as much as 6%, or it may strengthen by as much as 7%. The possible values
for the rate of change of the dollar–pound spot exchange rate are –7%, –5%, –3%, –
1%, 0%, 2%, 4%, and 6%. If these values are equally likely, what are the mean and
standard deviation of the future spot exchange rate if the current rate is $1.5845/£?
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10 Chapter 7: Speculation and Risk in the Foreign Exchange Market
Answer: The mean is the probability weighted average of the future possibilities. Because the
events are equally likely and there are 8 events, each gets weight 1/8 in the average. Thus the
mean is
(1/8)  [(-7%) + (-5%) + (-3%) + (-1%) + 0% + 2% + 4% + 6%] = -0.5%
Consequently, the expected future spot rate is (1 – 0.005)  $1.5845/£ = $1.5766/£. The
standard deviation is the square root of the variance. The variance is the probability weighted
average of the squared deviations of the possible realizations from their mean. Upon taking
the average of the squared deviations of the realizations from their means, we find the
standard deviation equal to 4.1533%. Because the standard deviation of the future exchange
rate is the current exchange rate times the volatility of the rate of change, we find the
volatility of the future exchange rate to be 4.1533%  $1.5845/£ = $0.0658/£.
2.
Consider the following hypothetical facts about Mexico: The peso recently lost over
40% of its value relative to the dollar. Over the course of the next 90 days, there is a
35% chance that the Mexican government will lose control of the economy. If it does,
the peso will lose 33% of its value relative to the dollar, and the Mexican stock market
will fall by 39%. Alternatively, the U.S. Congress may vote to help Mexico by offering
collateral for Mexican government loans. In that case, the peso will appreciate 27%
relative to the dollar, and the Mexican stock market will rise by 29%. As a U.S. investor
with no current assets or liabilities in Mexico, you have decided to speculate. Calculate
your expected dollar return from investing dollars in the Mexican stock market for the
next 90 days.
Answer: If you invest in the Mexican stock market, you must first convert dollars into pesos.
Then, you invest the pesos in the stock market. After receiving your stock return, you convert
back into dollars at the future exchange rate. The dollar return is therefore
S(t+90, $/MXN)
× R(t+90,MXN)
S(t, $/MXN)
where R(t+90,MXN) is the peso return in the Mexican stock market and S(t,$/MXN) is the
dollar-peso exchange rate. The realized dollar return has two possible values. Either the peso
will lose 33% of its value and the stock market will fall 39%, in which case the gross dollar
return is (1 - 0.33)  (1 – 0.39) = 0.4087, or the peso will gain 27% of its value and the stock
market will rise 29%, in which case the gross dollar return is (1 + 0.27)  (1 + 0.29) =
1.6383. The expected gross dollar return is the probability weighted average of these 2
events:
(0.35  0.4087) + (0.65  1.6383) = 1.2079.
The expected net dollar rate of return to investing in the Mexican stock market is therefore
20.79%.
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Chapter 7: Speculation and Risk in the Foreign Exchange Market 11
3. Suppose that the 90-day forward rate is $1.19/€, the current spot rate is $1.20/€, and
you expect the future spot rate in 90 days to be $1.21/€. What contract would you make
to speculate in the forward market by either buying or selling €10,000,000? What is
your expected profit? If the standard deviation of the 90-day rate of appreciation of the
euro relative to the dollar is 3%, what range covers 95% of your possible profits and
losses?
Answer: The forward rate of $1.19/€ is less than your expected future spot rate of $1.21/€.
Therefore, if you buy the euro forward, you expect to be able to sell euros at a higher dollar
price. You should speculate by buying the euro forward, in which case you want to sell
$10,000,000 forward. In the forward market, the euro value of $10,000,000 is $10,000,000 /
($1.19/€) = €8,403,361.34. If we contract to buy €8,403,361.34 forward, our expected dollar
profit is
 $1.21 $1.19 
 € - €  × €8,403,361.34 = $168,067.23
If the rate of appreciation of the euro is conditionally normally distributed, 95% of the
possible rates of appreciation will be between plus or minus 1.96 standard deviations of the
mean. The standard deviation of the level of the exchange rate is the current spot rate
multiplied by the standard deviation of the rate of appreciation or $1.20/€  0.03 = $0.036/€.
Therefore, 95% of the possible future exchange rates lie within a range from $1.1394/€ =
$1.21/€ - (1.96  $0.036/€) to $1.2806/€ = $1.21/€ + (1.96  $0.036/€). The 95% range of
dollar profits on our forward contract is therefore from
 $1.1394 $1.19 
× €8,403,361.34 = -$425,210.08
 €
€ 
to
 $1.2806 $1.19 
× €8,403,361.34 = $761,344.54
 €
€ 
4. Suppose the rate of appreciation of the dollar relative to the yen over the next 90 days
has a mean of –1% and a standard deviation of 3%. Use a spreadsheet program to
graph the distribution of the future yen–dollar exchange rate. If the current spot
exchange rate is ¥99/$, and the 90-day forward rate is ¥98.30/$, what is the expected
profit or loss in yen on a forward contract that sells $5,000,000 forward?
Answer: The graph should look like this:
©2017 Cambridge University Press
12 Chapter 7: Speculation and Risk in the Foreign Exchange Market
0.45
Probability of Yen-Dollar Exchange Rate
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
87.62
88.51
89.40
90.29
91.18
92.07
92.96
93.85
94.74
95.63
96.52
97.42
98.31
99.20
100.09
100.98
101.87
102.76
103.65
104.54
105.44
106.33
107.22
108.11
0
If the rate of appreciation of the dollar has a mean, which we denote by μ , of –1%, and a
standard deviation, which we denote by σ , of 3%, and if the innovation to the rate of
appreciation is normally distributed, then the future spot exchange rate is normally
distributed. In EXCEL, the command NORMDIST(x, μ , σ , 0) produces the value of the
PDF of a normal distribution with mean μ , and standard deviation, σ ,associated with the
value x. The graphic is based on 701 values of x.
Irrespective of the distribution, the expected future spot rate is
Et  S ( t + 90, ¥ / $ )  = S ( t , ¥ / $ )  (1 +  ) =
= ( ¥99 / $ )  (1 − 0.01) = ¥98.01/ $
Because this value is less than the 90-day forward rate of ¥98.30/$, there is an expected yen
profit from selling dollars forward. If we sell $5,000,000, the expected profit is
(¥98.30/$ - ¥98.01/$)  $5,000,000 = ¥1,450,000.
5. Suppose that the spot exchange rate is $1.55/£, that the beta on a forward contract to
buy pounds with dollars is 1.5, and that the expected dollar rate of return on the
market portfolio in excess of the dollar risk-free interest rate is 7%. What is the
expected profit or loss on a forward purchase of £1,000,000? Explain how this can be an
equilibrium.
Answer: The CAPM predicts that the expected profit on a contract to purchase pounds in the
forward market is
E t ( S(t+k) - F(t,k) )
= β F E t R M (t+1) - 1 + i ( t,$ ) 
S(t)


We are told that the beta is 1.5 and the expected return on the market portfolio in excess of
the risk-free return is 7%. Therefore, the right-hand side is 10.5%. The difference between
©2017 Cambridge University Press
Chapter 7: Speculation and Risk in the Foreign Exchange Market 13
the expected spot rate and the forward rate is 10.5% of the current spot or 10.5%  $1.55/£ =
$0.16275/£. If we purchase £1,000,000 forward, we have an expected gain of $0.16275/£ 
£1,000,000 = $162,750.
This would be an equilibrium result because the payoff on the forward purchase has a
positive covariance with the return on the market portfolio. In the problem, the pound tends
to weaken when the return on the market portfolio is down, as was observed in the financial
crisis. Investors who purchase pounds forward dislike this attribute of forward contracts, and
they demand a risk premium. Investors who sell the pound forward get a negative beta asset
which they willingly hold with an expected loss because of the portfolio insurance that it
provides.
6. Suppose the estimated slope coefficient in a regression of the rate of depreciation of the
dollar relative to the yen on a constant and the forward discount on the dollar is –2, and
the standard deviation of the forward discount, measured on an annualized basis, is
2.5%. What is a lower bound for the variability of the risk premium in the yen–dollar
forward market?
Answer: The unbiasedness regression is
s(t+30) = a + b fp(t) + ε(t+30)
where s(t+30) is the rate of appreciation of the dollar versus the yen and fp(t) is the forward
premium or discount on the dollar. We are told that the slope coefficient is -2. The forward
market return on the yen is s(t+30) – fp(t), so we know that
s(t+30) - fp(t) = a + ( b - 1) fp(t) + ε(t+30)
Because the fitted value of the regression is an estimate of the expected return, the volatility
of the risk premium in the yen forward market is predicted to be b −1 times the volatility of
the forward premium. Thus, we know that the estimated standard deviation of the risk
premium is at least 3 times the standard deviation of the forward premium, which is 2.5%.
The variability of the risk premium is therefore at least 7.5%. We say “at least” because using
more variables in the return regression will increase the variability of the expected forward
market return.
7. Suppose the British pound (GBP) is pegged to the euro (EUR). You think there is a 5%
probability that the GBP will be devalued by 10% over the course of the next month.
What interest differential would prevent you from speculating by borrowing GBP and
lending EUR?
Answer: If you speculate, you will borrow pounds at i(£) for one month. You will then
convert the borrowed pounds into euros at S(t,£/€), invest the euros for one month at i(€) and
sell the euro principal plus interest in the future spot market at S(t+30,£/€). You will expect
to break even if the expected pound return equals the pound borrowing cost:
©2017 Cambridge University Press
14 Chapter 7: Speculation and Risk in the Foreign Exchange Market
E t S ( t+30,£/€ ) 
S ( t,£/€ )
× 1 + i(€)  = 1 + i(£) 
Dividing this expression by 1 + i(€) and subtracting one from both sides gives
E t S ( t+30,£/€ ) - S ( t,£/€ ) 
S ( t,£/€ )
=
i(£) - i(€)
1 + i(€)
The left-hand side is the expected rate of appreciation of the euro, and the right-hand side is
the interest differential adjusted by the term that is only slightly greater that one in the
denominator. The information we have is that either the pound will remain pegged to the
euro with 95% probability in which case euro appreciation is zero, or the euro will strengthen
by 10% with a probability of 5%. Here, for simplicity, we are treating the devaluation of the
pound as equivalent to the strengthening of the euro even though this is only true with
continuously compounded rates of change.
The expected rate of euro appreciation is therefore (.05  0.1) + (.95  0) = 0.005. If we
approximate the denominator on the right-hand side of the equation as one (which should be
a very good approximation with monthly interest rates), 0.005 is also the monthly interest
differential that would prevent arbitrage. If we annualize this value and put it into percentage
per annum, we find 6%. Thus, if the one-month pound interest rate is 6% greater than the
one-month euro interest rate, we would not want to speculate on a possible devaluation of the
pound.
8. Argentina’s monetary stabilization plan in 1991 included introducing a currency board
that tied the Argentine peso (ARS) to the U.S. dollar at an exchange rate of
ARS1/USD1. On June 21, 2000, the 3-month interest rates quoted by Argentine banks
were 6.71% in USD and 7.33% in ARS. Suppose the difference reflected some
probability that the currency board would be abandoned and the peso devalued, and
investors think a 10% devaluation to ARS1.10/USD is possible. What is the probability
of this happening if uncovered interest rate parity holds? In early 2001, confidence in
the currency board eroded and interest differential soared to well over 10%. What is
the possibility of a 10% devaluation if the 3-month interest rates are 20% in ARS and
6.0% in USD?
Answer: We are told that uncovered interest rate parity is satisfied. Therefore, the expected
future spot rate is the current spot rate times the ratio of one plus the nominal interest rate for
that maturity.
1 + i (t ,90, ARS )
Et  S ( t + 90, ARS / USD )  = S ( t , ARS / USD ) 
1 + i (t ,90, USD)
By deannualizing the interest rates, we find
©2017 Cambridge University Press
Chapter 7: Speculation and Risk in the Foreign Exchange Market 15
1 + 7.33 / 400
= ARS1.001524 / USD
1 + 6.71/ 400
If p is the probability of a 10% devaluation, then the expected future spot rate is
Et  S ( t + 90 )  = (1 − p ) 1 + p  1.1 = 1.001524
Et  S ( t + 90, ARS / USD )  = ( ARS1/ USD ) 
Solving this we find p = 0.015244. Doing the same set of equations for the new three-month
interest rates of 20% for ARS and 6% for USD gives p = 0.034483.
9. The British bank Barclays has developed an Exchange Traded Note that pays off the
The Barclays Capital Intelligent Carry Index™. Look up information on this index on
the web. Explain why you like or dislike its strategy.
Answer: Here is the information from the Barclays Capital web site,
https://ecommerce.barcap.com/indices/index.dxml
The Intelligent Carry Index is intended to reflect the total return performance of the Intelligent
Carry Strategy, which is a systematic and a quantitative strategy. The Index aims at providing
investors with exposure to foreign money market instruments. The constituents of the
Intelligent Carry Index are Barclays EUR Overnight Index and the 1-month interbank money
market rates in 10 currencies, namely EUR, USD, GBP, CHF, JPY, NZD, AUD, SEK, NOK,
and CAD. The Index is designed to be within certain risk and return parameters.
The Intelligent Carry Strategy uses a quantitative approach to determine the Index
Composition. A systematic mean optimiser model is run to determine the core weights of each
of the currencies in the Index. This Mean Optimiser model generates “buy” or “sell” signals
based on the relative position of the ten money market instruments of the index. The mean
optimiser model allocates a greater weight to the money market instruments with a high yield.
The model is run every month to determine the optimal allocation.
The index can have only positive (long) positions in the Barclays Overnight Index. The index
can have positive (long) and negative (short) positions on any money market instruments
listed above.
The strategy tries to exploit deviations from unbiasedness and is essentially an “optimized”
version of the carry trade. While such a strategy has produced attractive Sharpe ratios over
relatively long periods of time, the strategy’s returns show fat tails and negative skewness
with negative realized returns during global equity crises as occurred in 1998 and 2008.
Standard mean variance optimization is not likely to change that property of the strategy. The
index peaked in July 2007 and remains below the peak 10 years later.
©2017 Cambridge University Press
Chapter
8
Purchasing Power Parity and Real
Exchange Rates
QUESTIONS
1. What does the purchasing power of a money mean? How can it be measured?
Answer: The purchasing power of a money is also known as its real value and indicates the
amount of goods and services that can be purchased with a given amount of the money. We
measure purchasing power by first calculating the price level, which is a weighted average of
the prices of the goods and services that people consume. The weights in the price level reflect
the shares of these goods and services in the consumption bundle of a typical individual. The
purchasing power of the money is then found by taking the reciprocal of the price level. The
units of the price level are an amount of money per consumption bundle, and the units of
purchasing power are consumption bundles per unit of money.
2. Suppose the government releases information that causes people to expect that the
purchasing power of a money in the future will be less than they previously had expected.
What will happen to the exchange rate today? Why?
Answer: Typically, when people think that the purchasing power of a money is going to decline
in the future, due to higher expected inflation, they try to sell that currency today to get into a
currency that will have more stable purchasing power. This reduced demand for the currency
causes that currency to weaken or depreciate immediately.
3. What is the difference between a price level and a price index?
Answer: The price level is a weighted average of the prices of the goods and services that
people consume. The price index is a ratio of a price level at one point in time to the price level
in some base year, with the ratio usually multiplied by 100. Thus, if the price level in a given
year is 30% higher than the price level in the base year, the price index would be 130. Price
levels give you information about the purchasing power of a currency. Price indexes give you
information about the rate of inflation between two points in time.
4. What do economists mean by the law of one price? Why might the law of one price be
violated?
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Chapter 8: Purchasing Power Parity and Real Exchange Rates
Answer: The law of one price says that the price of a good, when denominated in a particular
currency, is the same wherever in the world the good is being sold. The law of one price relies
on arbitrage in the goods market. If the good is being sold in one place at a low price and is
being sold in a different place at a high price, people have an incentive to arbitrage the two
markets. Therefore, anything that makes it difficult or costly to arbitrage in the goods market
can create a deviation from the law of one price. Clearly, transaction costs, such as the costs of
shipping, generate deviations from the law of one price that cannot be arbitraged. Tariffs and
quotas on imports and exports also create deviations. If markets are not competitive and firms
have some monopoly power, the corporation may decide to charge different prices in different
countries, but it must be able to segment the markets to prevent arbitrage. If arbitrage cannot
be done instantaneously, there will be a speculative element that enters the calculations and the
speculator may have to be compensated for the risk of loss with an expected profit from buying
in one market and selling in another market at a later point in time. Finally, various goods
markets are subject to a certain amount of price stickiness because of the costs of changing
prices. Because exchange rates are asset prices and freely flexible, unanticipated changes in
exchange rates will create deviations from the law of one price if goods prices are sticky.
5. What is the value of the exchange rate that satisfies absolute PPP?
Answer: Absolute purchasing power parity requires that the internal purchasing power of a
currency equals its external purchasing power. The internal purchasing power is calculated by
taking the reciprocal of the price level, and the external purchasing power is calculated by first
exchanging the domestic money into the foreign money in the foreign exchange market and
then calculating the purchasing power of that amount of foreign currency in the foreign
country. Hence, the prediction of absolute PPP for the exchange rate of domestic currency per
unit of foreign currency is found by equating the internal purchasing power of the domestic
currency to the external purchasing power of the domestic currency:
1
1
1
= PPP
P ( DC ) S P ( FC )
where P(DC) is the domestic price level, P(FC) is the foreign price level, and S PPP signifies the
exchange rate of domestic currency per unit of foreign currency that satisfies the PPP relation.
By solving for SPPP, we find
P (DC )
SPPP =
P (FC )
6. If the actual exchange rate for the euro value of the British pound is less than the
exchange rate that would satisfy absolute PPP, which of the currencies is overvalued and
which is undervalued? Why?
Answer: The terminology of “overvalued” and “undervalued” refers to the relationship of the
exchange rate to the PPP theory. If the actual exchange rate of euros per pound is less than the
PPP prediction, the euro is overvalued and the pound is undervalued. We know this is the
©2017 Cambridge University Press
Chapter 8: Purchasing Power Parity and Real Exchange Rates
3
correct answer because if the actual exchange rate were to move to the PPP prediction, the euro
would have to weaken, and the pound would correspondingly have to strengthen, on the foreign
exchange market. The weakening of the euro would correct its overvaluation, and the
strengthening of the pound would correct its undervaluation.
7. What market forces prevent absolute purchasing power parity from holding in real
economies? Which of these represent unexploited profit opportunities?
Answer: Any of the forces that create a deviation from the law of one price can also cause a
deviation from PPP. See the answer to question 4. In addition, even if the law of one price were
satisfied for all goods, if the consumption bundles in the two countries put different weights
on the goods because of taste differences across countries, relative price changes would be
reflected in deviations from PPP. It is our opinion that deviations from PPP do not represent
unexploited profit opportunities.
8. Why is it better to use a PPP exchange rate to compare incomes across countries than an
actual exchange rate?
Answer: When comparing incomes across countries, one is interested in comparing the quality
of life that occurs from earning such incomes and consuming in those countries. One way to
do such a comparison is to examine the real values of the nominal incomes, that is, to multiply
each of the nominal incomes times the respective purchasing powers of the currencies (which
is equivalent to dividing the nominal income by the price level). The real value of the income
tells you the command over goods and services that the nominal income provides when you
consume in that country. If the real incomes in countries A and B were the same, we would
have
nominal income in country A nominal income in country B
=
price level in country A
price level in country B
If we multiply this expression by the price level in country A, we get
nominal income in country A =
price level in country A
 nominal income in country B
price level in country B
In the above expression, the ratio of the price level in country A to the price level in country B
is the purchasing power parity exchange rate. Hence, if we multiply the nominal income in
country B by the purchasing power parity exchange rate we get a nominal income that is in the
units of the currency of country A and that can be compared to the nominal income in country
A. If the nominal income in country A is higher than the purchasing power exchange rate
multiplied by nominal income in country B, people in country A are better off in terms of their
ability to consume than those in country B.
If you use the actual exchange rate rather than the PPP exchange rate to convert the nominal
income in country B into currency of country A, you are effectively saying you would like to
earn the income in country B, but you want to consume it in country A. This can create
©2017 Cambridge University Press
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Chapter 8: Purchasing Power Parity and Real Exchange Rates
incorrect inferences about where is the best place to live. Suppose currency B is overvalued
relative to PPP. Then, the market exchange rate of currency A per unit of currency B, denoted
S, is greater than the PPP prediction,
SPPP =
price level in country A
price level in country B
That is S > SPPP. In such a situation, it can happen that
nominal income in country A > SPPP  nominal income in country B
in which case we know from the above discussion that we would prefer to earn income in
country A and consume there. Yet, when we compare incomes with the actual exchange rate,
we might find that the
nominal income in country A < S  nominal income in country B
The overvaluation of currency B causes us to think that the income in country B is preferred.
But, this is only correct if we earn the income in country B but consume in country A after
converting our income into the currency of country A.
9. What is relative PPP, and why does it represent a weaker relationship between exchange
rates and prices than absolute PPP?
Answer: The theory of relative PPP specifies that exchange rates adjust in response to
differences in inflation rates across countries to leave the deviation of the actual exchange rate
from absolute PPP unchanged. Intuitively, inflation is the rate of loss of the internal purchasing
power of a currency. Thus, if two currencies are losing internal purchasing power at different
rates because the rates of inflation in the two countries are not equal, the rate of change of the
exchange rate can offset the differential rates of inflation to leave the same absolute
relationship between the internal and external purchasing powers of the currencies. The relative
PPP theory is weaker than absolute PPP because relative PPP could be satisfied even though
there are deviations from absolute PPP. The requirement for relative PPP to hold is that the
deviations from absolute PPP do not change over time.
10. What is the real exchange rate, and how are fluctuations in the real exchange rate related
to deviations from absolute PPP?
Answer: The real exchange rate, say, of the dollar relative to the euro, is denoted RS(t,$/€). It
is defined to be the nominal exchange rate multiplied by the ratio of the price levels:
S(t,$/EUR)  P(t,EUR)
RS(t,$/EUR) =
P(t,$)
Notice that the real exchange rate would be 1 if absolute purchasing power parity held because
the nominal exchange rate, S(t,$/€), would equal the ratio of the two price levels, P(t,$)/P(t,€).
Similarly, if absolute PPP is violated, the real exchange rate is not equal to 1. Thus, fluctuations
in the deviations from absolute PPP are fluctuations in the real exchange rate.
©2017 Cambridge University Press
Chapter 8: Purchasing Power Parity and Real Exchange Rates
5
11. If the nominal exchange rate between the Mexican peso and the U.S. dollar is fixed, and
there is higher inflation in Mexico than in the United States, which currency experiences
a real appreciation and which experiences a real depreciation? Why? What is likely to
happen to the balance of trade between the two countries?
Answer: If the peso is pegged to the dollar and the rate of inflation in Mexico is greater than in
the rate of inflation in the United States, the peso is appreciating in real terms and the dollar is
experiencing a real depreciation. The logic is that the rate of inflation in Mexico measures the
loss of internal purchasing power, while because the exchange rate is pegged, the loss of the
peso’s external purchasing power is measured by the U.S. rate of inflation. If a currency’s loss
of internal purchasing power is greater than its loss of external purchasing power, that currency
experiences a real appreciation.
The real appreciation of the peso tends to make Mexican residents think that U.S. goods are
relative bargains, while the real depreciation of the dollar relative to the peso, makes U.S.
residents think that Mexican goods are relatively expensive. Thus, the balance of trade between
Mexico and the United States on the Mexican balance of payments should deteriorate with an
increase in imports from the United States and a decrease in exports to the United States.
PROBLEMS
1. If the consumer price index for the United States rises from 350 at the end of a year to
365 at the end of the next year, how much inflation was there in the United States during
that year?
Answer: Price indexes are ratios of the price level in a given year to the price level in a base
year. Because the base year is the same in the two price indexes under consideration, we can
take the ratio of the two price indexes and find the rate of inflation over that year. The ratio is
365/350 = 1.0429 or an inflation rate of 4.29%.
2. As a wheat futures trader, you observe the following futures prices for the purchase and
sale of wheat in 3 months: $3.00 per bushel in Chicago and ¥320 per bushel in Tokyo.
Delivery on the contracts is in Chicago and Tokyo, respectively. If the 3-month forward
exchange rate is ¥102/$, what is the magnitude of the transaction cost necessary to make
this situation not represent an unexploited profit opportunity?
Answer: The forward dollar price of wheat in Tokyo is the ratio of the futures price, ¥320 per
bushel, to the forward exchange rate, ¥102/$. This ratio is ¥320 per bushel / (¥102/$) = $3.14
per bushel. Since we can buy wheat for delivery in Chicago at $3 per bushel, if transaction
costs of shipping wheat from Chicago to Tokyo are smaller than $0.14 per bushel, we could
make an arbitrage profit. Thus, the minimum magnitude of the transaction cost necessary to
make this situation not represent an unexploited profit opportunity is $0.14 per bushel.
©2017 Cambridge University Press
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Chapter 8: Purchasing Power Parity and Real Exchange Rates
3. Suppose that the price level in Canada is CAD16,600, the price level in France is
EUR11,750, and the spot exchange rate is CAD1.35/EUR.
a. What is the internal purchasing power of the Canadian dollar?
Answer: It is probably best to calculate the purchasing power of CAD10,000. If we divide
this amount by the price level in Canada of CAD16,600, we find
CAD10,000
=0.6024 consumption bundles
CAD16,600 / consumption bundle
b. What is the internal purchasing power of the euro in France?
Answer: Performing a similar calculation to the one in part a., we find
EUR10,000
= 0.8511 consumption bundles
EUR11,750 / consumption bundle
c. What is the implied exchange rate of CAD/EUR that satisfies absolute PPP?
Answer: The implied PPP exchange rate equates the internal purchasing power of the CAD
to its external purchasing power. This implies that the PPP exchange rate is the ratio of the
Canadian price level in Canadian dollars to the French price level in euros:
SPPP (CAD/EUR ) =
CAD16,600 CAD1.4128
=
EUR11,750
EUR
d. Is the euro overvalued or undervalued relative to the Canadian dollar?
Answer: Because the actual exchange rate of CAD1.35/EUR is less than the PPP exchange
rate, the euro is undervalued on the foreign exchange market because it would have to
strengthen to move from CAD1.35/EUR to CAD1.4128/EUR.
e. What amount of appreciation or depreciation of the euro would be required to return
the actual exchange rate to its PPP value?
Answer: The exchange rate moves from the actual value of CAD1.35/EUR to the PPP value
of CAD1.4128/EUR for a percentage change of1.4128/1.35– 1 = 0.0466. This is a 4.66%
appreciation of the euro versus the Canadian dollar.
4. Suppose that the rate of inflation in Japan is 2% in 2017. If the rate of inflation in
Germany is 5% during 2017, by how much would the yen strengthen relative to the euro
if relative PPP is satisfied during 2017?
©2017 Cambridge University Press
Chapter 8: Purchasing Power Parity and Real Exchange Rates
7
Answer: The approximately correct answer is that the yen should strengthen by the differential
in the rates of inflation or 5% - 2% = 3%. The exact answer is found from equation (8.4) of
the text, which incorporates a denominator correction, and we get
DC  ¶ (t+1,DC ) - ¶ (t+1,FC )

s  t+1,
=
FC 
1 + ¶ (t+1,FC )

Since we are concerned about the strengthening of the yen, let the yen be the foreign currency
(FC), and let the euro be the domestic currency (DC). Then, the relative PPP formula states
that the rate of appreciation of the yen is
0.05 - 0.02
= 0.0294 or 2.94%
1 + 0.02
5. One of your colleagues at Deutsche Bank thinks that the dollar is severely undervalued
relative to the yen. He has calculated that the PPP exchange rate is ¥140/$, whereas the
current exchange rate is ¥105/$. Because interest rates are 3% p.a. lower in Japan than
in the United States, he thinks that this is a good time to speculate by borrowing yen and
lending dollars. What do you think?
Answer: Deviations from PPP are a weak reason to engage in speculation. While the data in
the problem indicate that the dollar is 33.33% undervalued, because that is the amount of dollar
appreciation that would be required to take the actual exchange rate from ¥105/$ to the PPP
prediction of ¥140/$, we know that the return to PPP will not be an overnight event.
The empirical analysis of the issue indicates that the half-life of PPP deviations is around 5
years. Thus, you might expect that the dollar will appreciate by 16.67% over the next 5 years.
But, uncovered interest rate parity actually suggests that the yen will appreciate in the short
run, because the yen interest rate is 3% less than the dollar interest rate. Notice, though, that
the correction back toward PPP can take place with differential rates of inflation in the two
countries. If Japanese rate of inflation falls below the U.S. rate of inflation, the PPP prediction
will begin falling toward the actual exchange rate. Finally, although the dollar is 33.33%
undervalued, there is no guarantee that the undervaluation will begin to be corrected now. It
may, in fact, get worse. If the undervaluation of the dollar goes to 50% over the next 2 years,
you would lose 16.67% in the foreign exchange market which would not be compensated by
the approximate 6% that you would earn by borrowing yen and lending dollars. Finally, do not
forget that your boss in proprietary trading at Deutsche Bank would not be happy with such a
situation.
6. Suppose that you are trying to decide between two job offers. One consulting firm offers
you $150,000 per year to work out of its New York office. A second consulting firm wants
you to work out of its London office and offers you £100,000 per year. The current
exchange rate is $1.65/£. Which offer should you take, and why? Assume that the PPP
exchange rate is $1.40/£ and that you are indifferent between working in the two cities if
the purchasing power of your salary is the same.
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Chapter 8: Purchasing Power Parity and Real Exchange Rates
Answer: We know from the extensive discussion in Question 8 that we should use the PPP
exchange rate to compare the pound salary to the dollar salary. If we do so, we find $1.40/£ 
£100,000 = $140,000. This is less than the $150,000 that you are being offered in New York.
The fact that the dollar is undervalued on the foreign exchange markets makes the perceived
salary of $1.65/£  £100,000 = $165,000, calculated with the spot exchange rate, seem more
attractive. But, the key point is that to achieve $165,000 of spending in the United States, you
would have to work in London and consume in New York.
7. Suppose that in 2017, the Japanese rate of inflation is 2%, and the German rate of inflation
is 5%. If the euro weakens relative to the yen by 10% during 2017, what would be the
magnitude of the real depreciation of the euro relative to the yen?
Answer: The real exchange rate is
RS(t, ´ / Û) =
S(t, ´ / Û)  P(t, Û)
P(t, ´ )
We also know that a real depreciation of the euro means that this real exchange rate decreases.
The new real exchange rate will be the old real exchange rate with each term multiplied by one
plus the respective percentage rate of change. Thus, one plus the percentage rate of change of
the real exchange rate is
1 + s(t, ´ / Û)  1 +  (t, Û) = 1 - 0.10   1 + 0.05 = 0.9265
1 + rs(t, ´ / Û) =
1 +  (t, ´ )
1 + 0.02
So, we conclude that the real depreciation of the euro is 7.35%.
8. Pick a particular brand of appliance, like a Bosch dishwasher with certain features, and
use the internet to compare its prices across countries. Be sure to have exactly the same
style of appliance in each country. How different are the prices when expressed in a
common currency?
We found the Bosch Model SHE3AR75UC on sale at Sears-Canada for CAD949.99. The
exact same model in the United States was available from Sears-US for USD539. The
exchange rate on June 6, 2017, when we investigated this, was CAD1.35/USD. Thus, a
Canadian could purchase the U.S. dishwasher for CAD1.35/USD x USD539 = CAD727.65.
Buying the dishwasher from Sears-Canada would have cost 30.56% more.
9. Go to the IMF’s web site at www.imf.org, find the Data and Statistics tab, locate World
Economic Outlook (WEO) data, and download the “Implied PPP conversion rate” for the
Indonesian rupiah and the Philippines peso versus the dollar. Calculate a rupiah per peso
PPP rate and compare it to the actual exchange rate. Which currency is overvalued, and
©2017 Cambridge University Press
Chapter 8: Purchasing Power Parity and Real Exchange Rates
9
by how much?
Go to the IMF’s WEO site at
http://www.imf.org/external/pubs/ft/weo/2017/01/weodata/index.aspx
Request data for Indonesia and the Philippines on their Implied PPP rates versus the U.S. dollar.
The 2016 rates were 4,091.831 for Indonesia and 17.945 for the Philippines. The ratio of these
two PPP values relative to the USD gives the Implied PPP rate of IDR/PHP:
IDR4,091.831/USD
IDR228.02
=
PHP17.945/USD
PHP
From Yahoo Finance, the 2016 average exchange rate of IDR/PHP, calculated as the ratio of
the averages of the IDR/USD to the PHP/USD, was IDR290.02/PHP. Hence, the Philippines
peso is overvalued relative to the Indonesian rupiah because the peso would have to weaken if
the actual exchange rate were to go to the Implied PPP rate. The peso would have to weaken
by 21.38%.
©2017 Cambridge University Press
Chapter
10
Exchange Rate Determination and
Forecasting
QUESTIONS
1. What is the difference between the ex ante and the ex post real interest rate?
Answer: The ex post interest rate corrects the nominal interest rate with the realized or ex
post rate of inflation; whereas the ex-ante (or expected) real interest rate corrects the nominal
interest rate for expected inflation.
As a lender, you care about the real return on your investment, which is the return that
measures your increase in purchasing power between two periods of time. If you invest $1,
$1
you sacrifice
real goods now, where P(t) is the price level. In 1 year, you get
P(t)
1+i
back
, where i is the nominal rate of interest. We calculate the real return by dividing
P(t+1)
the real amount you get back by the real amount that you invest. Thus, if rep is the ex post real
rate of return and ex post real interest rate, we have
 1+i 
 P(t+1) 
 = (1 + i )
ep
1+r = 
 1 
 P(t+1) 
 P(t) 
 P(t) 




Notice that the real rate of interest depends on the realization of the rate of inflation because
P(t + 1)/P(t) = 1 + π(t + 1), where π(t + 1) is the rate of inflation between time t and t + 1. For
simplicity, we drop the time notation and simply write
(1 + i)
1 + r ep =
(1 + π)
If we subtract 1 from each side, we have
(1 + i) (1 + π)
i-π
r ep =
=
(1 + π) (1 + π) (1 + π)
which is often approximated as
rep = i – π
©2017 Cambridge University Press
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Chapter 10: Exchange Rate Determination and Forecasting
The approximation involves ignoring the term (1 + π) in the denominator, which is close to 1
if inflation is not too high. Thus, the ex post real interest rate equals the nominal interest rate
minus the actual rate of inflation.
Because the inflation rate is uncertain at the time an investment is made, the lender cannot
know with certainty the real rate of return on the loan. By taking the expected value of both
sides of the equation, conditional on the information set at the time of the loan, we derive the
lender’s expected real rate of return, which is also called the expected real interest rate, or
the ex ante real interest rate, which we denote re:
r e = E t [r ep ] = i(t) - E t [π(t+1)]
2. Suppose that the international parity conditions all hold and a country has a higher
nominal interest rate than the United States. Characterize the country’s inflation rate
compared to the United States, the country’s expected exchange rate change versus the
dollar, the country’s currency forward premium (or discount) versus the dollar, and the
country’s real interest rate compared to the U.S. real interest rate.
Answer: When all the parity conditions hold, real interest rates are equalized across
countries, so the country’s real interest rate should equal that of the United States. The
country’s higher nominal interest rate therefore must reflect a higher expected rate of
inflation relative to the United States. Since the parity conditions hold, a higher expected rate
of inflation implies that country’s currency should be expected to depreciate relative to the
dollar, and the currency will trade at a forward discount relative to the dollar.
3. How do fundamental analysis and technical analysis differ?
Answer: Fundamental analysis typically uses formal economic models of exchange rate
determination and macroeconomic fundamental data such as money supplies, inflation rates,
productivity growth rates, and the current account of the balance of payments to predict
exchange rates. Technical analysis uses only past exchange rate data, and perhaps some other
financial data, such as the volume of currency trade, to predict future exchange rates.
4. Would technical analysis be useful if the international parity conditions held? Why or
why not?
Answer: If the parity conditions held, technical analysis would not be useful in the sense of
providing profitable trading information or information about expected exchange rates that
could not be obtained elsewhere. If the parity conditions held, the best predictor of the future
exchange rate would be the forward rate, and exchange rate forecasts based on other
indicators would not lead to systematic profits on currency speculation.
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5. Describe three statistics you should obtain from a currency-forecasting service in order
to judge the quality of its currency forecasts.
Answer: Three important statistics are the Root Mean Squared Error (RMSE) or Mean
Absolute Deviation of its forecasting record, which would provide information on accuracy;
the percentage of times they were on the correct side of the forward rate, which would
provide useful information on the profitability, and a risk–return statistic (such as the Sharpe
ratio), which would provide a characterization of the profitability of using their forecasts in a
real time trading strategy.
6. Does a large increase in the domestic money supply always lead to a depreciation of the
currency?
Answer: Most theories of the determination of exchange rates would predict that a large
increase in the money supply would imply a depreciation of the currency, definitely in the
long run, and especially as economists say when “everything else is equal.” However, it is
possible that the change in the money supply is accompanied by an increase in real income
that increases the demand for money and thus offsets the money supply’s effect on the
exchange rate.
7. Is a current account deficit always associated with a strong real exchange rate (that is,
one that is overvalued compared to the PPP prediction)?
Answer: Not necessarily. It is best to view the current account and the real exchange rate as
being determined in an equilibrium that depends on many forces, such as movements in net
foreign assets, government spending, productivity growth, and the expectations and risk
tolerances of domestic and foreign investors.
8. Describe how three macroeconomic fundamentals affect exchange rates.
Answer: According to the monetary exchange rate model, the domestic currency weakens
(strengthens) if the domestic (foreign) money supply increases today or if news arrives that
leads people to believe that the future domestic (foreign) money supply will increase. The
domestic currency also weakens if domestic real income falls, if foreign real income rises, or
if news arrives that causes people to expect lower domestic real growth or faster foreign real
growth. Finally, according to the equilibrium theory regarding the real exchange rate and the
current account, an increase in government spending or a decrease in taxes that causes a
budget deficit should increase the real exchange rate (and hence likely also the nominal
exchange rate). This is because an increase in government spending increases aggregate
demand in the economy, which causes the real interest rate to rise. The rise in the interest rate
reduces investment and encourages private saving.
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Chapter 10: Exchange Rate Determination and Forecasting
9. Which simple statistical model yields some of the best exchange rate predictions
available? What does this imply for the value of models of exchange rate determination
to multinational businesses?
Answer: It is surprisingly difficult to beat the forecasts of the random walk model. This
model uses the current exchange rate as the predictor of the future exchange rate. If this
model provided the best forecast, the unbiasedness hypothesis (which says the forward rate is
the best predictor) would be violated. If there were a forward premium on the foreign
currency, the forward rate would be above the expected future spot rate, and you would want
to sell the foreign currency in the forward market.
10. What is chartism?
Answer: Chartists graphically record the actual trading history of an exchange rate and then
try to infer possible future trends based on that information alone.
11. What is an x% filter rule?
Answer: An x% rule states that you should go long in the foreign currency (buy) after the
foreign currency has appreciated relative to the domestic currency by x% above its most
recent trough (or support level) and that you should go short in the foreign currency (sell)
whenever the currency falls x% below its most recent peak (or resistance level). Common
x% filter rules are 1% or 2%.
12. What is a moving-average crossover rule?
Answer: Moving-average crossover rules use moving averages of the exchange rate to
indicate trade directions. An n-day moving average is just the sample average of the last n
trading days, including the current rate. A (y, z) moving-average crossover rule uses averages
over a short period (y days) and over a long period (z days). The strategy states that you
should go long (short) in the foreign currency when the short-term moving average crosses
the long-term moving average from below (above). Common rules use 1 and 5 days (1, 5), 1
and 20 days (1, 20), and 5 and 20 days (5, 20).
13. Have currency traders been successful in exploiting their exchange rate forecasts?
Answer: While there exists scant empirical evidence on the forecasting ability of exchange
rate forecasting services, the number of active currency traders, mostly organized as hedge
funds, has grown considerably over the past decade. Because many of these currency traders
report returns to various indices, we can analyze their performance. If such funds fail to
forecast exchange rates, they should not consistently produce high returns! Pojarliev and
Levich (2008) conducted a study on the returns earned by currency managers reporting to the
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Chapter 10: Exchange Rate Determination and Forecasting
5
Barclay Currency Traders Index (BCTI) between January 1990 and December 2006. All of
these returns are reported net of fees. Hedge funds typically charge a fixed fee of 2% and a
variable fee of 20% on the performance over a benchmark (which can be zero or the Treasury
bill return). The study first tries to establish what techniques the currency managers use: Do
they use the carry strategy, do they follow trends, or do they trade based on fundamentals?
To do so, the investigators use historical data to create returns to carry-trade, trend-following,
and fundamental strategies for the major currencies, and they use regression analysis to
investigate whether the returns of the various managers correlate with these benchmark
returns. The majority of the funds (and the average index) appear to follow trend-following
strategies; many also show positive carry exposure, but there is not much of a link with the
return on fundamental strategies. Recent academic research has shown that the returns to
simplistic trend following strategies are no longer statistically significant, but currency
traders may follow more sophisticated strategies. The average excess return earned over 34
different managers with relatively long track records between 2001 and 2006 is 5.45%, and
the average (annual) Sharpe ratio is 0.47, which is higher than the Sharpe ratio generated by
the equity market. Pojarliev and Levich also check whether the managers outperform the
benchmark returns. Deutsche Bank, among others, has introduced easily tradable funds that
mimic the simple strategies represented by the benchmarks. For an investor, it would make
little sense to pay the heavy fees hedge funds charge for exposure to an index that can be
bought for a small fixed fee. Pojarliev and Levich find that only eight of the 34 managers
significantly outperform a combination of benchmark indices that best describes their
investment style.
More recent anecdotal evidence suggests that forecasting exchange rates and generating
profitable trades has become increasingly difficult. Many currency funds went belly-up in the
aftermath of the global financial crisis in 2008 (recall that the carry strategy had dramatically
negative returns then). Moreover, the marquee currency - focused hedge fund, FX Concepts,
which had more than $14 billion under management at its peak in 2007, closed in 2013. By then,
it was only left with $621 million in assets under management and then the San Francisco
Employees' Retirement System, which made up 66% of the fund's remaining assets, withdrew its
investment forcing FX Concepts to close down.
14. Are devaluations of pegged exchange rates totally unexpected?
Answer: While there is a debate about their predictability, some theories suggest that
devaluations may be partially predictable. These models argue that growing budget deficits,
fast money growth, and rising wages and prices usually precede devaluations. Increases in
nominal interest rates typically reflect a combination of the probability and magnitude of a
possible devaluation.
15. Construct a list of a country’s economic statistics you would assemble to help determine
the probability of a devaluation of its currency within the coming year.
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Chapter 10: Exchange Rate Determination and Forecasting
Answer: Based on theoretical and empirical work, the following economic variables should
prove useful predictors: PPP-based measures of currency overvaluation, current account
balances and monetary growth rates. In addition, if liquid financial markets exist, information
about forward rates or interest rates, currency option prices, and so on may prove useful in
terms of forecasting devaluations.
PROBLEMS
1. Suppose the 1-year nominal interest rate in Zooropa is 9%, and Zooropa’s expected
inflation rate is 4%. What is the real interest rate in Zooropa?
Answer: The expected real interest rate is approximately 9% - 4% = 5%. The correct
computation is: (1 + 0.09) / (1 + 0.04) – 1 = 0.0481 or 4.81%.
2. You were recently hired by the Doolittle Corporation corporate treasury to help
oversee its expansion into Europe. Blake Francis, the CFO, wants to hire a foreign
exchange forecasting company. Blake has asked you to evaluate three different
companies, and he has obtained information on their past performances. Out of a total
of 50 forecasts for the $/€ rate, the companies reported the number of times they
correctly forecast appreciations and depreciations:
Correct Down
Forecasts
Morrissey Forex
Advisors
Pixie Exchange Land
FOREX Cures
Correct Up
Forecasts
20
5
20
12
4
12
There are a total of 35 dollar appreciations (down periods) and 15 dollar depreciations
(up periods) in the sample. Blake wants to know two things:
a. Can anything be said about the companies’ forecasting ability with the available
data?
Answer: Yes, one can compute the number of correct “directional” forecasts. Morrissey has
the highest correct proportion with 25 out of 50 correct, whereas the other firms have less
than 50% correct. However, note that the dollar over this period was relatively strong and
appreciations (down forecasts for the $/€ rate) dominate. Hence, forecasts in the down period
may be more useful (see footnote 3 in the chapter). If we look at correct conditional
forecasts, we see that Morrissey is correct 20/35 or 57.14% of the time when the dollar
appreciates, but only 5/15 or 33.33% of the time when the dollar depreciates. According to
the Henriksson–Merton test, the sum of these two proportions should be over 1 for a firm to
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Chapter 10: Exchange Rate Determination and Forecasting
7
have market timing ability. However, the sum in this case is only 90.47%. While Morrissey
obviously dominates Pixie Land Exchange, it is not clear that it is better than FOREX Cures.
The proportions of correct conditional forecasts of FOREX Cures are 12/35 (34.29%) and
12/15 (80%) for a sum of 114.29%. Consequently, only FOREX Cures shows directional
forecasting ability.
b. What additional information should Blake try to obtain in order to form a better
judgment?
Answer: Directional forecasting ability in the foreign exchange market is not particularly
useful if the forecasts are to be used in speculative strategies. To this end, it would have been
more useful to know whether the forecasting firms were on the correct side of the forward
rate. Ideally, a full record of forecasts would be obtained. Then, accuracy statistics (like the
RMSE) and profitability statistics (like the Sharpe ratio) could be computed.
3. Mini-Case: Currency Turmoil in Zooropa
Fad Gadget has never worked so hard in his entire life. It is near midnight, and he is
still poring over statistics and tables. Fad recently joined Smashing Pumpkins, a
relatively young but fast-growing British firm. Smashing Pumpkins produces and
distributes an intricate device that turns fresh pumpkins into pumpkin pie in about 30
minutes. Recently, the firm has started exporting to Zooropa. Some of the largest and
tastiest pumpkins are grown in Zooropa, and Zooropa’s population boasts the highest
per capita pumpkin consumption in the world. A recent analysis of the pumpkin
market in Zooropa has left the company’s senior managers very impressed with the
profit potential.
Although Zooropa consists of 10 politically independent countries, their
currencies are linked through a system called the Currency Rate Linkage System
(CRLS) that works exactly like the former Exchange Rate Mechanism (ERM) of the
EMS worked before the currency turmoil started in September 1992. The anchor
currency is the banshee of Enigma, the leading country in Zooropa.
Initial contacts with importers in Zooropean countries indicated that they
typically insist on payment in their own local currency. About a week ago, Cab
Voltaire, the CEO of Smashing Pumpkins, expressed concerns about this development
and asked Fad to lead a research team to further examine the present state of the
currency system of Zooropa. Cab viewed the outlook for the banshee relative to the
pound quite favorably and did not predict any substantial depreciation of the banshee
against any other major currency. However, the precarious economic situation of some
of the countries in Zooropa and the growing importance of speculative pressures in
Zooropa’s currency markets last week suddenly made him suspicious about the
possibility of realignments within the system. He even doubted the long-term viability of
the system. Cab instructed Fad to examine the following issues:
 Which currencies in the system exhibit the highest realignment risk?
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Chapter 10: Exchange Rate Determination and Forecasting

If a currency realigns and gets devalued, what are the effects on our sales
and profit margins in this particular country? Can we take the realignment
possibility into account in our pricing?
 Suppose a currency is forced to leave the CRLS. What are the effects on
exchange rates, interest rates, and the outlook for sales in that country?
What is the likelihood of this occurring for the different countries?
Fad Gadget felt nervous. A meeting was scheduled with Cab the day after
tomorrow. He wanted to write a thorough and insightful report. At the last
management meeting, he had the uneasy feeling that some senior managers doubted his
abilities. Some managers were naturally suspicious of a young Australian newcomer
with his MBA. His earring and punk hairdo did not exactly help either. His team of
analysts had already assembled a table with relevant macroeconomic and financial data
(see Exhibit 10.11). “If only I could use this to rank the different countries according to
realignment risk,” he thought.
a) Realignment rankings
The data provided are a scrambled version of an Exhibit that appeared in the Economist of
September 19, 1992, when a currency crisis in Europe had just erupted. The Exhibit presented
macro-economic statistics for all the countries participating in the European system. To prevent
students guessing where the data are from, we scrambled the country names two ways. First, we
gave each European country another name that did provide a vague hint on the actual European
country of origin. However, we then randomly assigned the actual data to the fictitious
countries. Here is the “key”:
Zooropa
Country
Sinead
Carmen
Marquee
Fries
Ney
HelpIsink
Benfica
Che Ora
Vachement
Enigma
Reference to European Country
Ireland (Irish singer)
Carmen (Spanish opera)
UK (club in London)
Belgium (French fries are Belgian!)
Denmark (No in Danish)
the Netherlands (below sea level)
Portugal (soccer team)
Italy (only Italian Geert knows)
France (French stop word)
Germany (pop band)
Data
from
European
country
Ireland
France
Spain
Portugal
Denmark
Belgium
the Netherlands
UK
Italy
Germany
We now reproduce the original Economist table from the article “A Ghastly Game of
Dominoes.”
Who’s Next?
Legend for Chart:
A - Currency's ERM position Sept 15th
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Chapter 10: Exchange Rate Determination and Forecasting
9
B - Currency's over/under valuation, %
C - Reserves, import cover
D - Budget deficit as % of GDP, 1992
E - Inflation rate %, latest
F - GDP growth, %, 1992
G - Devaluation risk
Italy
Britain
Spain
Portugal
Denmark
Belgium
Holland
France
Ireland
Germany
A
27
-90
16
-3
-22
31
30
-36
-6
36
B
2
3
11
11
-2
-18
-16
-12
-10
--
C
0.5
2.6
8.2
11.7
2.5
1.3
1.5
3.1
2.9
1.7
D
-11.3
-4.6
-4.9
-5.4
-2.1
-5.5
-3.4
-2.3
-1.9
-3.4
E
5.2
3.6
5.7
9.5
2.2
2.1
3.5
2.7
3.6
3.5
F
1.3
-0.8
2.0
2.8
2.1
1.6
1.6
2.0
2.4
1.3
G
1
2
3
4
5=
5=
5=
8
9
--
Sources: OECD; IMF; government statistics; NatWest; The Economist poll of forecasters. A
indicates the currency value as a % of permitted divergence from the central rate, B indicates
the central rate against the DM relative to PPP, C indicates foreign-exchange reserves (mid
September estimates) in number of months' imports, D, E, and F are forecasts, G indicates
devaluation risk with 1=greatest risk, 9=least risk.
Based on this article, we can actually use the data given to come up with a realignment ranking.
For example, the position in the CRLS system (the divergence indicator in the EMS, a summary
measure of the currency’s position in its bands relative to all other currencies), and the reserves
import cover are direct indicators of devaluation pressure. An overvalued currency, a large
budget deficit, high inflation, and low GDP growth are “bad” economic fundamentals that may
contribute to speculative pressure on the currency. The Economist did a very simple exercise. It
ranked all the countries on these criteria from “worst” (most speculative pressure) to “best”
(least speculative pressure). It then added up the ranks and came up with an overall devaluation
risk ranking. Using the information provided on fundamentals leads to a surprisingly accurate
realignment risk ranking. These ranks are reproduced in the last column of the Economist table.
Italy and Britain were actually forced out of the EMS during the September currency crisis.
Spain was forced to devalue and Portugal later followed suit. The other countries with better
fundamentals duly survived.
Whether the currency crisis in 1992 was actually predictable is still a topic of academic debate.
Some important scholars in the area have argued that the crisis was almost completely
unpredictable. Our case seems to indicate otherwise, although more formal analysis is necessary
(and is still being conducted by many scholars). Finally, while some countries, such as France,
still looked relatively “safe,” another currency crisis erupted in July August 1993, which led to
rather drastic changes in the operation of the EMS, including a widening of the bands to 15%.
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10 Chapter 10: Exchange Rate Determination and Forecasting
b) Effects of Realignments/Exits for the Firm
Since the British firm agrees to local currency pricing, the risk is that the Zooropa currencies get
devalued. Two cases must be considered:
(1) The price remains fixed. In this case, the revenue per unit sold in pounds decreases and
the profit margin is squeezed because the firm’s costs are local (in pounds). Except for
“dynamic effects” (see below), there need not be any effect on the number of units sold.
Sales may remain unchanged in local currency, but sales go down in pounds.
(2) The firm tries to “pass through” the exchange rate change and raises the price of the
pumpkin device as in Chapter 9. The optimal response will be to raise the domestic price
because the firm does not want to sell as much quantity in that market. The amount of
the price increase depends on the elasticity of the demand curve. The price will rise less,
the more elastic is the demand – that is, the larger the percentage change in quantity with
a given percentage change in price.
These are the immediate effects. The realignment restores the general competitiveness of the
Zooropean country. If the devaluation is successful, it accomplishes a decrease in real wages
locally and shifts resources to the export sector. It should also make potential local competitors
to Smashing Pumpkins more competitive. However, the case seems to indicate that these are
non-existent. Initially, this may lower the demand for all imports (the whole idea of the
devaluation in the first place). If the economy was not producing at full capacity, the increased
competitiveness may spur considerable additional economic growth. This, in fact, happened in
Britain and Sweden after the 1992 devaluations. Higher growth may then lead to higher imports
and increase the demand for the pumpkin device. These are potential dynamic effects.
Eventually, this may cause inflationary pressures to creep back into the economy. Unsuccessful
devaluations will let higher import prices (if there is some pass through or most import products
are priced in foreign currency) affect wages and the general price level. In this case, there may
be only small effects for the British firm. Hence, the dynamic effects depend on the success of
the devaluation.
All of this analysis goes through for exits from the target zone. In fact, the effects for
realignments are probably considerably smaller, since an exit may lead to much lower exchange
rates and in some cases to lower interest rates, which further help the Zooropa economy become
more competitive.
It has to be said that these are all “elaborate guesses,” since in reality new shocks to the
economy may cause completely different outcomes.
c) Incorporating Realignment Risk into Pricing/Hedging
The market will anticipate the realignment. In fact, if UIRP holds, the interest differential with
Britain and the forward rate relative to the pound will reflect the expected currency depreciation
(the probability of a devaluation multiplied by the magnitude of the devaluation). Hence,
hedging the risk will automatically lead to lower pound revenue in the future. Ideally, one
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Chapter 10: Exchange Rate Determination and Forecasting 11
establishes a pricing scheme that takes potential realignments into account, for example using
forward rates. Your personal view on realignment risk may differ from the forward rates though.
Alternatively, a dynamic real exchange rate risk sharing formula as in the SAFE AIR case could
be proposed.
d) Effects of devaluation/exits on exchange and interest rates
Exchange rates fall, by definition. The effect on the interest rate however may be different in
both cases. With devaluation it is very likely that the interest rate will drop. That is because the
interest rate was most probably very high during the speculative attack preceding the
realignment and it now drops back to normal levels, which are still likely to be above the
interest rate of the anchor currency. With an exit, the pressure of a speculative attack gets
relieved as well, and now the interest rate need not exceed the rate on the anchor currency.
However, the exiting currency loses its “inflation credibility mechanism” by leaving the target
zone, and hence, interest rates may go up reflecting higher expected inflation in the future.
When Britain exited the ERM, its interest rates dropped substantially at the short end, but they
remained quite high at the long end. In Mexico, after the December 1994 crisis, peso interest
rates rose reflecting fears of future depreciation and inflation. Both market responses seem
justified both by the data at the time of the event and the subsequent experiences of the
economies.
4. Web Problem: Consider the Big MacPPP data from Chapter 8. Pick a currency that
was overvalued versus the dollar. Go to http://www.oanda.com/ to find the current
exchange rate and determine how much you would have made or lost by speculating on
the overvaluation?
While there are many possibilities, we examined data on June 9, 2017, to see what would have
happened if one had traded in 2015 on the MacPPP data from Chapter 8 for the Malaysian
ringgit (MYR) and the Swiss franc (CHF). In 2015, the ringgit was 56% undervalued against the
USD with a PPP exchange rate of MYR1.59/USD and an actual exchange rate of
MYR3.62/USD. Betting that the ringgit would strengthen would have been a losing proposition
as the actual exchange rate on June 9, 2017 from Oanda was MYR4.26/USD. On the other hand,
the Swiss franc was 57% overvalued in 2015 with a PPP exchange rate of CHF1.36/USD and an
actual exchange rate of CHF1.36. Betting that the CHF would weaken would have made money
as the actual exchange rate on June 9, 2017 from Oanda was CHF0.9696/USD.
Mini Case: Valuing Currency Management: TOM versus US Commerce Bank
On February 19, 2009, an arbitral tribunal found that USCB Analytics, a wholly owned
subsidiary of USCB Corporation, a large U.S. based bank, had breached an exclusivity
provision of its joint venture (JV) agreement with Trend Ontledings Maatschappij (TOM),
a Dutch currency management business. Consequently, TOM claimed USCB was obligated
to compensate the firm for lost earnings that would have accrued to TOM during the life of
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12 Chapter 10: Exchange Rate Determination and Forecasting
the joint venture. Established in 2006, the joint venture was to last for a minimum of four
years. USCB was responsible for marketing the JV to third party clients including central
banks, institutional investors and corporate clients. TOM was responsible for providing the
investment management expertise by delivering a low return, low volatility, alpha currency
investment product. TOM had a long history of quantitative trading in the currency
markets. In the 1970s, TOM was thought to be the first firm to apply computerized trading
to exchange rate markets. Successful partnerships with a number of U.S. banks in the
1980s and early-1990s made Geert Rijkaard, TOM’s founder, one of the richest men in the
world. Because the firm’s strategy focused on European currencies relative to the dollar,
the arrival of the euro in 1999 led to a suspension of TOM’s trading activities. However,
after adapting its models to focus on the euro/dollar pair, TOM started trading again in
2004, and began actively looking for partners that could help market the product.
However, as a result of the contract breach, TOM had terminated the JV on July 30,
2007. TOM claimed that it was owed in excess of $300 million from USCB. Both parties
assembled teams of experts to make their cases to the tribunal. The tribunal would then use
the information provided by these experts as the basis for making a decision as to the
amount of damages owed to TOM.
While all names used here are fictitious, the story above is based on a real-world
case. A Columbia CaseWorks case written by Geert Bekaert (2012) provides more details.
It lays out the analysis by TOM’s team to motivate the $300 million damages number,
relying largely on the detailed business plan at the time the JV was formed. The case
further describes several key exhibits assembled by USCB’s team. Its first task at hand was
to simply figure out what kind of currency manager TOM was: does it follow trends, trade
on fundamentals or run a carry strategy? The team also believed it would be important to
study the relative investment performance of the JV, and did so using actual data from the
Barclay Currency Traders Index. Given the large number of currency funds which were
available to investors, the JV’s ability to win clients and grow its AUM would undoubtedly
be closely linked to its performance, both in absolute terms and relative to other currency
funds. Finally, simply generating a plausible track record of returns suitable for use in
projections raised interesting issues. For example, TOM’s team had resorted to using paper
returns (meaning returns from a trading strategy that had not been used in actual trading
yet) to pull together a long return record. To learn more about this case, please go to.
http://www8.gsb.columbia.edu/caseworks/node/298
For instructors, the case also comes with a Teaching Note and PowerPoint slides which
contain the solution.
©2017 Cambridge University Press
Chapter
20
Foreign Currency Futures and Options
QUESTIONS
1. How does a futures contract differ from a forward contract?
Answer: Foreign currency futures contracts, or futures contracts for short, allow individuals
and firms to buy and sell specific amounts of foreign currency at an agreed-upon price
determined on a given future day. Although this sounds very similar to forward contracts,
there are a number of important differences between forward contracts and futures contracts.
The first major difference between foreign currency futures contracts and forward
contracts is that futures contracts are traded on an exchange, whereas forward contracts are
made by banks and their clients. Orders for futures contracts must be placed during the
exchange’s trading hours, and pricing occurs in the “pit” by floor traders or on an electronic
trading platform where demand is matched to supply. In contrast to forward contracts, where
dealers quote bid and ask prices at which they are willing either to buy or sell a foreign
currency, for each party that buys a futures contract, there is a party that sells the contract at
the same price. The price of a futures contract with specific terms changes continuously, as
orders are matched on the floor or by computer.
A second major difference is that futures exchanges standardize the amounts of
currencies that one contract represents. Thus, futures contracts cannot be tailored to a
corporation’s specific needs as can forward contracts. But the standardized amounts are
relatively small compared to a typical forward contract, and if larger positions are desired,
one merely purchases more contracts. Standardization with small contract sizes makes the
contracts easy to trade, which contributes to market liquidity.
A third major difference involves maturity dates. In the forward market, a client can
request any future maturity date, and active daily trading occurs in contracts with maturities
of 30, 60, 90, 180, or 360 days. The standardization of contracts by the futures exchanges
means that only a few maturity dates are traded. For example, IMM contracts mature on the
third Wednesday of March, June, September, and December. These dates are fixed, and
hence the time to maturity shrinks as trading moves from 1 day to the next, until trading
begins in a new maturity. Typically, only three or four contracts are actively traded at any
given time because longer-term contracts lose liquidity.
The final major difference between forward contracts and futures contracts concerns
credit risk. This issue is perhaps the chief reason for the existence of futures markets. In the
forward market, the two parties to a forward contract must directly assess the credit risk of
their counterparty. Banks are willing to trade with large corporations, hedge funds, and
institutional investors, but they typically don’t trade forward contracts with individual
investors or small firms with bad credit risk.
The futures market is very different. In the futures markets, a retail client buys a futures
contract from a futures brokerage firm, which in the United States is typically registered with
the Commodity Futures Trading Commission (CFTC) as a futures commission merchant
©2017 Cambridge University Press
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Chapter 20: Foreign Currency Futures and Options
(FCM). Legally, FCMs serve as the principals for the trades of their retail customers.
Consequently, FCMs must meet minimum capital requirements set by the exchanges and
fiduciary requirements set by the CFTC. In addition, if an FCM wants to trade on the IMM, it
must become a clearing member of the CME. In years past, clearing memberships used to be
tradable, and the prices at which they traded were indications of how profitable futures
trading on the exchange was expected to be. In 2000, the CME became a for-profit stock
corporation, and its shares now trade on the NYSE. To obtain trading rights, an FCM must
buy a certain amount of B-shares of CME stock and meet all CME membership
requirements.
When a trade takes place on the exchange, the clearinghouse of the exchange, which is an
agency or a separate corporation of a futures exchange, acts as a buyer to every clearing
member seller and a seller to every clearing member buyer. The clearinghouse imposes
margin requirements and conducts the daily settlement process known as marking to market
that mitigates credit concerns. These margin requirements are then passed on to the
individual customers by the futures brokers.
2. What effects does “marking to market” have on futures contracts?
Answer: The process of marking to market implies that futures contracts have daily cash
flows associated with them. One can be either long (having bought the contract) or short
(having sold the contract) in the futures market at a particular price. Since both sides are
treated symmetrically, let’s assume you are long. You must post funds in a margin account,
and if on subsequent days, the futures price moves in your favor, that is, the foreign currency
futures prices rises as the foreign currency strengthens; funds are placed into your margin
account and are taken out of the margin accounts of those who sold the foreign currency
futures contract. This process continues every day until the maturity date of the contract.
3. What are the differences between foreign currency option contracts and forward
contracts for foreign currency?
Answer: The primary difference between a foreign currency option contract and a forward
contract is that the option contract gives the purchaser of the option, the right, but not the
obligation to transact. If the state of the world in the future is favorable to the purchasers of
the option, they will transact. If the state of the world is unfavorable, the option is worthless.
Forward contracts are completely uncontingent on the state of the world in the future.
4. What are you buying if you purchase a U.S. dollar European put option against the
Mexican peso with a strike price of MXN10.0/$ and a maturity of July? (Assume that it
is May and the spot rate is MXN10.5/$.)
Answer: A European put option gives you the right to sell the underlying asset at the strike
price on the maturity date of the contract. Thus, you are buying the right to sell USD for
MXN at the price of MXN10.0/$ on the maturity date of the contract in July. This option is
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Chapter 20: Foreign Currency Futures and Options
3
currently said to be “out of the money” because the strike price is lower than the current
exchange rate.
5. What are you buying if you purchase a Swiss franc American call option against the
U.S. dollar with a strike price of CHF1.30/$ and a maturity of January? (Assume that it
is November and the spot rate is CHF1.35/$.)
Answer: American options can be exercised anytime between the purchase of the option and
the maturity date. Thus, a Swiss franc American call option against the U.S. dollar with a
strike price of CHF1.30/$ and a maturity of January gives the buyer the right, but not the
obligation, to purchase CHF with USD at a price of CHF1.30/$ between November and the
maturity date in January. The option is currently said to be “out of the money” because the
strike price (expressed in dollars per Swiss franc) is higher than the current exchange rate
[(1/1.30) > (1/1.35)] and you are purchasing CHF.
6. What is the intrinsic value of a foreign currency call option? What is the intrinsic value
of a foreign currency put option?
Answer: The immediate revenue from exercising an option is called the option’s intrinsic
value. Let K be the strike price, and let S be the current spot rate, both in domestic currency
per unit of foreign currency. Then, the intrinsic value per unit of foreign currency can be
represented as
Call option: max[S – K, 0]
Put option: max[K – S, 0]
where max denotes the operation that takes the maximum of the two numbers between square
brackets.
7. What does it mean for an American option to be “in the money”?
Answer: If an American option is “in the money,” its intrinsic value is positive. For a call
option, this means that the strike price is less than the current market price; while for a put
option, this means that the strike price is greater than the current market price.
8. Why do American option values typically exceed their intrinsic values?
Answer: The time value of an option is the current price or value of the option minus its
intrinsic value:
Time value of an option = Option price – Intrinsic value
Options have time value because the stochastic evolution of the underlying asset price
provides possibilities of even better payoffs in the future compared to the intrinsic value. If
this were not the case, the owner of the American option would exercise it.
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Chapter 20: Foreign Currency Futures and Options
9. Suppose you go long in a foreign currency futures contract. Under what circumstances
is your cumulative payoff equal to that of buying the currency forward?
Answer: The payoffs of futures contracts and forward contracts are only “essentially the
same” because a slight difference in payoffs arises due to the fact that interest is earned on
future profits, or interest must be paid on future losses, in the marking to market process.
Technically, if the path of short-term interest rates could be foreseen—that is, if there were
no random changes in future short-term interest rates—there would be an arbitrage
possibility if the forward exchange rate were different from the futures price because you
would know how you could invest the profits or borrow to finance your losses. However,
future interest rates are not known with certainty, so forward prices and futures prices can be
different, in theory. In practice, though, the price differentials are minimal, and they appear
to be within the transaction costs of the forward market. Therefore, we argue that futures
prices are “essentially the same” as forward prices.
10. What is basis risk?
Answer: The basis is the difference between the price of the futures contract at time t, for a
particular maturity in the future, and the spot rate at time t. At the maturity date, the basis is
zero. If the maturity of your foreign currency asset or liability does not match a settlement
date in the futures market, the relationship between the spot exchange rate at the time the
transaction takes place and the futures price of the foreign exchange is somewhat uncertain
(as the basis is not zero). To provide a perfect hedge, the price of the futures contract should
move one-for-one with the spot exchange rate. Then, being long in the foreign currency from
an underlying transaction can be hedged by going short in the corresponding futures contract.
If this is not the case, the hedge is said to suffer basis risk.
11. Your CEO routinely approves changes in the fire insurance policies of your firm to
protect the value of its buildings and manufacturing equipment. Nevertheless, he argues
that the firm should not buy foreign currency options because, he says, “We don’t
speculate in FX markets!” How could you convince him that his positions are mutually
inconsistent?
Answer: Options provide payoffs that are analogous to insurance and can be used in hedging
situations. With fire insurance, you pay the insurance premium, and if there is a fire, the
insurance company compensates you. The realization of the fire is a bad state of the world,
but if the fire does not occur, you needlessly paid for the insurance. If you are receiving
foreign currency, the bad state of the world is that the foreign currency weakens relative to
the domestic currency. To avoid this loss, you can buy a foreign currency put option, which
gives you the right but not the obligation to sell foreign currency at the strike price. If the bad
state of the world occurs, you exercise your put, but if the good state of the world occurs, you
ignore you option and sell the foreign currency which has appreciated in value relative to the
domestic currency. The put option places a floor on your domestic currency revenue, which
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Chapter 20: Foreign Currency Futures and Options
5
provides a hedge and is not speculation. Of course, ex post you will regret having hedged,
but this is exactly like the insurance situation in which the fire did not occur.
12. Why do options provide insurance against foreign exchange risks in bidding situations?
Why can’t you hedge with a forward contract in a bidding situation?
Answer: Let’s assume the bidding situation involves the company determining a particular
amount of foreign currency for providing a service or selling some goods. By bidding a fixed
amount of foreign exchange, a company incurs a contingent transaction foreign exchange
risk. If the company wins the bid and the foreign currency weakens relative to the domestic
currency, the domestic currency value of the contractual foreign currency revenue may
already have fallen such that the entire dollar profit could be eliminated before the project
begins. If the firm’s strategy is to get the contract and then hedge, it could be too late.
Foreign exchange options provide a hedging solution. Because the company ultimately
wants to sell the foreign currency if it wins the bid, the company should hedge by buying a
foreign currency put against the domestic currency. Then, if company wins the contract and
the foreign currency has weakened relative to the domestic currency, the loss of value on the
contract is offset by a gain in the value of the foreign currency put. The company can sell the
foreign currency from the contract at the exercise price, which is higher than the spot market.
If the company does not win the contract, the value of the foreign currency put is the
maximum that the firm can lose. This is exactly like an insurance contract.
If the company sells foreign currency forward, it acquires an uncontingent foreign
currency liability. No matter what happens at maturity, the company will have to sell a
specific amount of foreign currency to the bank. Everything will be fine if the company gets
the contract, but if the company does not get the contract, it will have to buy foreign currency
to fulfill the uncontingent commitment of the forward contract. If the foreign currency has
strengthened, the company will lose money, and the potential loss is essentially unbounded.
13. Suppose that you have a foreign currency receivable (payable). What option strategy
places a floor (ceiling) on your domestic currency revenue (cost)?
Answer: If you have a foreign currency receivable, you eventually want to sell foreign
currency. Purchasing the option that gives you the right to sell (a put option) provides a
hedge that places a floor on your domestic currency revenue. If you have a foreign currency
payable, you eventually want to buy foreign currency. Purchasing the option that gives you
the right to buy (a call option) provides a hedge that places a ceiling on your domestic
currency cost.
14. Describe qualitatively how changing the strike price of the option provides either more
or less expensive insurance.
Answer: Let’s consider hedging a foreign currency receivable with a put option. High-quality
insurance in this context means that the floor we create on our domestic currency revenue is
high. The floor is directly related to the strike price of the put option. The higher the strike
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Chapter 20: Foreign Currency Futures and Options
price of the option, the less the foreign currency must depreciate before we can exercise the
option and cut our losses. But, just as insurance that covers more losses is more expensive,
put options with higher strike prices are more expensive. Thus, if the foreign currency
strengthens and you do not need the insurance of the put option, you will have spent more
money on the insurance and will need more appreciation of the foreign currency before you
do better than you could have by locking in a forward contract.
15. Why does an increase in the strike price of an option decrease the value of a call option
and increase the value of a put option?
Answer: We know that holding constant the maturity date of two options implies that the
distribution of possible future exchange rates is the same for the two options. Hence, it
should be apparent that increasing the exercise price of a call option must decrease its value
because doing so removes possible states of the world over which the contract provides
revenue when the strike price is lower. Conversely, increasing the exercise price of a put
option must increase its value because doing so adds possible states of the world over which
the contract provides revenue compared to when the strike price is lower.
16. Why does an increase in the volatility of foreign exchange rates increase the value of
foreign currency options?
Answer: The easiest way to understand how an increase in variance affects option prices is to
place the strike price of a call option at the conditional mean of two probability distributions,
one with a low variance and the other with a high variance. The increase in the variance of
the possible future exchange rate clearly increases the possible range of future exchange
rates. But, because the conditional mean is the same, the probability that the option will
finish in the money is still one-half because one-half of the probability distribution remains
above the strike price. However, if the option does finish in the money, the distribution with
the larger variance yields possibly larger payoffs, and the option will cost more. A
symmetrical argument can be applied to a put option.
17. How does increasing time to maturity affect foreign currency option value?
Answer: Here, it is important to distinguish clearly between American-style and Europeanstyle options. For American options, the effect is unambiguous: Increasing the time to
maturity always increases an option’s value because it increases the uncertainty of the spot
exchange rate at maturity. When this effect is combined with the fact that the holder of a 6month option can always treat the option as a 3-month option, we clearly see that the
additional 3 months of maturity cannot hurt the payoff to the holder of the option as long as
the holder of the option can exercise it early.
For European options, the situation is not so simple. Although the effect of an increase in
time to maturity is technically ambiguous, in most situations, the effect of the increased
uncertainty of the spot exchange rate at maturity dominates, and option prices increase.
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Chapter 20: Foreign Currency Futures and Options
7
Nevertheless, this is not always true because it is possible for a European option that is
currently in the money to lose value as time evolves. You would like to be able to exercise the
option to lock in the revenue now, but you cannot do so prior to maturity.
18. What is the payoff on an average-rate pound call option against the dollar?
Answer: The payoff per pound on an average-rate pound call option against the dollar with a
strike price of K($/£) is max[0, S ($/£) – K($/£)], where S ($/£) defines the average dollarpound exchange rate between the initiation of the contract and the expiration date. To
calculate the average exchange rate, the counterparties to the option contract must agree on a
source for the data and a way of computing the average. They must decide on a time interval
for the observations entering the average, which could be daily, weekly, or monthly, and they
must decide whether the average is an arithmetic or geometric average. At the maturity of an
average-rate option, the seller of the option settles the contract by delivering the dollar value
of the option payoff to the buyer. Because an average of future exchange rates is less volatile
than the future spot rate at maturity, average-rate options are less expensive than standard
European options.
19. Suppose the current spot rate is $1.29/€. What is your payoff if you purchase a downand-in put option on the euro with a strike price of $1.31/€, a barrier of $1.25/€, and a
maturity of 2 months? When would someone want to do this?
Answer: For a down-and-in option, the exchange rate must first cross the barrier to activate
the contract. Then, the buyer of the option has the right to exercise at maturity. So, the payoff
on the option described above is max[0, $1.31/€ - S(T,$/€)], where S(T,$/€) is the exchange
rate at maturity, but only if the exchange rate falls from its current value of $1.29/€ to
$1.25/€ sometime during the 2 months between the initiation of the contract and the maturity
date. Such an option is less expensive than a standard put option. It might therefore be
purchased by someone who is bearish on the euro, believes initial volatility in the market will
activate the option with reasonably high probability, and wants to cut the cost of obtaining a
relatively high payoff, even in the currency strengthens somewhat towards the end of the
contract.
PROBLEMS
1. If you sold a Swiss franc futures contract at time t and the exchange rate has evolved as
shown here, what would your cash flows have been?
Day
t
Futures
Price
$/CHF
Change in
Futures
Price
Gain or
Loss
Cumulative
Gain or
Loss
0.7335
Margin
Account
$2,000.00
0.0056
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Chapter 20: Foreign Currency Futures and Options
t+1
0.7391
-$700.00
-$700.00
$2,000.00
t+2
0.7388
-0.0003
$37.50
-$662.50
$2,037.50
t+3
0.7352
-0.0036
$450.00
-$212.50
$2,487.50
t+4
0.7297
-0.0055
$687.50
$475.00
$3,175.00
Answer: Because you sold the Swiss franc futures contract, you will gain when the Swiss
franc depreciates versus the dollar and you will lose when the Swiss franc strengthens.
On the first day, we assume you established your account with an initial margin of
$2,000, and that you sold your contract at the closing price of $0.7335/CHF. The contract
size is CHF125,000. On the second day, the futures price increases from $0.7335/CHF to
$0.7391/CHF. You consequently lose $0.0056/CHF  CHF125,000 = $700, which would
take your margin account to $1,300. Because this is less than the maintenance margin of
$1,400, you would receive a margin call that would require you to bring you margin
account back to $2,000. On the third day, the futures price moves in your favor, and you
gain $0.0003/CHF  CHF125,000 = $37.50. You have the option of leaving this extra
money in your margin account or taking it out. We assume that you leave it in. On the
fourth day, the futures price again moves in your favor, and you gain $0.0036/CHF
 CHF125,000 = $450. Finally, on the fifth day, the futures price again moves in your
favor, and you gain $0.0055/CHF  CHF125,000 = $687.50. The cumulative gain is
$475.00.
2. Given the following information, how much would you have paid on September 16 to
purchase a British pound call option contract with a strike price of 155 and a maturity
of October?
Data for September 16
Calls
Puts
50,000 Australian Dollar Options (cents per unit)
64 Oct
—
0.48
65 Oct
—
0.90
67 Oct
0.22
—
31,250 British Pounds (cents per unit)
152½ Dec
—
4.10
155 Oct
1.50
3.62
155 Nov
2.35
—
Answer: The correct price on September 16 for a British pound call option with a strike
price of 155 and a maturity of October is 1.50. The units are cents per pound or
$0.0150/£. The contract size is £31,250. Therefore, you would have paid
$0.0150
× £31,250 = $468.75
£
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Chapter 20: Foreign Currency Futures and Options
9
3. Using the data in problem 2, how much would you have paid to purchase a Australian
dollar put option contract with a strike price of 65 and an October maturity?
Answer: The correct price on September 16 for an Australian dollar put option with a strike
price of 65 and a maturity of October is 0.90. The units are cents per Australian dollar or
$0.0090/AUD. The contract size is AUD50,000. Therefore, you would have paid
$0.0090
× AUD50,000 = $450
AUD
4. Suppose that you buy a €1,000,000 call option against dollars with a strike price of
$1.2750/€. Describe this option as the right to sell a specific amount of dollars for euros
at a particular exchange rate of euros per dollar. Explain why this latter option is a
dollar put option against the euro.
Answer: The €1,000,000 call option against dollars with a strike price of $1.2750/€ gives the
buyer the right, but not the obligation, to buy €1,000,000 at the strike price of $1.2750/€ in
which case the person would pay $1,275,000 for the euros. Clearly, this is the same as an
option to sell $1,275,000 at a strike price of [1 / ($1.2750/€)] = €0.784314/$. This latter
option is a $1,275,000 dollar put option against euros with a strike price of €0.784314/$.
5. Assume that today is March 7, and, as the newest hire for Goldman Sachs, you must
advise a client on the costs and benefits of hedging a transaction with options. Your
client (a small U.S. exporting firm) is scheduled to receive a payment of €6,250,000 on
April 20, 44 days in the future. Assume that your client can borrow and lend at a 6%
p.a. U.S. interest rate.
a. Describe the nature of your client’s transaction exchange risk.
Answer: Your client is scheduled to receive €6,250,000 in 44 days. If no hedging is done,
and the euro weakens in value relative to the dollar, the client will lose money. The
amount of the loss could be substantial if a major weakening occurs.
b. Use the appropriate American option with an April maturity and a strike price of
129¢/€ to determine the dollar cost today of hedging the transaction with an option
strategy. The cost of the call option is 3.93¢/€, and the cost of the put option is
1.58¢/€.
Answer: To hedge foreign currency revenue with an option, you must purchase a put
option that gives you the right to sell euros. This puts a floor on your revenue. The cost of
the option would be 1.58¢/€, or
$0.0158
× €6,250,000 = $98,750
€
c. What is the minimum dollar revenue your client will receive in April? Remember to
take account of the opportunity cost of doing the option hedge.
©2017 Cambridge University Press
10 Chapter 20: Foreign Currency Futures and Options
Answer: If the exchange rate is less than $1.29/€ in April, your client will be able to sell
euros at that value. If the future spot exchange rate is higher, the client will sell euros at
the future spot exchange rate. In either case, if they hedge with the option contract, they
will have less revenue. The future value of $98,750 at 6% for 44 days is

 6  44  
$98,750 × 1 + 

  = $99,474.17
 100  360  

Thus, the minimum net revenue that the client will have is
 $1.29

 € × €6,250,000  - $99,474.17 = $7,963,025.83


d. Determine the value of the spot rate ($/€) in April that would make your client
indifferent ex post to having done the option transaction or a forward hedge. The
forward rate for delivery on April 20 is $1.30/€.
Answer: If the client does the forward hedge, their revenue will be
$1.30
× €6,250,000 = $8,125,000
€
If the client does the option hedge and does not have to exercise the option, they will sell
the euros in 42 days, and their revenue will
(S(t+42,$/€) × €6,250,000 ) - $99,474.17
If this option revenue is to equal the forward revenue, we know
(S(t+42,$/€) × €6,250,000 ) - $99,474.17 = $8,125,000
Solving this equation gives S(t+42,$/€) =
$1.3159
.
€
6. Assume that today is September 12. You have been asked to help a British client who is
scheduled to pay €1,500,000 on December 12, 91 days in the future. Assume that your
client can borrow and lend pounds at 5% p.a.
a. Describe the nature of your client’s transaction exchange risk.
Answer: Your client is scheduled to pay €1,500,000 in 91 days. If no hedging is done, and
the euro strengthens in value relative to the pound, the client will lose money. The
amount of the loss could be substantial if a major strengthening occurs.
b. What is the option cost for a December maturity and a strike price of £0.72/€ to
hedge the transaction? The option premiums per 100 euros are £1.70 for calls and
£2.40 for puts.
Answer: To hedge foreign currency costs with an option, you must purchase a call option
that gives you the right to buy euros. This puts a ceiling on your costs. The cost of the
option would be £1.70 per 100 euro, or
©2017 Cambridge University Press
Chapter 20: Foreign Currency Futures and Options 11
£1.70
× €1,500,000 = £25,500
€100
c. What is the maximum pound cost your client will experience in December?
Answer: If the exchange rate is greater than £0.72/€ in December, your client will be able
to buy euros at that value. If the future spot exchange rate is lower than £0.72/€, the client
will buy euros at the future spot exchange rate. In either case, if they hedge with the
option contract, they will have higher costs. The future value of £25,500 at 5% for 91
days is

 5  91  
£25,500 × 1 + 

  = £25,817.88
 100  365  

Thus, the maximum net cost that the client will face is
 £0.72

 € × €1,500,000  + £25,817.88 = £1,105,817.88


d. Determine the value of the spot rate (£/€) in December that makes your client
indifferent ex post to having done the option transaction or a forward hedge if the
forward rate for delivery on December 11 is £0.70/€.
Answer: If the client does the forward hedge, their cost will be
 £0.70

 € × €1,500,000  = £1, 050, 000


If the client does the option hedge and does not have to exercise the option, they will buy
the euros in 91 days, and their cost will be
(S(t+89,£/€) × €1,500,000 ) + £25,817.88
If this option cost is to equal the forward cost, we know
(S(t+89,£/€) × €1,500,000 ) + £25,817.88 = £1, 050, 000
£0.6828
.
€
7. Assume that today is June 11. Your firm is scheduled to pay £500,000 on August 15, 65
days in the future. The current spot is $1.75/£, and the 65-day forward rate is $1.73/£.
You can borrow and lend dollars at 7% p.a. Suppose you think options are overpriced
because you think the dollar will be in a tight trading range in the near future. You
have been thinking about selling an option as a way to reduce the dollar cost of your
pound payable.
a. If an August pound option with a strike price of 175¢/£ costs 4.5¢/£ per pound for
the call and 4¢/£ for the put, what is the minimum that you will have to pay in
August to eliminate your pound payable? Over what range of future exchange rates
will this price be achieved?
Solving this equation gives S(t+89,£/€) =
Answer: You need to eventually buy pounds to eliminate your liability of £500,000 due in
65 days. If you think options are too expensive to buy, you can consider the speculative
©2017 Cambridge University Press
12 Chapter 20: Foreign Currency Futures and Options
strategy of selling someone an option that allows them to sell pounds to you. This is a
pound put option. You would take in 4¢/£ for the put, or
$0.04
× £500,000 = $20,000
£
The future value of this amount would be available to offset your costs in 65 days. This
future value is

 7  65  
$20,000 × 1 + 

  = $20,252.78
 100  360  

As long as the exchange rate is less than or equal to the strike price of $1.75/£, your costs
will be
 $1.75

 £ × £500,000  - $20,252.78 = $854,747.22


$1.7095
This is
, much less than the forward rate.
£
b. How much must the pound appreciate before your speculative option strategy ends
up costing you more than the forward rate?
Answer: At the forward rate of $1.73/£, you can lock in a dollar cost of
$1.73
× £500,000 = $865,000
£
The future spot rate that sets the speculative cost to the forward cost is found by equating
the two costs
(S(t+63,$/£) × £500,000 ) - $20,252.78 = $865,000
Solving this equation for the spot rate gives S(t+63,$/£) =
$1.7705
£
8. Upon arriving for work Monday, you observe a violation of put–call parity. In
particular, the synthetic forward price of dollars per yen is above the current forward
rate. How would you capitalize on this information?
Answer: Because the actual forward rate is below the synthetic forward rate, you would want
to contract to buy yen forward and then sell yen at the synthetic forward rate. Such an
arbitrage transaction is called a conversion. You synthetically sell yen forward by buying a
put and selling a call with the same strike price. If the exchange rate in the future is less than
the strike price, the call is worthless and you exercise your put and sell yen at the strike price.
If the exchange rate in the future is greater than the strike price, the put is worthless and the
person to whom you sold the call option exercises the option to buy yen from you at the
strike price. Therefore, you sell yen to them. By doing the two option contracts you have sold
yen forward at the strike price minus the future value of the put you purchased plus the future
value of the call you sold. This synthetic forward price is greater than the actual forward
price, so you make money.
9. Use interest rate parity to demonstrate that you can represent put-call parity as
©2017 Cambridge University Press
Chapter 20: Foreign Currency Futures and Options 13
P −C =
K
S
−
1 + i ($) 1 + i(€)
The derivation of put-call parity used the no-arbitrage idea that the forward rate should be equal
to the strike price of the options plus the future value of the cost of a call option minus the future
value of the cost of a put option because each of these sides is a way to purchase foreign
currency, in this case the euro, without any contingencies. Thus,
F = K + C[1 + i ($)] − P[1 + i ($)]
We first divide by [1 + i ($)] to get
F
K
=
+C −P
[1 + i ($)] [1 + i ($)]
From interest rate parity we know that
F =S
[1 + i ($)]
[1 + i (€)]
Substituting for F, cancelling [1 + i ($)] , and multiplying by -1 gives the desired result.
10. On April 28, 1995, the Paine Webber Group introduced a new type of security on the
NYSE: U.S. dollar increase warrants on the yen. At exercise, each warrant entitled the
holder to an amount of U.S. dollars calculated as
Greater of (i) 0 and (ii) $100 – [$100 × ¥83.65/$ / Spot rate)]
The “spot rate” in the formula refers to the yen/dollar rate on any day during the
exercise period, which extended until April 28, 1996. The 1-year forward rate on April
28 was ¥79.72/$, and the spot rate was ¥83.65/$.
a. What view on the future yen/dollar rate do investors in this security hold?
Answer: The investor gets the greater of (i) 0 and (ii) $100 – [$100 × ¥83.65/$ / S(¥/$)].
If the yen remains unchanged at ¥83.65/$, the payoff is zero. If the yen strengthens, the
ratio of ¥83.65/$ / S(¥/$) > 1, and you would be subtracting an amount greater than $100,
so the payoff would be zero. As the yen weakens, the payoff increases to a maximum of
$100. Thus, the investor must think that the yen is going to weaken.
b. This security was issued at a price of $5.50. To see whether the security is fairly
priced, which option prices would you want to examine?
Answer: While the payoff on the security is non-linear in the yen-dollar exchange rate, it
is linear in the dollar-yen exchange rate. The payoff is zero at exchange rates above 1 /
(¥83.65/$) = $0.0119546/¥ and it increases linearly along a forty-five degree angle at
exchange rates below $0.0119546/¥ until the payoff is $100 at an exchange rate of zero.
This payoff is identical to the payoff on a yen put with a strike price of $0.0119546/¥ for
an amount of yen equal to $100 × ¥83.65/$ = ¥8,365. Thus, you should examine the price
of a yen put for this amount against dollars to determine if $5.50 is the correct price for
the security.
©2017 Cambridge University Press
Chapter
20
Foreign Currency Futures and Options:
Appendix
ADDITIONAL QUESTIONS
1. Explain intuitively how foreign currency options can be replicated with portfolios of
borrowing and lending in the two currencies.
Answer: Consider a call option on a foreign currency. The value of the call option will increase if
the domestic currency weakens versus the foreign currency, and the value of the call option will
decrease if the domestic currency strengthens versus the foreign currency. This pattern of payoffs
can be replicated by borrowing domestic currency and lending foreign currency. For small units
of time, the change in the exchange rate can be modeled as a binomial process with up and down
increments. The precise amounts to borrow and lend then can be determined by solving a two
equation system in two unknowns such that the gain on the portfolio if the domestic currency
weakens and the loss on the portfolio if the domestic currency strengthens are equal to the gain
and the loss that would be experienced by holding the call option.
2. Why do the formulas for option prices not depend explicitly on the expected rate of
appreciation of one currency relative to another currency?
Answer: Option prices do not depend explicitly on the expected rate of appreciation because we
are able to price the option by a no-arbitrage argument. Essentially, the spot exchange rate, the
two interest rates, and the volatility of the process driving exchange rates implicitly characterize
the distribution of future spot rates.
3. What is the Garman-Kolhagen model of foreign currency option pricing?
Answer: The Garman-Kolhagen model of foreign currency option pricing is simply a version of
the Black-Scholes option pricing model that explicitly recognizes that interest is paid on the
foreign currency whereas stock options are often written on non-dividend paying stocks.
4. What is the delta of an option? Why is it useful?
Answer: The delta of a call option represents the change in the value of the derivative asset with a
small change in the value of the underlying asset. The expression is also sometimes called the
'hedge ratio' because it arises from the construction of the replicating portfolio. Delta is the
amount of foreign currency invested in the risk-free asset to replicate the payoff on the call
option.
5. What does it mean for a portfolio of options to be delta neutral?
Answer: A portfolio of options is delta neutral if the value of the portfolio is not exposed to risk
of loss from small changes in foreign exchange rates.
©2017 Cambridge University Press
Chapter 20: Foreign Currency Derivatives 169
6. What is the gamma of an option?
Answer: The gamma of a call option describes how the option’s delta changes with a
change in the underlying exchange rate
7. How does a change in the volatility of the rate of appreciation affect the pricing of foreign
currency options?
Answer: An increase in the volatility of the rate of appreciation increases the prices of both put
and call options on foreign currency.
8. Why does a change in the domestic interest rate affect the pricing of a foreign currency
option? Does a change in the foreign interest rate cause any change in option prices?
Answer: Changes in either the domestic interest rate or the foreign interest rate change foreign
currency option prices. An increase in the domestic interest rate increases the cost of borrowing
the domestic currency in the portfolio of borrowing and lending that replicates the option. This
increases the price of a call option. Similarly, an increase in the foreign interest rate increase the
return from lending the foreign currency and decreases the amount of foreign currency that must
be invested in the replicating portfolio, which decreases the cost of a call option.
9. What is the theta of an option?
Answer: The sensitivity of call option prices to the passage of time is often referred to as
the option's theta. The theta of a call option is simply the negative of the derivative of
the call option with respect to maturity, which is an increase in time. Theta describes
how the option price will evolve as the time remaining until maturity decays.
10. What is the implied volatility of an option?
Answer: Since all of the variables that determine foreign currency option prices are
observable except the volatility of the rate of appreciation of the foreign currency relative
to the domestic currency, option prices can be used in conjunction with an option pricing
model and the observations on the other variables to determine an implied volatility.
This is the unique value of volatility that sets the option price from the model equal to the
option price observed in the market.
Problems
1. Let the current spot rate be $1.25/€, and assume that one month from now the spot rate will
be either $1.30/€ or $1.20/€. Let the dollar interest rate be 0.4% per month, and let the
euro interest rate be 0.3% per month. Develop a portfolio that replicates the payoff on a
one-month euro call option with a strike price of $1.25/€. What is the corresponding price
of the euro put option with the same strike price?
Answer: To derive our replicating portfolio we invest in €X today, and we borrow $Y.
The initial dollar cost of our replicating portfolio is therefore
©2017 Cambridge University Press
170 Chapter 20: Foreign Currency Derivatives
 $1.25 

 €   €X  − $Y



We must buy €X in the spot market, but we borrow $Y, which partially offsets our dollar
cost. Remember that we will get interest on our €X at 0.3% per month no matter what
state of the world is realized in one month, and similarly we will owe interest at 0.4% per
month on our dollar borrowing.
If the dollar weakens, the value of a €100 call option is
 $1.30 
 $1.25  
 €  −  €    €100 = $5.00




Because we have the right to buy €100 at the strike price of $1.25/€ and we can sell the
€100 in the spot market for $1.30/€, we make $5.00. On the other hand, if the dollar
strengthens, the call option is worthless because no one wants to buy €100 at $1.25/€ if
the spot exchange rate is $1.20/€.
From the discussion of the two payoffs on the option, we want the value of our
replicating portfolio in one month to be
 $1.30 

 €   €X 1.003 − $Y 1.004 = $5.00



if the dollar weakens, and if the dollar strengthens, we want the value of the portfolio to
be
 $1.20 

 €   €X 1.003 − $Y 1.004 = 0



The above two equations are linear in two unknowns. Consequently, there is a unique
solution for €X and $Y.
Solving the second equation for €X gives
$Y 1.004
€X =
 $1.20 

 1.003
 € 
If we substitute this result into the first equation we get




 $1.30  $Y 1.004 

 

 − $Y 1.004 = $5.00



1.003

 €    $1.20  1.003 

 € 








Solving this equation for $Y gives
$Y = $59.76.
Substituting into the solution for €X gives
€X = €49.85.
Hence, the cost of the replicating portfolio is
 $1.25 

 €   €49.85 − $59.76 = $2.55



Consequently, because this portfolio replicates the payoff on the €100 call option, the
dollar cost of this option must be $2.55 to prevent arbitrage. An option on one euro
would therefore cost $0.0255.
©2017 Cambridge University Press
Chapter 20: Foreign Currency Derivatives 171
2. Suppose that the price of the euro call option in Problem 1 were $0.03/€. How would you
arbitrage between the market in risk-free assets and the foreign currency options market?
What would you do if the price of the call option were $0.02/€?
Answer: If the price of the option, $0.03, is greater than the cost of replicating the option,
$0.0255, then you would want to sell the option directly and hedge by replicating the purchase
of the option as in Problem 2. If the price of the option, $0.02, is less than the cost of replicating
the option, $0.0255, then you would want to buy the option and hedge by replicating a sale of
the option, which would take the opposite position to those in Problem 2.
3. Let the continuously compounded six-month USD interest rate be 3% p.a., let the analogous
JPY interest rate be 1% p.a., let the exchange rate be ¥98/$, and assume that the volatility
of the continuously compounded annualized rate of appreciation of the yen relative to the
dollar is 13%. Use the Garman-Kolhagen option pricing model to determine the yen price
of a six-month European dollar call option with a strike price of ¥100/$. How does your
answer change if the volatility were 16% p.a.?
Answer: The exchange rates are quoted as yen per dollar, and the option is on the dollar.
Consequently, the yen is the domestic currency in the Garman-Kolhagen model. With a 180 day
maturity, the value of the call option is ¥2.29/$. If volatility increases to 16% per annum, the
value of the call option increases to ¥3.07/$.
4. With the variables as in Problem 3, use put-call parity to determine the yen price of the
corresponding dollar put option with the same maturity and same strike price.
Answer: The value of the put option when volatility is 13% per annum is ¥5.23/$. If volatility
increases to 16% per annum, the value of the put option increases to ¥6.02/$.
5. Suppose a trader sells a call option on £500,000 with a delta of 0.35 and buys another call
option on £1,000,000 with different parameters whose delta is 0.55. What is his net
exposure to small movements in the exchange rate? How could he cover this position?
Answer: If a trader sells a call option on £500,000 with a delta of 0.35, and the exchange
rate increases by $0.0001/£, the value of the call option increases by 0.35 x $0.0001/£ x
£500,000 = $17.50. This is the trader’s loss. If the trader buys another call option on
£1,000,000 with different parameters whose delta is 0.55, his exposure to small
movements in the exchange rate is 0.55 x $0.0001/£ x £1,000,000 = $55. This is the
trader’s gain. The difference is $55 - $17.50 = $37.50. There are many ways that the
trader could offset this exposure. One way would be to sell an option on £1,000,000
with a delta of 0.375.
6. Assume you are looking at prices from the NASDAQ OMX PHLX and that the price of a
three-month European AUD call option with a strike price of 92 cents per Australian
dollar is 3.2¢/AUD. Suppose that the spot exchange rate is 90¢/AUD, the continuously
compounded annualized dollar interest rate is 2%, and the analogous AUD interest rate is
5%. What is the implied volatility of the continuously compounded annualized rate of
appreciation of the AUD relative to the dollar?
©2017 Cambridge University Press
172 Chapter 20: Foreign Currency Derivatives
Answer: From the Garman-Kolhagen spreadsheet, one finds that the implied volatility to be
24.63%.
7. Suppose the implied volatilities expressed in percent per annum of yen call options against
the dollar with maturities of one, two and three months are 9%, 10%, and 11%,
respectively. If you thought that the market would soon price options to have a common
volatility of 10%, what position would you take in the options to expect to profit from your
beliefs?
Answer: We know that pricing options with a common volatility of 10% will cause the price of
the one month option to rise and the three month option to fall. Thus, you would want to be long
the one month option and short the three month option. Structuring the position so that it is delta
neutral would allow you to bet strictly on the change in the pricing.
©2017 Cambridge University Press