CHAPTER 6 INTERNATIONAL PARITY RELATIONSHIPS AND FORECASTING FOREIGN
EXCHANGE RATES
ANSWERS & SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS
QUESTIONS
1. Give a full definition of arbitrage.
2. Discuss the implications of the interest rate parity for the exchange rate determination.
Answer: Assuming that the forward exchange rate is roughly an unbiased predictor of the future
spot rate, IRP can be written as:
3. Explain the conditions under which the forward exchange rate will be an unbiased predictor
of the future spot exchange rate.
Answer: The forward exchange rate will be an unbiased predictor of the future spot rate if (i) the
4. Explain the purchasing power parity, both the absolute and relative versions. What causes
the deviations from the purchasing power parity?
Answer: The absolute version of purchasing power parity (PPP):
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e = $ – £.
PPP can be violated if there are barriers to international trade or if people in different countries
have different consumption taste. PPP is the law of one price applied to a standard consumption
basket.
5. Discuss the implications of the deviations from the purchasing power parity for countries’
competitive positions in the world market.
Answer: If exchange rate changes satisfy PPP, competitive positions of countries will remain
unaffected following exchange rate changes. Otherwise, exchange rate changes will affect
6. Explain and derive the international Fisher effect.
Answer: The international Fisher effect can be obtained by combining the Fisher effect and the
relative version of PPP in its expectational form. Specifically, the Fisher effect holds that
7. Researchers found that it is very difficult to forecast the future exchange rates more
accurately than the forward exchange rate or the current spot exchange rate. How would you
interpret this finding?
Answer: This implies that exchange markets are informationally efficient. Thus, unless one has
8. Explain the random walk model for exchange rate forecasting. Can it be consistent with the
technical analysis?
Answer: The random walk model predicts that the current exchange rate will be the best
*9. Derive and explain the monetary approach to exchange rate determination.
Answer: The monetary approach is associated with the Chicago School of Economics. It is
where M denotes the money supply, V the velocity of money, and y the national aggregate
output. The theory holds that what matters in exchange rate determination are:
10. Explain the following three concepts of purchasing power parity (PPP):
a. The law of one price.
b. Absolute PPP.
c. Relative PPP.
Answer:
a. The law of one price (LOP) refers to the international arbitrage condition for the standard
consumption basket. LOP requires that the consumption basket should be selling for the same
11. Evaluate the usefulness of relative PPP in predicting movements in foreign exchange rates
on:
a. Short-term basis (for example, three months)
b. Long-term basis (for example, six years)
Answer.
PROBLEMS
1. Suppose that the treasurer of IBM has an extra cash reserve of $100,000,000 to invest for
six months. The six-month interest rate is 8 percent per annum in the United States and 7
percent per annum in Germany. Currently, the spot exchange rate is €1.01 per dollar and the
six–month forward exchange rate is €0.99 per dollar. The treasurer of IBM does not wish to bear
any exchange risk. Where should he/she invest to maximize the return?
Solution: The market conditions are summarized as follows:
2. While you were visiting London, you purchased a Jaguar for £35,000, payable in three
months. You have enough cash at your bank in New York City, which pays 0.35% interest per
month, compounding monthly, to pay for the car. Currently, the spot exchange rate is $1.45/£
and the three-month forward exchange rate is $1.40/£. In London, the money market interest
rate is 2.0% for a three-month investment. There are two alternative ways of paying for your
Jaguar.
(a) Keep the funds at your bank in the U.S. and buy £35,000 forward.
(b) Buy a certain pound amount spot today and invest the amount in the U.K. for three months
so that the maturity value becomes equal to £35,000.
Evaluate each payment method. Which method would you prefer? Why?
Solution: The problem situation is summarized as follows:
Option a:
Option b:
3. Currently, the spot exchange rate is $1.50/£ and the three-month forward exchange rate is
$1.52/£. The three-month interest rate is 8.0% per annum in the U.S. and 5.8% per annum in
the U.K. Assume that you can borrow as much as $1,500,000 or £1,000,000.
a. Determine whether the interest rate parity is currently holding.
b. If the IRP is not holding, how would you carry out covered interest arbitrage? Show all the
steps and determine the arbitrage profit.
c. Explain how the IRP will be restored as a result of covered arbitrage activities.
Solution: Let’s summarize the given data first:
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(2) Buy £1,000,000 spot using $1,500,000.
(3) Invest £1,000,000 at the pound interest rate of 1.45%;
maturity value will be £1,014,500.
(4) Sell £1,014,500 forward for $1,542,040
Arbitrage profit will be $12,040 (=$1,542,040 – $1,530,000).
c. Following the arbitrage transactions described above,
The dollar interest rate will rise;
The pound interest rate will fall;
The spot exchange rate will rise;
The forward exchange rate will fall.
These adjustments will continue until IRP is restored.
4. Currently, the spot exchange rate is $0.85/A$ and the one-year forward exchange rate
is $0.81/A$. One-year interest is 3.5% in the United States and 4.2% in Australia.
You may borrow up to $1,000,000 or A$1,176,471, which is equivalent to $1,000,000
at the current spot rate.
a. Determine if IRP is holding between Australia and the United States.
b. If IRP is not holding, explain in detail how you would realize certain profit in U.S. dollar
terms.
c. Explain how IRP will be restored as a result of arbitrage transactions you carry out above.
Solution:
5. Suppose that the current spot exchange rate is €0.80/$ and the three-month forward
exchange rate is €0.7813/$. The three-month interest rate is 5.60 percent per annum in the
United States and 5.40 percent per annum in France. Assume that you can borrow up to
$1,000,000 or €800,000.
a. Show how to realize a certain profit via covered interest arbitrage, assuming that you want to
realize profit in terms of U.S. dollars. Also determine the size of your arbitrage profit.
b. Assume that you want to realize profit in terms of euros. Show the covered arbitrage process
and determine the arbitrage profit in euros.
Solution:
6. In the October 23, 1999 issue, the Economist reports that the interest rate per annum is
5.93% in the United States and 70.0% in Turkey. Why do you think the interest rate is so high in
Turkey? Based on the reported interest rates, how would you predict the change of the
exchange rate between the U.S. dollar and the Turkish lira?
7. As of November 1, 1999, the exchange rate between the Brazilian real and U.S. dollar is
R$1.95/$. The consensus forecast for the U.S. and Brazil inflation rates for the next 1-year
period is 2.6% and 20.0%, respectively. What would you forecast the exchange rate to be at
around November 1, 2000?
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of McGraw-Hill Education.
to forecast the exchange rate.
E(e) = E($) – E(R$)
= 2.6% – 20.0%
= –17.4%
R$ is expected to depreciate by about 17.4% against the US dollar. Thus, the expected
exchange rate would be
E(ST) = So(1 + E(e))
= (R$1.95/$) (1 + 0.174)
= R$2.29/$
8. (CFA question) Omni Advisors, an international pension fund manager, uses the concepts of
purchasing power parity (PPP) and the International Fisher Effect (IFE) to forecast spot
exchange rates. Omni gathers the financial information as follows:
Base price level 100
Current U.S. price level 105
Current South African price level 111
Base rand spot exchange rate $0.175
Current rand spot exchange rate $0.158
Expected annual U.S. inflation 7%
Expected annual South African inflation 5%
Expected U.S. one-year interest rate 10%
Expected South African one-year interest rate 8%
Calculate the following exchange rates (ZAR and USD refer to the South African rand and U.S.
dollar, respectively).
a. The current ZAR spot rate in USD that would have been forecast by PPP.
b. Using the IFE, the expected ZAR spot rate in USD one year from now.
c. Using PPP, the expected ZAR spot rate in USD four years from now.
Solution:
9. Suppose that the current spot exchange rate is €1.50/₤ and the one-year forward exchange
rate is €1.60/₤. The one-year interest rate is 5.4% in euros and 5.2% in pounds. You can borrow
at most €1,000,000 or the equivalent pound amount, i.e., ₤666,667, at the current spot
exchange rate.
a. Show how you can realize a guaranteed profit from covered interest arbitrage. Assume that
you are a euro-based investor. Also determine the size of the arbitrage profit.
b. Discuss how the interest rate parity may be restored as a result of the above
transactions.
c. Suppose you are a pound-based investor. Show the covered arbitrage process and
determine the pound profit amount.
Solution:
10. Due to the integrated nature of their capital markets, investors in both the U.S. and U.K.
require the same real interest rate, 2.5%, on their lending. There is a consensus in capital
markets that the annual inflation rate is likely to be 3.5% in the U.S. and 1.5% in the U.K. for the
next three years. The spot exchange rate is currently $1.50/£.
a. Compute the nominal interest rate per annum in both the U.S. and U.K., assuming that the
Fisher effect holds.
b. What is your expected future spot dollar–pound exchange rate in three years from now?
c. Can you infer the forward dollar-pound exchange rate for one-year maturity?
Solution.
11. After studying Iris Hamson’s credit analysis, George Davies is considering whether he can
increase the holding period return on Yucatan Resort’s excess cash holdings (which are held in
pesos) by investing those cash holdings in the Mexican bond market. Although Davies would be
investing in a peso-denominated bond, the investment goal is to achieve the highest holding
period return, measured in U.S. dollars, on the investment.
Davies finds the higher yield on the Mexican one-year bond, which is considered to be
free of credit risk, to be attractive but he is concerned that depreciation of the peso will reduce
the holding period return, measured in U.S. dollars. Hamson has prepared selected economic
and financial data, given in Exhibit 3–1, to help Davies make the decision.
Selected Economic and Financial Data for U.S. and Mexico
Expected U.S. Inflation Rate 2.0% per year
Expected Mexican Inflation Rate 6.0% per year
U.S. One-year Treasury Bond Yield 2.5%
Mexican One-year Bond Yield 6.5%
Nominal Exchange Rates
Spot 9.5000 Pesos = U.S. $ 1.00
One-year Forward 9.8707 Pesos = U.S. $ 1.00
Hamson recommends buying the Mexican one-year bond and hedging the foreign currency
exposure using the one-year forward exchange rate. She concludes: “This transaction will result
in a U.S. dollar holding period return that is equal to the holding period return of the U.S. one–
year bond.”
a. Calculate the U.S. dollar holding period return that would result from the transaction
recommended by Hamson. Show your calculations. State whether Hamson’s conclusion
about the U.S. dollar holding period return resulting from the transaction is correct or
incorrect. After conducting his own analysis of the U.S. and Mexican economies, Davies
expects that both the U.S. inflation rate and the real exchange rate will remain constant over
the coming year. Because of favorable political developments in Mexico, however, he
expects that the Mexican inflation rate (in annual terms) will fall from 6.0 percent to 3.0
percent before the end of the year. As a result, Davies decides to invest Yucatan Resorts’
cash holdings in the Mexican one-year bond but not to hedge the currency exposure.
b. Calculate the expected exchange rate (pesos per dollar) one year from now. Show your
calculations. Note: Your calculations should assume that Davies is correct in his
expectations about the real exchange rate and the Mexican and U.S. inflation rates.
c. Calculate the expected U.S. dollar holding period return on the Mexican one-year bond.
Show your calculations. Note: Your calculations should assume that Davies is correct in his
expectations about the real exchange rate and the Mexican and U.S. inflation rates.
Solution:
Solving for YUS:
b. The expected exchange rate one year from now is 9.5931. The rate can be calculated by
using the formula:
c. The expected U.S. dollar holding period return on the Mexican one-year bond is 5.47%. The
return can be calculated as shown below, using the formula in Part A and the current spot
exchange rate and expected one-year spot exchange rate calculated in Part B.
12. James Clark is a foreign exchange trader with Citibank. He notices the following quotes.
Spot exchange rate SFr1.2051/$
Six-month forward exchange rate SFr1.1922/$
Six-month $ interest rate 2.5% per year
Six-month SFr interest rate 2.0% per year
a. Is the interest rate parity holding? You may ignore transaction costs.
b. Is there an arbitrage opportunity? If yes, show what steps need to be taken to make
arbitrage profit. Assuming that James Clark is authorized to work with $1,000,000, compute
the arbitrage profit in dollars.
Solution:
13. Suppose you conduct currency carry trade by borrowing $1 million at the start of each year
and investing in New Zealand dollar for one year. One-year interest rates and the exchange rate
between the U.S. dollar ($) and New Zealand dollar (NZ$) are provided below for the period
2000 – 2009. Note that interest rates are one-year interbank rates on January 1st each year, and
that the exchange rate is the amount of New Zealand dollar per U.S. dollar on December 31
each year. The exchange rate was NZ$1.9088/$ on January 1, 2000. Fill out the columns (4) –
(7) and compute the total dollar profits from this carry trade over the ten-year period. Also,
assess the validity of uncovered interest rate parity based on your solution of this problem. You
are encouraged to use Excel program to tackle this problem.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Year
iNZ$
i$
SNZ$/$
iNZ$ – i$
eNZ$/$
(4)-(5)
$ Profit
2000
6.53
6.50
2.2599
2001
6.70
6.00
2.4015
2002
4.91
2.44
1.9117
2003
5.94
1.45
1.5230
2004
5.88
1.46
1.3845
2005
6.67
3.10
1.4682
2006
7.28
4.84
1.4182
2007
8.03
5.33
1.2994
2008
9.10
4.22
1.7112
2009
5.10
2.00
1.3742
Data source: Datastream.
Solution:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Year
iNZ$
i$
SNZ$/$
iNZ$ – i$
eNZ$/$
(4)-(5)
$ Profit
2000
6.53
6.50
2.2599
0.03
18.40
–18.37
–183655
2001
6.70
6.00
2.4015
0.7
6.27
–5.57
–55680
2002
4.91
2.44
1.9117
2.47
–20.40
22.87
228676
2003
5.94
1.45
1.5230
4.49
–20.33
24.82
248220
2004
5.88
1.46
1.3845
4.42
–9.10
13.52
135159
2005
6.67
3.10
1.4682
3.57
6.05
–2.48
–24790
2006
7.28
4.84
1.4182
2.44
–3.40
5.84
58438
2007
8.03
5.33
1.2994
2.7
–8.38
11.08
110810
2008
9.10
4.22
1.7112
4.88
31.69
–26.81
–268106
2009
5.10
2.00
1.3742
3.1
–19.69
22.79
227922
Notes:
1. Interest rates are interbank 1-year rates on January 1st of each year and measured in percent
terms.
2. Spot exchange rates, SNZ$/$, are measured on December 31st of each year and spot
exchange rates was
NZ$1.9088 per US$ on January 1, 2000.
3. All data are from Datastream.
If uncovered interest rate parity holds, profit from carry trade should be insignificantly different
from zero. But since the profit in column (7) substantially differs from zero each year, uncovered
IRP does not appear to hold.
Mini Case: Turkish Lira and the Purchasing Power Parity
Veritas Emerging Market Fund specializes in investing in emerging stock markets of the world.
Mr. Henry Mobaus, an experienced hand in international investment and your boss, is currently
interested in Turkish stock markets. He thinks that Turkey will eventually be invited to negotiate
its membership in the European Union. If this happens, it will boost the stock prices in Turkey.
But, at the same time, he is quite concerned with the volatile exchange rates of the Turkish
currency. He would like to understand what drives the Turkish exchange rates. Since the
inflation rate is much higher in Turkey than in the U.S., he thinks that the purchasing power
parity may be holding at least to some extent. As a research assistant for him, you were
assigned to check this out. In other words, you have to study and prepare a report on the
following question: Does the purchasing power parity hold for the Turkish lira-U.S. dollar
exchange rate? Among other things, Mr. Mobaus would like you to do the following:
1. Plot past annual exchange rate changes against the differential inflation rates between
Turkey and the U.S. for the last 20 years.
2. Regress the annual rate of exchange rate changes on the annual inflation rate differential to
estimate the intercept and the slope coefficient, and interpret the regression results.
Data source: You may download the annual inflation rates for Turkey and the U.S., as well as
the exchange rate between the Turkish lira and US dollar from the following source:
http://data.un.org. For the exchange rate, you are advised to use the variable code 186 AE ZF.
Solution:
Data obtained from http://data.un.org
Inf_TK (%)
(1)
Inf_US (%)
(2)
∆Inf
(1)-(2)
S(TL/$)
End-of-year rate
∆St/St-1 (%)
:= et
1989
0.0023
1990
60.3127
5.3980
54.9147
0.0029
26.6406
1991
65.9694
4.2350
61.7344
0.0051
73.3720
1992
70.0728
3.0288
67.0440
0.0086
68.5938
1993
66.0971
2.9517
63.1454
0.0145
68.9838
1994
106.2630
2.6074
103.6556
0.0387
167.5833
1995
88.1077
2.8054
85.3023
0.0597
54.0309
1996
80.3469
2.9312
77.4157
0.1078
80.6790
1997
85.7332
2.3377
83.3955
0.2056
90.7724
1998
84.6413
1.5523
83.0890
0.3145
52.9457
1999
64.8675
2.1880
62.6795
0.5414
72.1660
2000
54.9154
3.3769
51.5385
0.6734
24.3785
2001
54.4002
2.8262
51.5740
1.4501
115.3493
2002
44.9641
1.5860
43.3781
1.6437
13.3485
2003
25.2964
2.2701
23.0263
1.3966
-15.0307
2004
10.5842
2.6772
7.9070
1.3395
-4.0912
2005
10.1384
3.3928
6.7457
1.3451
0.4143
2006
10.5110
3.2259
7.2851
1.4090
4.7545
2007
8.7562
2.8527
5.9035
1.1708
-16.9056
2008
10.4441
3.8391
6.6050
1.5255
30.2913
2009
6.2510
-0.3555
6.6065
1.4909
-2.2649
Solution:
1. In the current solution, we use the annual data from 1990 to 2009.
2. We regress the rate of exchange rate changes (e) on the inflation rate differential and
estimate the intercept ( ) and slope coefficient ( ):
The estimated intercept is insignificantly different from zero, whereas the slope coefficient is
positive and significantly different from zero. In fact, the slope coefficient is insignificantly