Test Questions & Answers Chapter 4 The International Parity

Chapter 4 The International Parity Conditions and Their Consequences

Notes to instructors:

Answers to non-numeric multiple choice questions are arranged alphabetically, so that answers are randomly assigned to the five outcomes.

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1. The law of one price states that “Equivalent assets sell for the same price.”

2. Pure or riskless arbitrage is defined as a profitable position obtained with no net investment and no risk.

3. Speculators make their profit from situations in which they have no net investment and no risk.

They typically have an investment at risk.

4. The actions of arbitrageurs promote the law of one price in currency markets.

5. Arbitrageurs pursuing covered interest arbitrage are hoping to make a profit by putting their own money at risk.

Pure (or riskless) arbitrage involves no risk.

6. Real assets are more likely to conform to the law of one price than financial assets.

Because market frictions are generally lower for financial assets than for real assets, financial assets are more likely to conform to the law of one price.

7. Prices in two currencies are related through the equation V^d = V^f S^d/f.

8. Prices in two currencies are related through the equation V^d = V^f / S^d/f.

Prices are related through the equation Vd = Vf Sd/f or, equivalently, Vd / Vf = Sd/f.

9. Prices in two currencies are related through the equation V^d = V^f / S^f/d.

Vd = Vf Sd/f = Vf / Sf/d.

10. Locational arbitrage ensures that bilateral exchange rates are in equilibrium.

11. Even if quoted exchange rates do not allow arbitrage, banks quoting the lowest offer prices in a currency will attract the bulk of customer purchases in that currency.

12. A bank is in a long euro and short yen position when it has purchased euros and sold yen.

13. A bank is in a short euro and long yen position when it has purchased euros and sold yen.

Purchasing (selling) an asset creates a long (short) position in that asset.

14. A currency cross rate does not involve the domestic currency.

15. The following exchange rates are in equilibrium: S^SFr/€ = SFr1.60/€, S^€/¥ = €0.0125/¥, and S^¥/SFr = ¥50.00/SFr.

SSFr/€ S€/¥ S¥/SFr = (SFr1.60/€)(€0.0125/¥)(¥50/SFr) = 1.

16. The following exchange rates are in equilibrium: S^SFr/€ = SFr1.7223/€, S^€/¥ = €0.009711/¥, and S^¥/SFr = ¥61.740/SFr.

SSFr/€ S€/¥ S¥/SFr = 1.0326 > 1. Triangular arbitrage would yield a profit so long as transactions costs are less than about 3.26 percent.

17. Suppose S^SFr/€ = SFr1.50/€ and S^¥/€ = ¥135/€. The spot rate should be S^¥/SFr = ¥90/SFr.

SSFr/€ S€/¥ S¥/SFr = (SFr1.5000/€)(¥135/€)1(¥90/SFr) = 1.

18. Interest rate parity (IRP) holds within the bounds of transactions costs in the international currency and Eurocurrency markets.

19. Forward premiums and discounts depend on interest rate differentials.

20. Covered interest arbitrage ensures that the ratio of forward to spot exchange rates is determined by the inflation differential between two currencies.

Forward/spot ratios are determined by interest rate differentials.

21. Empirical evidence indicates that forward parity holds in the international currency and Eurocurrency markets.

Interest rate parity is the only reliable international parity condition.

22. Empirical evidence indicates that relative purchasing power parity holds in the international currency and Eurocurrency markets.

Interest rate parity is the only reliable international parity condition.

23. Empirical evidence indicates that relative purchasing power parity can be used to successfully predict short-term changes in spot exchange rates.

Inflation differentials are not good predictors of short-term changes in spot exchange rates.

24. The spot exchange rate between Swiss francs and dollars is S₀^SFr/$ = SFr1.7122/$. Expected inflation is 0 percent in Switzerland and 11 percent in the United States. Using relative purchasing power parity, the expected spot rate in one period is E[S₁^SFr/$] = SFr1.5425/$.

E[S1SFr/$] = S0SFr/$(1 + E[pSFr])/(1 + E[p$]) = (SFr1.7122/$)(1.00/1.11) = SFr1.5425/$.

25. Deviations from purchasing power parity in real exchange rates can persist for several years.

26. Because currencies are standardized assets that are actively traded in international markets, real deviations from purchasing power parity are rare and fleeting.

Real deviations from purchasing power parity can be large and persistent.

27. Real changes in currency values reflect changes in relative purchasing power.

28. Real exchange rate changes are determined by inflation differentials.

Real changes equal nominal changes adjusted for inflation differentials.

29. Real (inflation adjusted) exchange rate changes have little economic significance.

Real exchange rate changes reflect changes in purchasing power.

30. A real appreciation in the value of a currency increases the purchasing power of that currency relative to other currencies.

31. A nominal appreciation of the domestic currency raises the price of domestic goods relative to foreign goods.

Only a real appreciation of the domestic currency raises the price of domestic goods relative to foreign goods.

32. For daily measurement intervals, both nominal and real exchange rates are close to a random walk.

33. A real appreciation of the domestic currency lowers the relative price of foreign goods.

34. An increase in the real value of foreign currencies helps the domestic economy as imported goods and raw materials cost less.

Although an appreciation of a foreign currency may help domestic exporters, it will hurt importers and consumers as foreign goods will cost more.

35. An increase in the nominal exchange rate helps the domestic economy as imported goods and raw materials cost less.

Changes in real exchange rates are much more important influences.

36. There is no correlation between annual changes in real exchange rates.

Real exchange rates are autocorrelated (i.e., autoregressive).

37. Relative purchasing power parity holds over the short run.

RPPP may hold in the long run, but generally does not hold in the short run.

38. Empirical tests indicate that persistent inflation differentials between two currencies eventually have an impact on nominal exchange rates.

39. Real exchange rates are nominal exchange rates minus the forward/spot premium or discount.

Real exchange rates are nominal exchange rates adjusted for differential inflation.

40. International parity conditions are useful for predicting changes in the real exchange rate.

The international parity conditions predict that nominal exchange rates change to accommodate inflation and interest rate differentials. Real exchange rates remain the same.

41. Empirical tests of forward parity over short forecasting periods suggest that forward exchange rates are poor predictors of future spot exchange rates.

42. Empirical tests of forward parity suggest that forward exchange rates improve as predictors of future spot exchange rates over long forecasting periods.

43. Empirical studies indicate that forward parity holds at each point in time.

Deviations from forward parity are positively autocorrelated. At any point in time, the deviation from forward parity is likely to be of the same sign as in the previous period.

44. Technical analysis uses past price patterns to forecast exchange rates.

45. Technical analysts believe that the currency markets are weak form efficient.

Weak form efficient markets have no memory. Technical analysts search for exchange rate patterns that would not be apparent in a weak form efficient market.

46. Fundamental analysis uses macroeconomic data to forecast exchange rates.

47. Fundamental analysts believe that the currency markets are semi-strong form efficient.

Semi-strong form efficient markets are informationally efficient with respect to publicly available information. Fundamental analysts base their forecasts on relationships between exchange rate movements and publicly available macroeconomic data such as balance of payments statistic, inflation, and real and nominal interest rate movements.

Multiple Choice Select the BEST ANSWER

1. The relation between interest rate differentials and expected inflation differentials is called ____.

a. forward parity

b. interest rate parity

c. relative purchasing power parity

d. the international Fisher relation

e. none of the above

2. The relation between expected future changes in the spot rate and expected inflation differentials is called ____.

a. forward parity

b. interest rate parity

c. relative purchasing power parity

d. the international Fisher relation

e. none of the above

3. The relation between the forward exchange rates and expected future spot rates is called ____.

a. forward parity

b. interest rate parity

c. relative purchasing power parity

d. the international Fisher relation

e. none of the above

4. The relation between the forward/spot ratio and interest rate differentials is called ____.

a. forward parity

b. interest rate parity

c. relative purchasing power parity

d. the international Fisher relation

e. none of the above

5. S^USD/ARS = $0.35/ARS and S^ARS/ZAR = ARS0.31/ZAR. What is S^ZAR/USD?

a. ZAR 0.886/USD

b. ZAR 1.129/USD

c. ZAR 3.226/USD

d. ZAR 3.459/USD

e. ZAR 9.217/USD

6. Annual interest rates are 6 percent in euros and 5 percent in dollars. The spot rate is S₀^$/€ = $1.20/€. What is the one-year forward exchange rate?

a. $1.18/€

b. $1.19/€

c. $1.20/€

d. $1.21/€

e. $1.22/€

7. If annual interest rates are 10 percent in the U.S. and 4 percent in Switzerland with quarterly compounding and the 90-day forward rate for the Swiss franc is $0.3864/SFr, at what current spot rate will interest rate parity hold?

a. $0.3910/SFr

b. $0.3897/SFr

c. $0.3807/SFr

d. $0.3762/SFr

e. $0.3648/SFr

8. If the expected inflation rate is 5 percent and the real required return is 6 percent, then the Fisher equation says that the nominal interest rate should be exactly ____.

a. 1.0%

b. 6.0%

c. 11.0%

d. 11.3%

e. none of the above

9. If expected inflation is 10 percent and the real required return is 8 percent, then the Fisher equation says that the nominal interest rate should be exactly ____.

a. 80.0%

b. 18.8%

c. 18.0%

d. 2.0%

e. none of the above

10. Suppose annual inflation rates in the United States and Mexico are expected to be 6 percent and 80 percent, respectively, over the next several years. If the current spot rate for the Mexican peso is $0.005/Ps, then relative purchasing power parity suggests that the peso’s value in 3 years will be approximately ____.

a. $0.01088/Ps

b. $0.00346/Ps

c. $0.00321/Ps

d. $0.00102/Ps

e. $0.00024/Ps

11. Suppose that S₀^$/SFr = $1.27/SFr and F₁^$/SFr = $1.28/SFr for exchange between U.S. dollars and Swiss francs. These prices indicate that ____.

a. nominal interest rates are higher in the United States than in Switzerland

b. the inflation rate in Switzerland is declining

c. the Swiss franc has recently risen in relation to the dollar

d. the Swiss franc is expected to fall in value relative to the dollar

e. it is not possible to claim that any of the above are without additional information

12. The current U.S. dollar value of the Hong Kong dollar is $0.1250/HK$. The 180-day forward rate is $0.12148/HK$. The difference between the two rates suggests that ____.

a. inflation in the United States during the past year was lower than in Hong Kong

b. interest rates are rising faster in Hong Kong than in the United States

c. prices in Hong Kong are expected to rise more rapidly than in the United States

d. the Hong Kong dollar’s value is expected to rise against the U.S. dollar

e. more than one of the above

13. Which of the following is an effect of a real appreciation of the domestic currency?

a. It helps hold down domestic inflation.

b. It helps domestic importers as imported goods and raw materials cost less.

c. It shifts resources within the domestic economy from export-oriented firms toward import-oriented firms.

d. All of the above are effects of a real appreciation of the domestic currency.

e. None of the above are effects of a real appreciation of the domestic currency.

14. A real depreciation of the domestic currency has which of the following consequences?

a. a decrease in the domestic cost of living

b. higher domestic prices for imported goods

c. higher domestic unemployment

d. lower domestic inflation

e. none of the above

15. Which of statements (a) through (c) is ?

a. Attempts to forecast short term changes in exchange rates generally fail to beat a naive guess of today’s spot exchange rate.

b. Long run forecasts of nominal exchange rates have difficulty beating the current spot rate as predictors of future spot rates of exchange.

c. Prices in the interbank foreign exchange markets are difficult to predict.

d. More than one of the above is

e. All of the above are

16. Which of statements (a) through (c) is ?

a. Real changes in exchange rates can be forecast by the international parity conditions.

b. Real changes in exchange rates reflect changes in purchasing power.

c. Real exchange rates vary over time.

d. All of the above are

e. More than one of the above is

17. Market-based exchange rate forecasts include which of (a) through (c)?

a. forward/spot differentials

b. inflation differentials

c. interest rate differentials

d. more than one of the above

e. none of the above

18. A technical analyst is likely to use which of the following in forecasting exchange rates?

a. growth in gross national product (GNP)

b. the federal government deficit

c. the past history of spot exchange rate movements

d. the trade deficit

e. balance of payments statistics

19. A fundamental analyst is likely to use which of the following in forecasting exchange rates?

a. forward exchange rates

b. inflation differentials

c. nominal interest rate differentials

d. real interest rate differentials

e. all of the above

Problems (Some of these can be converted into Multiple Choice questions.)

1. Calculate the following cross exchange rates.

a. If exchange rates are 1.2 dollars per euro and 1.5 dollars per pound, what is the euro per pound exchange rate?

b. If the euro trades at ¥125/€ and HK$10/€, what is the yen per HK$ exchange rate?

2. How large would transactions costs have to be as a percentage of the principal to make triangular arbitrage between the exchange rates S^$/€ = $0.9000/€, S^$/¥ = $0.009000/¥, and S^¥/€ = ¥101.00/€ unprofitable?

3. Given S₀^$/£ = $1.5000/£ and the one-year forward rate F₁^$/£ = $1.3500/£, what is the dollar forward premium or discount (a) in basis points and (b) as a percentage of the spot rate? Based on the unbiased forward expectations hypothesis, by how much is the dollar expected to appreciate or depreciate over the next year? Forecast the spot exchange rate one year into the future.

4. The euro is quoted at “€0.008300/¥ Bid and €0.008200/¥ Ask” in Tokyo. The yen is quoted at “¥121.20/€ Bid and ¥121.10/€ Ask” in Paris. Convert the Tokyo quote into a ¥/€ quote and then answer these questions.

a. Calculate the bid/ask spread on the euro (in the denominator) as a percentage of the bid price from the Japanese and from the French perspectives.

b. Is there an opportunity for profitable arbitrage? If so, describe the necessary transactions using a €1 million starting amount.

5. The real rate of interest on three-month government securities is 3 percent in both Switzerland and the United States. Inflation is 2 percent in Switzerland and 4 percent in the United States. In equilibrium, what are the nominal required returns on three-month government securities in Switzerland and in the United States?

6. The current spot exchange rate is S₀^$/€ = $1.10/€ and the one-year forward rate is F₁^$/€ = $1.11/€. The prime rate in the United States is 5 percent.

a. What is the prime rate in euros if the international parity conditions hold?

b. According to forward parity, by how much should the euro change in value during the next year?

c. By how much should the dollar change in value during the next year?

d. If this forward premium persists into the future, what should be the euro’s value in five years?

7. Price indices in the United States and the United Kingdom are currently V₀^$ = $2000 and V₀^£ = £10, respectively. The spot rate of exchange is S₀^£/$ = £0.80/$. Inflation rates are expected to be i^$ = 2 percent and i^£ = 3 percent per period, respectively, over the foreseeable future.

a. Looking one year into the future, what are the expected levels of the price indices E[V₁^$] and E[V₁^£] and the expected nominal spot rate of exchange E[S₁^£/$]?

b. Looking two years into the future, what are the expected price levels [V₂^$] and E[V₂^£] and the expected nominal spot rate of exchange E[S₂^£/$]?

8. A foreign exchange dealer provides the following quotes for the dollar against the Brazilian real.

Bid (BR/$) Ask (BR/$)

Spot 4.0040 4.0200

One month forward 3.9920 4.0090

Three months forward 3.9690 3.9888

Six months forward 3.9360 3.9580

a. Six-month Eurodollars yield 5 percent per year with semiannual compounding. What should be the annualized yield on six-month Brazilian real Eurocurrency interest rates with semiannual compounding? Use ask quotes for the U.S. dollar when calculating these rates from interest rate parity.

b. Verify your answer to part a with a hypothetical investment of $10 million for six months in each country. Use only ask quotes for simplicity and ignore other fees, charges, and taxes.

9. Quotes for the dollar and euro are as follows:

Spot contract midpoint S₀^€/$ = €0.8890/$

One-year forward contract midpoint F₁^€/$ = €0.8960/$

One-year Eurodollar interest rate i^$ = 3% per year

a. Your newspaper does not quote one-year Eurocurrency interest rates on EU euros. Make your own estimate of i^€.

b. Suppose that you can trade at the prices for S^€/$, F^€/$ and i^$ just given and that you can also either borrow or lend at a Thai Eurocurrency interest rate of i^€ = 4 percent per year. Based on a $1 million initial amount, how much profit can you generate through covered interest arbitrage?

10. Expected inflation over the next year is E[p] = 10 percent. What nominal interest rate i should investors charge on the following assets?

a. Investors require a real rate of return of ʀ = 2 percent on a one-year corporate bond.

b. Investors require a real return of ʀ = 6 percent on a portfolio of stocks.

c. Investors require a real return of ʀ = 10 percent on an investment in an oil field.

11. Suppose the spot rate ends the year exactly where it began at S₀^¥/$ = S₁^¥/$ = ¥100/$. Inflation in Japan was 2 percent during the year. Inflation in the United States was 3 percent. Calculate the percentage change in the real exchange rate x^¥/$. What does this change indicate?

12. Suppose the Japanese yen appreciates against the Indian rupee from R0.30/¥ to R0.33/¥ during a year. Inflation during the year is 5 percent in India and 3 percent in Japan. What is the real depreciation or appreciation of the yen during the year?

Problem Solutions

1. a. S^€/£ = S^$/£ / S^$/€ = ($1.50/£) / ($1.20/€) = €1.25/£

b. S^¥/HK$ = S^¥/€ / S^HK$/€ = (¥125/€) / (HK$10/€) = ¥12.5/HK$.

2. S^$/¥ S^¥/€ S^€/$ = S^$/¥ S^¥/€ / S^$/€ = 1.01 > 1. Triangular arbitrage would yield a profit of about 1 percent of the starting amount. For triangular arbitrage to be profitable, transactions costs on a round turn cannot be more than this amount.

3. The one-year forward price is at a 15 basis point discount to the spot price. This is a discount of (F₁^$/£/S₀^$/£) – 1 = ($1.35/£)/($1.50/£) – 1 = –0.10, 10 percent. Forward parity suggests F_t^$/£/S₀^$/£ = E[S_t^$/£]/S₀^$/£, so the spot rate is expected to depreciate by 10 percent.

4. In Tokyo’s quote of “€0.008300/¥ Bid and €0.008200/¥ Ask,” the bid is higher than the ask. Consequently, the currency being quoted must be the euro in the numerator of the quotes. Converting the Tokyo quote yields a direct quote for the euro of “¥121.21/€ Bid and ¥121.36/€ Ask.” Similarly, the Paris quote of “¥121.20/€ Bid and ¥121.10/€ Ask” indicates that euros can be purchased at ¥121.10/€ and sold at ¥121.20/€.

a. Percentage bid-ask spreads on the euro are as follows:

Tokyo quote for the euro: (¥121.36/€ – ¥121.21/€)/(¥121.21/€) = 0.121%

Paris quote for the euro: (¥121.20/€ – ¥121.10/€)/(¥121.10/€) = 0.083%

b. The winning strategy is to buy euros from the Paris bank at the ¥121.20/€ euro ask price and sell euros to the Tokyo bank at the ¥121.21/€ bid price. Buying €1 million in Paris yields (€1 million)/(¥121.20/€) = €121.20 million. Selling €1 million in Tokyo yields (€1 million)/(¥121.21/€) = €121.21 million. Your arbitrage profit is €0.01 million, or €10,000.

5. According to the Fisher relation, i^SFr = (1 + p^SFr)(1 + q^SFr) – 1 = (1.03)(1.02) – 1 = 5.06% and i^$ = (1 + p^$)(1 + q^$) – 1 = (1.03)(1.04) – 1 = 7.12%.

6. a. From interest rate parity, ($1.11/€)/($1.10/€) = (1 + i^¥)/(1.05) ⇒ i^¥ = 0.05955,

or 5.955%.

b. The euro should appreciate by ($1.11/€)/($1.10/€) – 1 = 0.009091, or 0.9091%.

c. (1/(1 – 0.009091)) – 1 = (1/(1 – 0.009091)) – 1 = 0.009174, or 0.9174%.

d. E[S₅^$/€] = S₀^$/€ (F₁^$/€/S₀^$/€)⁵ = ($1.10/€) [($1.11/€)/($1.10/€)]⁵ = $1.1509/€.

7. a. E[V₁^$] = V₀^$(1 + p^$) = $2000(1.02) = $2040

E[V₁^£] = V₀^£ (1 + p^£) = £100(1.03) = £10.3

E[S₁^£/$] = S₀^£/$ (1 + p^£) / (1 + p^$) = (£0.80/$) (1.03/1.02) = £0.80784/$

b. E[V₂^$] = V₀^$(1 + p^$)² = $2000(1.02)² = $2080.8

E[V₂^£] = V₀^£ (1 + p^£)² = £10(1.03)² = £10.609

E[S₂^£/$] = S₁^£/$ [(1 + p^£) / (1 + p^$)]² = (£0.80/$) [(1.03/1.02)]² = £0.81576/$

8. a. A 5 percent annualized rate with semiannual compounding is equivalent to 5%/2 = 2.5 percent per six months. From interest rate parity, the six-month real interest rate is (1 + i^BR)/(1 + i^$) = F^BR/$/S^BR/$ ⇒ i^BR = (1.025) [(BR3.9580/$)/(BR4.0200/$)] – 1 = 0.0091915, or 0.91915 percent per six months. Annualized, this is equivalent to (0.91915%)2 = 1.83830 percent per year with semiannual compounding. Alternatively, the effective rate is (1.0091915)² – 1 = 0.0184676, or 1.84676 percent per year.

b. $10,000,000 invested at the six-month U.S. rate yields $10,250,000. Converting into BR at the six-month forward rate yields ($10,250,000)(BR3.9580/$) = BR40,569,500. You can finance this investment by borrowing ($10,000,000)(BR4.0200/$) = BR40,200,000. Your obligation on this contract will be (BR40,200,000)(1.0091915) ≈ BR40,569,500, which is exactly offset by the proceeds from your forward contract.

9. a. F_t^€/$/S₀^€/$ = (1 + i^€)^t/(1 + i^$)^t ⇒ i^€ = (1.03)(€0.8890/$)/(€0.8960/$) – 1 = 3.81102%

b. F₁^€/$/S₀^€/$ = (€0.896/$)/(€0.889/$) = 1.007874 < 1.009709 = (1 + i^€)/(1 + i^$) = (1.0381102)/(1.03). So, borrow at i^$ and lend at i^€.

This leaves a net gain at time 1 of $1,031,875 – $1,030,000 = $1,875, which is worth $1,875/1.03 = $1,820 in present value.

10. Nominal interest rates are calculated from the Fisher equation (1 + i) = (1 + p)(1 + ʀ).

a. (1 + i) = (1 + p)(1 + ʀ) = (1.10)(1.02) = 1.122, or i = 12.2 percent

b. (1 + i) = (1 + p)(1 + ʀ) = (1.10)(1.06) = 1.166, or i = 16.6 percent

c. (1 + i) = (1 + p)(1 + ʀ) = (1.10)(1.10) = 1.210, or i = 21 percent

11. x^¥/$ = (S₁^¥/$/S₀^¥/$) (1 + i^$)/(1 + i^¥) 1 = [(¥100/$)/(¥100/$)](1.03/1.02) 1 = 0.0098, or a 0.98 percent real appreciation of the dollar in the denominator of the foreign exchange quote. Change in the real exchange rate indicates change in the purchasing power of the two currencies. In this example, the purchasing power of the dollar rose slightly less than 1 percent against the yen during the period.

12. x^R/¥ = (S₁^R/¥ /S₀^R/¥)(1 + i^¥)/(1 + i^R) 1 = [(R0.33/^¥)/(R0.30/^¥)](1.03/1.05) 1 = 0.07905, or a 7.905 percent real yen appreciation in the denominator of the quote.

Appendix 4A Continuous Time Finance

Multiple Choice Select the BEST ANSWER

1. The yen rises from $0.0080/¥ to $0.0100/¥. Stated in continuously compounded returns, what percentage change is this in the value of the yen?

a. –22.3%

b. –20%

c. + 20%

d. + 22.3%

e. cannot be determine from the information given

2. The yen rises from $0.0080/¥ to $0.0100/¥. Stated in continuously compounded returns, what percentage change is this in the value of the dollar?

a. –22.3%

b. –20%

c. + 20%

d. + 22.3%

e. cannot be determine from the information given

Problems (Some of these can be converted into Multiple Choice questions.)

1. Suppose the spot rate starts at S₀^$/¥ = $0.0100/¥ and appreciates by 25.86 percent.

a. What is the percentage appreciation of the yen in continuously compounded returns?

b. Calculate the holding period depreciation of the dollar from the continuously compounded appreciation of the yen.

2. Suppose continuously compounded changes in the spot rate S_t^$/¥ are independently and identically distributed (iid) as normal. The relation between the variance ² of continuously compounded iid normal returns over a single period (e.g., one year) and the variance of return _T² over T periods is given by _T² = T².

a. If the standard deviation of continuously compounded spot rate changes is 40 percent per year, what is the standard deviation of three-month (T = 0.25) changes in the spot rate in continuously compounded returns? If the current spot rate is $0.0100/¥, find the exchange rates that are plus or minus three standard deviations from this exchange rate after one year.

b. Suppose you collect 52 weeks of spot exchange rates S^$/SFr. The standard deviation of continuously compounded spot rate changes is 1.5 percent per week.

b-1. Assuming instantaneous (continuously compounded) spot rate changes are normally distributed with constant variance, what is the annual standard deviation of continuously compounded spot rate changes?

b-2. The current spot rate is S^SFr/$ = SFr1.50/$. What spot rates are ±2σ after one year?

b-3. What are the prices for ±2σ stated in terms of the S^$/SFr spot rate?

Problem Solutions

1. a. s^$/¥ = ln(1 + s^$/¥) = ln(1.2586) = 23.00 percent appreciation

b. Continuously compounded changes in exchange rates are symmetric in that an appreciation in one currency is mirrored by a depreciation of the same magnitude in another currency. The 23 percent continuously compounded yen appreciation means that the dollar depreciated 23 percent in continuous returns. The holding period depreciation of the dollar is s^$/¥ = e^.23 1 = 20.55%. As a check: S₁^¥/$ = S₀^¥/$ (1 + s^¥/$) = (1/$0.0100/¥) (1 .2055) = (¥100/$) (0.7945) = ¥79.45/$, which yields the 25.86 percent appreciation of the yen S₁^$/¥ = (S₁^¥/$)^1 = 1/(¥79.45/$) = $0.012586/¥.

2. a. σ_T = √T(σ) = √0.25 (0.4) = 0.20, or 20 percent per quarter. From s^$/¥ = ln(1 + s^$/¥) ⇔ (1 + s^$/¥) = e^s$/¥:

+ 3σ: S₁^$/¥ = S₀^$/¥(1 + s^$/¥) = S₀^$/¥e^3 = ($0.01/¥)(e^3.40) = ($0.01/¥)(3.32) = $0.0332/¥

3σ: S₁^$/¥ = S₀^$/¥(1 + s^$/¥) = S₀^$/¥e^3 = ($0.01/¥)(e^3.40) = ($0.01/¥)(0.30) = $0.0030/¥

b-1. σ_T = √T(σ) = (52)^1/2 (0.015) = 10.8167% per year

b-2. Plus or minus two standard deviations in the value of the dollar (in the denominator of the exchange rate) is:

+ 2σ: S₁^SFr/$ = S₀^SFr/$(1 + S^SFr/$) = S₀^SFr/$e^2 = (SFr1.5/$)(e^20.108167)

= (SFr1.5/$)(1.2415) = SFr1.8623/$

2σ: S₁^SFr/$ = S₀^SFr/$(1 + s^SFr/$) = S₀^SFr/$e^2 = (SFr1.5/$)(e^20.108167)

= (SFr1.5/$)(0.8055) = SFr1.2082/$

b3. This can be done either of two ways. First, note that an appreciation of the franc against the dollar results in a depreciation of the dollar against the franc. Taking reciprocals of the time t = 1 SFr/$ spot rates yields plus or minus two standard deviations at:

+ 2σ: 1/S₁^SFr/$ = S₁^$/SFr = 1/(SFr1.2082/$) = $0.8277/SFr

2σ: 1/S₁^SFr/$ = S₁^$/SFr = 1/(SFr1.8623/$) = $0.5370/SFr

Taking reciprocals of the SFr/$ rates yields the spot rate S₀^$/SFr = 1/S₀^SFr/$ = $0.6667/SFr. Because changes in continuously compounded returns are symmetric, ±2 in the value of the SFr are at:

+ 2σ: S₁^$/SFr = S₀^$/SFr(1 + s^$/SFr) = S₀^$/SFr e^2 = ($0.6667/SFr)(e^20.108167)

= ($.6667/SFr)(1.2415) = $0.8277/SFr

2σ: S₁^$/SFr = S₀^$/SFr(1 + s^$/SFr) = S₀^$/SFr) e^2 = ($0.6667/SFr)(e^20.108167)

= ($.6667/SFr)(0.8055) = $0.5370/SFr

These methods are equivalent.