Extra Practice Problems
Instructions: The problems are similar to the ones on your previous problem sets. All interest rates and rates of inflation given in the problems are annualized (i.e., stated as Annual Percentage Rates) and each problem has information about the compounding schedule.
1. Suppose the Japanese Yen per Dollar exchange rate is 87.60 in the spot market ($1=¥87.60). Meanwhile the Japanese interest rate is 2% (compounded annually) for a one-year maturity, and the US rate is 1% (compounded annually). Calculate the equilibrium exchange rate for one year forward (so that there would be no arbitrage opportunity).
2. Suppose the 1-year interest rate in Switzerland is 1%, while the expected inflation rate in Switzerland is zero. If the expected inflation rate in England were 2%, what would be the equilibrium nominal interest rate in England? (State your answer as an APR with annual compounding.)
3. Suppose the Swiss Franc per Dollar exchange rate is 0.90 in the spot market. Meanwhile the price of gold is CHF 1440 per ounce in the spot market and CHF 1429.12
per ounce in the futures market (for a 6-month maturity). The price of gold in New York is $1600 per ounce in the spot market and $1624 per ounce in the futures market (for a 6-month maturity). Find the equilibrium currency exchange rate for 6 months forward, for exchanging Swiss Franc and Dollar. (Note: Assume storage cost for gold is negligible, and ignore taxes.) 4. If you borrow money at a nominal interest rate of 4% for a purpose that allows
the interest to be tax-deductible, you are in the 35% marginal income tax bracket, and inflation is 1%, what is the real rate of interest you are paying, after tax? (State your answer as an APR with annual compounding.)
5. Suppose that at present exchange rates one Euro buys 132.45 Yen, and one Dollar buys 0.66 Euro. In equilibrium, how many Yen should one Dollar buy?
6. What is the present value of $1,000 to be received 5 years from now if the required real rate of return is 3% compounded annually and the expected rate of inflation is 1% compounded annually?
8. The following interest rates are observed (compounding is daily): • Instruments maturing on day 30 yield 0.6%
• Instruments maturing on day 90 yield 1.8%
Calculate the equilibrium discount rate for bond futures that mature on day 30, and call for delivery on day 30 of instruments with 60 days remaining until maturity.
9. The following prices are observed:
• Treasury bonds with 7% coupon and 20 years remaining to maturity are selling at $90 per $100 of face value (net of accrued interest).
• Treasury bonds with 5% coupon and 20 years remaining to maturity are selling at $70 per $100 of face value (net of accrued interest).
Calculate the equilibrium price for Treasury bonds with 6% coupon and 20 years remaining to maturity (find the price so that there would be no arbitrage opportunity)
10. Option A is a currency call option written in New York to buy Euros in exchange for U.S. Dollars, with expiration in three months. Option B is a put option written in Frankfurt to sell dollars in exchange for Euros with expiration in three months. Both options would exercise at an exchange rate of $1 = € 0.67. Which of the following is a correct equilibrium relationship, assuming both options involve the same amount of currency at exercise (say, both options involve trading $100 for € 67)?
a. Option A is worth more than Option B, comparing local prices at the spot exchange rate.
b. Option B is worth more than Option A, comparing local prices at the spot exchange rate.
c. Option A is worth the same as Option B, comparing local prices at the spot exchange rate.
11. The following facts are observed:
• UK government bonds with 0% coupon and 2 years remaining to maturity cost £90.00 per £100 of face value.
• Exchange rates are $1.00 = £0.60 spot, and $1.00 = £0.6333 for 2-year forward. Calculate the equilibrium price for US Treasury bonds with 0% coupon, 2 years
12. The following prices are observed:
• Treasury bonds with 6% coupon and 7 years remaining to maturity are selling at $93 per $100 of face value (net of accrued interest).
• Treasury bonds with 5% coupon and 7 years remaining to maturity are selling at $87.50 per $100 of face value (net of accrued interest).
• Treasury bonds with 4% coupon and 7 years remaining to maturity are selling at $82 per $100 of face value (net of accrued interest).
• Treasury bonds with 3% coupon and 7 years remaining to maturity are selling at $76.50 per $100 of face value (net of accrued interest).
Calculate the equilibrium price for an equivalent-risk instrument with 0% coupon and 7 years remaining to maturity (find the price so that there would be no arbitrage opportunity)
13. The following prices are observed:
• Euro per Dollar exchange rate is 0.6647 spot ($1=€0.6647). • Euro interest rate is 1.00% compounded daily.
• US interest rate is 0.5% compounded daily.
Find the equilibrium forward exchange rate for 180 days forward (so that there would be no arbitrage opportunity). Note: Use a 365-day year in your calculations.
14. The following prices are observed:
• London gold price per ounce is £960 spot and £941.92 for 180-day forward. • New York gold price per ounce is $1600 spot and $1624 for 180-day
forward.
• London Dollar exchange rate is $1=£0.60 spot.
15. You are an expatriate working for Bank America in Hong Kong, and observe the following prices.
• Swiss Franc per Dollar exchange rate is 0.90 spot ($1 = CHF 0.90) • Swiss interest rate is 0.25% APR with daily compounding.
• US stock market index is 1,390 today.
• The US stock market index 180-day futures price is 1,380.
• At today's level of the index, the average annual dividend yield on the stocks in the index is 2% (for simplicity, assume the dividends for your six-month holding period will all be paid at the end of 180 days, so the yield for this holding period would be 1%).
Find the equilibrium180-day forward exchange rate for the U.S. Dollar and the Swiss Franc (rounded to four decimal places).
16. Suppose you are a multi-national company with supplies of crude oil stored in several nations, including ports in the United States and Japan. The following pricing opportunities are available:
• Yen per Dollar exchange rate is 79.50 spot and 78.00 for 180-day forward ($1 = ¥ 79.50 spot and $1 = ¥ 78 forward).
• Crude oil for immediate delivery is $85.00
per barrel today in the US and ¥6700 in Japan.
• Forward price for oil (180-days) is $87.00
in the US and ¥6800.00
in Japan. • In order to keep it simple, assume storage costs are the same in both
countries.
Mini-Cases for Class Discussion
17. Dialog Semiconductor is a British company producing custom chipsets and doing research into new application-specific integrated circuits (ASICs). Dialog needs to finance some new expansion and would like to borrow £10 million at a fixed rate for five years, but the lowest rate available to them in England is 10%—which management considers too high. So, Dialog has decided to borrow for five years at a variable rate 2% over the rate for 1-year British Treasury Notes (the British Treasury rate (BT) is now 6%). Meanwhile Fluid Devices, an American company, needs $20 million for only a year; and can borrow in the U.S. for that maturity at 7%. If it wanted, Fluid Devices could borrow for five years at a fixed rate of 8% in the U.S. market. Currency exchange rate is £1=$2 spot, and also £1=$2 in the 1-year forward market. Suppose you work for Bankers Trust. Can you figure out an alternative borrowing and swap arrangement that would make both Dialog Semiconductor and Fluid Devices Corp better off?
18. A U.S. company expects substantial Yen revenues for the next ten years from a licensing agreement with a Japanese affiliate (expected to equal approximately $10 million per year at today’s exchange rate). This company fears a decline in the value of the Yen relative to the U.S. dollar, and is considering issuing bonds in the Japanese market (to be retired from the licensing fees) in order to immediately convert the principal into U.S. dollars and use it to finance growth of its domestic operations.
Meanwhile, a French electric utility company needs a large amount of financing to expand its generating capacity, but fears that the French financial markets cannot at the moment absorb an issue of the size anticipated. It has considered a Yen financing in the Japanese market, but would prefer keeping its obligations denominated in European currencies. It also had considered issuing bonds denominated in European Currency Units (ECUs), but was concerned that it may have “worn out its welcome” in that market, due to its large amount of such obligations already outstanding. Nevertheless, its revenue prospects were quite good, and generally considered sufficient to comfortably cover the anticipated new obligations
Solutions: Answer Steps 1. $1 = ¥88.4673 ¥1 = $0.0113 In US $1.00 grows to $1.01 In Japan, ¥87.60 grows to ¥89.3520 Forward rate must equal ¥89.3520/$1.01 Reference: Problem Set 1, problem 3
2. 3.02% Real rate in Switzerland is 1%, so the equilibrium situation in England would be: (R–2%)/1.02 = 1%
So, R = 1.02% + 2% = 3.02%
Reference: Problem Set 1, problems 4 & 5, plus 10-18 3. $1=CHF0.88
CHF1=$1.1364
1429.12/1624 = 0.88
Reference: Problem Set 1, problem 7 4. 1.58% r =( 4%(1–.35) – 1%)/1.01 = 1.58%
Reference: Problem Set 1, problems 16 & 17 5. ¥87.42 .66 * 132.45 = 89.40
Reference: Problem Set 1, problems 1 & 2
6. $820.74 n=5, %I=4.03, FV=1000, 1 compounding period per year, compute PV
Reference: Problem Set 1, problems 12 & 13 7. $5.00 £3 / 0.6 = $5.00
Reference: Problem Set 2, problems 8 & 9
8. 2.4% To begin, borrow $1,000,000 for 30 days at 0.6%. So PV is $1million, %i is 0.6, P/YR is 365, n is 30, compute FV. You will owe $1,000,493.27 on day 30 (store this result).
Invest this money for 90 days at 1.8%. So PV is $1million, %i is 1.8, n is 90, compute FV. You will receive $1,004,448.11, but you will need to sell a futures contract so you can recover the money on day 30 when you will need to repay the debt. Leave this result as FV.
Reclaim the earlier result you stored in memory, and enter it as PV (with negative sign). Now n is 60. Compute %i
9. $80.00 From the two bonds listed, one can see that an extra $2 per year of income for 20 years adds $20 to the price of the bond (so an extra $10 per year of income for 20 years should add $10 to the price). Add $10 to the price of the bond with 5% coupon (or subtract $10 from the price of the bond with 7% coupon) in order to find the answer.
Reference: Problem Set 3, problems 45 & 46
10. C With currency options, a New York call on the Euro is the same as a Frankfurt put on the dollar. When rendered on the same scale, the premia should match when compared at the spot exchange rate. Reference: Problem Set 2, problems 8&9
11. At the forward exchange rate, it would take £63.33 in two years to get $100. Today these future Pounds would cost 0.6333 * £90 = £ 56.997. At the spot exchange rate, this equals $95.
Reference: Problem Set 3, problems 51 & 52
12. From the bonds listed, one can see that $1 of extra income per year for 7 years adds $5.50 to the price of the bond. So we subtract 3* $5.50 from the price of the bond with 3% coupon. The result is $60
Reference: Problem Set 3, problem 45 13. $1 = €0.6663 (€1 = $1.5007)
This is because €66.47 would grow to €66.80, and $100 would grow to $100.25. Therefore at the forward exchange rate $100.25 should equal €66.80. Then 66.80/100.25 = 0.6663
Reference: Problem Set 1, problem 3 14. $1 = £0.58 (£1 = $1.7241)
The spot prices for gold follow the spot exchange rate. So should the forward exchange rate reflect the forward prices for gold. Thus £941.92 should equal $1624. Then £1 should equal 1624/941.92 = 1.6557 and $1 = 941.92 /1624 = £0.58
Reference: Problem Set 1, problem 7. 15. Answer: $1 = CHF 0.8986 (CHF 1 = $1.1129)
In the US, one unit of the stock market index, hedged in the futures market, would become $1380 + $13.90 = $1393.90. In Switzerland it would take 1390*0.90 = CHF1251.00 to buy a unit of the index today. If that amount were invested in Switzerland at 0.25% compounded daily for 180 days, it would grow to CHF1252.5433. Thus in equilibrium $1393.90 should equal CHF 1252.5433. The exchange rate would be $1 = 1393.90 / 1252.5433 = CHF 0.9816 (the Dollar would weaken forward).
16. Sell 100,000 bbl in the US for $8,500,000. Convert to ¥675,750,000. Spend ¥6,700,000,000 to buy 100,000 bbl of oil in Japan, and keep ¥5,750,000 profit (or leave $72,327.04 profit in the US).
Sell the oil forward in Japan and buy forward in the US. Then at settlement receive ¥680,000,000, convert to $8,717,948.72. Spend $8,700,000 to buy 100,000 bbl of oil in the US (restoring inventory there) and keep $17,948.72 profit (or leave ¥1,400,000 profit in Japan).
So profits are ¥5,750,000 ($72,327.04) immediately, plus $17,984.72 (¥1,400,000) six months in the future, with global oil inventory unchanged.
