Homework 1 Solution Chapter 3 1. Suppose the Canadian dollar is currently traded at C$ 1.40/$. The Deutsche mark is traded at DM 1.39/$. Ignoring transaction costs: a. Determine the C$/DM exchange rate consistent with these direct quotations. b. Suppose the C$/DM cross rate in the market was at C$ 1.05/DM. Is there any arbitrage opportunity? c. How would you take advantage of any arbitrage situation? d. What is your profit? SOLUTIONS: a. Spot is C$ 1.0072/DM = (1.40 [C$/$] / 1.39 [DM/$]) b. Arbitrage opportunity: DM cheaper with combination of direct rates than using the cross rate. c. Buy US $ with C$ at 1.40, buy DM with US $, sell DM at the market's cross rate of C$ 1.05/DM. d. Gain is C$ 0.0425 for each C$ 1.0 that can be arbitraged or 4.25% [(1/C$1.4/$) * DM1.39/$ * C$1.05/DM = C$1.0425; 1.0425 ‐ 1 = 0.0425] 2. Suppose the Mexican Peso is currently traded at 7 MP/$. The yen is traded at Yen 90/$. a. Determine the MP/Yen cross rate. b. Suppose the MP/Yen cross rate in the market was at MP 0.1/Yen. Is there any arbitrage opportunity? c. How would you take advantage of any arbitrage situation? d. What is your profit? SOLUTIONS: a. Spot is MP 0.078/Yen = (MP7/$ / Yen90/$) b. Arbitrage opportunity: Yen cheaper using the direct rates than the cross rate. c. Buy $ with MP, sell $ for Yen, sell Yen at the market's cross rate of 0.1 MP/Yen. d. Gain is MP 0.2857. [ (1/MP7/$) * Yen90/$ * MP0.1/Yen = MP1.2857; 1.2857 ‐ 1 =0.2857] 6. 7. 10. Suppose the spot rate is $ 0.60/DM, i$,6 is 6.5% per annum and iDM,6 is 9% per annum. a. What is your estimate of today's six‐month forward $/DM rate? b. Suppose the six‐month forward is quoted at $ 0.60/DM. What would you do to take advantage of the arbitrage opportunity? Where would you borrow and lend? SOLUTIONS: a. b. Ft = St * (1 + i$,6/2) / (1 + iDM,6/2) = $0.60/DM * (1 + .065/2) / (1 + .09/2) = $ 0.5928/DM Borrow in US$, buy DM, invest in DM security, sell DM forward. Profit: ($1 / $0.60/DM) (1 + 0.09/2) * $0.60/DM ‐ $1 * (1 + 0.065/2) = $0.0125; or 1.25% gain on transaction Suppose the spot rate is Yen 100/$, i$,6 is 6.5% per annum and iYen,6 is 2.5% per annum. a. What is your estimate of today's six‐month forward rate? b. Suppose the forward is currently quoted at Yen 95/$. What would you do to take advantage of the arbitrage opportunity? Where would you borrow and lend? SOLUTIONS: a. b. Ft = St * (1 + iYen,6/2) / (1 + i$,6/2) = Yen100/$ * (1 + 0.025/2) / (1 + 0.065/2) = Yen 98.0629/$ Borrow in $, buy Yen, invest in Yen securities, sell Yen forward. Profit: $1 * Yen100/$ * (1 + .025/2) / Yen95/$ ‐ $1 * (1 + 0.065/2) = $0.0333; or 3.33% gain on transaction Suppose a German firm wishes to issue commercial paper in DM, but it is unable to do so in the German market. a. What can the firm do to replicate commercial paper (CP) securities without using German securities? Describe the transactions. b. Assume that the spot rate is $0.60/DM. The three‐month forward rate is $0.58/DM. The three‐month US$ CP rate is 8%. At what rate can the German firm expect to issue synthetic DM three‐month CP? SOLUTIONS: a. b. The German firm could borrow in the US$ CP market and swap its dollar obligation into DM, that is by buying US$ forward to match its future CP payments (principal plus interest) and selling DM forward. The German firm can secure the following rate: 1 + iDM/4 = St/Ft * (1 + i$/4); 1 + iDM/4 = 0.60/0.58 * (1 + .08/4); which implies that iDM = 22.07% Chapter 4 5. Suppose the current spot rate is $ 1.55/£ on the first of January. By year's end, the US CPI is expected to climb from 144 to 150 and the UK CPI is expected to climb from 120 to 130. According to PPP, what is the expected spot rate on December 31? SOLUTIONS: SPPP,t+1 = St,$/£ * (CPIt+1,US/CPIt,US)/(CPIt+1,UK/CPIt,UK); SPPP,Dec = $1.55/£ * (150/144)/(130/120) = $ 1.4904/£ 6. Consider the following data for the U.S. and Surlandia for the years 1975 ‐ 1980. 1975 1976 1977 1978 1979 1980 Pengo/$ 8.5 17.4 28.0 34.0 39.0 39.0 Surlandia CPI 100 312 599 838 1118 1511 U.S. CPI 161.2 170.5 181.5 195.4 217.4 246.8 a. According to the Purchasing Power Parity Theory, by how much is the Pengo over‐ or under‐valued at the end of 1980? b. On what basis could someone refute your calculation? SOLUTIONS: a. One way to calculate the PPP rate in 1980 is SPPP, 1980 = S1975 * [ PSUR, 1980 / PU.S., 1980] / [ PSUR, 1975 / PU.S., 1975] = 8.5 peso/$ * [1511/246.8] / [ 100/161.2] = 83.89 peso/$ 83.89 peso/$ = 2.15 ===> peso is 115% overvalued 39.00 peso/$ b. We could refute the calculation by questioning the selection of 1975 as the base year, the comparability of the consumption baskets, or whether there had been changes in tariff barriers or real productivity changes that would cause a change in the equilibrium real exchange rate. For example, if 1980 is assumed to be the base year for PPP, then by assumption, the peso is neither over‐ or under‐valued in 1980. 7. In the above table of numbers for Surlandia and the US, the nominal bi‐lateral exchange rate at the end of 1980 was reported as 39.0 Pengos/$. What was the real bi‐lateral exchange rate? SOLUTIONS: The real bi‐lateral exchange rate is St / SPPP = 39/83.89 = 0.46 [Surlandian goods / US goods] Chapter 5 1. Suppose the US and UK three‐month interest rates are respectively 6% and 8% per annum and that the spot rate is $1.55/£. a. Calculate the forward premium (or discount) on the £ expressed on a per annum basis. b. What value of the three‐month forward rate establishes Interest Rate Parity? SOLUTIONS: a. Forward Premium = (i$/4 ‐ i£/4)/(1+i£/4) = (0.06/4 ‐ 0.08/4) / (1 + 0.08/4) = ‐0.004902; which implies ‐1.96% per annum. £ is at a forward discount. b. (F‐S)/S = (i$/4 ‐ i£/4)/(1+i£/4) ; which implies F = S + S * (i$/4 ‐ i£/4)/(1+i£/4); F = $1.55/£ + $1.55/£ * ‐0.004902 = $1.542402/£. 4. The following data were taken from the July 28, 1994 issue of the Currency and Bond Market Trends by Merrill Lynch: JAPAN BRITAIN U.S. Spot exchange rates: 98.75 ¥/$ $1.53/£ ‐‐‐‐ 5‐year bonds: 3.73% 7.94% 6.88% 10‐year bonds: 4.34 8.24 7.24 20‐year bonds: 4.70 8.26 7.40 Compute the break‐even exchange rate for investors weighing the choice between $‐bonds and Yen‐bonds, and between $‐bonds and Pound sterling bonds for each of the three maturities. (Note: Assume that interest is paid twice yearly.) SOLUTIONS: A "break‐even" exchange rate is the exchange rate that would make a risk‐neutral investor indifferent between the US$ bond and the foreign currency denominated bond. In other words, it is the exchange rate that makes the Fisher International effect (i.e. uncovered interest parity) hold: E(St+n) = St [(1+i/2) / (1+i*/2)]2n where n = number of years to maturity of the bond. E(St+n) n = 5 years n = 10 years n = 20 years 5. ¥/$
84.70
74.50
58.47
$/£
1.4538
1.3896
1.2966
The point of this question is first, to get you accustomed to working with prices in $ per foreign currency and foreign currency per $. The second point is the shock value of seeing what a "small" interest differential of 1, 2, or 3% implies about exchange rates when compounded for 5, 10 or 20 years. As you can see, the impact is considerable. In 1986, The Seagram Company Ltd. (Canada) issued Swiss franc bonds (SFr 250,000,000) due September 30, 2085 with a 6% coupon. Assume that a similar bond denominated in $ would have required a 9% coupon and that the spot rate on issue day was $0.50/SFr. a. Compute the break‐even exchange rate for the redemption of the Seagram's bond at maturity. b. Discuss why Seagram's may have issued this bond rather than a US$ denominated bond. SOLUTIONS: a. This question is really identical to #4, except that the maturity of the instrument is even longer. With annual coupons, the calculation is: E(St+99) = $0.50/SFr [(1+.09) / (1+.06)]99 = $7.92/SFr With semi‐annual coupons, the calculation would be: E(St+99) = $0.50/SFr [(1+.09/2) / (1+.06/2)]198 = $8.75/SFr Again, the purpose of making the calculation is to see the power of compounding and the shock value of the number. b. Seagram's very likely issued the bond in order to exploit the feeling that uncovered interest parity reflects a bias: interest rates may be "too high" (relative to the actual exchange rate change) on weak currencies, and "too low" (relative to the actual exchange rate change) on strong currencies. If so, corporate treasurers should issue bonds in low interest rate currencies; and portfolio managers should invest in bonds with high interest rates. Both would be betting that the interest differential more than compensates for the exchange rate change. The Fisher International effect predicts that the interest differential will be an exact offset for the exchange rate change. 6. Suppose that the interest rates in question #5 reflect a 0.5% per annum currency risk premium for bond investors to willingly hold US$‐denominated bonds. a. Compute the expected exchange rate on the maturity date of the bond in this case. b. How does the currency risk premium affect the choice by Seagram's to issue a US$ or SFr denominated bond? SOLUTIONS: a. With annual coupons, the calculation is: E(St+99) = $0.50/SF [(1+.09 ‐ 0.005) / (1+.06)]99 = $5.03/SF With semi‐annual coupons, the calculation would be: E(St+99) = $0.50/SFr [(1+.085/2) / (1+.06/2)]198 = $5.45/SFr b. According to this calculation, the market expects a stronger US$ and weaker SFr than in Question 5. This tilts the choice toward borrowing SFr, which Seagram's can repay with fewer US$ than indicated by the calculation in Question 5.