International Financial Management Prerequisites 1. The quoted interest rate is 5% p.a. What is the effective interest rate for 6 months if the quoted interest rate is a) simple, b) annually compounded, c) quarterly compounded or d) continuously compounded? 2. A deposit earns interest of 12% p.a., compounded quarterly. At which continuously compounded interest rate would a bond yield the same effective return as this deposit over a period of a) half a year and b) one year? Spot Exchange Markets 3. You have just graduated from the University of Florida and are leaving on a whirlwind tour of Europe. You wish to spend USD 1,000 each in Switzerland, France, and Great Britian (USD 3,000 in total). Your bank offers you the following bid-ask quotes: USD/ CHF 0.84 0.86, USD/EUR 1.30 1.32, and USD/GBP 1.95 1.97. a) If you accept these quotes, how many CHF, EUR, and GBP do you have at departure? b) If you return with CHF 300, EUR 1,00, and GBP 75, and the exchange rates are unchanged, how many USD do you have? c) Suppose that instead of selling your remaining CHF 300 once you return home, you want to sell them in Paris. At the train station, you are offered EUR/CHF 0.64 0.69, while a bank three blocks from the station offers EUR/CHF 0.66 0.68. At what rate are you willing to sell your CHF 300? How many EUR will you receive? 4. Bank X quotes the following exchange rate: Spot rate / : 0.6862-0.6878 Spot rate /$: 0.9180-0.9290 Bank Y quotes: Spot rate /$: 0.6374-0.6385 a) Is there an arbitrage or least-cost dealing opportunity? b) If there is an arbitrage opportunity, how would you profit from it? c) If there is a least-cost dealing opportunity, how would investors behave in this market? 1
5. Please use these quotes to answer the following exercises: Bid Ask GBP/USD 0,5276 0,5383 USD/EUR 1,2797 1,3055 a) Suppose that you convert 10 EUR into USD and finally into GBP and then use the proceeds to do the reverse conversion. What is the amount you end up with? b) In your answer to a), verify the "Law of the Worst Rate"! c) Please compute the synthetic cross rate. d) Suppose that a bank quotes as stated above. Would you require additional information in order to check possibilities for least cost dealing or arbitrage? If so, which information? e) Suppose that there is another bank quoting EUR/GBP 1.4850-1.4900. Is there a triangular arbitrage opportunity? f) Given that there are three currencies involved, how many conditions must be satisfied such that there are no arbitrage opportunities? Forward Contracts in Perfect Markets 6. You are given the following data: the spot exchange rate is CAD/EUR 1.617; the p.a. simple interest rate on a six-month deposit is 3% in Canada and 4% in Austria. Compute: a) The forward rate for a six-month forward contract. b) The time-t CAD value of a CAD(t) 1 investment. c) The time-t EUR value of a EUR(T) 1 loan. d) The time-t EUR value of a CAD(T) 1 forward sale, given the forward rate computed in a). e) The time-t EUR value of a CAD(T) 1 spot sale (hint: use the certainty equivalent). 7. The following information is given: 6-month interest rate = 5% p.a., 6-month interest rate = 4% p.a., Spot / rate = 0.6873. A British firm sold goods to a German importer. The invoice is payable within 180 days. The British firm wants to set the Euro price such that the present value equals 100,000. Which price should be charged? 8. Suppose that the spot exchange rate between the Japanese Yen and the US Dollar is JPY/USD 107.615. The forward rate for 3 months is JPY/USD 105.5 and the 90-day interest rates are 7/32% p.a. in Japan and 6 1/8% p.a. in the US. Are there any arbitrage opportunities? If so, how would you make a risk-free profit and what would happen if a very large number of investors try to do likewise? 2
9. Suppose that the expected spot rate between the Canadian Dollar and the US Dollar in one year is CAD/USD 1.30. Moreover, let the current forward rate for 1 year equal this expected future spot rate. Draw a diagram which depicts the payoff from a) a forward sale and b) a forward purchase of 1 million USD for 1 year when the realized spot rate differs from the expected rate. The Value of a Forward Contract and Its Implication 10. Which of the following statements are correct? a) A forward purchase contract can be replicated by borrowing foreign currency, converting it into domestic currency, and investing the domestic currency. b) A forward purchase contract can be replicated by borrowing domestic currency, converting it into foreign currency, and investing the foreign currency. c) A forward sale contract can be replicated by borrowing foreign currency, converting it into domestic currency, and investing the domestic currency. d) A forward sale contract can be replicated by borrowing domestic currency, converting it into foreign currency, and investing the foreign currency. 11. Please solve the following exercises: a) Given the following data, compute the value today of an outstanding forward purchase contract initiated at t 0 for 100.000 units of foreign currency (where the exchange rate is HC/FC). Does the new value represent a gain or a loss to the holder of the old contract? (Hint: First compute the new forward rate.) Spot Rate Old Forward Rate r t,t r* t,t (1) USD/GBP 1,96 1,99 6,35% 5,10% (2) JPY/USD 107,62 109,05 3,20% 6,35% b) Repeat the previous question using the same data, but with an outstanding forward sale contract. 12. A US company expects to receive EUR 1m in one year. The treasurer of this company wants to value this A/R. So, she looks up a consensus forecast which states an average of some experts' opinions about the spot rate in one year's time. This consensus forecast is USD/EUR 1.1300. Moreover, she looks up the interest rate on USD CDs for one year which turns out to be 5.92% p.a. Finally, she does some calculations and reports the value of the A/R as USD 1066843. a) How did she come up with this number? Did she make a mistake? At lunch, the treasurer talks to a friend and tells him about the interesting consensus forecast she found in a newspaper this morning. In response, her friend wants to show off his own knowledge of international finance: he replies that nobody should really care about some experts' forecasts since the forward rate, as the "consensus" of the market, is the best forecast one may ask for. According 3
to her friend, the treasurer should therefore redo her calculations and replace the consensus forecast of the future spot rate by the forward rate. b) Is the friend a real expert in international finance? c) Suppose that the treasurer follows her friend's suggestions. Would she make a mistake? (Suppose that the forward rate is USD/EUR 1.1200.) d) Can you think about an alternative way to determine the value of the A/R? 13. Suppose that you sold forward (360 days) GBP 2m at the forward rate CAD/GBP 2.05 to hedge a payment from a British customer. Ten months later the GBP trades at CAD/GBP 2.20 and compounded interest rates for 60 days are 5% p.a. for the CAD and 7% p.a. for the GBP. Unexpectedly, the customer pays 2 months earlier than schedule. Consider the following alternatives. You may: (1) Invest the GBP 2m for 2 months, deliver the principal to the bank with which you signed the forward contract, and sell the interest you earn from your investment through a forward contract. (2) Sell the GBP 2m spot and negotiate an early termination of the outstanding forward sale. (3) Sell the GBP 2m spot and buy them forward 60 days so that you can deliver the required amount to your bank. a) Analyze each alternative. Which alternative is the best one? b) Comment on your answers from part a. Is this what you would expect? Why (not)? Forward Contracts and Market Imperfections 14. An American company plans to send an employee to Vienna to learn German. The employee will need EUR 5,000 in 3 months time when the classes begin, and an additional EUR 2,000 in 6 months to cover his living expenses. Today, the American company wants to lock in the costs of training his employee. Given the following information, what amount of USD will the American company have to pay in 3 and in 6 months time? Bid Ask USD/EUR 1,15 1,27 R 6,1% p.a. 6,7% p.a. r* 3,8% p.a. 4,3% p.a. 15. After completion of your studies, you work as the foreign exchange risk manager of a big company. One day, you enter into a one-year forward purchase of EUR 10 m at USD/EUR 1.1013. An hour later, you see that your bank quotes that the rate is now at USD/EUR 1.1015-1.1020. a) How could you "lock in" your gain? b) At which price could you sell your contract from one hour ago to your bank if the US interest rate is 6.00-6.25 % p.a.? c) Instead of selling to the bank, you talk to a friend who is the foreign exchange risk manager of the company across the street. It turns out that this company is about to enter into a 4
one-year forward purchase contract of EURO 10 m at the current forward rate. However, after talking to you, your friend considers whether to buy your old forward purchase contract. What is the maximum amount that you could receive from your friend? What are the implications for the value of your old forward purchase contract? 16. Suppose you are a Canadian and are presented with the following information: Investing Borrowing Interest rate in Canada 6% p.a. 7% p.a. Interest rate in USA 4,5% p.a. 5,5% p.a. CAD/USD bid CAD/USD ask Spot rate 1,185 1,191 Forward rate 1.189 1.201 a) Where should you borrow? b) Where should you invest? c) If interest income is taxed at 50%, but foreign exchange earnings are untaxed, would this affect your answer to part b)? d) If there exists no direct quotes for the forward rates and you have to calculate them synthetically, would this affect your answer to part a) and b)? Currency Futures Markets 17. An Englishman wishes to hedge an accounts receivable of USD which will become due in 3 weeks time. He contacts his broker and sells 50 futures contracts, each of which is for the sale of USD 125,000 at GBP/USD 0.5250. The contracts expire in 3 weeks. Assume that today is Monday, and that the settlement prices at the end of each day during the first week are as follows: Settlement GBP/USD Mon 0,5263 Tue 0,5245 Wed 0,5266 Thu 0,5311 Fri 0,5327 The initial margin is set at GBP 5,000 per contract and the maintenance margin at GBP 4,000 per contract. No interest is paid on the margin account. Assume that the man withdraws money from his margin account whenever he may do so, but that the first time he must make a margin-call, he fails to do so. (i.e. he does pay the initial margin as required). a) Show the value of any withdrawals which the man made, and the value of all margin-calls. b) If the man fails to pay the first margin-call, what does the broker do? What is the final settlement? 5
18. A German firm wants to hedge the EUR value of a CAD 1m inflow using futures contracts. On Germany's exchange, there is a futures contract for USD 100,000 at EUR/USD 0.75. a) Your assistant runs a bunch of regressions: (1) S [ EUR / CAD] = α 1 + β1 f [ USD / EUR] (2) S [ EUR / CAD] = α 2 + β 2 f [ EUR / USD] S CAD EUR = α + β f EUR / USD (3) [ / ] 3 3 [ ] (4) S [ CAD EUR] = α + β f [ USD / EUR] / 4 4 Which regression is relevant to you? b) If the relevant β is 0.85, how many contracts do you buy? Sell? 19. A German exporter wants to hedge an outflow of NZD 1m. She decides to hedge the risk with a EUR/USD contract and a EUR/AUD contract. The regression output is, with the t-statistics shown in parentheses, and R2 = 0.59: [ EUR / NZD] = + 0.15 f [ EUR / USD] + 0.7 f [ EUR AUD] S α / (1.57) (17.2) a) How will you hedge if you use both contracts, and if the face value of a USD contract is for USD 50,000 and AUD contract for AUD 75,000? b) Should you use the USD contract, in view of the low t-statistic? Or should you only use the AUD contract? 20. Consider three points in time and the following information: t 0 t 1 t 2 Forward rate: F t0, t2 0.65 -- -- Futures rate: f t0, t2 0.65 0.66 -- Spot rate: S t0 -- -- 0.67 A firm will have to cover an account payable of FC 100,000 at t 2. Given the above information, what are the cash flows at t 0, t 1, and t 2 if a) the firm does not hedge? b) the firm hedges, using a forward contract? c) the firm hedges, using a futures contract? 21. When is a currency futures contract more valuable than a comparable forward contract? Why? 6
Forward Rate Agreement 22. Suppose you enter into a nine-to-twelve-month forward rate agreement with a notional deposit of EUR 5m. The forward interest rate is 12% p.a.. Nine months later, the 3-month Euro interest rate turns out to be 10% p.a.. What is your profit/loss from the forward rate agreement? 23. Suppose you enter into a nine-to-twelve-month forward rate agreement with a notional deposit of EUR 5m. The forward interest rate is 12% p.a.. 3 months later, the six-month and ninemonth Euro interest rates are 10% and 10.5% p.a., respectively. What is the value of your forward rate agreement? Currency and Interest Rate Swaps 24. Currency Swap A few years ago, a UK company and an Austrian company carried out a swap. The UK company needed 180 million EUR, and the spot rate at the time was GBP/EUR 0.52. The Austrian company borrowed EUR at a swap rate of 10% p.a. with quarterly payments. The UK company borrowed GBP at a swap rate of 8% p.a. payable every 6 months. The principals were then swapped. The swap ends in 14 months. - The current GBP swap rate for 14 months is 9.5% p.a. - The current EUR swap rate for 14 months is 7.5% p.a. - The current spot rate is GBP/EUR 0.49. a) Calculate the EUR-value of the EUR-loan. b) Calculate the GBP-value of the GBP-loan. c) What is the value of the swap for the Austrian company? 25. Interest rate swap Some time ago, a German company speculated on a rise in fixed-rate interest rates, and swapped EUR 5m fixed rate (at 8% p.a.) into floating rate (at the p.a. EUR LIBOR). That is, the contract stipulates that every year, the firm pays 8% on a notional EUR 5m and receives the EUR LIBOR rate on the same notional amount. Suppose that the interest rates have risen above the 8% p.a. and that the company wants to sell the contract to lock in its gain. Which amount will it receive? Current conditions are as follows: The swap has 3 years and 1 month left until maturity. The current three-year EUR swap rate is 9% p.a. The EUR LIBOR rate, set 11 months ago for the current period, is 11% p.a. The current one-month EUR LIBOR rate is 10.5% p.a. 26. Combined currency and fixed/floating rate swap Two and three quarter years ago, an Austrian company wanted to borrow Canadian dollars. It found it was cheapest to borrow EUR and carry out a swap with a Canadian company which wanted EUR. The Austrian company took out a loan of EUR 500.000 at a swap rate of 13% and gave the principal to the Canadian company which paid interest to the Austrian company. The 7
Canadian company took out a loan of CAD 1.000.000 at a swap rate of LIBOR. It gave the principal to the Austrian company which paid interest to the Canadian company. The spot rate at that time was 0,5 EUR/CAD. The term of the swap was for 6 years and payments are made at the end of each year. Now, with three and a quarter years to go, the Austrian company would like to cancel the swap i.e. stop servicing the CAD loan and to resume servicing the EUR loan. How much would the Austrian company be willing to pay to the Canadian company to be able to do this? Additional information: The current 3 1/4 year swap rate is 11% p.a. 1 year LIBOR at the last reset date 9 months ago was 10.5% p.a. The current 3 month LIBOR is 8% p.a. The current spot rate is EUR/CAD 0,6391. 27. Combined currency and fixed/floating rate swap Starapopoulos, a Greek shipping company, needed 500 million Yen to buy a ship from Mitisitua Heavy Industries in Japan. At the same time, Mitisitua wanted Euros to buy Greek engines for its products. The two companies performed a swap over 4 years with payments made at six monthly intervals, such that the first payment was made 6 months after the swap was set up. Starapopoulos borrowed EUR at a fixed swap rate of 12.6% p.a. Mitisitua borrowed JPY at 6-month LIBOR plus a spread of 2% p.a.. At the time the swap was set up, the spot rate was JPY/EUR 135,55. 13 months after the deal was set up, the following quotes were observed: The EUR swap rate for the same maturity as that above was 11.2% p.a. The 6 month JPY LIBOR rate at the last reset date was 3.5% p.a. The 5 month JPY LIBOR rate was 2.7% p.a. The spot rate was JPY/EUR 138,25. What value in JPY did the swap have for Mitisitua? Currency Options 28. Is the following statement true? Proof your answer graphically. The payoff from buying a put and selling a call can be replicated by borrowing domestically and investing abroad. 29. Assume that the put and the call both have a strike price equal to X, a domestic T-Bill has a face value equal to X, and both a foreign T-Bill and forward contract pay off one unit of foreign currency at expiration. All instruments expire on the same date. A call can be replicated by a) buying a foreign T-Bill and selling a put b) buying foreign currency forward and buying a put c) buying a put, selling a domestic T-Bill, and buying a foreign T-Bill d) all of the above e) none of the above 8
30. Assume that the put and the call both have a strike price equal to X, a domestic T-Bill has a face value equal to X, and both a foreign T-Bill and forward contract pay off one unit of foreign currency at expiration. All instruments expire on the same date. A forward sale can be replicated by a) buying a put and selling a call b) selling a foreign T-Bill and buying a domestic T-Bill c) selling a put and buying a call d) both (1.) and (2.) e) all of the above 31. A cylinder option on the sale of foreign currency is a contract defined as follows: If S T <X 1, you sell foreign exchange at X 1, the floor If S T >X 2, where X 2 >X 1, you sell foreign exchange at X 2, the cap If X 1 S T X 2, you sell at the uncertain value of S T. a) Show the payoff of the contract graphically. b) Show that it can be viewed as a combination of European options. c) Illustrate the value of a foreign currency claim hedged with such a contract. 32. A straddle is a contract defined as follows: If ST < X you sell foreign exchange at X. If ST > X you buy foreign exchange at X. Show the payoff of the contract graphically. Can it be viewed as a combination of European options? Is a straddle a useful instrument for hedging or for speculating? What do you hedge respectively on what do you speculate? Explain your answers! 33. Derive the CF diagram at T for a Butterfly Spread: Long one Call at K1 Short two calls at K2 Long one call at K3 K1 < K2 < K3 What is the purpose of a Butterfly Spread? 34. Which options do you buy/sell to generate this CF? 8 7 6 5 4 3 2 1-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 9
35. The Mexican company TACO has debt of EUR 100,000, which is repayable in 12 months. TACO s controller is worried because the debt is unhedged. The current MXM/EUR exchange rate is 14, and p.a. interest rates are 12% in Mexico and 2.0% in EUR. The controller is considering a forward hedge at the current forward rate F t,t. However, the controller of another Mexican company tells her that he recently bought a call on EUR 100,000 with strike price X=14, and is willing to sell it to her at the historic cost, MXN 1 per EUR or 100000 MXN for the total contract. What should TACO s controller do? 36. An Austrian firm, VOGEL IN NOOT AG (VIN), sells its goods at home at EUR 15,000 when the EUR/HUF exchange rate is high. However, when this exchange rate decreases below the "critical" value of EUR/HUF 0.006, then VIN faces competition from Hungary: in order to sell its goods, VIN must then match the price charged by its Hungarian competitor. a) Illustrate how the EUR/HUF exchange rate determines VIN's profit (in EUR) from the sale of one unit. b) How could VIN eliminate its exposure to the EUR/HUF rate? 37. Mister X is a currency speculator for a private banking house in Boston. His base currency is therefore the U.S. dollar. He is currently convinced that the EUR is overvalued and wishes to take a speculative position to profit from his expectations. He will invest $1 Million in his position (spent entirely on option premiums). The current spot rate is 1.0650 USD/EUR. The current premiums on call and put options on Euro with a strike price of 1.06 USD/EUR are the following: 90 days Call options on euro: premiums (USD) 0.0401 Put options on euro: premiums (USD) 0.0865 a) What should Mister X do buy a call, sell a call, buy a put, or sell a put on the EUR? b) What would be his profits or losses if he bought a 90-day put with a strike price of 1.06 USD/EUR, and the actual spot exchange rate in 90 days turned out to be 1.02 USD/EUR? Pricing Currency Options: Using the Binomial Model 38. Suppose that the current EUR/USD spot rate is 0.6, the effective risk-free rates of return are 6% in EUR and 8% in USD. The spot rate will either increase to 0.62 or decrease to 0.57 at time 1. a) What is the risk-adjusted probability of a spot rate increase? Decrease? b) What is the risk-adjusted expected value of a European call with a strike price of 0.59? c) What is the time-0 value of the call? d) What is the factor u (d) by which the spot rate increases? Decreases? e) What is the option s exposure? f) Would the option s value change if it were American? 10
39. Assume all the information from the previous example. Assume further that you read in the newspaper that a European Call at a strike price of 0.59 is traded at a price of 0.05 (i.e., the call is overpriced). Given this information, can you find a strategy to realize arbitrage profits? If yes, show the arbitrage strategy and explain how it works. 40. Suppose you want to buy a put option on the USD with an exercise price of EUR 1.00 and a maturity of 6 months. An equivalent call option on the USD is trading at 0.25. The riskless USD interest rate is 3% p.a. and the riskless EUR interest rate is 4% p.a.. The current spot rate equals 1.2 EUR/USD. What is the price of the put option? 41. Consider a one-period call option on the British Pound. Suppose that the current exchange rate is USD/GBP 2, the exercise price is USD/GBP 1.9, the one-period risk-free rate on the USD is 5%, and the one-period risk-free rate on the GBP is 10%. Suppose that the spot rate can either go up by a factor of 1.1 or down by 0.9. a) Write down the two equations that show how one can replicate the cash-flow from the option by investing in the foreign currency and borrowing domestically. What is the value of the call option, using the replication approach? b) Compute the risk-neutral probabilities and use these to value the above call option. 42. Please solve the following exercises: a) An Austrian is offered a European put option to sell GBP 100.000 at a strike price of EUR/GBP 1.70. Use the binomial option pricing model to calculate the fair price of this option, given the following information: The option expires in 8 months. The simple interest rate in Austria is 5.2% p.a. The simple interest rate in the UK is 4.0% p.a. S 22 = 1.9602 S 11 = 1.7820 S 00 = 1.6200 S 21 = 1.6038 S 10 = 1.4580 S 20 = 1.3122 Note that each time interval above is 4 months, i.e. from time 0 to time 2 is 8 months. The above-stated exchange rates are expressed in EUR/GBP. b) Calculate the present value of an "old" forward sale contract, for the sale of GBP 100.000 at a forward-price of EUR/GBP 1.70 and with a maturity date 8 months from today. c) Use your answer to parts a and b to find the value of an 8 month European call option on GBP 100.000 with an exercise price of EUR/GBP 1.70. d) What would be the present value of the option described in part a) if it were American? 11
43. Recall the binomial tree and all the related information given in the previous exercise. Describe the dynamic hedging approach that you use to hedge the uncertain payoffs from a European Put option with a strike price of EUR/GBP 1.70. 44. Consider a four-month call option on the British Pound. Suppose that the current exchange rate is USD/GBP 1.6, the exercise price is USD/GBP 1.6, the risk-free rate on the USD is 8% p.a., and the risk-free rate on the GBP is 11% p.a., and the volatility of the spot rate (and the forward rate) is 10%. Translate the volatility into an up and down factor, then solve the following problems: a) What is the value that you would be willing to pay for this American call option if you used the one-period binomial approach to value it? b) What would you be willing to pay for this option if the volatility were 14.1%? 45. Consider the following binomial tree model. The exchange rate process follows the nodes of a recombinant tree (as in all our exercises) which moves from value s at some particular node along the next up/down branch to the new values: s exp µ δt + σ δt, if up ( ) ( µ δt σ δt ), if down s exp Assume the following parameter values: µ=0.16, σ=0.25, and the time of one period δt=1/250. Furthermore the current spot rate equals 123.40, the effective domestic rate for one period equals 1.25% and the effective foreign rate for one period equals 1%. What is the price of a European call option with maturity T=8/250 (i.e., you have to consider a binomial tree for 8 periods) and strike price X=127? Hint: You can save much time by exploiting the fact, that the number of ups follows a binomial distribution, instead of drawing the full tree. Pricing European Options: The Lognormal Model 46. An US investor has an eight months European put option on GBP with an exercise price of 1.9 USD/GBP. The spot rate is 1.6 USD/GBP and the continuously compounded risk free rate of interest in the US is 18% while that in the UK is 12%. If the standard deviation of the exchange rate is 0.141 per year, at what price would this put option sell? 47. An Austrian owns a zero coupon bond (i.e. there are no coupon payments; at maturity you receive the face value of the bond) with a face value of USD 1000. The bond has a time to maturity of 4 months. The Austrian has the right to get the principal in either USD or EUR at a fixed exchange rate of USD/EUR 1.25. The current spot rate is EUR 1=USD 1.2. The continuously compounded, risk free rate of interest is 4% in the US and 3% in the Euro zone. The volatility (standard deviation) of the exchange rate is 0.15. 12
a) Which FX-option is included in this bond if you consider the EUR as your HC? Give a detailed explanation. b) Determine the value of the bond for the investor in EUR. 48. An American owns a 15% GBP 1000 bond from a UK company. The company can decide whether to pay the coupon in USD or GPB. a) In how many different ways can one interpret this option? b) What is the value of the bond, given the following information? The continuously compounded risk-free interest rate in the US is 6% p.a. and in the UK it is 10% p.a. The standard deviation of the change in the exchange rate is 0.1. The bond has 12 months to maturity. Coupons are paid annually. The current spot rate is USD/GPB 1.5. The exercise price of the option is USD/GBP 1.5. c) At what coupon rate would the bond trade at par? 13