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Finance 6324 FINN 6210 / BPHD 8240 Spring Semester 2021 Problem Set 2: Forwards and Futures Assignment Due: March 9 by 11:59 pm. You can work in a group with no more than two students. Please hand in one copy per group and list the names of the students in the group. Email your write up to me (
[email protected]) as an email attachment. Send only Word or PDF documents. 1. (Lecture Note 1) Answer the following questions. a. Describe the profit from the following portfolio: a short forward contract on an asset and a long European call option on the asset with the same maturity as the forward contract and a strike price that is equal to the forward price of the asset at the time the portfolio is set up. b. Describe the profit from the following portfolio: a long forward contract on an asset and a long European put option on the asset with the same maturity as the forward contract and a strike price that is equal to the forward price of the asset at the time the portfolio is set up. 2. (Lecture Note 2) The current spot price of a barrel of oil, ?0, is $70.63. The per year continuously compounded risk-free rate of interest, ?, is 3%, storage cost, ?, is 2%, and convenience yield, ?, is 8%. The expected return of oil, ??, is 12%. Answer the following questions for a futures contract with a maturity, ?, of 6 months. a. What is the expected spot price on the maturity date of the contract? b. What is the no arbitrage futures price? c. Is the futures price in part b in contango or normal backwardation? d. Give your answer in part c, would a speculator go long or short in the future contract? Explain your answer. 3. (Lecture Note 2) In the Spring of 1999, the U.S. Dollar-Deutschemark (DM) exchange rate was $0.5405 per DM. The U.S. and German interest rates (annualized, continuously compounded) were r = 6% and fr = 7.5%, respectively. Answer the following questions. a. What was the no-arbitrage forward price of DM for a 3 month forward contract? b. Suppose the actual quoted price for a 3 month forward contract was $0.5632 per DM. Explain whether or not there was an arbitrage opportunity. If one did exist, use an arbitrage table to demonstrate how you could have profited. The arbitrage table should have the following column titles: “Transaction”, “Payoff (now)”, and “Payoff (at T)”. mailto:
[email protected] 2 c. Suppose the actual quoted price for a 3 month forward contract was $0.5102 per DM. Explain whether or not there was an arbitrage opportunity. If one did exist, use an arbitrage table to demonstrate how you could have profited. The arbitrage table should have the following column titles: “Transaction”, “Payoff (now)”, and “Payoff (at T)”. d. In March of 1999, suppose that one month earlier (i.e., in February) a foreign currency trader entered into a 3-month long forward contract at the forward price that you calculated in part a. The contract was for the delivery of 1 million Marks. The March spot Dollar-DM exchange rate was $0.6125. What was the value in March of this long forward contract? 4. (Lecture Note 2) Consider a 9-month forward contract on Amazon.com Inc. (AMZN). The current price of one share is $1,670, and the annual continuously compounded risk-free interest rate is 6%. Answer the following questions. a. Compute the no-arbitrage forward price of the stock for a 9-month contract. b. Suppose a small private investor can borrow money at 8% per year with quarterly compounding and can lend money at 5.5% per year with semiannual compounding. Suppose the actual quoted forward price for a 9-month contract is 1,850.25 per share of AMZN. Explain whether there is an arbitrage opportunity. In your explanation, use an arbitrage table with the following column titles: “Transaction”, “Payoff (now)”, and “Payoff (9 months)”. c. What would the borrowing or lending rate have to be to eliminate the arbitrage opportunity in part b of this question? Your answer should report the rate in the compounding interval of the borrowing (quarterly) or lending (semiannual) rate. 5. (Lecture Note 3) Fidelity Investments has hired you to help them develop enhanced equity index funds using index futures contracts. Fidelity’s initial idea is to offer an S&P 500 index fund that promises twice (200%) the S&P 500 index. Consider the following information. The company plans to start the portfolio with $250 million invested to replicate the S&P 500 index. Of course, before using derivatives the portfolio will have a beta coefficient of 1.0. (Assuming the beta is computed relative to the S&P 500 index.) For the computations below, assume the current level of the S&P 500 index is 1,288 and the dividend yield of the index is 1.2% per year. The risk-free rate of interest is 5% per year. Both the dividend yield and the risk-free rate are continuously compounded. Answer the following questions. a. Currently the December S&P 500 stock index futures price is 1,310.50. Assuming a remaining maturity of exactly four months, compute the no arbitrage December S&P 500 futures price. 3 b. Given the no arbitrage index futures price that you calculated in a, is there an arbitrage opportunity? If there is, use an arbitrage table to illustrate the transactions and the payoffs now and in four months. The arbitrage table should have the following column titles: “Transaction”, “Payoff (now)”, and “Payoff (at T)”. c. What would the risk-free rate of interest have to be to eliminate the arbitrage opportunity? d. How many December S&P 500 futures contracts would you have to establish a position in to make the portfolio twice as volatile as the S&P 500 index? Should you be short or long? Use the current December futures price of 1,310.50. e. Suppose you establish the futures position in part d. If the S&P 500 index turns out to be 1,416.80 in three months, compute the value of the portfolio plus the index futures position. Show that the combined position has a beta coefficient of about 2.0. Assume in three months the December futures contract has exactly 1 month to maturity. f. Suppose you establish the futures position in part d. If the S&P index turns out to be 1094.80 in three months, compute the value of the portfolio plus index futures position. Show that the combined position has a beta coefficient of about 2.0. Assume in three months the December futures contract has exactly 1 month to maturity. 6. (Lecture Note 2) The accounting department is preparing the firm’s quarterly financial statements. As part of that process, they would like you to value some forward contracts. Answer the following questions. a. Compute the value of a long forward contract on the Chinese Yuan with a remaining maturity of 8 months. The current spot exchange rate is 0.15 U.S. dollars per Yuan, the U.S. domestic risk-free rate of interest (with continuous compounding) is 3 percent per year, and the Chinese risk-free rate of interest (with continuous compounding) is 4.25 percent per year. The delivery price in this previously-negotiated forward contract is 0.12 U.S. dollars per Yuan and the size of the contract is 25,000,000 Yuan. b. What is the value of a short forward contract on Chinese Yuan with exactly the same contract terms as the long forward contract in part a of this question? c. Compute the value of a long forward contract on rhodium with a remaining maturity of 9 months. The current spot price of rhodium is $2,500 per ounce and the risk-free rate of interest (with continuous compounding) is 3.5% percent per year. The delivery price in the previously-negotiated forward contract is $2,475 per ounce. Storage costs are $0.15 per ounce every 3 months and are payable in advance. The size of the contract is 100,000 ounces. d. What is the value of a short forward contract on rhodium with exactly the same contract terms as the long contract in part c of this question? 4 7. (Lecture Note 2) Consider a 6-month forward contract on gold. The current spot price is $1,000 per ounce and the annual continuously compounded risk-free interest rate is 5%. Assume storage costs and the convenience yield are zero. Answer the following questions. a. Compute the no-arbitrage forward price of gold for a 6-month forward contract. Round your answer to two places after the decimal place, e.g. 1.0153 rounds to 1.02. b. Suppose you can borrow money at 8 percent per year with monthly compounding and can lend money at 4.5 percent per year with monthly compounding. Compute the continuously compounded borrowing and lending rates. Round your answer to four places after the decimal place, e.g., 0.076562 rounds to 0.0766 or 7.66%. c. Suppose the actual quoted forward price for a 6-month contract is $1,027.50 per ounce of gold. Explain whether there is an arbitrage opportunity. If one does exist, use an arbitrage table to demonstrate how you can make a riskless arbitrage profit. The arbitrage table should have the following column titles: “Transaction”, “Payoff (now)”, and “Payoff (6 months)”. 8. (Lecture Note 2) Suppose the spot price of winter wheat is $5.98 per bushel. Storage costs are $0.15 per bushel every 3 months and are payable in advance. The interest rate is 5 percent per year with continuous compounding. Assume that winter wheat is a pure consumption commodity. Answer the following questions. a. Suppose the actual quoted price for a nine-month forward contract is $6.55 per bushel. Explain whether there is an arbitrage opportunity. If one does exist, use an arbitrage table to demonstrate how you can make a riskless arbitrage profit. The arbitrage table should have the following column titles: “Transaction”, “Payoff (now)”, and “Payoff (9 months)”. b. Suppose the actual quoted price for a nine-month forward contract is $6.80 per bushel. Explain whether there is an arbitrage opportunity. If one does exist, use an arbitrage table to