3.1. A bank quotes you a rate of interest of 14% per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?
3.2. Explain what happens when an investor shorts a certain share.
3.3. Suppose that you enter into a 6-month forward contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 12% per annum. What is the forward price?
3.4. A stock index currently stands at 350. The risk-free interest rate is 8% per annum (with continuous compounding) and the dividend yield on the index is 4% per annum. What should the futures price for a 4-month contract be?
3.5. Explain carefully why the futures price of gold can be calculated from its spot price and other observable variables, while the futures price of copper cannot.
3.6. Explain carefully the meaning of the terms "convenience yield" and "cost of carry". What is the relationship between the futures price, the spot price, the convenience yield, and the cost of cany?
3.7. Is the futures price of a stock index greater than or less than the expected future value of the index? Explain your answer.
3.8. An individual receives $1,100 in 1 year in return for an investment of $1,000 now. What is the percentage return per annum with
(a) annual compounding?
(b) semiannual compounding?
(c) monthly compounding?
(d) continuous compounding?
3.9. What rate of interest with continuous compounding is equivalent to 15% per annum with monthly compounding?
3.10. A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?
3.11. A 1-year-long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.
(a) What are the forward price and the initial value of the forward contract?
(b) Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract?
3.12. A stock is expected to pay a dividend of $1 per share in 2 months and again in 5 months. The stock price is $50 and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a 6-month forward contract on the stock.
(a) What are the forward price and the initial value of the forward contract?
(b) Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract?
3.13. The risk-free rate of interest is 7% per annum with continuous compounding and the dividend yield on a stock index is 3.2% per annum. The current value of an index is 150. What is the 6-month futures price?
3.14. Assume that the risk-free interest rate is 9% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, the dividend yield is 5% per annum. In other months, it is 2% per annum. Suppose that the value of the index on July 31, 1992 is 300. What is the futures price for a contract deliverable on December 31, 1992?
3.15. Suppose that the risk-free interest rate is 10% per annum with continuous compounding and the dividend yield on a stock index is 4% per annum. The index is standing at 400 and the futures price for a contract deliverable in 4 months is 405. What arbitrage opportunities does this create?
3.16. Estimate the difference between risk-free rates of interest in Germany and the United States from the information in Table 3.3.
3.17. The 2-month interest rates in Switzerland and the United States with continuous compounding are 3% and 8% per annum, respectively. The spot price of the Swiss franc is $0.6500. The futures price for a contract deliverable in 2 months is $0.6600. What arbitrage opportunities does this create?
3.18. The current price of silver is $9 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. Assuming a flat term structure with a continuously compounded interest rate of 10%, calculate the futures price of silver for delivery in 9 months.
3.19. A bank offers a corporate client a choice between borrowing cash at 11% per annum and borrowing gold at 2% per annum. (If gold is borrowed, interest and principal must be repaid in gold. Thus 100 ounces borrowed today would require 102 ounces to be repaid in 1 year.) The risk-free interest rate is 9.25% per annum and storage costs are 0.5% per annum. Discuss whether the rate of interest on the gold loan is too high or too low in relation to the rate of interest on the cash loan. The interest rates on the two loans are expressed with annual compounding. The risk-free interest rate and storage cost are expressed with continuous compounding.
3.20. Suppose that F\ and F2 are two futures contracts on the same commodity with maturity dates of /j and t2 and t2 > t\. Prove that
where r is the risk-free interest rate (assumed to be constant) between t\ and t2 and U is the cost of storing the commodity between times t\ and t2 discounted to time t\
at the risk-free rate. For the purposes of this problem, assume that a futures contract is the same as a forward contract.
3.21. When a known cash outflow in a foreign currency is hedged by a company using a forward contract, there is no foreign exchange risk. When it is hedged using futures contracts, the marking to market process does leave the company exposed to some risk. Explain the nature of this risk. In particular, consider whether the company is better off using a futures contract or a forward contract when
(a) the value of the foreign currency falls rapidly during the life of the contract
(b) the value of the foreign currency rises rapidly during the life of the contract
(c) the value of the foreign currency first rises and then falls back to its initial level
(d) the value of the foreign currency first falls and then rises back to its initial level Assume that the forward price equals the futures price.
3.22. It is sometimes argued that a forward exchange rate is an unbiased predictor of future exchange rates. Under what circumstances is this so?
3.23. A company that is uncertain about the exact date when it will pay a foreign currency sometimes wishes to negotiate with its bank a forward contract where there is a period during which delivery can be made. The company wants to reserve the right to choose the exact delivery date to fit in with its own cash flows. Put yourself in the position of the bank. How would you price the product that the client wants?
3.24. What is the difference between the way in which prices are quoted in the foreign exchange futures market, the foreign exchange spot market, and the foreign exchange forward market?
3.25. The forward price on the German mark for delivery in 45 days is quoted as 1.8204. The futures price for a contract that will be delivered in 45 days is 0.5479. Explain these two quotes. Which is more favorable for an investor wanting to sell marks?
3.26. The Value Line Index is designed to reflect changes in the value of a portfolio of over 1,600 equally weighted stocks. Prior to March 9, 1988, the change in the index from one day to the next was calculated as the geometric average of the changes in the prices of the stocks underlying the index. In these circumstances, does Equation (3.12) correctly relate the futures price of the index to its cash price? If not, does the equation overstate or understate the futures price?
3.27. A company has a $10 million portfolio with a beta of 1.2. How can it use futures contracts on the Major Market Index to hedge its risk? The index is currently standing at 270.
3.28. "When the convenience yield is high, long hedges are likely to be particularly attractive to a company that knows it will require a certain quantity of a commodity on a certain future date." Discuss.
3.29. A U.S. company is interested in using the futures contracts traded on the IMM to hedge its German mark exposure. Define r as the interest rate (all maturities) on the U.S. dollar and rf as the interest rate (all maturities) on the mark. Assume that r and rf are constant and suppose that the company uses a contract expiring at time T to hedge an exposure at time t (T > t). Show that the optimal hedge ratio is e(rf-r)(T-t)
APPENDIX 3A: A PROOF THAT FORWARD AND FUTURES PRICES ARE EQUAL WHEN INTEREST RATES ARE CONSTANT
In this appendix, we show that forward and futures prices are equal when interest rates are constant. Suppose that a futures contract lasts for n days and that F, is the futures price at the end of day / (0 < i < n). Define <5 as the risk-free rate per day (assumed constant). Consider the following strategy.2
1. Take a long futures position of es at the end of day 0 (i.e., at the beginning of the contract).
2. Increase long position to e2S at the end of day 1.
3. Increase long position to e3s at the end of day 2.
This strategy is summarized in Table 3.5. By the beginning of day i, the investor has a long position of es'. The profit (possibly negative) from the position on day i is
Assume that this is compounded at the risk-free rate until the end of day n. Its value at the end of day n is
The value at the end of day n of the entire investment strategy is therefore
[(F„ - F„_i) + (F„_! - F„_2) + • ■ • + (Fi - F0)]enS = (Fn - F0)enS
Since F„ is the same as the terminal asset price, St, the terminal value of the investment strategy can be written
2This strategy was proposed by J. C. Cox, J. E. Ingersoll, and S. A. Ross, "The Relationship between Forward Prices and Futures Prices," Journal of Financial Economics, 9 (December 1981), 321-46.
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