CHAPTER 5
Determination of Forward and Futures Prices
Practice Questions
Problem 5.8.
Is the futures price of a stock index greater than or less than the expected future value of the index? Explain your answer.
The futures price of a stock index is always less than the expected future value of the index. This follows from Section 5.14 and the fact that the index has positive systematic risk. For an alternative argument, letbe the expected return required by investors on the index so that. Becauand, it follows that.
Problem 5.9.
A one-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?
a) The forward price, , is given by equation (5.1) as:
or $44.21. The initial value of the forward contract is zero.
b) The delivery price in the contract is $44.21. The value of the contract, f, after six months is given by equation (5.5) as:
i.e., it is $2.95. The forward price is:
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新世纪大学英语综合教程3or $47.31.
Problem 5.10.
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 the index is 150. What is the six-month futures price?
Using equation (5.3) the six month futures price is
corenor $152.88.
maProblem 5.11.
ladykillersAssume 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, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppo that the value of the index on July 31 is 1,300. What is the futures price for a contract deliverable on December 31 of the same year?
The futures contract lasts for five months. The dividend yield is 2% for three of the months and 5% for two of the months. The average dividend yield is therefore
The futures price is therefore
or $1,331.80.
Problem 5.12.
Suppo that the risk-free interest rate is 10% per annum with continuous compounding and that 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 four months is 405. What arbitrage opportunities does this create?如何提高英语口语
The theoretical futures price is
The actual futures price is only 405. This shows that the index futures price is too low relative to the index. The correct arbitrage strategy is
a) Buy futures contracts
b) Short the shares underlying the index.
Problem 5.13.
Estimate the difference between short-term interest rates in Japan and the United States on July 13, 2012 from the information in Table 5.4.
The ttlement prices for the futures contracts are to
Sept: 1.2619
Dec: 1.2635
The December 2012 price is about 0.13% above the September 2013 price. This suggests that the short-term interest rate in the United States exceeded short-term interest rate in Japan by about 0.13% per three months or about 0.52% per year.
Problem 5.14.
The two-month interest rates in Switzerland and the United States are 2% and 5% per an
num, respectively, with continuous compounding. The spot price of the Swiss franc is $0.8000. The futures price for a contract deliverable in two months is $0.8100. What arbitrage opportunities does this create?
The theoretical futures price is
The actual futures price is too high. This suggests that an arbitrageur should buy Swiss francs and short Swiss francs futures.
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Problem 5.15.
The current price of silver is $30 per ounce. The storage costs are $0.48 per ounce per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of silver for delivery in nine months.
The prent value of the storage costs for nine months are
0.12 + 0.12e−0.10×0.25 + 0.12e−0.10×0.5 = 0.351
or $0.351. The futures price is from equation (5.11) given bywhere
F0 = (30 + 0.351)e0.1×0.75acquisitive= 32.72
i.e., it is $32.72 per ounce.
Problem 5.16.
Suppo that and are two futures prices on the same commodity where the times to maturityof the contracts are and with. Prove that
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where is the interest rate (assumed constant) and there are no storage costs. For the purpos of this problem, assume that a futures contract is the same as a forward contract.
If
multi touchan investor could make a riskless profit by
a) taking a long position in a futures contract which matures at time; and
b) taking a short position in a futures contract which matures at time
When the first futures contract matures, the ast is purchad forusing funds borrowed at rate r. It is then held until time at which point it is exchanged forunder the cond contract. The costs of the funds borrowed and accumulated interest at time is. A positive profit of
