HullFund8eCh05ProblemSolutions

HullFund8eCh05ProblemSolutions - CHAPTER 5 Determination of...

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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, let be the expected return required by investors on the index so that ( ) 0 ( ) q T T E S S e . Because r and ( ) 0 0 r q T F S e , it follows that 0 ( ) T E S F . 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, 0 F , is given by equation (5.1) as: 0 1 1 0 40 44 21 F e   or $44.21. The initial value of the forward contract is zero. b) The delivery price K in the contract is $44.21. The value of the contract, f , after six months is given by equation (5.5) as: 0 1 0 5 45 44 21 f e     2 95 i.e., it is $2.95. The forward price is: 0 1 0 5 45 47 31 e    or $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 (0 07 0 032) 0 5 150 152 88 e     or $152.88. Problem 5.11.
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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, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose 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 1 (3 2 2 5) 3 2 5 % The futures price is therefore 80 . 331 , 1 300 , 1 4167 . 0 ) 032 . 0 09 . 0 ( e or $1,331.80. Problem 5.12. Suppose 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?
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