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Question 1
An investor could like to buy a futures contract on the ALCOA share. Today's price of the ALCOA share is $17. The maturity of the futures contract is in 6 months and the risk-free interest rate is 6% per annum.
(a) What is the 6-month futures price of the ALCOA share?
(b) If the actual futures price were higher than the no-arbitrage 6-month futures price, which arbitrage could you implement?
Question 2
An investor would like to buy a futures contract on the ALCOA share. Today's price of the ALCOA share is $17. The maturity of the futures contract is in 1 year and the risk-free interest rate is 6% per annum. ALCOA pays a $2 dividend in 3 months from now and a $1 dividend in 9 months from now.
(a) What is the 1-year futures price of the ALCOA share?
(b) If the actual futures price were lower than the no-arbitrage 1-year futures price, which arbitrage could you implement?
Question 3
We are at t=0. A stock is expected to pay a dividend of $10 per share at t=2 months and t=5 months. The stock price is $80, and the risk-free rate is 3.5% per annum. 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?
We are now at t=3 (months), the price of the stock is $68 and the risk-free rate of interest is now 3% per annum.
(b) What is the forward price?
(c) What is the value of the short position in the forward contract at this point in time?
Question 4
On BB you will find the file "Q5HW1.xls". This is a Microsoft Excel file. The file contains daily prices on a commodity ("Und") and on the corresponding futures ("Futures").
(a) Both for the underlying asset and the futures, compute the price changes ΔS and ΔF.
(b) Compute means, standard deviations, and correlation of the price changes.
(c) Using the results in (b), calculate the optimal hedge ratio which minimizes risk. Comment on your results.
(d) Suppose that the price exposure is for 1,000,000 units of "Und" and each futures contract is for 55,000 units of the underlying. How many futures contracts should be used to hedge?
Question 5
On BB you will find the file "Q6HW1.xls". This is a Microsoft Excel file. The file contains two time series. "Index" represents the value of an index, and "Portfolio" represents the value of a portfolio in millions of dollars.
(a) For both the portfolio and the index compute the rate of return R_{L,T} = log(P_{L,T}/P_{L,T-1}) , where P_{i,t} is either the price at time t of the portfolio (i=Port) or the price at time t of the index (i=index).
(b) Compute the CAPM-β of the portfolio with respect to the market. (Assume that the market is represented by the index.)
(c) On December 27^{th}, 2007, the value of the portfolio is $243.54 million. Assume that you wish to use futures contracts on the index to hedge the portfolio's risk. The index on that day is standing at 130.89, and each contract is for the delivery of $250 times the index. What is the hedge (i.e., number of contracts) that minimizes risk?