Problem 1: An airline knows that it will need to purchase 20,000 metric tons of jet fuel in three months. It wants some protection against an upturn in prices using futures contracts. The company can hedge using heating oil futures contracts traded on NYMEX. The notional for one contract is 42,000 gallons. There is no futures contract on jet fuel, the risk manager wants to check if heating oil could provide an efficient hedge. The current price of jet fuel is $300/metric ton. The futures price of heating oil is $0.7/gallon. The standard deviation of the rate of change in jet fuel prices over three months is 20%, that of futures is 15%, and the correlation is 0.8.
a) You decide to hedge your price exposure. What is your optimal strategy? Explain.
b) What are the payoffs of hedging?
Problem 2: Assume today is September 9, 2009. Today's close price of the MSFT share is $35. The risk-free interest rate is 5% per annum, continuously compounded. Consider a six-month forward contract to purchase the stock.
a) What is the forward price, assuming zero dividends?
[Hint. To calculate the forward price, set up two strategies and then make use of the arbitrage principle].
b) If the 6-month forward price is $35.5, what is the implied continuously compounded dividend yield? What is the forward and annualized forward premium?
c) One month later, the price of the stock is $40 and the risk-free interest rate declined to 4.0% per annum. What is the forward price?
[Hint. Once again, to calculate the forward price, set up two strategies and then make use of the arbitrage principle].
d) What is the value of a long position in the original forward contract? Comment on your results.
Problem 3: Compute Macaulay and modified durations for the followings bond?
a) A 5-year bond paying annual coupons of 4.432% and selling at par?
b) An 8-year bond paying semiannual coupons with a coupon rate of 8% and a yield of 7%.
c) A 10 year bond paying annual coupons of 6% with a price of $92 and face value of $100.
Problem 4: Suppose that in order to hedge interest rate risk on your borrowing, you enter into an FRA that will guarantee a 6% effective annual interest rate for 1 year on $500,000,000.00. On the date you borrow the $500,000,000.00, the actual interest rate is 5%. Determine the dollar settlement of the FRA assuming
a) Settlement occurs on the date the loan initiated
b) Settlement occurs on the date the loan is paid.
Problem 5: Suppose the September Eurodollar futures contract has a price of 96.4. You plan to borrow $50m for 3 months in September at LIBOR, and you intend to use the Eurodollar contract to hedge your borrowing rate.
a) What rate you can secure?
b) Will you be long or short the futures contract?
c) How many contracts will you enter into?
d) Assuming the true 3-month LIBOR is 1% in September, what is the settlement in dollars at expiration of the futures contract?
Problem 6: The current price of oil is $83.27 per barrel. Forward prices for 3, 6, 9, and 12 months are $87.24, $88.82, $89.70 and $90.66. Assuming 2% compounded annual risk-free rate, what is annualized lease rate for each maturity? Is this an example of contango or backwardation? (Lease rate can be considered as the negative of storage cost)
Problem 7: The following table gives the price of bonds
The following table gives the price of bonds
a) Calculate the zero rates for maturities 0.5, 1.0, 1.5 and 2.0 years. Comment on your results.
b) What are the forwards rates for the periods: 6 months to 12 months, 12 months to 18 months, 18 months to 24 months? Comment on your results.
c) Estimate the price and yield of a 2-years bond providing a semiannual coupon of 7% per annum.
d) Assume that the zero rates you computed in part a) are bond yields. Describe and discuss the shape of the yield curve.
Bond Principal Time to Maturity (years) Annual Coupon ($) Bond Price ($)
100 0.5 0 98
100 1 0 95
100 1.5 6.2 101
100 2 8 104