Referencing Styles : Harvard Question 1 (20 Marks) Consider a portfolio consisting of $15 million invested in the S&P 500 and $15 million invested in U.S. Treasury bonds. The S&P 500 has an expected return of 8 percent and standard deviation of 16 percent. The Treasury bonds have an expected return of 4 percent and standard deviation of 8 percent. The correlation between the S&P 500 and the bonds is 0.35. All figures are stated on annual basis. (i) Find the VAR for one year at a probability of 0.01. [10 marks] (ii) Use the historical method and the following information for the last 120 days of returns to calculate an approximate VAR for a portfolio of $30 million using a probability of 0.05: Less than -10% 5 -10% to -5% 18 -5% to 0% 42 0% to 5% 36 5% to 10% 15 Greater than 10% 4 [10 marks] Question 2 (10 Marks) (i) The following term structure of LIBOR is given Term Rate 90 days 6.00% 180 days 6.20% 270 days 6.30% 360 days 6.35% (a) Find the rate on a 6x9 FRA. [2 marks] (b) Consider an FRA that was established previously at a rate of 5.2% with a notional principal of $30 million. The FRA expires in 180 days, and the underlying is 180-day LIBOR. Find the value of FRA from the perspective of the party paying fixed and receiving floating as of the point in time at which the above term structure applies. [3 marks] (ii) You are the treasurer of a firm that will need to borrow $10 million at LIBOR plus 2.5 points in 45 days. The loan will have a maturity of 180 days, at which time all the interest and principal will be repaid. The interest will be determined by LIBOR on the day loan is taken out. To hedge the uncertainty of this future rate, you purchase a call on LIBOR with a strike of 9 percent for a premium of $32,000. Determine the amount you will pay back and the annualized cost of borrowing for LIBORs of 6 percent and 12 percent. Assume the payoff is based on 180 days and a 360-day year. The current LIBOR is 9 percent. [5 marks] Question 3 (20 Marks) A corporation enters into a $35 million notional amount interest rate swap. The swap calls for the corporation to pay a fixed rate and receive a floating rate of LIBOR. The payments will be made every 90 days for one year and will be based on the adjustment factor 90/360.The term structure of LIBOR when the swap is initiated as follows: Days LIBOR 90 7.00% 180 7.25% 270 7.45% 360 7.55% (i) Determine the fixed rate on the swap. [6 marks] (ii) Calculate the first net payment on the swap [4 marks] Assume that is now 30 days into the life of the swap. The new term structures of LIBOR as follows: Days LIBOR 60 6.80% 150 7.05% 240 7.15% 330 7.20% (iii) Calculate the value of the swap [10 marks] Question 4 (10 Marks) On June 17 of a particular year, an American watch dealer decided to import 200,000 Swiss watches. Each watch costs SF325. The Dealer would like to hedge against a change in the dollar/Swiss franc exchange rate. The forward rate was $0.3881. Required Determine the outcome from the hedge if it was closed on August 16, when the spot rate was $0.4434. Question 5 (10 Marks) Assume there is a forward market for a commodity. The forward price of the commodity is $45. The contract expires in one year. The risk-free rate is 10 percent. Now, six months later, the spot price is $52. What is the forward contract worth at this time? Explain why this is the correct value of the forward contract in six months even though the contract does not have a liquid market like a futures contract. Question 6 (20 Marks) The following option prices were observed for calls and puts on a stock on July 6 of a particular year. The stock was priced at 165.13. The expirations are July 17, August 21, and October 16. The continuously compounded risk-free rates associated with the three expirations are 0.0503, 0.0535, and 0.0571, respectively. The standard deviation is 0.21. Calls Puts Strike Jul Aug Oct Jul Aug Oct 160 6.00 8.10 11.10 0.75 2.75 4.50 165 2.70 5.25 8.10 2.40 4.75 6.75 170 0.80 3.25 6.00 5.75 7.50 9.00 (i) A slight version of a straddle is a strap, which uses two calls and one put. Construct a long strap using the October 165 options. Hold the position until expiration. Determine the profits and graph the results. Identify the breakeven stock prices at expiration and the minimum profit. [10 marks] (ii) A strip is a variation of a straddle involving two puts and one call. Construct a short strip using the August 170 options. Hold the position until the options expire. Determine the profits and graph the results. Identify the breakeven stock prices at expiration and minimum profit. [10 marks] Question 7 (10 Marks) A stock is priced at $50 with a volatility of 35 percent. A call option with an exercise price of $50 has expiration in one year. The risk-free rate is 5 percent. (i) Compute the Black-Scholes price of the call and European lower bound for stock prices of $5, $50 and $100 and verify that the former is at least as large as the latter. [3 marks] (ii) Can implied volatilities be expected to vary for options on the same stock with the same exercise prices but different expirations? [1 marks] (iii) Can implied volatilities be expected to vary for options on the same stock with the same expirations but different exercise prices? [1 marks] (iv) Why and how are implied volatilities used to quote options prices? [5 marks]