Silver Mining Professor André Farber Solvay Business School Université Libre de Bruxelles

October 31, 2005 Silver Mining - Solution |2 Question 1 - Data Three months ago the company entered into a forward contract to sell 10,000 ounces of silver, the quantity that Silver Mining expected to produce in the first half of 2006 in one of their mines. The forward contract matures in 9 months from now and the delivery price had been set at $6 per ounce. As a consequence of a major earthquake, silver extraction had to be stopped. Production is not expected to resume in the near future. The forward contract is no longer necessary. The current price of silver is $7 per ounce and the current 9-month interest rate is 4% per annum with continuous compounding. To offset the initial forward contract, you are asked to enter into a new forward contract to buy silver in 9 months. Assume first that the cost of storing silver is zero.

October 31, 2005 Silver Mining - Solution |3 1. What is the forward price of this new contract? Forward price: future value of spot price Underlying assumption: no arbitrage Value of new forward contract is 0: f = S – Fe -rT = 0

October 31, 2005 Silver Mining - Solution |4 2. How would you proceed to create a synthetic forward contract if a counterparty for the new forward contract is impossible to find? Silver Mining is SHORT on a 9-month forward contract for 10,000 oz with delivery price K = $6. To close the position, they should go LONG (buy forward). If no forward contract is available, they would create a SYNTHETIC foward. NowAt maturity Buy spot-70,00010,000 S T Borrow+ 70,000-72,132 Total010,000 (S T – )

October 31, 2005 Silver Mining - Solution |5 3. What is the value of your net position? At maturity Short position10,000 (6 – S T ) Synthetic forward10,000 (S T – ) Total-12,132 The value of the position today is the present value of -12,132 = -11,773 Remember that the unit value of a long forward contract with delivery price K is: f = (F – K) e -rT As Silver Mining is short, the value of of their position is: 10,000 (6 – ) e -3%×0.75 = -11,773

October 31, 2005 Silver Mining - Solution |6 4. You receive a fax from Mineral Trading confirming that they are ready to buy or sell forward silver in 6 months at $6.25 per ounce. What, if any, arbitrage opportunity does this create? The trader at Mineral Trading should follow a class in Derivatives! You make money by buying 6.25 from Mineral Trading and selling 7.21, the current 6-month forward price. You might have to create a synthetic short forward contract: Short silver (borrow silver and sell spot) Invest the proceed at the risk free rate Note: 1. Taking a short position is easy on paper – but you have to find someone willing to lend silver for 6 months. 2. Beware of credit risk. What if Mineral Trading doesn’t deliver at maturity?

October 31, 2005 Silver Mining - Solution |7 5. Assume now that the storage costs are $0.25 per ounce per year payable quarterly in advance. Calculate the futures price of silver for delivery in 9 months. where U is the present value of the cost of storage. The cost of storage is $0.25 / 4 = $ per quarter to be paid at time t = 0, t = 0.25 and t = 0.50 U = e -4%× e -4%×0. 50 = 0.186

October 31, 2005 Silver Mining - Solution |8 Question 2 - Data Silver Mining will have to invest in the coming months to repair its mining installations. The Treasurer plans to borrow $1 million in 6 months from now for a period of 6 months. He is considering taking a position on a 6×12 FRA to hedge the interest rate risk. The 6-month LIBOR rate is 3.5% per annum and the 12-month USD LIBOR rate is 4.3% (both with continuous compounding).

October 31, 2005 Silver Mining - Solution |9 6. Calculate the fixed rate on the 6×12 FRA. The fix rate on the FRA is equal to the 6 ×12 forward rate with simple compounding. Where does this formula come from? A quick review Consider a forward contract on a zero-coupon with face value 1+R(T*-T) and forward price = 1. What should be R in order for the value of the contract to be zero? Spot price of Zero Coupon Forward price Solve:

October 31, 2005 Silver Mining - Solution |10 7. What position (long or short) should Silver Mining take? Explain. The payoff on the FRA at time T is: LONG FRA:receives Floating rate r T pays Fix rate R Silver Mining should go LONG on an FRA

October 31, 2005 Silver Mining - Solution |11 8. Suppose that, 6 months later, the 6-month LIBOR rate (with continuous compounding) is 4.5% per annum. Verify the effectiveness of the hedge. 6-month Libor with simple compounding: =4.55% – 5.17% At time T* At time T Small difference due to rounding