Silver Mining

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Silver Mining

• Professor André Farber
• Solvay Business School
• Université Libre de Bruxelles

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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.

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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
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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.
Now At maturity
Buy spot -70,000 10,000 ST
Borrow 70,000 -72,132
Total 0 10,000 (ST 7.2132)
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3. What is the value of your net position?
At maturity
Short position 10,000 (6 ST)
Synthetic forward 10,000 (ST 7.2132)
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 7.2132) e-30.75 -11,773
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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
forward _at_ 6.25 from Mineral Trading and selling
forward _at_ 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 doesnt deliver at
maturity?
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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
0.0625 per quarter to be paid at time t 0, t
0.25 and t 0.50 U 0.0625 0.0625 e-40.25
0.0625 e-40. 50 0.186
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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 612 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).

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6. Calculate the fixed rate on the 612 FRA.

• The fix rate on the FRA is equal to the 612
  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 1R(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
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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 rT pays Fix
rate R
Silver Mining should go LONG on an FRA
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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
Small difference due to rounding
4.55 5.17
At time T
At time T
At time T