Silver Mining
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
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
F  Se
rT
 7 1.03  7.21
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
10,000 (6 – ST)
Short position
10,000 (ST – 7.2132)
Synthetic forward
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-3%×0.75 = -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 @ 6.25 from Mineral Trading
and selling forward @ 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?
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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.
F  ( S  U )erT
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-4%×0.25 + 0.0625 e-4%×0. 50 = 0.186
F  (7  0.186)e4%0.75  $7.40
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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
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).
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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.
dT
1
1 e3.50%0.50
R
(
 1) 
( 4.50%1  1)  5.17%
T * T dT *
0.5 e
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
S  [1  R (T * T )]e  r*T *
Forward price
F  1  Se rT
Solve:
e r*T * dT
 1  R(T * T )  rT 
e
dT *
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7. What position (long or short) should Silver Mining
take? Explain.
• The payoff on the FRA at time T is:
CFFRA
LONG FRA:
M (rT  R)(T * T )

1  rT (T * T )
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:
1  r  0.50  e4.50%0.50  r  4.55%
Long 6x12 FRA
Notional amount
* (rSpot-Rfra)
* Contract period
=
PV
Interest paid
Payoff FRA
Total
Interest rate per annum
October 31, 2005
$100,000
-0.61% =4.55% – 5.17%
0.5 (year)
At time T*
-$307
At time T
-$300
-$2,276
-$307
At time T*
-$2,583
5.17%
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Small
difference
due to
rounding