A mine can produce 80 million pounds of copper one year from now. The two possible copper prices then are $0.90 per pound under high demand and $0.40 per pound with low demand. The 1 year ahead forward price for copper is now $0.80 per pound. The risk free one-year interest rate is 2%. Extraction costs are $0.70 per pound. The firm can shut down without cost if needed. What is the mine worth?

Scenario 1 (High Demand)

Cash Flow = (Price of copper - Extraction Costs) x How much the mine can produce

= (.90 - .70) x 80

= 16

The equation in the tracking portfolio when demand is high is

x(.90 - .80) + y(1.02) = 16

where

x = pounds of copper purchased forward

y = dollars invested in zero-coupon bonds today that mature in one year

Scenario 2 (Low Demand)

Cash Flow = (Price of copper - Extraction Costs) x How much the mine can produce

= (.40 - .70) x 80

= -24 (The mine would lose money so it would shutdown production at no cost)

= 0

The equation in the tracking portfolio when demand is low is

x(.40 - .80) + y(1.02) = 0

Simultaneously solve the equations for the scenario 1 and 2 gives the tracking portfolio

x(.90 - .80) + y(1.02) = 16

x(.40 - .80) + y(1.02) = 0

.5X = 16

x = 32

Substitute 32 for x in one of the above equations and solve for y:

32(.90 - .80) + y(1.02) = 16

3.2 + 1.02y = 16

y = 12.549 M

The value of the tracking portfolio is $12.549 M so the value of the copper mine must be $12.549 M