A gold mine will produce 2000 pounds of gold next year and 1500 pounds in two years. After that the mine will be close forever. Suppose for simplicity that there are no operating expenses. The forward price of gold delivered one year from now is $10,000/pound, and no forward market is available for delivery in two years. The risk free rate is 5% and the cost of capital for gold mines is 10%. Suppose that, in the second year, the gold price fluctuates so it can be $12,000/pound or $9,000/pound with equal probabilities.

a. Find the value of the gold mine. (Hint: if you are really clueless, you can do part b first and then go back to part a.)

b. Suppose now that the mine can be operated for one year only (i.e., ignore any information for year two). Form the tracking portfolio for the gold mine and value the mine in this case.

Answer:

The Cost of Capital in this case is 10%. We shall use this rate to be the discounting rate. And since no operating expenses have been specified, I believe we can value the mine on the amount of gold it will produce times the forward price or the probable forward price discounted at 10%.

So for part a, we have a scenario where the mine will produce gold for 2 years, we have the forward rates of gold for the first year, and probable forward rates for the second year. And we will be required to discount the inflows by the cost of capital, which is 10%.

Stating otherwise, the Gold produced by the mine in the first year can be sold as a surety for $10,000, while there are equal chances of it being sold for either $12,000 or $9,000 in the second year.

Since there are no operating expenses or initial outlays involved, We can value the mine in the following manner:

The Price of Gold will be $10,000 within the first year, while the price for second year is probable at either $12,000 or $9,000 with 50% probability for each price. Thus, the average price for second year will be $(12,000+9,000)*50%, or $10,500.00.

As for Part b, it asks that the same mine will now be operated for only the first year, and we are required to value the mine in this case. However, given that we have already calculated the discounted value for the first year above, we are not required to calculate the same again!

Thus, we can take only the discounted cash flows arising in the first year from the screenshot above to be the value of the mine, which you can see will be $18,181,818.18 or $18.18 millions.

PFA