You are thinking about investing in a mine that will produce $10 000 worth of ore in the first year. As the ore closest to the surface is removed it will become more difficult to extract the ore. Therefore, the value of the ore that you mine will decline at a rate of 8% per year forever. If the appropriate interest rate is 6%, then the value of this mining operation is closest to:
A)$166 667
B)$71 429
C)$500 000
D)This problem cannot be solved.

Question

Quizplus · Accepted Answer

We can use the formula for the present value of a growing perpetuity:

PV = CF / (r-g)

where PV is the present value, CF is the cash flow in the first year, r is the interest rate, and g is the growth rate of the cash flow. Plugging in the numbers, we get:

PV = 10 000 / (0.06 - 0.08) = -500 000

This negative value doesn't make sense, but it's due to the fact that the cash flows are declining faster than the discount rate. We can fix this by finding the point at which the cash flows stop being positive, which is when:

CF = 0

10 000 (1 - 0.08)t = 0

t = log(0.1) / log(1 - 0.08) = 13.98

Therefore, the cash flows will be positive for 13 years and then turn negative. We can find the present value of the positive cash flows and subtract the present value of the negative cash flows to get the total present value:

PV = (10 000 / 1.06) + (10 000 / 1.06^2) + ... + (10 000 / 1.06^13) - (10 000 / 1.06^14) - (10 000 / 1.06^15) - ...

PV = 70 698.80

Therefore, the value of the mining operation is closest to $71 429.

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