You own a coal mining company and are considering opening a new mine. The mine itself will cost 

$ 119.6$119.6 

million to open. If this money is spent​ immediately, the mine will generate 

$ 21.7$21.7 

million for the next 10 years. After​ that, the coal will run out and the site must be cleaned and maintained at environmental standards. The cleaning and maintenance are expected to cost 

$ 1.7$1.7 

million per year in perpetuity. What does the *IRR* rule say about whether you should accept this​ opportunity? If the cost of capital is 

8.1 %8.1%​, 

what does the *NPV* rule​ say?

To decide whether to accept this mining opportunity, we can use both the **Internal Rate of Return (IRR)** and the **Net Present Value (NPV)** methods. Here's how we can approach both.

### 1. **Internal Rate of Return (IRR)**

The IRR is the discount rate at which the NPV of the project's cash flows equals zero. The IRR rule states that if the IRR is greater than the cost of capital, you should accept the project. If it's less, you should reject it.

#### Cash Flows:
- **Initial investment (Year 0):** $-119.6 \, \text{million}$
- **Cash inflows (Years 1-10):** $+21.7 \, \text{million/year}$
- **Perpetual outflow after year 10 (Years 11 onward):** $-1.7 \, \text{million/year in perpetuity}$

The goal is to find the IRR, where the NPV of these cash flows equals zero.

The cash flows can be broken down as:
\[
NPV = -119.6 + \sum_{t=1}^{10} \frac{21.7}{(1 + IRR)^t} + \sum_{t=11}^{\infty} \frac{-1.7}{(1 + IRR)^t}
\]

For the perpetual outflow starting in year 11, we use the formula for the present value of a perpetuity:

\[
\text{Present Value of perpetuity} = \frac{\text{Annual perpetuity}}{\text{Discount rate}} = \frac{1.7}{IRR}
\]

But this perpetuity starts in year 11, so we need to discount it back 10 years:

\[
PV_{10}

You own a coal mining company and are considering opening a new mine. The mine itself will cost $119.6 million to open. If this money is spent immediately, the mine will generate $21.7 million for the next 10 years. After that, the coal will run out and the site must be cleaned and maintained at environmental standards. The cleaning and maintenance are expected to cost $1.7 million per year in perpetuity. What does the IRR rule say about whether you should accept this opportunity? If the cost of capital is 8.1%, what does the NPV rule say?

Explore how to calculate the Internal Rate of Return (IRR) and Net Present Value (NPV) for a coal mining project. The mine costs $119.6 million to open, generates $21.7 million annually for 10 years, and incurs perpetual cleaning costs of $1.7 million annually. Understand the IRR rule and the NPV rule with a cost of capital of 8.1%.

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