A project costs $50,000 today and returns $18,000 per year for 4 years at a 9% discount rate. What is the NPV and should the project be accepted?

What we are trying to measure

Net present value (NPV) compares the project’s cost today to the present value of its future cash inflows discounted at the required return (9%). If $NPV>0$, the project is expected to increase value relative to earning 9% elsewhere.

Present value of the 4-year annuity

The project pays $18{,}000$ at the end of each year for 4 years, so it is an ordinary annuity. Use the annuity present value factor:

$$PV=PMT\left(\frac{1-(1+r)^{-n}}{r}\right)$$

Substitute $PMT=18{,}000$, $r=0.09$, $n=4$:

$$PV=18{,}000\left(\frac{1-(1.09)^{-4}}{0.09}\right)$$

Compute the factor:

$$1.09^4\approx 1.41158 \quad\Rightarrow\quad (1.09)^{-4}\approx 0.70843$$ $$\frac{1-0.70843}{0.09}=\frac{0.29157}{0.09}\approx 3.23972$$

So,

$$PV\approx 18{,}000\times 3.23972\approx 58{,}315$$

NPV and the accept/reject decision

Now subtract the initial cost:

$$NPV=-50{,}000+58{,}315=8{,}315$$

Because $NPV\approx +\$8{,}315$ is positive, the project should be accepted.