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**Net Present Value**

For a project with one investment outlay, made initially, the net present value (NPV) is the present value of the future after-tax cash flows minus the investment outlay, or

$NPV=\sum _{t=1}^{n}\frac{C{F}_{t}}{(1+r{)}^{\mathit{t}}}-Outlay$

Where

CFt = after-tax cash flow at time t

r = required rate of return for the investment

Outlay = investment cash flow at time zero

To illustrate the net present value criterion, we will take a look at a simple example. Assume that Gerhardt Corporation is considering an investment of €50 million in a capital project that will return after-tax cash flows of €16 million per year for the next four years plus another €20 million in Year 5. The required rate of return is 10 percent

For the Gerhardt example, the NPV would be

$NPV=\frac{16}{(1.10{)}^{1}}+\frac{16}{(1.10{)}^{2}}+\frac{16}{(1.10{)}^{3}}+\frac{16}{(1.10{)}^{4}}+\frac{20}{(1.10{)}^{5}}-50$

NPV = 14.545 + 13.223 + 12.021 + 10.928 + 12.418 − 50

NPV = 63.136 − 50 = €13.136 million

The investment has a total value, or present value of future cash flows, of €63.136 million. Since this investment can be acquired at a cost of €50 million, the investing company is giving up €50 million of its wealth in exchange for an investment worth €63.136 million. The investor’s wealth increases by a net of €13.136 million. Because the NPV is the amount by which the investor’s wealth increases as a result of the investment, the decision rule for the NPV is as follows:

Invest if NPV > 0

Do not invest if NPV < 0

Positive NPV investments are wealth-increasing, while negative NPV investments are wealth-decreasing.

Many investments have cash flow patterns in which outflows may occur not only at time zero, but also at future dates. It is useful to consider the NPV to be the present value of all cash flows:

$NPV=CF0+\frac{CF1}{(1+r{)}^{1}}+\frac{CF1}{(1+r{)}^{2}}+\frac{CF1}{(1+r{)}^{3}}+.......+\frac{CFn}{(1+r{)}^{n}}$ or

$NPV=\sum _{t=1}^{n}\frac{C{F}_{t}}{(1+r{)}^{t}}$

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2017-03-20

Thanks for the template. Very nice