 # Net Present Value, NPV By Yuriy Smirnov Ph.D.

## Definition

The net present value (NPV) method is widely used in capital budgeting and investment decisions. It is also considered as the best single screening criterion to reject or accept a project because the NPV method takes into account the time value of money concept. Its value reflects an expected change in shareholders’ value caused by a project.

## Formula

The net present value of a project is the sum of the present value of each expected cash flow (both inflows and outflows) discounted at a discount rate. The equation of NPV is as follows:

 NPV = CF0 + CF1 + CF2 + … + CFN (1 + r)1 (1 + r)2 (1 + r)N

or

 NPV = N CFt Σ (1 + r)t t = 0

Here, CF is an expected cash flow at the end of designated year t, r is discount rate, and N is life of the project in years.

## Discount Rate

It is important to remember that the discount rate takes into account not only the time value of money concept but also the risk of uncertainty of expected cash flows! That is the reason why the project’s weighted average cost of capital (WACC) should be used as the discount rate. In other words, a project’s WACC is the required rate of return on capital invested in the project. Accordingly, the greater the risk of uncertainty of expected cash flows, the higher the discount rate, and vice versa.

## Decision rule

The decision rule in using the NPV method is rather straightforward. The threshold value of zero indicates that a project’s cash flows exactly cover the cost of invested capital and provide the required rate of return on invested capital. The general rules can be stated as follows:

1. A stand-alone project should be accepted if its NPV is positive, rejected in case it is negative, and stay indifferent if zero.
2. In the case of considering a number of independent projects, all those that have a positive NPV should be accepted.
3. Among several mutually exclusive projects, the one with the highest positive net present value should be accepted.

## Example

A company is considering two projects with the same initial cost of \$20,000,000. They are equally risky, have the same cost of capital of 11.5%, and the same total expected cash flows. The only difference is that cash flows from Project Y come in relatively sooner and relatively later from Project Z. The detailed information about expected cash flows is presented in the table below.

 Initial Cost, CF0 Cash flows at the end of relevant year, CFt 0 1 2 3 4 5 Project Y -\$20,000,000 \$10,000,000 \$8,000,000 \$6,000,000 \$4,000,000 \$2,000,000 Project Z -\$20,000,000 \$2,000,000 \$4,000,000 \$6,000,000 \$8,000,000 \$10,000,000

Let’s put all the data available in the formula above or use the calculator. So the net present value of Project Y is \$3,480,385.27 and \$318,148.89 for Project Z.

 NPV of Project Y = -\$20,000,000 + \$10,000,000 + \$8,000,000 + \$6,000,000 + \$4,000,000 + \$2,000,000 = \$3,480,385.27 (1 + 0.115)1 (1 + 0.115)2 (1 + 0.115)3 (1 + 0.115)4 (1 + 0.115)5

 NPV of Project Z = -\$20,000,000 + \$2,000,000 + \$4,000,000 + \$6,000,000 + \$8,000,000 + \$10,000,000 = \$318,148.89 (1 + 0.115)1 (1 + 0.115)2 (1 + 0.115)3 (1 + 0.115)4 (1 + 0.115)5

Discounted cash flows of both projects are schematically presented in the figure below. If the projects are independent, the company should accept both because of positive NPV. If the projects are mutually exclusive, the company should reject Project Z and accept Project Y because it has a higher net present value.

## NPV in Excel

The net present value of a project can also be calculated in Excel as in the example below. 1. Select output cell H6.
2. Click fx button, select All category, and select NPV function from the list.
3. In field Rate, select cell B1.
4. In field Value1, select the data range C6:G6, leave field Value2 empty, and press the OK button.

We didn’t take into account the initial cost of Project Z. Select output cell H6, and add cell B6 to the formula in the Formula Bar. 