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## How to Calculate Net Present Value

Net present value (NPV) is the present value of all future cash flows of a project. Because the time-value of money dictates that money is worth more now than it is in the future, the value of a project is not simply the sum of all future cash flows. Those future cash flows must be discounted because the money earned in the future is worth less today. In order to calculate NPV, we must discount each future cash flow in order to get the present value of each cash flow, and then we sum those present values associated with each time period.

###### Where:
• = Cash Flow at time t
• = discount rate expressed as a decimal
•  = time period

You can think of NPV in different ways, but I think the easiest way is to think of it is as the sum of the present value all cash inflows, i.e. cash you earn from the project, less the present value of all cash outflows, i.e. cash you spend on the project. This way of thinking about NPV breaks it down into two parts, but the formula takes care of both of these parts simultaneously. The way we calculate the present value is through our discount rate, r, which is the rate of return we could expect from alternative projects. Say you have a dollar. If you don’t invest that dollar, you will still have that same dollar bill in your pocket next year; however, if you invest it, you could have more than that dollar one year from now. The alternative project is investing the dollar, and the rate of return for that alternative project is the rate that your dollar would grow over one year.

Just by thinking of things intuitively by the time value of money, if you have a time series of identical cash flows, the cash flow in the first time period will be the most valuable, the cash flow in the second time period will be the second most valuable, and so forth. This means that the present value of the cash flows decreases. Now, this is not always the case, since cash flows typically are variable; however, we must still account for time. The way we do this is through the discount rate, r, and each cash flow is discounted by the number of time periods that cash flow is away from the present date. This means that our cash flow for the first time period of the project would be discounted once, the cash flow in the second time period would be discounted twice, and so forth. To discount a cash flow, simply divide the cash flow by one plus the discount rate, raised to the number of periods you are discounting. This methodology follows from compound interest. Let’s take a look at an example.

#### Example

Suppose you as the investor are looking at investing in a project for your company that would extend to you the ownership of a new piece of machinery that may help your business produce widgets more efficiently. This new piece of machinery costs \$500,000 for a three-year lease, but your hope is that your company will operate more efficiently and generate higher cash flows as a result of this new machine. This machine operates differently than the one your company currently uses to produce widgets, so it may take time for your employees to get used to operating the new equipment. Thus, you expect cash flows to increase over time as your employees become more familiar with the equipment. Your analysts are projecting that the new machine will produce cash flows of \$210,000 in Year 1, \$237,000 in Year 2, and \$265,000 in Year 3. The rate of return of an alternative project is 6%. What is the net present value of your potential investment?

We have our rate at 6% listed first, and you can see below each year and the cash flow associated with that year. Remember, at time 0 (the present day), you must outlay \$500,000 in order to receive the new piece of machinery. The following years you will receive more cash due to an increase in production of widgets. Now that we have a good visual of what the project looks like financially, let’s begin our NPV calculation.

We discount our first cash flow, a cash outflow to be precise, by zero years. This just means that we really aren’t discounting the first cash flow because you would be paying for the project at the present time, so the present value of the first cash flow is just that, the first cash flow at face value. The other cash flows will need to be discounted by the number of years associated with each cash flow. We discount our cash flow earned in Year 1 once, our cash flow earned in Year 2 twice, and our cash flow earned in Year 3 thrice. Once we calculate the present value of each cash flow, we can simply sum them, since each cash flow is time-adjusted to the present day. Once we sum our cash flows, we get the NPV of the project. In this case, our net present value is positive, meaning that the project is a worthwhile endeavor. Be careful, however, because the projected cash flows are estimates typically, as is the discount rate. Our final calculation is only as good as its inputs.