# Net Present Value

Net Present Value, or NPV, is a tool used in financial analysis to decide whether or not a project is worth investing in. It's particularly useful for a business which is trying to decide whether to make a particular investment, or to decide which of two comparable investments to make.

There are several pieces of information you need to calculate the NPV. These are:

- The initial investment amount
- The expected returns on the the investment. This is likely to be in the form of a series of cash flows throughout the life of the project, for example the expected profit per year.
- The discount rate. This is how you take account of the time value of money.

Both the expected returns and the discount rate have to be determined ahead of time. So as you can see, there's quite a lot of prediction (or even guesswork) involved in calculating the NPV for a project.

## Discount Rate

The reason we need to take the discount rate into account is because otherwise our investment can look better than it really is. A simple example should illustrate this.

If we invest $1000 in a project, and we know that in a year's time we'll get back $1030, that looks like quite a good investment because we will make $30 profit. But if we could get 5% interest on that money by putting it in the bank, we could get back $1050 instead.

To properly evaluate the future $1030 in today's money, we need to discount it back to its present value, using 5% as the discount rate, with the formula:

`PV = R_t / (1 + i)^t`

where `PV` is the present value, `R_t` is the return we get at time `t`, and `i` is the discount rate (expressed as a decimal amount).

So plugging in the numbers from our example we get:

`PV = ($1030) / (1 + 0.05)^1`

`PV = $980.95`

This gives us a present value of $980.95. Since this is less than the amount we're planning to invest, we can see that we're going to lose money, $19.05 in this case.

There are several possible ways that a company might determine the discount rate:

- A company may have a standard cost of capital rate that they use.
- A company may have a typical expected return rate from a historical list of projects that they've carried out in the past. In which case it makes sense to evaluate new projects against this rate, as it reflects the likely rate of return they could get by investing in other projects.
- For more personal projects, you could use the best interest rate you could get from the bank or another safe investment.
- In other cases it may make sense to use the rate of inflation, as this is the rate at which the value of the money invested is declining over time.

## NPV Formula

We've actually already calculated the Net Present Value in the simple example above; that -$19.05 is the NPV: it's just the present value of the expected return minus the initial investment amount. But in that case, there was a single initial investment, and a single return after 1 year. Most investments aren't going to be quite that simple, of course. The profits from the project might come in over several years, and there might be some profit or expense at the end of the project to deal with.

So in these cases, the NPV is simply the sum of all the expected returns converted into present values, minus the initial investment amount:

`NPV = R_1 / (1 + i)^1 + R_2 / (1 + i)^2 + ... + R_n / (1 + i)^n - I_0`

Here, `R_1` is the expected return from year 1, `R_2` is the expected return from year 2 (and so on), `i` is the discount rate and `I_0` is the initial amount invested. This formula can also be written using a summation over `N` years as follows:

`NPV = (sum_(n=1)^NR_n / (1 + i)^n) - I_0`

## Example

Here's a more complex example of calculating and using the Net Present Value.

A toy business is considering investing in a new production line for their factory, to make a new type of toy racing car. The new production line will cost $20000 to build and install. They estimate that they can make $5000 per year in profit for the first year, $6000 in the second year and $7000 per year thereafter, until the production line reaches the end of its useful life 5 years later. (I've given it a rather short life so the calculations don't get too big!) They will be able to sell the equipment for scrap at the end of its life for $1000. Since this is a large, well-established company, they have a standard discount rate of 8%, based on many previous projects this company has undertaken.

So to calculate the NPV, and decide whether or not it's worth installing this new production line, they need to discount the profit values back to present values. The NPV is the sum of all these values, minus the initial investment:

`NPV = 5000 / (1 + 0.08)^1 + 6000 / (1.08)^2 + 7000 / (1.08)^3 + 7000 / (1.08)^4 + (7000 + 1000) / (1.08)^5 - 20000`

Note that the last return here includes $7000 profit and $1000 scrap value for the production line equipment.

`NPV = 4629.63 + 5144.03 + 5556.83 + 5145.21 + 5444.67 - 20000`

`NPV = 25920.36 - 20000`

`NPV = $5920.36`

The result is that the NPV for this project is $5920.36. This is a positive amount, so it looks like they should proceed with the project.

## See Also

For an easy way to calculate the Net Present Value, you can use the NPV Calculator.