Net Present Value (NPV) and Internal Rate of Return (IRR)
Deciding Whether to Invest
Imagine that your organization is considering buying a machine costing $450,000.
The machine is expected to yield $100,000 a year for five years, but your board decides not to go ahead with it. The investment is pretty much risk-free, the money is available, and the business is almost guaranteed a profit of $10,000 a year. So, why have they decided not to invest?
At first glance, investment decisions like these can seem to be "no brainers" – make the investment, and then collect the profit. But the value of money received in the future is less than the value of that money now (because of inflation and interest rates). So it's possible that the return from an investment might actually be worth less than the investment itself, once you have compared the value of money today, and in the future.
There are also other investment opportunities to consider. Would investing the money elsewhere provide a better return?
In this article, we'll look at three key approaches used in making investment decisions:
- Present value (PV).
- Net present value (NPV).
- Internal rate of return (IRR).
These approaches are the foundations of capital investment decisions. By learning about them, you can better appreciate why your organization makes certain investment decisions. You'll then be able to participate in financial discussions with more interest and more confidence.
Don't worry if you find some of the formulas in this article complicated. We have included them for reference, but you don't need to learn them to understand the key principles of PV, NPV and IRR.
If you're new to finance, you'll find some useful definitions in our article Words Used in Corporate Finance.
Present Value (PV)
Present Value (PV) is the value today of an amount of money that you'll receive in the future. You can use it to predict what the future returns from a potential investment are worth in "today's money."
The basis behind PV is that money "loses value" the further into the future you receive it – if you were offered $100 to take now or in a year's time, you would be wise to take it now, because you can invest the money today and start earning interest on it immediately. So, if the interest rate is 2 percent, and you had the $100 now, you'd have $102 in a year's time. But equally, $100 in one year's time would only be worth just over $98 in today's money, because you would expect to receive nearly $2 in interest during that year.
To calculate PV, you need a comparable interest rate, and the simplest one to use is often the rate you'd get by putting the cost of the investment in the bank.
For example, you may have an investment that's expected to return $15,000 one year from now. You also know that comparable investments are earning a 3 percent return. So, at a 3 percent rate of return, how much is that $15,000 worth today?
Here's the formula we can use to answer this question:
Present Value (PV) = C1/(1 + r)
C1 = Cash flow (return) after one year
r = One year rate of return for comparable, alternative investments, expressed as a decimal (i.e. 0.03 not 3 percent)
Using our example, we calculate PV as follows:
C1 = 15,000
r = 0.03 (3 percent)
15,000/(1 + 0.03) = 14,563.11
So the return has a present value of $14,563.11, meaning that $15,000 one year from now is worth the same to us as $14,563.11 in today's money.
Often, there will be cash flows every year for several years in the future, and we therefore have to adjust the PV equation to reflect this. Here we use the full present value formula for each future cash flow:
Present Value (PV) = Ct/(1 + r)^t
Ct = Cash flow (return) t years into the future
r = Rate of return expressed as a decimal
t = Number of periods in the future
If the cash flow in the example above was due in four years, we'd calculate PV as follows:
Ct = 15,000
r = 0.03 (3 percent)
t = 4
15,000/(1 + 0.03)^4 = 15,000/1.13
15,000/1.13 = 13,327.31
So the return has a present value of $13,327.31, meaning that $15,000 four years from now is worth $13,327.31 in today's money.
The rate of return used to calculate the discount rate is often called the opportunity cost of capital. An opportunity cost is the cost of an investment, compared to another investment. If you purchase investment A, then you don't purchase investment B – so the opportunity cost is the difference between the actual return of investment B versus the actual return of investment A.
Net Present Value
The Net Present Value (NPV) of an investment is the difference between what the investment costs you, and its present value. NPV is the net contribution it makes to the overall value, and helps you decide if an investment is likely to give a sufficiently large return to be worth pursuing.
We calculate NPV as follows:
NPV = PV – I
PV = Present Value
I = Cost of investment
In our simplified one-year example above, let's say that to get the $15,000 return in one year, we have to buy equipment that costs $14,000. Then:
NPV = 14,563.11 (the PV, see earlier) – 14,000.00 (the investment)
NPV = 563.11
So the investment would actually provide a net increase in value of $563.11.
The formula for the NPV in this case is therefore:
NPV = (C1/(1 + r)^t) – I
NPV = (15,000/(1 + 0.03)^1) – 14,000
NPV = 563.11
In our example, we would probably consider making the investment of $14,000 as it has the potential to return a surplus of $563.11 in today's money, compared to investing the money in the bank.
To calculate NPV for investments that give returns over several years, calculate the PVs for each year, and add these up. Then subtract the initial investment.
Here, the NPV formula looks like this:
NPV = ∑ (Ct/(1 + r)^t ) – CØ
CØ = Initial cash (out)flow
Ct = Cash flow for time period
t = Time period (year number)
r = Interest rate in the time period
(The ∑ sign means "the total of all", so here it's telling you to add together the present values for each of the future years for which you'll get a return.)
Overall, when the projected NPV is greater than zero, the investment will be profitable. When the NPV is less than zero, the investment won't make a profit. Of course, these calculations assume that your estimated cash flows are reasonably close to reality!
In these examples, we've assumed that the rate of return remains constant over the life of the investment. In the real world, some investments have a term structure of interest rates, but we've used one interest rate to keep things simple in this article. Similarly, we've assumed that all investment is up-front. Where this isn't the case (and this is often true) you'll need to base your NPV on net cash flows for each period.
Net Present Value Example
You have an investment opportunity that will pay $1,000 per year for the next three years. The initial cost of the investment is $2,700, and a comparable interest rate is 3.5 percent. Should you make the investment?
Calculate each annual cash flow for the project in today's money by using the PV formula:
Year 1 = PV = (C1/(1 + r)^1) = 1,000/1.0350 = 966.18
Year 2 = PV = (C2/(1 + r)^2) = 1,000/1.0712 = 933.51
Year 3 = PV = (C3/(1 + r)^3) = 1,000/1.1087 = 901.94
Add all of the PVs from each of the three years:
Total PV = 966.18 + 933.51 + 901.94 = 2,801.63
Work out the NPV:
NPV = 2,801.63 – I
NPV = 2,801.63 – 2,700
NPV = 101.63
Since the NPV is positive at $101.63, it may be worth making the investment. However, as it's only slightly positive, you should carefully check how robust your assumptions about expected revenues and the comparable interest rate are. A small, unfavorable change in these could easily wipe out your positive return.
Always remember that PV and NPV assessments are only as good as the accuracy of the estimates, and changes in interest rates can also affect the outcome significantly.
Internal Rate of Return
Another common investment assessment approach is to calculate the Internal Rate of Return (IRR), which is also called the Discounted Cash Flow method.
Essentially, the IRR is the rate at which the NPV of an investment equals zero.
When you calculate IRR, you treat it as a cut-off point for investment decisions. For example, your organization may specify that it will only run projects if they exceed a "hurdle rate" of 11 percent. Some people refer to the IRR as the break even interest rate.
IRR is expressed as a percentage – if it's greater than the rate of return on other investments, the investment may be worthwhile.
This method is popular with investors who have a certain rate of return they want to achieve with their investments. A quick calculation shows what rate a particular investment will return over a specific time period.
Working from the NPV formula, r becomes IRR. And when we set NPV = 0, we get the following:
0 = ∑ (Ct/(1 + IRR)^t) – I
∑ (Ct/(1 + IRR)^t)) = I
Calculating the value of the IRR can clearly become very complicated! However, you can set up a spreadsheet, or use a free online IRR calculator like this one to work out IRRs painlessly.
Internal Rate of Return Example
You've found an opportunity that will pay $3,500 for each of the next three years on a $10,000 investment. Other opportunities are providing a rate of return of 3 percent for the same investment. So, is this a worthwhile project in which to invest? Or should you invest in one of the other opportunities?
Using an IRR calculator, you discover that this project has an IRR of 2.48 percent. It would therefore be better to invest the $10,000 in one of the other opportunities rather than this one.
IRRs are particularly useful when used in conjunction with NPVs. IRRs help you understand the rate of return of the investment, while NPVs help you assess the absolute size of the return.
This article explains IRRs and NPVs on a very simple level. In practice, these calculations can be extremely complex, and your organization will probably have standards that you need to follow to assess project return. Get the help of your finance department to make sure that any project evaluation that you carry out conforms to these rules.
For much, much more on this topic, take a look at Principles of Corporate Finance by Richard A. Brealey and Stewart C. Myers. This is the standard textbook on evaluating projects from a financial perspective.
Valuing investments is a key part of financial decision making, and understanding the time value of money is key to this process.
While you may never have the need or opportunity to calculate present value, net present value, or internal rate of return, understanding what the terms mean and why they're important can be very useful.
Financial decisions and capital projects rely on investing money wisely. The tools discussed here provide a starting point for making those crucial decisions.
There are more exercises on calculating present value and IRR in our Bite-Sized Training session on Project Evaluation and Financial Forecasting.