NPV, IRR and payback: judging a project by its cash
A project spends money now to earn it later, so its worth depends on timing. Net present value turns that into one number, the internal rate of return turns it into a percentage, and payback says how long the money is at risk. This lesson builds all three, shows where they disagree, and reads Apple's lease payments as a borrowing with a rate of return.
- Level
- Beginner
- Reading time
- About 28 minutes
- Know this first
- Bond pricing and yield
By the end you will be able to:
- compute the net present value of a project and apply the accept-or-reject rule
- find the internal rate of return and explain what it means
- compute payback and discounted payback, and say what they leave out
- choose between projects when NPV and IRR disagree
Is it worth spending money now?
A business spends money before it earns any. A factory is built, a machine is bought, a shop is fitted out, and only then do the sales come. Whether to go ahead is a question about timing. The outlay is today and the benefits are scattered over years, so the amounts cannot just be added up. The tools of the last four lessons settle it: discount every cash flow to today and compare.
Three questions are asked of any project. Does it create value, once the cost of waiting is counted? What rate of return does it earn? How long does it take to get the money back? The first is answered by net present value, the second by the internal rate of return, and the third by payback. They are related, and they can disagree, and the disagreement is where careful judgment is needed.
Net present value
The net present value (NPV) of a project is the present value of all the cash it brings in, less the cash it costs. The discount rate is the return required for taking on a project of that risk, often called the hurdle rate or cost of capital. If the NPV is positive, the project returns more than the hurdle and adds value. If it is negative, it returns less, and the money would be better used elsewhere.
The rule is simple: accept projects with an NPV above zero. An NPV of $13,724 does not mean the project earns $13,724 in profit. It means that, after earning the required return on the money and getting it back, the project leaves value equal to $13,724 received today.
Test a project
Imagine spending a sum now on a project that pays cash each year. Set the outlay, the cash received in the first year, how much that cash grows or shrinks each year, the project's life, and the return you require. The left page shows the total cash and its present value, with the NPV at the bottom. The right page shows the internal rate of return, the profitability index, and two payback measures. All of it is made up for teaching and is not a recommendation.
Spending $100,000 for $30,000 in the first year, level, for 5 years, has a net present value of $13,724 at 10.0%. It clears the hurdle: the project earns about 15.24% a year, above the 10.0% required. Discounted, the outlay is recovered in 4.3 years.
Four experiments. First, raise the required return step by step. The NPV falls as the rate rises, and it reaches zero at 15.24%, which is the internal rate of return shown on the right. Above it the NPV is negative: at 20% it is −$10,282. Second, set the project life to 8 years. The NPV rises to $60,048, but the simple payback stays at 3.3 years, because payback ignores everything after the money is recovered. Third, make the cash shrink 10% a year. The NPV is −$4,997 and the outlay is never recovered once the cash is discounted. Fourth, set the project life to 4 years with the other defaults. Simple payback is still 3.3 years, comfortably inside the life, yet the NPV is −$4,904, negative. The money comes back, but not with the return that was required.
The NPV profile
NPV depends on the discount rate, and the table shows the default project, $100,000 for $30,000 a year for five years, at six rates. It falls as the rate rises. The rate where it crosses zero is the internal rate of return.
| Required return | Net present value |
|---|---|
| 0% | $50,000 |
| 5% | $29,884 |
| 10% | $13,724 |
| 15% | $565 |
| 20% | −$10,282 |
| 25% | −$19,322 |
At 0% the NPV is simply the total cash less the outlay, $50,000. At 15% it is $565, still just positive, and at 20% it is negative. The crossing lies between, at 15.24%.
Internal rate of return
The internal rate of return (IRR) is the discount rate at which a project's NPV is exactly zero. It is the project's own yearly rate of return, in the way that the yield to maturity is a bond's. The decision rule is to accept the project if its IRR is above the required return. For the default project the IRR is 15.24% and the required return is 10%, so it passes, consistent with its positive NPV.
There is no formula for the IRR of a project with several cash flows. It is found by trying rates until the NPV is zero, and spreadsheets and calculators do this. The calculator above uses 200 steps of halving the search range. Many people prefer the IRR to the NPV because it is a percentage, which is easy to compare with a borrowing rate. That is a real advantage, and it is also the source of its problems.
Payback and discounted payback
The payback period is the time taken for the cash received to add up to the outlay. For the default project, $100,000 ÷ $30,000 a year is 3.3 years. It is simple and tells you how long your money is at risk, and for that reason it is widely used as a rough screen. It has two defects. It ignores the time value of money, since it treats a dollar in year 3 as equal to a dollar in year 1. And it ignores everything after the payback date, so a project that returns the money in three years and then stops looks the same as one that continues for twenty.
The discounted payback repairs the first defect by adding up the discounted cash. For the default project it is 4.3 years against 3.3 years undiscounted. It is always at least as long as the simple payback, and for a project with a positive NPV it falls within the life of the project. It still ignores what happens after, so it is a measure of risk and liquidity, not of value.
When the tests disagree
NPV is the test that measures value created in dollars, so when the methods disagree, NPV is normally the one to follow. Two situations show why.
Projects of different size. Project A costs $10,000 and returns $14,000 in a year, an IRR of 40%. Project B costs $100,000 and returns $125,000 in a year, an IRR of 25%. At a required return of 10%, A has an NPV of $2,727 and B has an NPV of $13,636. The IRR prefers A. NPV prefers B, which creates five times as much value. If only one can be chosen, the larger dollar gain is what matters.
Cash flows that change sign more than once. Suppose a project costs $1,000 now, pays $5,000 in year 1, then needs $6,000 for clean-up in year 2. Such a project has two IRRs: 100% and 200%. At both rates the NPV is zero. At 150% the NPV is positive, so the usual rule of accepting when the IRR exceeds the hurdle breaks down. The check is simple: after any project with an unusual pattern, compute the NPV at the hurdle rate, which always has one answer.
| Discount rate | NPV of: -$1,000 now, +$5,000 in year 1, -$6,000 in year 2 |
|---|---|
| 0% | −$2,000 |
| 50% | −$333 |
| 100% | $0 |
| 150% | $40 |
| 200% | $0 |
| 300% | −$125 |
A related issue is the assumed use of the money received. NPV assumes the cash can be reinvested at the discount rate. IRR assumes it can be reinvested at the IRR itself, which is hard to believe when the IRR is 40%. NPV's assumption is more cautious and more realistic.
Apple: the rate behind a lease
Every borrowing has an IRR. If you receive a sum now and make a stream of payments, the rate that makes the present value of the payments equal to the sum is the interest rate you are paying, the project's IRR seen from the borrower's side. Apple's lease note lets us see this. Apple has received the use of property worth $12,430 million, valued at the present value of its lease payments, and it will pay $14,790 million in total over the following years. The payments for fiscal 2025 to 2029 are given, and the $6,400 million after that is not dated.
To find the IRR we have to assume when that last amount falls. The table spreads it evenly from fiscal 2030 to a final year, tries four different final years, and finds the rate at which the present value of all payments equals $12,430 million.
| Last year of payments | Years the later payments are spread over | Yearly amount after fiscal 2029 | Rate that matches the liability |
|---|---|---|---|
| Fiscal 2036 | 7 years | 914 | 3.37% |
| Fiscal 2038 | 9 years | 711 | 3.14% |
| Fiscal 2039 | 10 years | 640 | 3.03% |
| Fiscal 2040 | 11 years | 582 | 2.94% |
The rate lies between 2.94% and 3.37% over the whole range, and it is 3.14% if the payments end in fiscal 2038, the case from the lesson on present value. Apple's disclosed weighted-average discount rate is 3.1%. The two agree, which is what we should expect: a lease discount rate and the IRR of the lease payments are the same thing, with the unknown timing the only gap.
Seen as the cost of borrowing, the figure invites a comparison. Apple's notes issued in 2025 carry effective interest rates of 4.07% to 4.83%. Its commercial paper cost 4.19% at the end of fiscal 2025. The lease rate of about 3.1% is lower, probably because it is an average of rates set when each lease was signed, some of them in years when rates were lower, and because a lease is backed by the property itself. All three are the IRR of a different borrowing.
This is also how the hurdle rate for a project is set in practice. A project paid for with borrowing at 4.5% must earn more than 4.5% before it adds value, and in practice more again, to reward the owners for the risk they carry. The full cost of capital blends debt and the owners' own money, and is a topic of its own.
What these tests cannot tell you
- The cash flows are forecasts. Every test is only as good as the numbers put into it. A small error in the first-year cash or the life of the project can change the sign of the NPV.
- The discount rate is a judgment. Different rates for the same project give different answers, as the profile table shows.
- They leave out options. A project may open the door to later projects, or can be stopped early. Neither is in a simple NPV.
- Non-financial effects are missed. Safety, reputation and legal requirements often decide a project regardless of its NPV.
- Limited money. If funds are limited, the profitability index, the NPV per dollar spent, helps rank projects, but needs care.
Investment decisions at global scale
Every large company is making investment decisions all the time, and the totals are large. The table shows how much of its operating cash flow each of the three spends on capital investment in a year.
| Latest year, millions of each currency | Cash from operations | All capital spending | Capital spending as a share of cash from operations |
|---|---|---|---|
| Apple | $111,482 | $12,715 | 11.4% |
| Microsoft | $136,162 | $64,551 | 47.4% |
| Toyota | ¥3,696,934 | ¥4,903,731 | 132.6% |
Apple spent 11.4% of its operating cash flow on property and equipment, Microsoft 47.4%, and Toyota 51.6% on its own plants and equipment. Once the equipment that Toyota buys to lease to customers is added, the total is 132.6% of operating cash flow, which is consistent with its heavy borrowing. At these scales a wrong discount rate or an optimistic forecast changes billions, and net present value is the tool a company uses to check that each commitment is worth its cost.
One project, two currencies
A multinational often has to judge a project whose cash flows are in a foreign currency. The example is hypothetical. A project costs ¥15,000 million today and returns ¥5,800 million a year for three years. The company's required return in yen is 6.0%. The yen interest rate is 1%, the dollar rate 4% and the exchange rate ¥150 per dollar.
- In yen: discount at 6.0%. The net present value is ¥503 million, which is $3.36 million at the current exchange rate.
- In dollars: convert each flow at the forward exchange rate for its year and discount at the equivalent dollar required return of 9.15%, which is higher because dollar interest rates are higher. The result is the same $3.36 million.
- The mistake: discounting the yen flows at the dollar rate of 9.15% gives ¥−357 million, which turns a project that adds value into one that appears to destroy it.
The conclusion is practical. A required return belongs to a currency, and a company that uses one rate for every country will reject good projects in low-rate currencies and approve poor ones in high-rate currencies. The internal rate of return has the same problem: it should be compared with the required return in the currency of the cash flows.
Common mistakes
Following payback alone
A short payback can come with a negative NPV, as in the four-year experiment. Payback is a liquidity screen.
Choosing between projects by IRR
The project with the higher IRR can add less value. Compare NPVs for projects that exclude each other.
Discounting the wrong cash
Use cash that comes in and goes out because of the project, after tax. Interest paid on a loan is accounted for by the discount rate and should not be subtracted again.
Forgetting the outlay's timing
If the spending is spread over several years, discount each part. Treating all of it as today overstates the cost.
Using the IRR where cash flows change sign repeatedly
There may be no IRR, or several. Use NPV.
Try it yourself
1. A project costs $50,000 and returns $20,000 at the end of each of three years. The required return is 8%. Find the NPV.
Show the working
The present value of three payments of $20,000 at 8% is 20,000 × 2.5771 = $51,542. NPV = 51,542 − 50,000 = $1,542. It is positive, so the project clears the hurdle.
2. You spend $1,000 and receive $1,210 two years later, nothing in between. What is the IRR?
Show the working
(1,210 ÷ 1,000) to the power 1/2, minus 1, is 10.0%. With only two cash flows, the IRR is the same arithmetic as the compound annual growth rate.
3. A project costs $100,000 and pays $40,000 a year for 3 years. What is the payback? Why might it still be a poor project?
Show the working
Payback is 100,000 ÷ 40,000 = 2.5 years. But the project lasts only 3 years and returns $120,000 in total, so at a required return of 12% the NPV is 40,000 × 2.4018 − 100,000 = −$3,927, negative. A quick payback does not mean the required return was earned.
4. If Apple's lease payments were spread over a longer period than assumed, would the IRR needed to match the liability of $12,430 million be higher or lower?
Show the working
Lower. The same total of payments arrives later, so a smaller rate is needed to bring its present value down to the liability. The table shows this: the rate falls from 3.37% to 2.94% as the final year moves later.
Where this is used
- A company ranks proposals for factories, systems or acquisitions by NPV before it commits capital.
- A private equity or venture investor states an expected IRR for a fund or a deal.
- A bank or an analyst computes the IRR of a loan or a lease to find its true cost.
- A government compares public projects, such as a road or a power station, by discounting their costs and benefits.
Key terms
- Net present value (NPV)
- The present value of a project's cash inflows less its outlay, at the required rate of return.
- Internal rate of return (IRR)
- The discount rate at which a project's NPV is zero.
- Hurdle rate
- The minimum return a project must earn to be accepted, usually the cost of capital adjusted for the project's risk.
- Payback period
- The time taken for a project's cash receipts to equal its outlay.
- Discounted payback
- The time taken for the discounted cash receipts to equal the outlay.
- Profitability index
- The present value of cash received divided by the outlay. Above 1 means a positive NPV.
- Cost of capital
- The return that providers of money, lenders and owners together, require for funding a business.
- Mutually exclusive
- Projects of which only one can be chosen.
Questions people ask
Which is better, NPV or IRR?
NPV measures the value created and always gives a clear answer, so it is preferred for decisions. IRR is a useful summary percentage, but it can mislead when comparing projects of different size or timing.
What discount rate should I use for a project?
The return required for money put into something as risky as the project. For a company that is usually its cost of capital, raised for riskier projects.
Can the NPV be positive when the payback looks long?
Yes. A project with a slow start and long, steady returns, such as infrastructure, can have a long payback and a large NPV.
Is the IRR the same as the return on my investment?
It is a yearly rate that accounts for the timing of every cash flow, so it is a fuller measure than a simple profit-over-cost percentage. It assumes the projected cash flows are received as planned.
Sources and review
- Apple Inc., Form 10-K for the fiscal year ended 28 September 2024, filed with the US Securities and Exchange Commission. Source of the lease payments by year, the lease liability and the weighted-average discount rate.
- Apple Inc., Form 10-K for the fiscal year ended 27 September 2025. Source of the commercial paper rate.
- EDGAR Tools, Apple long-term debt disclosures, a data site that reproduces the debt note, for the effective interest rates on the 2025 notes.
- EDGAR Tools, Apple lease disclosures, a data site that reproduces the lease note.
- Wikipedia, Internal rate of return, for the definition, the multiple-IRR problem and the reinvestment assumption.
- AnalystPrep, Measures of profitability: NPV, IRR, AAR and PI, for the decision rules and the profitability index.
- Apple Inc., Form 8-K, Exhibit 99.1: fourth quarter and full-year fiscal 2025 results, filed with the US Securities and Exchange Commission. Source of Apple's statements.
- Microsoft Corporation, Form 8-K, Exhibit 99.1: fourth quarter and fiscal year 2025 results, filed with the US Securities and Exchange Commission. Source of Microsoft's statements.
- Toyota Motor Corporation, FY2025 Financial Summary (IFRS, unaudited, in millions of yen), furnished to the US Securities and Exchange Commission on Form 6-K. Source of Toyota's statements.
How the numbers were checked: the NPV and IRR calculator was tested against separate arithmetic for more than five hundred combinations of outlay, cash, growth, life and rate. At the IRR the NPV was shown to be zero. A positive NPV was shown to occur exactly when the IRR is above the required return. Discounted payback was shown never to be shorter than simple payback. The two-IRR example was solved by hand and by code. The lease IRR table was computed by repeated halving and each rate was checked by discounting the payments at it and matching the liability.