How NPV and IRR work

Net present value and internal rate of return look like rival tests. They are not. Both start from the same list of dated cash flows and the same present-value sum. What changes is which side of the equation you hold still.

NPV fixes a discount rate $r$ and solves for a value in today's money:

$$NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$$

IRR fixes that value at zero and solves for the rate:

$$0 = \sum_{t=0}^{n} \frac{CF_t}{(1 + IRR)^t}$$

That is one equation. Ask it for a number of dollars and you have NPV. Ask it for a rate and you have IRR. The calculator above does the first. The IRR calculator does the second on the same series.

The ordinary shape is money out at the start and money in afterwards: a conventional cash flow series with one sign change. For that shape the two tests cannot disagree on accept or reject. Discount below the IRR and NPV is positive. Discount above it and NPV is negative. At the IRR, NPV is zero by construction. So "accept when NPV is above zero at your hurdle rate" and "accept when IRR is above your hurdle rate" are the same sentence, spoken in money and in percent.

A machine that costs $10,000 today and returns $3,000 at the end of each of the next five years is the series this page works throughout. At 8 percent its NPV is $1,978.13, so it clears an 8 percent hurdle. Its IRR is 15.24 percent, so it also clears that hurdle read as a rate. Those are not two facts. They are one fact stated two ways.

Turn the series around, money in first and out afterwards, and the picture inverts. That shape is a borrowing rather than an investment. Its NPV rises as the discount rate rises, and the rule becomes accept when your cost of capital sits above the IRR rather than below it. A rate quoted without the shape of the series does not tell you which case you are in. Write the signs down first.

Ranking several projects is not that test. A rate is silent on size, timing and how long the money is tied up. The NPV vs IRR comparison is the at-a-glance version of the split. The engine underneath is the time value of money: discounting is compounding run backwards.

The sum starts at $t = 0$ for a reason. At time zero the divisor is $(1+r)^0 = 1$. Money that moves today is already in today's units. It is not discounted, not because it is special, but because there is no wait to price.

That is the outlay. It enters the sum at full size, with a minus sign. Discount it anyway, at 8 percent, and you shrink the cost by about 7.4 percent. Every dollar taken off the cost lands straight in NPV, so on the machine above the surplus would swell by about 37 percent: enough, on a close project, to flip a rejection into an acceptance. Year zero means no discounting at all.

The receipts are the opposite. Each is divided by $(1+r)^t$, once for every year of waiting. At 8 percent the factor for year 1 is $1/1.08 = 0.9259$, for year 2 it is $1/1.08^2 = 0.8573$, and by year 5 it is down to $0.6806$. The same $3,000 counts for less every year further out.

Add the five present values and the receipts are worth $11,978.13 today. The cost is at year zero, so it is not discounted. Subtract it whole: $11,978.13 minus $10,000 leaves an NPV of $1,978.13. Every year of cash has already been charged 8 percent for the waiting, and a surplus is still left in today's money, so the machine clears the bar.

When every year pays the same amount there is a shortcut, the annuity factor $\frac{1 - (1+r)^{-n}}{r}$. At 8 percent over five years that factor is 3.992710, and $3,000 times 3.992710 is the same $11,978.13 the year-by-year sum produces. Round the factor to 3.9927 first and the product comes out three cents light, which is why the digits you carry are part of the answer.

Cash is assumed to arrive at the end of each year. Money arriving at the start of a year is discounted one period less. Write the series out with a year number against every figure, year zero included, before you divide anything.

The $r$ in the NPV formula is an argument you supply. It is not a market quote the project comes with. It is the return the same money could earn in its next best use at similar risk, which is why two people can study one project, do the arithmetic correctly, and still disagree.

A company often takes that rate from its weighted average cost of capital, which the WACC calculator works out from the mix of equity and debt. A person might use the rate on a debt they would otherwise pay down. Riskier cash flows deserve a higher rate, and the choice matters more the longer the project runs. The cost of capital is the long form of that choice.

Raise the rate and NPV falls on any project that spends first and collects later, because every future receipt is divided by a bigger number, and the far years take most of the damage. The machine worth $1,978.13 at 8 percent is worth $814.33 at 12 percent, less than half as much, without a cent of the cash flows changing. At 12 percent the five receipts are worth $10,814.33 today, so the room above the $10,000 cost is much thinner, and year 5 is now being divided by $1.12$ five times over.

NPV is a function of the rate. Push the argument up and the value falls, crosses zero at the IRR, and turns negative beyond it. That crossing is not a second test. It is the same curve read at a different point.

The exception is a project carrying a large cost in a late year, a mine to fill in or a plant to strip out. The higher rate discounts that cost too, so it can lift the NPV rather than cut it.

Match the rate to the cash flows. Amounts written in today's prices need a rate with inflation stripped out; amounts that already include price rises need a rate that still contains it. Mix the two and you charge for inflation on the way out without adding it on the way in. The real return calculator converts between the two versions of a rate.

IRR is defined by an equation rather than computed by one. For the machine you are looking for the rate $r$ at which the discounted receipts add back to the outlay:

$$10000 = \sum_{t=1}^{5} \frac{3000}{(1+r)^t}$$

There is no rearrangement that isolates $r$ for this series. Substitute $x = 1/(1+r)$ and the equation becomes an ordinary polynomial of degree 5 in $x$. The Abel-Ruffini theorem says a general polynomial equation of degree 5 or higher has no solution in radicals: no finite combination of roots, plus, minus, times and divide that works for every such equation the way the quadratic formula works for every degree 2 equation. Degree 5 is not an exotic case. It is five yearly receipts after an outlay, the series this page opened on.

Shorter series can have closed forms. Degree 2, 3 and 4 have general radical formulas, unused in practice because searching is a better way to get a number. The one shape worth writing down is a single amount out and a single amount back, which collapses to a growth rate: the CAGR calculator is that special case, and the last worked example on this page is exactly that shape.

So in practice every IRR is found by searching. Guess a rate, price the series, adjust. At 10 percent the five receipts of $3,000 are worth $11,372.36 today, more than the $10,000 outlay, so 10 percent is too low and the NPV is $1,372.36. At 20 percent the receipts are worth less than the outlay, so 20 percent is too high. At 15 percent the NPV is a little above zero; at 16 percent it is a little below. Keep halving the gap. It settles at $r = 0.1524$, which is 15.24 percent a year, or 15.238237 percent before rounding. At the unrounded rate the NPV is 0 by definition.

Against a 10 percent cost of capital the same project is worth $1,372.36 in today's money. The IRR of 15.24 percent and that positive NPV are one fact: the project beats 10 percent. For a conventional series, accept when the IRR is above your cost of capital. That is the NPV rule, spoken in percent.

IRR breaks as a ranking tool on size and on the reinvestment reading, and it can fail as a number when the series changes sign more than once.

Several roots, or none. A polynomial of degree $n$ has up to $n$ roots. Descartes' rule of signs bounds how many of them are positive: at most the number of sign changes in the cash-flow series, and short of that only by an even number. One sign change leaves one positive root, so a conventional project has one IRR above minus 100 percent.

Series that change sign more than once are ordinary. A mine restored at the end of its life, a plant needing a mid-life rebuild, a job paid in advance that then spends and collects at handover: the money turns over twice. Two sign changes allow two rates that each set NPV to zero, both genuine, with nothing in the mathematics to rank them. The same shape can leave no real rate at all. NPV still returns one number, because it evaluates an expression rather than searching for a root. A non-conventional series is a question to put to NPV.

Size. A rate says nothing about how much money is at stake. Scale every cash flow by two and NPV doubles while IRR does not move. A 15.83 percent return on $5,000 is not more money than a 15.24 percent return on $10,000. At a 10 percent cost of capital the larger project is worth $1,372.36 in today's money. Rank by IRR and the small project wins. Rank by NPV and the large one wins, because NPV is denominated in money. The last worked example is that smaller series: $5,000 out today, $9,000 back at the end of year four.

Reinvestment. The usual complaint is that IRR assumes every interim receipt is put back to work at the IRR itself, while NPV assumes the same at the discount rate. Neither assumption sits in the arithmetic. Both measures are present-value sums, and discounting states an equivalence between money at two dates, not a forecast about where a receipt ends up.

What is true is narrower. Read an IRR as the compound return you will actually earn across the whole life of the project and you have made that reinvestment assumption yourself. If the project earns 15.24 percent and everything else you own earns 8 percent, treating 15.24 percent as a holding-period yield flatters it. The modified internal rate of return, MIRR, makes the assumption explicit: outflows discounted at your cost of capital and inflows compounded at a reinvestment rate you name.

Rates also do not add. Two NPVs of independent projects can be summed. Two IRRs cannot be averaged into anything meaningful unless the outlays and the lives happen to match, which they rarely do.

For a single conventional project judged on its own, the two measures agree and the choice between them is which unit you want to read. Disagreement on ranking surfaces once you are choosing between things.

Three questions decide whether the rankings can split. Are the projects mutually exclusive, so that taking one rules out the other? Do they differ in size? Do they differ in timing or in length? Any yes, and IRR is silent on all three. Independent projects stay easy: take every one whose NPV is positive at the rate your money costs, which is the same list as those whose IRR beats that rate, provided each series is conventional. Conflict starts when you can take only one of them.

The split has a location. The crossover rate is the discount rate at which two projects have equal NPV. For a pair whose profiles cross once, that rate sits below both IRRs. Above the crossover the project with the higher IRR also has the higher NPV, so the rankings agree. Below it they part, and the higher rate belongs to the project worth less. IRR does not misrank at random: it ranks the pair the way NPV would if your discount rate sat above the crossover, whatever rate you actually face.

When they split, follow the NPV. NPV measures what a decision adds to your wealth at the rate you actually pay for money, and a rate cannot be spent. The 15.83 percent project puts $5,000 to work; the 15.24 percent project puts $10,000 to work. At 10 percent the larger one adds $1,372.36 in today's money. Doubling every cash flow of either project would double that contribution and leave the IRR where it was.

Where capital is genuinely rationed, so that you cannot take every project worth taking, what maximises value is NPV per unit of the scarce resource rather than NPV alone. That is still an NPV rule, and it is still not an IRR rule.

IRR earns its keep even so. It states a return without a discount rate having to be agreed first, and it measures headroom: a project whose IRR sits 5.24 points above a 10 percent hurdle can absorb just over five points of increase in that rate before the decision turns over. Compute both, decide on the NPV, quote the IRR.

The calculator above is the NPV tool, because ranking is the decision this page is for once accept or reject has been settled. For the rate on a single series, use the IRR calculator. For the hurdle a company actually faces, use the WACC calculator. For the side-by-side in a table, see NPV vs IRR.