Internal rate of return (IRR)

Quick refresher

IRR is the root of the equation $\mathrm{NPV}(i) = 0$ , that is, the rate $i$ at which the equation holds. On the NPV profile (the graph of NPV against the discount rate), it is the point where the curve crosses the rate axis. For the project of the lessons on cash flow and NPV, IRR ≈ 29.6%.
IRR depends only on the project's cash flow: not on the company's rate and not on the time origin (the date to which all the flows are brought). We compare it with the required rate. For an ordinary project, IRR > $i$ exactly when NPV( $i$ ) > 0.
There is no formula for IRR. You find it numerically, step by step: by trial and error, by halving the interval or by Newton's method. The IRR function of a spreadsheet works in the same way, by iterations (repeated rounds that improve a guess).
A calculation at a single rate adds interest at the IRR to every intermediate balance of the “project account” (a running balance of the money between the project and the investor). If the money set aside for late costs earns less, the actual return is lower. We estimate it with the sinking fund method (the money for the late costs is set aside in a separate fund) or with the modified IRR (MIRR).
If the sign of the flow changes more than once, there can be several IRRs or none. Then we decide by NPV at the company's rate.
IRR ignores the size of the project and is sensitive to how the money is spread over time. Among mutually exclusive options (only one of them can be chosen), we choose by NPV. The IRR of the incremental flow (the flow of the extra investment) shows whether the extra investment earns the required return.

Key relationships

\sum_{t=0}^{T} \frac{\mathrm{NCF}_t}{(1+\mathrm{IRR})^{t}} = 0

definition of the internal rate of return

\mathrm{IRR} > i \iff \mathrm{NPV}(i) > 0

decision rule for an ordinary project (one change of sign in the flow)

i_{k+1} = i_k - \frac{\mathrm{NPV}(i_k)}{\mathrm{NPV}'(i_k)}

one step of Newton's method

B_t = B_{t-1}\,(1 + \mathrm{IRR}) + \mathrm{NCF}_t, \quad B_T = 0

at a rate equal to IRR, the project account ends at zero

\mathrm{MIRR} = \left(\frac{FV^{+}}{|PV^{-}|}\right)^{1/T} - 1

modified IRR, with explicit finance and reinvestment rates

Common pitfalls

Choosing, among mutually exclusive options, the one with the larger IRR.
Trusting the single number from the IRR function of a spreadsheet when the sign of the flow changes several times: the function returns the root to which the starting guess leads.
Comparing an IRR calculated on a nominal flow (in the money of each year) with a real rate (a rate for constant prices), or the other way round.
Taking a high IRR as a sign of a big gain: 50% on an investment of US$1 million creates less value than 20% on US$100 million.
Including loans and the interest on them in the flow: the result is the return on the owners' own capital, not the return of the project.

After this lesson you can

define IRR through NPV and through the “project account”, and explain why it depends neither on the discount rate nor on the time origin
find IRR by trial and error, by halving the interval and by Newton's method, and check the answer of a spreadsheet
explain what a calculation at a single rate assumes and how late costs, such as abandonment (closing the field down), affect IRR, and allow for them with the sinking fund method or the modified IRR
recognise a cash flow that has several IRRs or none, and decide by NPV
explain why ranking by IRR and by NPV can disagree, and choose between options by incremental analysis

IRR is the rate at which NPV is zero

The lesson on NPV shows that the NPV of an ordinary project (first the investment, then the receipts) falls as the discount rate rises. At some rate it becomes zero: the project's receipts and its investments, both discounted at this rate, are exactly equal. This rate is called the internal rate of return.

Internal rate of return

\mathrm{NPV}(\mathrm{IRR}) = \sum_{t=0}^{T} \frac{\mathrm{NCF}_t}{(1+\mathrm{IRR})^{t}} = 0

where:

$\mathrm{IRR}$ internal rate of return, fraction per year
$\mathrm{NCF}_t$ net cash flow at time $t$ : inflow “+”, outflow “−”, US$ million
$t$ time from the origin to the flow, years
$T$ time of the last flow of the project, including abandonment, years

Units: the currency of the cash flow; IRR is a fraction per period between the flows (a fraction per year for annual flows)

Holds when

The flow is the project's net cash flow after tax, without loans and the interest on them. Otherwise the result is the return on the owners' own capital, not the return of the project.
An ordinary flow (first minuses, then pluses: one change of sign) has exactly one root, and it is positive if the sum of the flow is greater than zero. If the sign changes several times, there can be several roots or none (see the section below).
All flows are placed at moments by one convention. Shifting all the flows in time by the same amount does not change IRR.

For an ordinary project, IRR is the highest rate at which the project still does not reduce the value of the company. It is the return that the money earns while it is invested in the project. Unlike NPV, IRR is a percentage, not a sum of money, and you do not need a discount rate to calculate it.

NPV of the example project at different rates, US$ million

Rate $i$: 0%
NPV: 210.4

Rate $i$: 10%
NPV: 100.1

Rate $i$: 20%
NPV: 36.9

Rate $i$: 25%
NPV: 15.5

Rate $i$: 30%
NPV: −1.3

Rate $i$: 40%
NPV: −25.3

Rate $i$	NPV
0%	210.4
10%	100.1
20%	36.9
25%	15.5
30%	−1.3
40%	−25.3

The same project as in the lessons Project cash flow and Net present value (NPV): the time origin is the middle of year 1, and the flows are in the middle of each year. The data in this example are assumed for illustration.

NPV changes sign between 25 and 30%, and the exact root is IRR = 29.57%. On the NPV profile this is the point where the curve crosses the rate axis (the figure in the lesson on NPV).

Why “internal”. IRR depends only on the project's own cash flow: you do not need a discount rate to calculate it, and the time origin does not change it. Shifting the origin by $\Delta$ years multiplies NPV at any rate by $(1+i)^{\Delta}$ , and zero multiplied by anything is still zero. For the same reason, IRR does not change if all the annual flows are moved from the middle of the year to the end: they all shift by the same time. So the IRR function of a spreadsheet gives the right IRR without the correction that the NPV function needs (see the lesson on NPV).

IRR > $i$ : the project pays back the investment with a return higher than required. NPV( $i$ ) > 0, and the project creates value.
IRR = $i$ : NPV( $i$ ) = 0. The project only just earns the cost of capital.
IRR < $i$ : NPV( $i$ ) < 0. It is better to put the money where the return $i$ is available.

This rule is correct for an ordinary project. The flow of the example project changes sign once more at the end, because of abandonment (closing down the wells and facilities) in year 10. But it still has only one positive IRR, so the rule applies to it (the reason is in the advanced section). The rate $i$ that IRR is compared with is called the hurdle rate. It is usually the company's discount rate, sometimes with an extra margin for risk (Choosing the discount rate). The difference $\mathrm{IRR} - i$ shows the margin in the rate. For the example project at a rate of 10%, it is about 20 percentage points.

How IRR is found

The equation $\mathrm{NPV}(\mathrm{IRR}) = 0$ is a polynomial of degree $T$ in $x = 1/(1 + \mathrm{IRR})$ (a sum of powers of $x$ with different coefficients). For three payments it is a quadratic equation. But for a flow five years long or more, there is no general formula for the root. So IRR is found numerically: you try a rate and improve it step by step. All the methods look for the rate at which NPV becomes zero. They differ in how they choose the next trial rate.

Trial with interpolation. Calculate NPV at several rates and find two neighbouring rates where NPV has different signs. Join the two points by a straight line (the example in the lesson on NPV). The profile curves like a bowl (it is convex), so the chord, the straight line between the two points, slightly overestimates IRR. We improve the estimate by narrowing the interval.
Halving the interval (bisection). If NPV has different signs at the two ends of an interval, take the middle point and keep the half whose ends still have different signs. Each step halves the interval: 20 steps give an accuracy of about one millionth of its width. The method is slow but sure: the root cannot be lost inside the interval.
Newton's method. The next rate is found from the tangent (the straight line that just touches the curve) to the NPV profile. The method gets to the root very fast, but it needs a good starting guess. It can jump to another root or to a meaningless rate.

One step of Newton's method for IRR

i_{k+1} = i_k - \frac{\mathrm{NPV}(i_k)}{\mathrm{NPV}'(i_k)}, \qquad \mathrm{NPV}'(i) = -\sum_{t} \frac{t\,\mathrm{NCF}_t}{(1+i)^{t+1}}

where:

$i_k$ rate at step $k$ , fraction
$\mathrm{NPV}(i_k)$ NPV at this rate, US$ million
$\mathrm{NPV}'(i)$ derivative of NPV with respect to the rate: the slope of the NPV profile, US$ million per unit of rate

Units: rates are fractions; NPV and its derivative are in the currency of the flow

Holds when

The NPV profile is smooth and not horizontal near the root.
The starting guess is close enough to the root. If there are several roots, the method finds the one to which the starting rate leads.

The tangent to the profile at the point $i_k$ crosses the rate axis at the point $i_{k+1}$ . Near the root, the number of correct digits roughly doubles with each step.

Worked exampleIRR of the project by Newton's method

Given (assumed data)

Flow of the project in years 1–10: −50.0, −90.0, 32.8, 78.4, 78.4, 65.6, 52.8, 33.6, 20.8, −12.0 US$ million (lesson Project cash flow)
Starting guess: 10%: this is the value that the IRR function of a spreadsheet takes by default

Solution

At $i_0 = 0.10$ : $\mathrm{NPV} = 100.13$ million dollars, and the slope of the profile is $\mathrm{NPV}' = -816.6$ million dollars per unit of rate.
First step: $i_1 = 0.10 + 100.13/816.6$ $= 0.2226$ , which is 22.26%.
At $i_1$ : $\mathrm{NPV} = 26.57$ , $\mathrm{NPV}' = -431.4$ . The second step gives $i_2 = 28.42$ %.
Next: $i_3 = 29.54$ %, $i_4 = 29.566$ %. At $i_4$ NPV is 0.000 million dollars, practically zero.

Answer: IRR = 29.57%. Four steps were enough to get it to within a hundredth of a per cent.

What it means. Each step replaces the NPV profile by its tangent. The profile of this project is convex, so from a rate below the root the tangents approach it from below and do not jump over it. The IRR and XIRR functions of a spreadsheet work in the same way, by iterations. If the calculation does not settle on a root, give it a different starting guess. The data in this example are assumed for illustration.

What IRR means: the project account

Imagine that all the project's flows go through an account on which the balance, of either sign, is multiplied by $1 + \mathrm{IRR}$ every year. Investment makes the balance negative: the project “owes” the investor. Receipts pay off this debt together with the interest. At a rate equal to IRR, the balance is exactly zero at the end of the project. The project has paid back all the investment and has paid IRR a year for the use of the money, no more and no less.

Balance of the project account at a rate equal to IRR

B_t = B_{t-1}\,(1 + \mathrm{IRR}) + \mathrm{NCF}_t, \qquad B_0 = \mathrm{NCF}_0, \qquad B_T = 0

where:

$B_t$ balance of the account after the flow at time $t$ ; a minus means that the investor's money is still in the project, US$ million
$\mathrm{NCF}_t$ net cash flow at time $t$ , US$ million
$\mathrm{IRR}$ internal rate of return, fraction per year

Units: the currency of the cash flow

Holds when

Interest is added at one rate, IRR, on a balance of either sign.
The flows are one period apart, and IRR refers to this period.

A negative balance is capital that is still invested in the project. IRR is the return on exactly this capital, not on the original sum of the investment over the whole life of the project: as money comes back, the invested capital gets smaller.

The project account at a rate equal to IRR (29.57%), US$ million

Year: 1
NCF: −50.0
Interest for the year: —
Balance at the end of the year: −50.0

Year: 2
NCF: −90.0
Interest for the year: −14.8
Balance at the end of the year: −154.8

Year: 3
NCF: 32.8
Interest for the year: −45.8
Balance at the end of the year: −167.7

Year: 4
NCF: 78.4
Interest for the year: −49.6
Balance at the end of the year: −138.9

Year: 5
NCF: 78.4
Interest for the year: −41.1
Balance at the end of the year: −101.6

Year: 6
NCF: 65.6
Interest for the year: −30.0
Balance at the end of the year: −66.1

Year: 7
NCF: 52.8
Interest for the year: −19.5
Balance at the end of the year: −32.8

Year: 8
NCF: 33.6
Interest for the year: −9.7
Balance at the end of the year: −8.9

Year: 9
NCF: 20.8
Interest for the year: −2.6
Balance at the end of the year: 9.3

Year: 10
NCF: −12.0
Interest for the year: 2.7
Balance at the end of the year: 0.0

Year	NCF	Interest for the year	Balance at the end of the year
1	−50.0	—	−50.0
2	−90.0	−14.8	−154.8
3	32.8	−45.8	−167.7
4	78.4	−49.6	−138.9
5	78.4	−41.1	−101.6
6	65.6	−30.0	−66.1
7	52.8	−19.5	−32.8
8	33.6	−9.7	−8.9
9	20.8	−2.6	9.3
10	−12.0	2.7	0.0

Interest is added to the balance of the previous year. A negative balance is the project's debt to the investor. The data in this example are assumed for illustration.

The project's debt grows until the end of year 3 (167.7 million dollars): the receipts of year 3 do not even cover the interest. Then the debt is paid off, 8.9 million dollars remain at the end of year 8, and after year 9 the balance becomes positive: 9.3 million dollars. The investor has already got everything back with a return of IRR. But abandonment is still ahead, 12.0 million dollars in year 10. The calculation quietly assumes that these 9.3 million will grow by 29.6% in a year and will exactly cover it: $9.26\cdot 1.2957 = 12.0$ .

Late costs and the assumption about the rate

For the capital invested in the project, adding interest at the rate IRR is the very definition of this rate. But for money that the project has already earned and holds while it waits for late costs, it is an assumption. In real life, the money set aside for abandonment earns the return of safe investments, not 30%. The same applies to any calculation at one constant rate: the intermediate balances of both signs are valued at this rate.

It is more realistic to set the money for abandonment aside explicitly. In the sinking fund method (in English-language literature such a calculation is also called the extended yield), we move the abandonment costs back in time by discounting them at the return $s$ of the fund. We pay for them out of the nearest earlier positive flows. These flows become smaller, and the late costs disappear. The IRR of such a “normalised” flow no longer assumes that the fund grows at the rate IRR.

Worked exampleIRR with an explicit abandonment fund

Given (assumed data)

Flow: the example project; abandonment 12 million dollars in year 10
Return of the abandonment fund $s$: 5% a year

Solution

To have 12 million dollars in year 10, we must set aside in year 9: $12/1.05 = 11.43$ million dollars.
The flow of year 9 (20.8 million dollars) covers this. It falls to 9.37 million dollars, and the flow of year 10 becomes zero.
The IRR of the normalised flow is 29.48%, against 29.57% for the original flow. If the fund earns nothing at all ( $s = 0$ ), it is 29.45%.

Answer: Allowing for the abandonment fund explicitly lowers IRR from 29.57 to 29.48%, by 0.09 percentage points.

What it means. For a small and distant abandonment, the effect is small: at a rate of about 30%, late flows weigh little. For an expensive abandonment, for example of offshore structures, the difference is larger and worth checking. And with very large late costs, the flow can get a second IRR (the next section). The data in this example are assumed for illustration.

Another way to replace a hidden assumption by an explicit one is the modified IRR (MIRR). The investments are discounted back to the start at a finance rate (the cost of the money that pays for them). The receipts are compounded forward to the end of the project (grown with interest) at a reinvestment rate (the rate at which they are put to work again). MIRR is the constant growth rate that turns the first amount into the second. The formula and an example are in the advanced section.

Several IRRs or none

An ordinary (conventional) project has one change of sign in its flow, from minus to plus, and its NPV profile crosses the rate axis once. If the sign changes several times, there can be several crossings (no more than the number of sign changes), or there can be none. Such flows are called non-conventional. In oil and gas projects they appear in acceleration projects (projects that bring production forward), with very expensive abandonment, and with large repeat investments in the middle of the project's life.

Worked exampleTwo IRRs of an infill well

Given (assumed data)

Incremental flow, years 0–7: −10, +11, +6, +4, −3, −3, −3, −3 US$ million
Where this flow comes from: the infill well (a new well drilled between existing ones) speeds up production: the neighbouring wells will later produce less than they would have without it
Company's rate: 10%

Solution

The sign of the flow changes twice: minus → plus → minus. The undiscounted sum is $\mathrm{NPV}(0) = -1.0$ million dollars.
NPV is 0.82 at 10%, 1.15 at 20% and −0.84 at 60% (million dollars): the profile first rises, then falls.
The roots of the equation $\mathrm{NPV} = 0$ , found by halving the interval on the ranges 0–20% and 20–60%, are 4.0% and 46.0%.
Newton's method from a starting guess of 10% arrives at 4.0%, and from 30% at 46.0%. From 20%, where the profile is almost horizontal, it jumps to a meaningless rate of −166%.

Answer: There are two IRRs, about 4.0 and 46.0%, and NPV at the company's rate of 10% is positive: 0.82 million dollars.

What it means. Neither of the two IRRs is the “return” of the well. The flow combines an investment (−10 now, then +21) with a loan from the future: the oil that the neighbouring wells would have produced later is received now and is “paid back” by lower production later. At low rates the loan is the costly part, at high rates the investment is. We decide by NPV at the company's rate. Newton's method from 10%, which is the default starting guess of the IRR function of a spreadsheet, gives a root of about 4%. So a project that creates value at 10% would wrongly be rejected on IRR. The data in this example are assumed for illustration.

CalculatedNPV profile of the incremental flow of an infill well: the curve crosses the rate axis twice, at about 4% and 46%. At the company's rate of 10%, NPV is positive.

Count the changes of sign in the flow. If there is more than one, IRR may not be unique (tests for this are in the advanced section).
Build the NPV profile over a wide range of rates, instead of looking for one root from one starting guess.
Decide by NPV at the company's rate. If you need a measure of efficiency in per cent, use MIRR or the sinking fund method: their answer is unique.

Sometimes there is no IRR at all. Take the flow “+5 now, −12 in a year, +8 in two years”: for example, an advance payment from a customer, then the contractor's costs and the final settlement. It has a positive NPV at any rate: the equation $\mathrm{NPV} = 0$ has no roots. Such a deal is profitable at any rate, but IRR is not defined for it.

A project of the ordinary kind can also get two IRRs if its late costs are very large. Suppose the abandonment of the example project cost not 12 but 230 million dollars. Its undiscounted total would become negative (−7.6 million dollars), and the NPV profile would cross the rate axis twice, at about 1.9 and 16.7%. And yet NPV at 10% is positive (7.7 million dollars): the rate “reduces the value” of the distant costs. This conclusion is right only if the money for abandonment can really be invested at this rate. That is again the question of the sinking fund.

IRR and NPV when choosing between options

For independent projects with ordinary flows, IRR and NPV give the same answer to “accept or reject”. They start to disagree when only one of several options can be chosen (the options are mutually exclusive), and when the options differ in the size of the investment or in how the money is spread over time.

Worked exampleHydraulic fracturing or a sidetrack

Given (assumed data)

Option A: hydraulic fracturing (cracking the rock around the well so that more oil flows): −3.0 now, then +2.2, +1.4, +0.8 and +0.4 in years 1–4, US$ million
Option B: a horizontal sidetrack (a new branch drilled sideways from the existing well): −8.0 now, then +2.8, +2.8, +2.4, +2.0, +1.6 and +1.2 in years 1–6
Condition: only one of the two can be carried out in the well; the company's rate is 10%

Solution

IRR: 30.7% for A and 18.1% for B. By IRR, A is better.
NPV at 10%: 1.03 for A and 1.70 for B (million dollars). By NPV, B is better.
The incremental flow “B minus A” for years 0–6 is −5.0, +0.6, +1.4, +1.6, +1.6, +1.6, +1.2 million dollars. Its IRR is 14.1%, and its NPV at 10% is 0.67 million dollars.
The NPV profiles of the two options cross at a rate of 14.1%, where the NPV of both is 0.78 million dollars. At lower rates, B has the larger NPV. At higher rates, A has.

Answer: At a rate of 10%, choose B: the additional 5 million dollars of investment earn 14.1%, which is more than the required 10%.

What it means. The high IRR of option A does not mean a bigger gain. It reflects a quick return of a small sum. For a comparison by IRR to be fair, we would have to assume that the early receipts of A can again be invested at 30%. In reality they will be invested at the company's rate. Incremental analysis answers the question directly: does the additional investment earn the required return? The data in this example are assumed for illustration.

CalculatedNPV profiles of option A (hydraulic fracturing) and option B (sidetrack). They cross at 14.1%. At the company's rate of 10%, B has the larger NPV, although A has the higher IRR (30.7% against 18.1%).

What NPV and IRR show

Question: Does the project create value at the company's rate?
NPV: yes, directly
IRR: yes, if the flow is ordinary: IRR is above the rate

Question: How much value does it create?
NPV: yes, in money
IRR: no

Question: Which of the mutually exclusive options is better?
NPV: the option with the larger NPV
IRR: only through the IRR of the incremental flow

Question: What rate can the project stand?
NPV: needs the NPV profile
IRR: yes: this is what IRR is

Question: Is the answer unique for a non-conventional flow?
NPV: yes
IRR: no: there can be several roots or none

Question	NPV	IRR
Does the project create value at the company's rate?	yes, directly	yes, if the flow is ordinary: IRR is above the rate
How much value does it create?	yes, in money	no
Which of the mutually exclusive options is better?	the option with the larger NPV	only through the IRR of the incremental flow
What rate can the project stand?	needs the NPV profile	yes: this is what IRR is
Is the answer unique for a non-conventional flow?	yes	no: there can be several roots or none

IRR is useful as a clear measure of the margin in the rate, for a first selection of independent projects, in negotiations (“the project can stand a cost of capital of up to 29%”) and for comparing projects of similar size and profile. When capital is limited, the NPV index (the NPV for each unit of money put in) shows the efficiency of investments better (Net present value (NPV)). Acceleration projects are covered in the lesson Acceleration projects and incremental analysis.

Terms and symbols

IRR: The discount rate at which the NPV of a project is zero; the return on the money while it is invested in the project.
Hurdle rate: The minimum return that is required, with which IRR is compared; usually the company's discount rate.
Ordinary flow: A cash flow with one change of sign, first investment and then receipts; it has one positive IRR.
Non-conventional flow: A flow whose sign changes more than once; there can be several IRRs or none.
Incremental IRR: The IRR of the difference between the flows of two options; it equals the rate at which their NPVs are the same.
Sinking fund method: A way to allow for late costs with money that is set aside in advance from the earlier receipts, at a stated return of the fund.
MIRR: The modified IRR, with stated rates for financing the investments and for reinvesting the receipts; it always has one answer.

Common mistakes and misconceptions

Wrong

“IRR is 30%, so all the invested money earns 30% a year over the whole period.”

In fact

IRR is the return on capital while it is invested in the project. As money comes back, the invested capital gets smaller. The calculation also assumes that the spare money that builds up along the way grows at the rate IRR. MIRR tests this assumption more realistically.

Wrong

“Of two development options, the one with the higher IRR is better.”

In fact

Among mutually exclusive options, we choose by NPV at the company's rate. IRR ignores the size of the project and favours options that return money quickly. We check the difference between the options by the IRR of the incremental flow.

Wrong

“The IRR function returned 4%, so the project does not pass the 10% rate.”

In fact

First check how many times the sign of the flow changes. A non-conventional flow can have several roots, and the function returns the one to which the starting guess led. We decide by NPV.

Wrong

“IRR, like NPV, has to be corrected when the time origin is moved, or when the flows are moved from the middle of the year to the end.”

In fact

A shift of all the flows by the same time does not change IRR: NPV is multiplied by one factor, and zero stays zero. Only the gaps between the flows matter. If they are unequal, use a calculation by dates.

Wrong

“The nominal IRR is 25% and the company's real rate is 10%, so the margin is huge.”

In fact

We can compare only quantities of the same kind: a nominal IRR with a nominal rate. With inflation of 5%, a real 10% corresponds to a nominal $1.10\cdot1.05 - 1 = 15.5$ %.

In depth: advanced detail

How many roots there can be

The equation $\mathrm{NPV} = 0$ is a polynomial in $x = 1/(1+i)$ with the coefficients $\mathrm{NCF}_t$ . By Descartes' rule of signs, the number of its positive roots (that is, of IRRs greater than −100%) is no larger than the number of changes of sign in the series $\mathrm{NCF}_t$ , and it differs from that number by an even number. One change of sign gives exactly one root. Two changes give two roots or none.

A finer test looks at the cumulative flow (the running total). If it changes sign exactly once, starting from negative values, then there is exactly one positive IRR. This condition is sufficient (it guarantees one root) but not necessary (a flow can have one root without meeting it). In the example project the sign of NCF changes twice, because of abandonment. But the cumulative flow (−50.0, −140.0, −107.2, −28.8, 49.6, 115.2, 168.0, 201.6, 222.4, 210.4) changes sign once, and there is one positive IRR. The second root, which the rule of signs says may exist, lies at a negative rate, about −69%, and has no economic meaning. For the infill well, the cumulative flow (−10, 1, 7, 11, 8, 5, 2, −1) changes sign twice, and there really are two roots.

The modified IRR

Modified internal rate of return

\mathrm{MIRR} = \left(\frac{FV^{+}}{|PV^{-}|}\right)^{1/T} - 1

where:

$FV^{+}$ sum of the positive flows, compounded to the time of the last flow at the reinvestment rate, US$ million
$PV^{-}$ sum of the negative flows, discounted to the time of the first flow at the finance rate, US$ million
$T$ number of periods from the first flow to the last, years

Units: the currency of the flow; MIRR is a fraction per period

Holds when

The positive flows are invested at the reinvestment rate until the end of the project.
The negative flows are covered in advance by money invested at the finance rate.
This is how the MIRR function of a spreadsheet defines it. Other variants exist, so state the rates next to the result.

MIRR always has one answer. If both rates are equal to the company's rate $i$ , then MIRR > $i$ exactly when NPV( $i$ ) > 0. The conclusion is the same as with NPV.

Worked exampleMIRR of the example project

Given (assumed data)

Flow: the example project, years 1–10 ( $T = 9$ )
Finance and reinvestment rates: 10% a year

Solution

The negative flows discounted to year 1: $PV^{-} = -50 - 90/1.1 - 12/1.1^{9}$ $= -136.9$ million dollars. Abandonment is included here too: the money for it is set aside in advance.
The positive flows of years 3–9, compounded to year 10: $FV^{+} = 558.9$ million dollars.
$\mathrm{MIRR} = (558.9/136.9)^{1/9} - 1$ $= 16.9$ %.
If the receipts can be invested only at 5% (the finance rate stays at 10%), MIRR falls to 14.2%.

Answer: MIRR ≈ 16.9% against an IRR of 29.6%. MIRR is the average growth rate of all the investor's money up to year 10, if the free receipts earn only 10%. IRR is the return only on the capital that is still invested in the project.

What it means. MIRR is above the company's rate, so the project is profitable, as NPV also says. Its value shows the growth of the investor's money more honestly if the investor can invest the free cash of the project only at 10%. The data in this example are assumed for illustration.

IRR in real and in nominal money

Suppose a nominal flow (in the money of each year) is made from a flow in constant prices by multiplying by $(1+f)^{t}$ , where $f$ is the rate of inflation. Then $1 + \mathrm{IRR}_{\text{nom}} = (1 + \mathrm{IRR}_{\text{real}})(1 + f)$ . This is the same relation as for interest rates (Interest and compounding). The IRR of the example project in constant prices is 29.6%. With inflation of 5% a year, the IRR of the same flow in nominal money would be 36.0%. We can compare IRR only with a rate of the same kind: real with real, nominal with nominal. Taxes are calculated in nominal money, so in a full model this simple relation can break down (Project cash flow).

Connections to other topics

Net present value (NPV) — the NPV profile, the decision rule and the NPV index when capital is limited.
Discounting and present value — discount factors and the conventions on the timing of flows.
Interest and compounding — compound interest, effective and real rates, the reinvestment rate.
Project cash flow — what the cash flow that IRR is calculated on consists of.
Acceleration projects and incremental analysis — acceleration projects, whose IRR is often ambiguous.
Choosing the discount rate — which rate to compare IRR with.
Sensitivity analysis and scenarios — how IRR and NPV respond to price, costs and timing.

Check yourself

1
The flow of a project is −100 now and +121 in two years. What is the IRR?
1. A.11%
2. B.10.5%
3. C.21%
4. D.10%
Show the answer
Correct answer: D. 10%
$(1+\mathrm{IRR})^{2} = 121/100 = 1.21$ , so IRR = 10%. 21% is the growth over two years, not a yearly rate. 10.5% is what you get by dividing 21% by two years, as with simple interest.
2
The NPV of an ordinary project at 12% is +8 million dollars. What can we say about IRR?
1. A.IRR is more than 12%
2. B.Nothing: IRR is not related to NPV
3. C.IRR is less than 12%
4. D.IRR is equal to 12%
Show the answer
Correct answer: A. IRR is more than 12%
For an ordinary project, NPV falls as the rate rises and becomes zero at IRR. Since NPV at 12% is still positive, zero comes at a higher rate.
3
A flow: −100 now, +60 in a year, +60 in two years. Find IRR and check it with the project account.
Show the answer
With $x = 1/(1+\mathrm{IRR})$ : $60x^{2} + 60x - 100 = 0$ , so $x = 0.8844$ and IRR ≈ 13.1%. The project account: $-100\cdot1.131 + 60 = -53.1$ ; $-53.1\cdot1.131 + 60 \approx 0$ . The debt is paid off exactly at the end of the second year. At a rate of 10%, the NPV of this flow is +4.1 (lesson on NPV), which agrees with an IRR above 10%.
4
Two mutually exclusive options: A has an IRR of 35% and an NPV at 10% of 4 million dollars; B has an IRR of 22% and an NPV at 10% of 9 million dollars. The company's rate is 10%. Which should we choose?
1. A.A: it has the higher IRR
2. B.B: it has the larger NPV at the company's rate
3. C.Both: they are both profitable
4. D.We need to know the payback period
Show the answer
Correct answer: B. B: it has the larger NPV at the company's rate
Only one option can be carried out, and we choose the one that creates more value at the company's rate: B. The higher IRR of option A shows a quick return of a smaller sum, not a bigger gain. If the incremental flow “B minus A” is ordinary (first an extra investment, then extra receipts), its IRR is above 10%, because the NPV of the incremental flow is $9 - 4 = 5 > 0$ .
5
A flow: +5 now, −12 in a year, +8 in two years. Find IRR.
Show the answer
With $x = 1/(1+i)$ : $\mathrm{NPV} = 8x^{2} - 12x + 5$ . The discriminant is $144 - 160 < 0$ , so there are no roots. The polynomial is positive at $x = 0$ and $x = 1$ , so NPV is above zero at any rate. IRR does not exist, but the deal is profitable at any discount rate.
6
Why does the IRR function of a spreadsheet need no correction for the time origin, while the NPV function does?
Show the answer
The NPV function discounts the first value of the range by a whole period, so its result depends on the moment to which the flows are assigned, and it has to be corrected. IRR is the rate at which NPV is zero. Shifting all the flows by the same time multiplies NPV by one positive factor, and zero stays zero. So the IRR function gives the same IRR for a series of annual flows with any time origin and with the mid-year convention.

Updated: 30 September 2026