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Internal Rate of Return Calculator

Find the yearly return an investment's cash flows work out to, the rate where their net present value is zero. See every IRR, an NPV chart and an optional modified IRR.

Level: Grade 12 to college

  • Common Core standard HSA.REI.D.11Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Printable worksheet

Internal Rate of Return Calculator

Know the answer you need? Pick an input to work backward.

Inputs

Net cash for period 1, 2, 3 … in order, separated by commas. Use a minus sign for a period with a net outflow.

Each period is a

Optional. The rate the project's incoming money can earn once you receive it. Turns on the modified IRR.

Result

Internal rate of return (per year)

16.34%

The cash flows return about 16.34% per year: discounting them at that rate makes the NPV zero.

Number of IRRs found
1
Undiscounted net cash flow (NPV at 0%)
$4,000.00
NPV profile: NPV at each discount rate
$0$1K$2K$3K$4K01020304050

Discount rate per year (%)

See the step-by-step solution ↓

NPV profile (the IRR is where the NPV crosses zero)
NPV profile (the IRR is where the NPV crosses zero)
Rate per year (%)NPV
0$4,000.00
5$2,540.76
10$1,307.29
15$255.61
20-$648.15
25-$1,430.40
30-$2,111.97
35-$2,709.44
40-$3,236.15
45-$3,702.90
50-$4,118.52

Step-by-step solution

  1. Look for the rate r where the marked-down cash flows add up to exactly your investment, so the NPV is zero. The flows switch sign 1 time, so there can be at most 1 such rate

    NPV(r)=−I + Σ CFₖ ÷ (1 + r)^k=0

  2. No formula gives r directly, so search for it. Find two rates where the NPV switches sign, then close in between them until the NPV is zero

    $3,000.00(1+0.1634056)1+$4,200.00(1+0.1634056)2+$6,800.00(1+0.1634056)3−$10,000.00=$0.00

  3. The rate that does this, as a percent per year

    IRR=16.3406%

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

Please note: Results are estimates for planning and comparison. Lenders, banks, and tax authorities may round differently or include fees not modeled here, so confirm figures with your provider before deciding.

How to solve this on a calculator

The keys to press to get this page's worked-example answer on an ordinary calculator, with what the display should show after each step. A named-model tab follows that calculator's official guidebook.

BA II Plus

Internal rate of return with the BA II Plus cash flow worksheet

Problem
A project costs $10,000 today and returns $3,000, $4,200 and $6,800 at the end of the next three years. What is its IRR?
Formula
IRR: the rate r where −10,000 + 3,000 ÷ (1 + r) + 4,200 ÷ (1 + r)² + 6,800 ÷ (1 + r)³ = 0
Example values
CFo = −10,000 (initial cash flow (paid out, so negative)); C01, C02, C03 = 3,000; 4,200; 6,800 (cash flows at the end of years 1–3, each once (F = 1)). These are the fixed values from the worked example on this page, so the keys stay the same whatever you type into the calculator above.
Why the BA II Plus
The BA II Plus finds the IRR from the cash flow worksheet by searching for the rate that makes the NPV zero.
The whole sequence
CF2ND[CLR WORK]10000+|-ENTER↓3000ENTER↓↓4200ENTER↓↓6800ENTERIRRCPT

Step by step

  1. CF2ND[CLR WORK]

    Press CF to open the cash flow worksheet, then 2ND [CLR WORK] to clear any cash flows left from before.

    Display: CFo=0.00 (our simulation; the guidebook doesn't show this screen)

  2. 10000+|-ENTER

    10000 +|- ENTER: the cost is paid out today, so CFo is negative.

    Display: CFo=-10,000.00

  3. ↓3000ENTER

    ↓ to C01; 3000 ENTER.

    Display: C01=3,000.00

  4. ↓

    ↓ shows F01 = 1: C01 happens once. Leave it.

    Display: F01=1.00

  5. ↓4200ENTER

    ↓ to C02; 4200 ENTER.

    Display: C02=4,200.00

  6. ↓↓6800ENTER

    ↓ passes F02 (leave it at 1), ↓ to C03; 6800 ENTER.

    Display: C03=6,800.00

  7. IRR

    IRR shows the internal rate of return variable; it hasn't been computed yet.

    Display: IRR=0.00

  8. CPT

    CPT searches for the rate: about 16.34% a period (a year here).

    Display: IRR=16.34

If your answer looks wrong

  • Sign convention: CFo is paid out, so negative. If every cash flow has the same sign (CFo keyed in as +10000), there is no IRR and CPT shows Error 5: press CE|C and key CFo in with +|-.
  • P/Y and BGN don't apply here: the IRR is per cash flow period (a year here), whatever P/Y is, and END/BGN doesn't affect this worksheet. For monthly cash flows the IRR is a monthly rate.
  • Cash flows left from an earlier problem change the answer: with an old C04 of 500 still there, CPT shows IRR=17.76. Press 2ND [CLR WORK] inside the CF worksheet first.
  • If the cash flows change sign more than once, there can be several IRRs; the calculator shows the one closest to zero, so check the NPV at your required rate too.
  • Decimals: these displays are at 2 decimal places. If yours shows more or fewer, check DEC with 2ND [FORMAT].

Checked against the official guidebook, pp. 2, 4, 37–38, 40–44, 79: BA II PLUS™ Calculator (guidebook) (English PDF, copyright © 2011–2025 Texas Instruments Incorporated (as printed on page ii)). Every key and setting above follows those pages; the instructions are written in our own words.

How this is checked: an automated test replays these keys on this site's simulation of the BA II Plus, built from that guidebook, checks every display along the way, and confirms the answer matches the calculator on this page (last checked 2026-10-19). It has not been tried on a physical BA II Plus. Displays are shown at 2 decimal places, the setting after a reset; the calculator's own display also has small status indicators that aren't shown here. Second functions are written in brackets: press the second-function key (2ND, or 2nd on some models), then the key with that name printed above it.

Calcapedia is independent and is not affiliated with or endorsed by Texas Instruments. TI and BA II Plus are trademarks of their owners and are used here only to say which calculator these steps are for.

What this calculator does

Find the yearly return an investment really earns, based on the money you put in and the money you get back. That return is the internal rate of return, or IRR. It's the one interest rate that the project's cash flows act like. Future money is worth less than money today, so each future amount gets marked down at that rate (this is called discounting). At the IRR, the marked-down amounts add up to exactly what you invested. In other words, the net present value (NPV), what the project is worth in today's money after its cost, is zero. That makes the IRR easy to compare with a target return, a loan rate or another investment.

Enter the starting investment and the cash flow for each later period. The calculator finds the IRR with a careful step-by-step search and shows the working. It turns quarterly or monthly results into yearly ones. It also draws the NPV profile, a chart showing where the NPV crosses zero. Sometimes cash flows switch between money in and money out more than once, and there's more than one IRR. Then the calculator reports them all, instead of quietly picking one. Add a reinvestment rate to get the modified IRR too.

Where it's used

Why this math matters outside the classroom:

  • Project hurdle rates. A company accepts a project only if its IRR beats a 12% hurdle, after checking the cash flows change sign just once.
  • Private investments. An investor compares a fund's IRR on its yearly capital calls and payouts with the return on a stock index fund.
  • Comparing loans and leases. Finding the rate that equates a lease's payments with a purchase price reveals the lease's implied interest rate.
  • Algebra by approximation. Students find the rate where −100 + 60 ÷ (1 + r) + 60 ÷ (1 + r)² = 0 by narrowing a table of values, landing near 13.07%.
  • Real estate returns. A landlord computes the IRR of a purchase, five years of net rent and a final sale to compare with a savings account.

Grade level and standards: Grade 12 to college; Common Core State Standards for Mathematics HSA.REI.D.11. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Enter the starting investment you pay today, as a positive number.
  2. List the net cash flow for each later period, in order, separated by commas. Use a minus sign for any period where more money goes out than comes in.
  3. Choose whether the periods are years, quarters or months. If you like, enter a reinvestment rate and a finance rate for the modified IRR.
  4. Read the IRR and the steps. Then check the NPV profile table: the IRR is where the NPV switches from positive to negative.

What the inputs mean

Initial investment (paid today)
The cost you pay at the start (time 0). Enter 0 if the first cash flow in your list is the money you put in.
Cash flows, one per period
The net cash for each period, starting with the end of period 1. Every period needs a number, even if it's 0. That keeps the timing of later flows right.
Each period is a
The IRR is a rate per period. For quarterly or monthly cash flows, the calculator also turns it into a yearly rate, with compounding (growth on top of growth).
Reinvestment rate for MIRR (per period)
Optional. The return the project's incoming money can realistically earn once you receive it. The standard IRR quietly assumes it earns the IRR itself. That can make high-IRR projects look better than they are.
Finance rate for MIRR (per period)
Optional. The rate used to discount the money going out when working out the MIRR, such as your cost of borrowing. Leave it blank to use the reinvestment rate.

Formula and method

0 = −I + CF₁ ÷ (1 + IRR) + CF₂ ÷ (1 + IRR)² + … + CFₙ ÷ (1 + IRR)^n
annual IRR = (1 + IRR per period)^p − 1
MIRR = (FV of inflows at the reinvestment rate ÷ PV of outflows at the finance rate)^(1/n) − 1

I is the starting investment, and CFₖ is the cash flow at the end of period k. The letter n is the number of periods, and p is the number of periods in a year. FV means future value, what money grows to later. PV means present value, what it's worth today. MIRR is the modified IRR. Once there are more than a few periods, there's no formula that solves for the IRR directly. So it has to be found by searching.

First, the calculator rewrites the NPV using x = 1 ÷ (1 + r), where r is a trial rate. That turns it into a polynomial, a sum of powers of x. Then it checks a wide range of rates, from −99.99% to 1,000,000% per period. It looks for every place the NPV switches sign. It narrows each one with Brent's method, a root-finder that mixes safe halving with faster guessing. The answer is accurate far beyond the decimals shown.

The number of IRRs can't be more than the number of times the cash flows switch sign. This is known as Descartes' rule of signs. Ordinary projects, with money going out first and coming in after, switch sign once and have one IRR. A project with a large cost at the end, such as a cleanup, can have two. Then the IRR isn't a reliable way to rank projects.

Worked example

A project costs $10,000 today. It returns $3,000, $4,200 and $6,800 at the end of the next three years.

  1. The flows switch sign once (one outflow, then inflows), so there is at most one IRR.
  2. At 15% the NPV is $255.61, and at 20% it is −$648.15. So the IRR lies between them.
  3. Narrowing the search gives r ≈ 16.3406%, where $3,000 ÷ 1.163406 + $4,200 ÷ 1.163406² + $6,800 ÷ 1.163406³ − $10,000 = $0.00.
  4. Modified IRR with a 10% reinvestment rate: the inflows grow to $3,000 × 1.1² + $4,200 × 1.1 + $6,800 = $15,050 by year 3. So MIRR = (15,050 ÷ 10,000)^(1/3) − 1 ≈ 14.60%.

Result: The IRR is about 16.34% a year. With the incoming money reinvested at 10%, the modified IRR is about 14.60%.

Understanding the result

If the IRR is above the return you require, the project's NPV at that required return is positive. So, taken on its own, it's worth doing. In the example, a 16.34% IRR clears a 10% target easily. That matches the positive NPV of $1,307.29 at 10%.

The IRR assumes each cash flow you receive can be reinvested at the IRR itself. When that's unrealistic, the MIRR is usually closer to the return you'll really get, because it uses a reinvestment rate you choose. In the example, it's 14.60% rather than 16.34%.

When there are several IRRs, none of them is 'the' return. Judge the project by its NPV at your required return instead. Or use the MIRR, which always has a single value.

Assumptions and limitations

  • Cash flows arrive at the end of each period, equally spaced. The starting investment is paid at the start.
  • IRRs are searched between −99.99% and 1,000,000% per period.
  • Cash flows are entered after whatever taxes and costs you want to count.
  • In a rare case, the NPV only touches zero without crossing it (a repeated root). That IRR may not be reported.
  • IRR doesn't show how big a project is. A small project can have a higher IRR but create less value than a large one. Compare NPVs for that.
  • Cash flows on specific, uneven dates aren't supported. Every period must be the same length.

Frequently asked questions

Why does the calculator say there is no IRR?

An IRR needs at least one outflow and one inflow. Otherwise, the NPV is positive (or negative) at every rate. An IRR can also fail to exist when the flows switch sign but the NPV never actually reaches zero. That happens with some patterns of later costs. In those cases, use the NPV at your required return.

What does it mean to have two IRRs?

Say the cash flows switch sign twice: an investment, then profits, then a large closing cost. The NPV can then cross zero twice. Both rates make the NPV zero, so neither is a meaningful return. The NPV profile shows the range of rates where the project adds value.

Is a monthly IRR the same as an annual one divided by 12?

No. A monthly IRR compounds, so the yearly equivalent is (1 + monthly IRR)^12 − 1. A 1% monthly IRR is about 12.68% a year, not 12%. The calculator shows this yearly figure for quarterly and monthly cash flows.

How is IRR related to CAGR?

CAGR is the compound annual growth rate, the steady yearly growth that gets you from start to finish. For a single investment with one payoff at the end and nothing in between, the IRR equals the CAGR. IRR stretches the same idea to any series of money in and out.

How this calculator is checked

Method: IRR as the root of the NPV equation with end-of-period flows, found by a logarithmic sign-change scan and Brent's method; MIRR from the terminal value of inflows and the present value of outflows. Tested against Microsoft's documented IRR and MIRR examples and closed-form cases. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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