Enter the initial investment and the cash flow for each year starting from year 1 (the money received that year). The calculator finds the discount rate that makes NPV (net present value) exactly 0, which is the IRR (internal rate of return).
Enter amounts in dollars as plain numbers. Enter the initial investment as a positive amount (it is treated as a negative outflow). If you invested more in a later year, enter a negative amount for that year (for example, -50000); leave years with no money moving blank or 0. You can add rows with "Add a row" up to 20 years (20 years is enough for most investment decisions).
Result and graph
Enter the initial investment and the cash flow for each year on the left and press "Calculate". The result and a graph will appear here.
Table of Contents
-
What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formulas and graphs
-
Symbols and terms
-
Good to know before you start
-
How to calculate it in Excel
-
How to calculate it in Google Sheets
-
How to calculate it in Python
-
How to write it in LaTeX and other math languages (copy and paste)
-
How to have ChatGPT do the calculation
-
DataChef Features
-
Related Features
-
NumberChef Calculators List
What you can do on this page
- Enter the initial investment and the cash flow for each year (the money received that year), and get the IRR (internal rate of return, per year)
- The investment period, total return (gain or loss over the whole period) and gross return are shown at the same time
- The yearly amounts can all be different, and additional investments in later years (negative cash flows) are supported
- A graph of NPV (net present value) shows at a glance where NPV becomes 0, which is the IRR
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This page assumes one cash flow per year. For the rate of an investment with equal payments each period, "TVM Calculator" in the related pages (solving for I/Y) is handy. IRR here is a math calculation that does not include fees or taxes, and this page does not recommend any investment.
What is this calculation used for?
Deciding on an equipment investment (capital budgeting and the hurdle rate)
"If we install a $4,000,000 machine, we expect cost savings of $1,000,000, $2,000,000 and $3,000,000 over 3 years." With this estimate, the IRR is about 19.4% a year. In company investment decisions, the standard use is to compare this IRR with the minimum required return (the hurdle rate, such as the borrowing rate or the cost of capital) and accept the project if it is higher. Every corporate finance textbook covers it.
Future cash flows are only estimates, though, so it is good practice to also run the numbers with a cautious estimate.
Evaluating a rental property (yearly rent plus the final sale)
The money in a real estate investment flows like this: you buy the property first, rent comes in every year, and the sale price comes in at the end, so the yearly amounts are all different. IRR's strength is turning uneven cash flows like these into one yearly rate, and real estate investors widely use it to compare properties.
Unlike the gross yield (rent ÷ price), it also includes the gain or loss on the sale and years when vacancies cut the rent.
Checking whether solar panels or energy-saving equipment pay off
Home solar power has a cash flow that suits IRR: you pay for the installation first, then every year you save on your electric bill and may earn credits for power sent to the grid (such as net metering, where available). For example, if the installation costs $20,000, list the yearly savings and calculate the IRR, and you can compare the system with a savings account or an index fund on the same scale: "this equipment is like an investment returning X% a year".
The result changes when assumptions change, such as electricity rates, credit rules or the life of the equipment, so it is a good idea to calculate a few scenarios.
Measuring fund and M&A performance (finance careers)
For investment funds such as venture capital and private equity, IRR is the standard measure of fund performance. When money is called from investors and when it is paid back differs from fund to fund, so only IRR, which takes time into account, gives a fair comparison.
Investment banks and companies considering M&A (mergers and acquisitions) also calculate IRR from the purchase price and the future profit plan to decide whether to go ahead. It is one of the first calculations people learn when they aim for a career in finance.
Seeing in numbers that money returned sooner is worth more
Suppose investments A and B both cost $100,000 and both pay back $150,000 in total over 5 years (a total return of $50,000 and a gross return of 50%). A pays more early ($5,000, $20,000, $25,000, $40,000, $60,000), while B pays mostly at the end ($0, $10,000, $30,000, $30,000, $80,000).
The totals are the same, but A's IRR is 11.290% a year and B's is 10.259%, so A, which returns money sooner, comes out ahead. Money that comes back can be invested again. IRR shows you this time value of money in numbers.
Formulas and graphs
Turning money in \(t\) years into its value today (present value)
Graph
Standard notation
(the usual math form)
\(PV\)
\(=\)
\(CF_t\)
\(\div\)
\((1+r)\)
\(t\)
In words
(symbols replaced with words)
④
\(PV\): present value
\(=\)
①
\(CF_t\): cash flow in year \(t\)
\(\div\)
②
discount factor for one year \((1+r)\)
③
\(t\): years
The formula in words
①
Take the
\(CF_t\): cash flow in year \(t\)
②
and divide it by the
discount factor for one year \((1+r)\)
(\(r\) is the discount rate)
③
multiplied by itself once for each of the
\(t\): years
④
and you get the
\(PV\): present value
Quick example
At a 10% discount rate, the present value of $1,210,000 received in 2 years is (in units of $10,000)
\(PV\): present value
\(=\)
money in 2 years ($1,210,000)
\(\div\)
discount factor (1.1)
years (2)
\(1.1 \times 1.1 = 1.21\)
\(121 \div 1.21 = 100\)
Key idea
If money can earn 10% a year, "$1,000,000 today" and "$1,210,000 in 2 years" are worth the same. Turning future money into what it is worth today is called discounting, and the yearly rate \(r\) used is called the discount rate. Money has a time value: this is the idea that IRR is built on.
Formula for NPV (net present value)
Figure
Standard notation
(the usual math form)
\(\mathrm{NPV}\)
\(=\)
\(CF_0\)
\(+\)
\(\displaystyle\sum_{t=1}^{n} \dfrac{CF_t}{(1+r)^{t}}\)
In words
(symbols replaced with words)
③
NPV: net present value
\(=\)
①
\(CF_0\): initial investment (a negative value)
\(+\)
②
sum of the present values of the cash flows from year 1 to year \(n\)
The formula in words
①
To the
\(CF_0\): initial investment
(money paid, so it is negative)
②
add the
sum of the present values of each year's cash flow
(each year's money turned into today's value with formula 1, then all added up)
③
and you get the
NPV: net present value
Quick example
At a 10% discount rate, the NPV of an investment of $1,000,000 that returns $550,000 in year 1 and $605,000 in year 2 is (in units of $10,000)
NPV: net present value
\(=\)
initial investment (−$1,000,000)
\(+\)
sum of the present values of $550,000 and $605,000
\(55 \div 1.1 = 50\)
\(60.5 \div 1.21 = 50\)
\(-100 + 50 + 50 = 0\)
Key idea
NPV is what you get when you turn all the money in and out of an investment into today's value and add it up. If it is positive, the investment does better than earning the discount rate \(r\); if negative, worse; if exactly 0, exactly the same as earning \(r\). The NPV in the example is exactly 0 because this investment returns exactly 10% a year.
Definition of IRR (internal rate of return)
Graph
Standard notation
(the usual math form)
\(\mathrm{NPV}(\mathrm{IRR})\)
\(=\)
\(0\)
In words
(symbols replaced with words)
①
NPV calculated with \(\mathrm{IRR}\) as the discount rate
\(=\)
②
exactly \(0\)
The formula in words
①
The
NPV calculated with \(\mathrm{IRR}\) as the discount rate
②
is
exactly \(0\)
. The discount rate that makes this true is the internal rate of return, IRR, and it shows the return per year of the investment
Quick example
For the investment of $1,000,000 that returns $550,000 in year 1 and $605,000 in year 2, NPV is exactly 0 at a 10% discount rate, as in the example for formula 2, so
NPV (at a 10% discount rate)
\(=\)
exactly 0
\(\mathrm{IRR} = 0.10\ \ (10\%)\)
Key idea
This equation cannot be rearranged into the form \(\mathrm{IRR}=\cdots\) (it becomes an equation of degree \(n\)). Instead, a computer changes the discount rate \(r\) little by little, calculates NPV, and searches for where it becomes 0 (a numerical method). The calculator on this page solves it the same way.
One caution: when the signs (plus and minus) of the cash flows change twice or more, for example "invest → receive → invest a large amount again", more than one discount rate can make NPV equal to 0 (the multiple IRR problem). In that case the calculator shows a note. For investments with multiple IRRs, the standard practice is to decide by NPV, not IRR.
Total return and gross return
Standard notation
(the usual math form)
\(TR\)
\(=\)
\(S\)
\(-\)
\(C_0\)
\(G\)
\(=\)
\(TR\)
\(\div\)
\(C_0\)
In words
(symbols replaced with words)
③
\(TR\): total return
\(=\)
①
\(S\): total money received
\(-\)
②
\(C_0\): initial investment
⑤
\(G\): gross return
\(=\)
\(TR\): total return
\(\div\)
④
\(C_0\): initial investment
The formula in words
①
From the
\(S\): total money received
②
subtract the
\(C_0\): initial investment
③
to get the
\(TR\): total return
(the gain or loss over the whole period). Divide that again by the
④
\(C_0\): initial investment
⑤
and you get the
\(G\): gross return
(multiply by 100 to write it as a %)
Quick example
If you invest $4,000,000 and receive $1,000,000 in year 1, $2,000,000 in year 2 and $3,000,000 in year 3 (in units of $10,000)
\(TR\): total return
\(=\)
total received ($6,000,000)
\(-\)
initial investment ($4,000,000)
\(100 + 200 + 300 = 600\)
\(600 - 400 = 200\)
\(200 \div 400 = 0.5\ \ (50\%)\)
Key idea
Gross return is the % you gained compared with the investment over the whole period, and it ignores time. A 50% gain over 3 years and a 50% gain over 10 years are very different. IRR takes the time value into account and turns the result into a return per year, and it is the main value on this page. In fact, the gross return in this example is 50%, but the IRR is 19.438% a year.
IRR (internal rate of return) is the discount rate that makes NPV (net present value) exactly 0, which is the return per year of the investment. It cannot be solved by rearranging, so it is found numerically by changing the discount rate little by little. Even with the same total return, the sooner the money comes back, the higher the IRR.
Symbols and terms
Symbols
| IRR | I R R | Internal rate of return. The discount rate that makes NPV exactly 0; it shows the return per year of the investment. (Example - invest $4,000,000 and receive $1,000,000, $2,000,000 and $3,000,000 over 3 years, and the IRR is 19.438% a year.) |
| NPV | N P V | Net present value. All the money in and out of an investment, turned into today's value and added up. If it is positive, the investment does better than the discount rate. |
| \(CF_t\) | C F sub t | The cash flow in year \(t\). The money received that year (negative if it was money paid). |
| \(CF_0\) | C F sub zero | The cash flow in year 0 (now). The initial investment is money paid, so it is negative. On this calculator you enter it as a positive amount, and it is treated as negative. |
| \(r\) | r | The discount rate. The yearly rate used to turn future money into today's value. IRR is the special \(r\) where NPV is 0. |
| \(t\) | t | The year number. It counts which year the money is in, with the time of the investment as year 0. |
| \(n\) | n | The investment period in years. The year of the last cash flow. |
| \(PV\) | P V | Present value. Future money turned into its value today. |
| \(S\) | S | The total money received. All the cash flows from year 1 to year \(n\) added up (additional investments are added as negative amounts). |
| \(TR\) | T R | Total return. The total received minus the initial investment, the gain or loss over the whole period. |
| \(G\) | G | Gross return. The total return as a % of the initial investment, a whole-period % that ignores time. Note that in the fund industry, "gross return" usually has a different meaning, the return before fees. |
Terms
| cash flow | Money coming in or going out. Money received is a positive number and money paid is a negative number. For IRR, when and how much the money moves both matter. |
| discount rate | The yearly rate used to turn future money into today's value (to discount it). For example, $110,000 one year from now, discounted at 10% a year, is $100,000 today. |
| present value | What future money is worth today. Because money has a time value, the later you receive the same $1,000,000, the smaller its present value. |
| NPV (net present value) | The sum of the present values of all the cash flows of an investment. A positive NPV says the investment earns more than the discount rate. It is the most basic yardstick for investment decisions. |
| IRR (internal rate of return) | The discount rate that makes NPV exactly 0. It shows in one number what return per year the investment is equal to, so it is widely used to evaluate business projects and real estate. |
| hurdle rate | The minimum return required from an investment (such as the cost of capital). In practice, a project is accepted if its IRR is above the hurdle rate. |
| multiple IRR problem | A mathematical property - when the signs of the cash flows change twice or more, two or more discount rates can make NPV equal to 0. IRR alone cannot decide in that case, so you decide by NPV. |
| numerical method | A way to solve an equation that cannot be solved by rearranging, where a computer tries values little by little and gets closer to the answer. IRR is found this way (bisection, Newton's method and others). |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Percents (Grades 6–7) |
|
| Positive and negative numbers (Grades 6–7) |
|
| Exponents (Grades 6–8) |
|
| The idea of an equation (Grades 7–8) |
|
| Compound interest and the time value of money (high school personal finance and math) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table for the present value of money in t years
| Money received in t years ($) | 110000 |
| Discount rate (%) | 10 |
| Years t | 1 |
| Present value ($) | =B1/(1+B2/100)^B3 |
Table for NPV (net present value)
| Initial investment (enter as negative) | -1000000 |
| Cash flow in year 1 | 550000 |
| Cash flow in year 2 | 605000 |
| Discount rate (%) | 10 |
| NPV ($) | =NPV(B4/100,B2:B3)+B1 |
Table for IRR (internal rate of return)
| Initial investment (enter as negative) | -4000000 |
| Cash flow in year 1 | 1000000 |
| Cash flow in year 2 | 2000000 |
| Cash flow in year 3 | 3000000 |
| IRR (%) | =IRR(B1:B4)*100 |
Table for total return and gross return
| Initial investment | 4000000 |
| Total money received | 6000000 |
| Total return ($) | =B2-B1 |
| Gross return (%) | =(B2-B1)/B1*100 |
Excel has a function just for IRR. As in the third table, pass the list of cash flows, with the initial investment entered as negative, to "=IRR(B1:B4)", and you get about 19.438 (%) in the example.
The NPV function in the second table has a well-known trap. Excel's NPV function should include only the money from year 1 onward in its range; the initial investment (year 0) is added outside the function (the example gives exactly 0).
The first table gives exactly 100,000, and the fourth gives a total return of 2,000,000 and a gross return of 50%. Just replace column B with your own numbers.
The NPV function in the second table has a well-known trap. Excel's NPV function should include only the money from year 1 onward in its range; the initial investment (year 0) is added outside the function (the example gives exactly 0).
The first table gives exactly 100,000, and the fourth gives a total return of 2,000,000 and a gross return of 50%. Just replace column B with your own numbers.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table for the present value of money in t years
| Money received in t years ($) | 110000 |
| Discount rate (%) | 10 |
| Years t | 1 |
| Present value ($) | =B1/(1+B2/100)^B3 |
Table for NPV (net present value)
| Initial investment (enter as negative) | -1000000 |
| Cash flow in year 1 | 550000 |
| Cash flow in year 2 | 605000 |
| Discount rate (%) | 10 |
| NPV ($) | =NPV(B4/100,B2:B3)+B1 |
Table for IRR (internal rate of return)
| Initial investment (enter as negative) | -4000000 |
| Cash flow in year 1 | 1000000 |
| Cash flow in year 2 | 2000000 |
| Cash flow in year 3 | 3000000 |
| IRR (%) | =IRR(B1:B4)*100 |
Table for total return and gross return
| Initial investment | 4000000 |
| Total money received | 6000000 |
| Total return ($) | =B2-B1 |
| Gross return (%) | =(B2-B1)/B1*100 |
The same formulas and the same IRR and NPV functions as in Excel work as is. Copy the whole table, paste it into cell A1, and replace column B with your own numbers.
How to calculate it in Python
initial_investment = 4000000 # initial investment ($)
cash_flows = [1000000, 2000000, 3000000] # cash flow for each year from year 1 ($)
# NPV (net present value): turn every cash flow into today's value and add them up
def npv(rate, cfs):
return sum(cf / (1 + rate) ** t for t, cf in enumerate(cfs))
cfs = [-initial_investment] + cash_flows
# Find the discount rate where NPV is 0 (the IRR) by bisection
# (a simple version for ordinary investments whose signs change only once)
low, high = -0.9999, 10.0
for _ in range(200):
mid = (low + high) / 2
if npv(low, cfs) * npv(mid, cfs) <= 0:
high = mid
else:
low = mid
irr = (low + high) / 2
total_return = sum(cash_flows) - initial_investment
gross_return = total_return / initial_investment * 100
print(f"IRR: {irr * 100:.3f}%")
print(f"Total return: ${total_return}")
print(f"Gross return: {gross_return:.3f}%")
Runs with the standard library only. IRR cannot be solved by rearranging, so it is found by bisection (a numerical method that cuts the range in half again and again to close in on the answer). Change the initial investment and the list of cash flows at the top and run it (the example gives 19.438%).
How to write it in LaTeX and other math languages (copy and paste)
Turning money in \(t\) years into its value today (present value)
PV = CFₜ ÷ (1 + r)ᵗ
PV = \dfrac{CF_t}{(1+r)^{t}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>PV</mi>
<mo>=</mo>
<mfrac>
<msub><mi>CF</mi><mi>t</mi></msub>
<msup>
<mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mo>)</mo></mrow>
<mi>t</mi>
</msup>
</mfrac>
</mrow>
</math>
PV = (CF_t) / ((1 + r)^t)
cf/(1 + r)^t
PV := cf/(1 + r)^t;
PV = cf/(1 + r)^t;
PV = CF_t/(1 + r)^t
Formula for NPV (net present value)
NPV = CF₀ + CF₁/(1 + r)¹ + … + CFₙ/(1 + r)ⁿ
\mathrm{NPV} = \sum_{t=0}^{n} \dfrac{CF_t}{(1+r)^{t}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>NPV</mi>
<mo>=</mo>
<munderover>
<mo>∑</mo>
<mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow>
<mi>n</mi>
</munderover>
<mfrac>
<msub><mi>CF</mi><mi>t</mi></msub>
<msup>
<mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mo>)</mo></mrow>
<mi>t</mi>
</msup>
</mfrac>
</mrow>
</math>
NPV = sum_(t=0)^n (CF_t) / ((1 + r)^t)
Sum[cf[t]/(1 + r)^t, {t, 0, n}]
NPV := sum(cf(t)/(1 + r)^t, t = 0 .. n);
NPV = sum(cf ./ (1 + r).^(0:n));
NPV = ∑_(t=0)^n CF_t/(1 + r)^t
Definition of IRR (internal rate of return)
CF₀ + CF₁/(1 + IRR)¹ + … + CFₙ/(1 + IRR)ⁿ = 0
\sum_{t=0}^{n} \dfrac{CF_t}{(1+\mathrm{IRR})^{t}} = 0
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<munderover>
<mo>∑</mo>
<mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow>
<mi>n</mi>
</munderover>
<mfrac>
<msub><mi>CF</mi><mi>t</mi></msub>
<msup>
<mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>IRR</mi><mo>)</mo></mrow>
<mi>t</mi>
</msup>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</mrow>
</math>
sum_(t=0)^n (CF_t) / ((1 + IRR)^t) = 0
Solve[Sum[cf[t]/(1 + x)^t, {t, 0, n}] == 0, x]
solve(sum(cf(t)/(1 + x)^t, t = 0 .. n) = 0, x);
solve(sum(cf ./ (1 + x).^(0:n)) == 0, x)
∑_(t=0)^n CF_t/(1 + IRR)^t = 0
Total return and gross return
G = (S − C₀) ÷ C₀
G = \dfrac{S - C_0}{C_0}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>G</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>S</mi><mo>−</mo><msub><mi>C</mi><mn>0</mn></msub></mrow>
<msub><mi>C</mi><mn>0</mn></msub>
</mfrac>
</mrow>
</math>
G = (S - C_0) / C_0
(s - c0)/c0
G := (s - c0)/c0;
G = (s - c0)/c0;
G = (S − C_0)/C_0
How to have ChatGPT do the calculation
You are an investment calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). I invested $4,000,000 and received $1,000,000 in year 1, $2,000,000 in year 2 and $3,000,000 in year 3. Find each of the following: 1. The IRR (internal rate of return, in %, to 3 decimal places) = the r that makes NPV(r) = -4000000 + 1000000/(1+r) + 2000000/(1+r)^2 + 3000000/(1+r)^3 equal to 0 2. The total return (in dollars) 3. The gross return (in %, to 3 decimal places) Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
DataChef Features
Easy and Free
Unlimited conversions for free.
No technical knowledge required.
Intuitive and user-friendly operation.
No technical knowledge required.
Intuitive and user-friendly operation.
No Registration Required
Available immediately after access.
Can be used without registering personal information.
Can be used without registering personal information.
Safe and Secure
Fully SSL encrypted communication.
Automatic file deletion by clicking "download".
Automatic file deletion by clicking "download".
Fast
High-speed site access
and rapid file conversion.
and rapid file conversion.
No Watermark
No watermark.
No attribution required.
No attribution required.
Commercial Use Available
Free for commercial use.
No need to contact us for commercial use permission.
No need to contact us for commercial use permission.
Related Features
Payback Period Calculator (Simple and Discounted Payback)
From the same yearly cash flows, finds how many years it takes to get your money back. A standard measure used together with IRR
ROI Calculator (Return on Investment and Annualized ROI)
Finds ROI and annualized ROI from the amount invested and returned. Easier for a simple investment with a single payback
TVM Calculator (Time Value of Money Finance Calculator)
To solve for the rate (I/Y) of an investment with equal payments each period. A five-key calculator like a financial calculator
Present Value Calculator (Discounting and Present Value of an Annuity)
Discounting, turning future money into its value today. Learn the idea that IRR is built on
Average Return Calculator (Annualized Geometric Average and Cumulative Return)
For the annual rate when you know the return (%) for each period. Finds the geometric mean that smooths out uneven results
Ad 
