How do I calculate project internal rate of return?

People type this into search every day: how do I calculate project internal rate of return.

Internal rate of return is the discount rate that sets net present value to zero. The US Department of Energy’s retrofit financial analysis uses that definition. On the project IRR calculator the cash shape is one outlay, then equal inflows at the end of each year.

Outlay = annual cash × [1 − (1 + r) ^ (−n)] ÷ r

The r that solves that equation is the IRR. NPV at a hurdle you type uses the same present-value sum. Investor.gov’s compound interest formula, future value = present value × (1 + r) ^ n, is the discount inside that sum. These examples compare a rate with a hurdle you chose. They are not a funding decision.

The NPV post starts from a rate and returns a dollar leftover. This page starts from the leftover you want, zero, and returns the rate.

Example 1. $50,000 in, $15,000 a year for five years

Calculator defaults:

• Project outlay: $50,000
• Annual net cash inflow: $15,000
• Years of inflows: 5
• Comparison hurdle rate: 8%

Undiscounted inflows: 15,000 × 5 = $75,000.00.

IRR: 15.2%.

NPV at the 8% hurdle: $9,890.65.

That $9,890.65 is the same leftover the NPV calculator prints at 8%. The IRR sits above the hurdle, and the NPV at the hurdle is positive. Those two readings are one comparison.

The rate that zeros NPV is 15.238%. The calculator rounds that to 15.2%. Typing 15.2% into the NPV page leaves $45.14, because the rounded rate still sits a little under the solved rate.

Example 2. Change the hurdle, and the IRR stays put

The hurdle is a comparison rate. The solve uses the outlay, the annual cash, and the years.

Hurdle 12%. IRR is still 15.2%. NPV at 12% is $4,071.64.

Hurdle 18%. IRR is still 15.2%. NPV at 18% is −$3,092.43, printed as -$3,092.43.

At 12% the IRR is above the hurdle and the NPV is positive. At 18% the IRR is below the hurdle and the NPV is negative. The project cash stays the same. The comparison rate is what moved. A hurdle of 8%, 12%, or 18% has to come from the alternative you are measuring against. The calculator will use whatever percent you type.

Example 3. What if the same cash runs for ten years?

• Outlay: $50,000
• Annual cash: $15,000
• Years: 10
• Hurdle: 8%

Undiscounted inflows: $150,000.00.

IRR: 27.3%.

NPV at 8%: $50,651.22.

Five more years of $15,000 raise the IRR from 15.2% to 27.3%. The outlay is covered sooner relative to a longer stream, and the later cash still counts in the rate. NPV at 8% rises from $9,890.65 to $50,651.22, the same pair of results as the NPV page.

Example 4. Four years, just under the hurdle

• Outlay: $50,000
• Annual cash: $15,000
• Years: 4
• Hurdle: 8%

Undiscounted inflows: $60,000.00.

IRR: 7.7%.

NPV at 8%: −$318.10, printed as -$318.10.

The inflows add to $10,000 more than the outlay, and the IRR is still under an 8% hurdle. The $10,000 arrives in the later years, and 7.7% is the rate that uses up that cushion. One more year, the example 1 life, jumps the IRR from 7.7% to 15.2% because year 5 is another $15,000 on a series that had almost broken even.

Example 5. Three years, and the inflows never cover the outlay

Three years of $15,000 is $45,000, which is $5,000 short of $50,000 even at a 0% discount.

IRR: −5.1%. The label is negative internal rate of return.

NPV at 8%: −$11,343.55, printed as -$11,343.55.

A negative IRR means a 0% discount still leaves a hole. You would need a negative discount rate, which inflates later cash, to push the present value up to $50,000. Simple payback on these inputs is 50,000 ÷ 15,000 = 3.33 years, so a 3-year life ends before payback. The IRR sign agrees with that.

Cash of $8,000 a year for five years totals $40,000 against a $50,000 outlay.

IRR: −7%.

NPV at 8%: −$18,058.32, printed as -$18,058.32.

Undiscounted inflows: $40,000.00. The project is short $10,000 before any discount, so the rate that “zeros” NPV has to be negative.

Annual cash of $0. The calculator prints No IRR. A zero inflow has no rate that recovers the outlay.

Example 6. The inflows exactly match the outlay

• Outlay: $50,000
• Annual cash: $10,000
• Years: 5
• Hurdle: 8%

Undiscounted inflows: 10,000 × 5 = $50,000.00.

IRR: 0%.

NPV at 8%: −$10,072.90, printed as -$10,072.90.

At a 0% rate, NPV is 50,000 − 50,000 = $0, so the solved rate is 0%. An 8% hurdle still discounts the $10,000 payments, and the present value falls short by $10,072.90. A 0% IRR clears a hurdle only when the hurdle is 0% or negative. Against 8% this project is behind.

Example 7. One year, which you can check by hand

• Outlay: $10,000
• Annual cash: $12,000
• Years: 1
• Hurdle: 8%

IRR: (12,000 − 10,000) ÷ 10,000 = 20%.

For a single end-of-year inflow, IRR and the holding-period return are the same division. The ROI page on $10,000 invested and $12,000 received over one year prints that 20%.

NPV at 8%: 12,000 ÷ 1.08 − 10,000 = 11,111.11 − 10,000 = $1,111.11.

The rate is above the hurdle, and the NPV is positive. This is the cleanest case of the rule in example 1.

Example 8. What if the yearly cash doubles to $30,000?

• Outlay: $50,000
• Annual cash: $30,000
• Years: 5
• Hurdle: 8%

Undiscounted inflows: $150,000.00.

IRR: 52.8%.

NPV at 8%: $69,781.30.

Doubling the inflow from $15,000 to $30,000 does more than double the IRR. It moves the rate from 15.2% to 52.8%. The outlay is fixed, and a much fatter annuity has to be discounted very hard before its present value falls to $50,000.

The machine bar is example 10: $40,000 out, $16,000 a year, five years, IRR 28.6%. It sits next to the ten-year version of the default project, 27.3%, so you can see two different shapes landing near the same rate.

Example 9. IRR is not the CAGR of the undiscounted total

A common swap treats the $75,000 of inflows as if it were one ending balance, five years after a $50,000 start. The CAGR calculator does that job, and Investor.gov’s CAGR glossary defines it as the constant annual rate from a starting value to an ending value.

Start $50,000, end $75,000, five years.

CAGR: (75,000 ÷ 50,000) ^ (1/5) − 1 = 1.5 ^ 0.2 − 1 = 8.45%.

Total change: 50%. Growth multiple: 1.5×.

The IRR on the same outlay and the same five $15,000 inflows is 15.2%.

The 8.45% figure pretends the $75,000 arrives as a lump on the last day, and that you did not receive $15,000 at the end of years 1, 2, 3, and 4. IRR gives those earlier deposits credit. CAGR connects two balances and ignores cash in the middle. The CAGR post is the right page when you really do have only a start and an end.

Example 10. The payback machine, and a second outlay

The payback defaults are a $40,000 outlay and $16,000 a year. Simple payback is 2.5 years. Extend the inflows to five years and solve the IRR at an 8% hurdle.

IRR: 28.6%.

NPV at 8%: $23,883.36.

Undiscounted inflows: $80,000.00.

Ten years of that $16,000, same $40,000 outlay, same 8% hurdle:

IRR: 38.5%.

NPV at 8%: $67,361.30.

Undiscounted inflows: $160,000.00.

Payback stayed 2.5 years in both rows, because payback ignores cash after the outlay is back. IRR and NPV both rise when you count the later years. A short payback and a high IRR can describe one project. They answer different questions.

A second cash outlay breaks the one-rate solve. Take the example 1 project and spend another $20,000 in year 3, so year 3’s net cash is 15,000 − 20,000 = −$5,000. Discount each year at 8%, the same way the NPV post does:

NPV = −$5,985.99.

Cash goes out at the start and again in year 3. Two sign changes can create two IRRs. The calculator’s solver assumes cash out once, then inflows, so it is the wrong tool for this list. The −$5,985.99 at 8% is the comparison I would keep.

Ten IRR cases side by side

Open the IRR calculator when the project is one outlay and a flat yearly inflow. Read the rate next to the NPV at your hurdle. If the cash is lumpy or you spend again halfway through, discount the year list on the NPV calculator and use that dollar leftover. If you only have a starting balance and an ending balance, the rate you want is CAGR, and the return on investment page puts a holding-period percent on those same two amounts.