Actuarial Mathematics for Modelling · Project appraisal

How to Calculate Internal Rate of Return (IRR)

Updated 11 October 2026 · Fact-checked

The internal rate of return (IRR) is the interest rate i at which the net present value of a project's cash flows is zero. Solve the equation of value for i. If no neat solution exists, calculate NPV at two rates either side of zero and use linear interpolation. Then check that the root is unique.

Understand Internal Rate of Return

A project has cash flows at different times. To compare money at different times you need an interest rate. The net present value (NPV) at rate i is the present value of all inflows minus the present value of all outflows.

The internal rate of return is the rate i that makes NPV equal to zero. It is the rate at which the project exactly earns its money back with interest. You are not given a rate. You find it from the cash flows themselves.

If the cash flows are an outlay followed by only inflows, NPV falls steadily as i rises. Then there is one IRR. If the signs of the cash flows change more than once, the equation can have several roots, or none. Always check this before you quote an answer.

For simple cases (two or three payments) you can solve the equation exactly, often as a quadratic in v = 1 ÷ (1 + i). For other cases you try two rates and use linear interpolation between them. This gives an approximation, because NPV is a curve, not a straight line.

A common decision rule: if the IRR is above the rate you require (the investor's target or cost of funds), the project is acceptable. This rule is reliable for the usual outlay-then-inflows pattern. It can mislead when cash flows change sign several times, or when you compare projects of different size or timing.

Key rules to remember

Definition of IRR: NPV(i) = Σ c(t) × (1 + i)^(-t) = 0; c(t) is the net cash flow at time t (inflows positive, outflows negative). The IRR is the i that solves this.
Equation of value form: PV of inflows at i = PV of outflows at i; This is the same equation. Use whichever way you find easier to write.
Linear interpolation: i ≈ i₁ + (i₂ − i₁) × NPV(i₁) ÷ (NPV(i₁) − NPV(i₂)); NPV(i₁) and NPV(i₂) must have opposite signs, with the root between i₁ and i₂. Keep the two rates close together.
Discount factor: v = 1 ÷ (1 + i); Useful for turning the equation into a polynomial in v, such as a quadratic.
Sign-change rule (Descartes): Number of positive roots for v ≤ number of sign changes in the cash flows; The number of positive roots is the number of sign changes or less by an even number. One sign change gives at most one positive root. It is a check, not proof that the root exists.
Decision rule: Accept if IRR > required rate of return; Valid for an outlay followed by inflows. State the condition when you use it.

How to solve Internal Rate of Return questions

Use this method for any IRR question, whether the answer is exact or approximate.

1List every cash flow with its time and sign. Outflows are negative, inflows positive. Make sure all times use the same unit, usually years.
2Write the equation of value: NPV(i) = 0, or PV of inflows = PV of outflows at rate i.
3Count the sign changes in the cash flows. One sign change means a unique positive IRR is likely. More changes mean you must check for several roots.
4If the equation is simple, solve it exactly. Put v = 1 ÷ (1 + i), solve the polynomial for v, reject impossible roots (v ≤ 0), then find i = 1 ÷ v − 1.
5Otherwise choose a first trial rate. Calculate NPV. Pick a second rate on the other side, so the NPV has the opposite sign.
6Apply the interpolation formula. Keep the working to enough decimal places, at least 4 significant figures in the NPVs.
7If asked, check by calculating NPV at your answer. It should be close to zero.
8State the answer as a percentage and say whether it is exact or approximate. Add a comment on uniqueness or the decision if the question asks for one.

Quickest way: Bracket and interpolate with close rates

When to use it: Use when the equation cannot be solved exactly in a minute or two, and a calculator is allowed.

Estimate the IRR. Total inflows divided by outlay gives a rough idea of whether it is low or high.
Try a round rate such as 8% or 10%. Note the sign of NPV. Positive NPV means the IRR is higher than your trial rate.
Move 1% or 2% in the right direction until the sign flips.
Interpolate between the two rates. Use the stored discount factors so you do not recalculate them.
Do one more NPV at your answer if time allows, to confirm the sign is near zero.

Common mistakes in Internal Rate of Return

Interpolating between two rates with NPVs of the same sign.
Students pick two trial rates quickly and do not check the signs.
Fix: Check that one NPV is positive and one negative before using the formula. If not, you are extrapolating, and the answer is unreliable.
Assuming there is only one IRR.
Most textbook examples have a single outlay then inflows, so students forget other patterns exist.
Fix: Count the sign changes in the cash flows. If there are two or more, look for several roots. For example, cash flows of −100 at time 0, +230 at time 1 and −132 at time 2 give IRRs of 10% and 20%.
Mixing up the sign in the interpolation formula, giving an answer outside the two trial rates.
The formula is memorised without understanding.
Fix: The answer must lie between i₁ and i₂. Use the absolute NPV sizes: the root is closer to the rate with the smaller NPV in absolute value.
Choosing trial rates that are far apart, such as 5% and 20%.
The student wants to be sure of getting opposite signs.
Fix: NPV is curved, so wide gaps give a poor estimate. Narrow the bracket to rates 1% or 2% apart.
Accepting any project with a positive IRR.
The decision rule is half remembered.
Fix: Compare the IRR with the required rate of return. A positive IRR alone means little.
Using wrong times for cash flows, for example putting the first receipt at time 0.
Careless reading of the timeline, especially with payments at the start or end of a year.
Fix: Draw a timeline before writing the equation. Label each cash flow with its exact time.

Worked examples

Example 1

A project costs ₹1,00,000 now. It returns ₹60,000 at the end of year 1 and ₹60,000 at the end of year 2. Find the internal rate of return exactly.

Show the solution

Equation of value: 1,00,000 = 60,000 v + 60,000 v², where v = 1 ÷ (1 + i).
Divide by 20,000: 5 = 3v + 3v², so 3v² + 3v − 5 = 0.
Quadratic formula: v = (−3 ± √(9 + 60)) ÷ 6 = (−3 ± √69) ÷ 6.
√69 = 8.3066. The positive root is v = 5.3066 ÷ 6 = 0.88444. The negative root is rejected because v must be positive.
i = 1 ÷ 0.88444 − 1 = 1.13066 − 1 = 0.13066.
There is one sign change (−, +, +), so this is the only positive IRR.

Answer: IRR ≈ 13.07% per year, and it is unique.

Example 2

A project costs ₹1,00,000 now and returns ₹30,000 at time 1, ₹50,000 at time 2 and ₹40,000 at time 3 (years). Use linear interpolation between 8% and 10% to estimate the IRR.

Show the solution

NPV at 8%: 30,000 ÷ 1.08 = 27,777.78. 50,000 ÷ 1.1664 = 42,866.94. 40,000 ÷ 1.259712 = 31,753.29. Sum = 1,02,398.01. NPV = +₹2,398.01.
NPV at 10%: 30,000 ÷ 1.1 = 27,272.73. 50,000 ÷ 1.21 = 41,322.31. 40,000 ÷ 1.331 = 30,052.59. Sum = 98,647.63. NPV = −₹1,352.37.
The signs are opposite, so the root lies between 8% and 10%.
Interpolate: i ≈ 8% + 2% × 2,398.01 ÷ (2,398.01 + 1,352.37) = 8% + 2% × 0.63942.
i ≈ 8% + 1.279% = 9.279%.
Check: only one sign change, so the IRR is unique. The true value, from NPV at 9% (+₹494.57) and 10%, is about 9.27%, so the estimate is close.

Answer: IRR ≈ 9.28% per year (about 9.3%).

Exam tips

Write the equation of value clearly before any numbers. Marks are often given for the correct equation even if the arithmetic goes wrong.
In written questions, always comment on uniqueness. Count sign changes and say what it means for the number of roots.
For MCQs, test the answer options by calculating NPV at each rate. The correct rate gives NPV close to zero.
Show full working for interpolation: both NPVs with signs, the formula and the final rate.
In computer-based questions, state the cash flow vector, the method (for example a root-finder or a goal seek), and the result, and check the NPV at the answer.

Practice questions from Project appraisal

Internal Rate of Return in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Internal Rate of Return: frequently asked questions

How do I calculate internal rate of return by linear interpolation?

Find NPV at two rates where the NPV signs are opposite. Then use i ≈ i₁ + (i₂ − i₁) × NPV(i₁) ÷ (NPV(i₁) − NPV(i₂)). Keep the two rates close together for a better estimate.

Can a project have more than one IRR?

Yes. If the cash flows change sign more than once, the NPV equation can have several positive roots. The sign-change rule gives the maximum possible number. Always check this before relying on one IRR.

What is the difference between NPV and IRR?

NPV is a value in rupees at a stated interest rate. IRR is the rate at which NPV is zero. NPV depends on the rate you choose, while IRR comes from the cash flows alone. Use IRR by comparing it with the required return.

Does the IRR always exist?

No. For some cash flow patterns, NPV never reaches zero for any sensible rate. For example, if all cash flows are positive, no positive IRR exists. Check the signs first.