Management Accounting · Asset budgeting and investment appraisal

Internal Rate of Return (IRR) and How to Calculate It by Hand

Q: What is the IRR interpolation formula in ACCA?

IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L). L is the lower rate with a positive NPV and H is the higher rate with a negative NPV. The result is an estimate that lies between L and H.

Q: How do I calculate IRR by hand?

Calculate NPV at one rate, then at a second rate so that one NPV is positive and one is negative. Put these into the interpolation formula. If the cash inflows are level, you can also divide the outlay by the annual inflow and find the matching annuity factor in the tables.

Q: What is the difference between NPV and IRR?

NPV is a dollar figure showing the gain in wealth at the cost of capital. IRR is a percentage showing the discount rate at which NPV is zero. NPV is generally preferred because it shows the size of the gain, while IRR ignores project size.

Q: Is the IRR from interpolation exact?

No. Interpolation assumes NPV falls in a straight line between the two rates, but the true relationship is a curve. The estimate is closer to the true IRR when your two rates are near each other.

Updated 11 October 2026 · Fact-checked

The **internal rate of return (IRR)** is the discount rate at which a project's NPV is zero. To estimate it, calculate NPV at two rates, one giving a positive NPV and one a negative NPV, then interpolate between them. If the IRR is higher than the cost of capital, accept the project.

Understand Internal Rate of Return (IRR)

Every project has a discount rate that makes its NPV exactly zero. That rate is the IRR. It is the project's own break-even return on the money invested in it.

Think of it this way. NPV tells you the dollar gain at a rate you choose (the cost of capital). IRR flips the question: how high could the cost of capital go before the project stops adding value? If the IRR is 22% and your cost of capital is 10%, there is a wide safety margin.

The decision rule for a single, conventional project is simple. If IRR is greater than the cost of capital, NPV at the cost of capital is positive, so accept. If IRR is lower, reject.

You cannot solve IRR directly by algebra when there are several cash flows. So in the exam you estimate it. Calculate NPV at one rate, then at a second rate, so that one NPV is positive and one is negative. The IRR lies between them. You then assume NPV changes in a straight line between the two rates and read off where it crosses zero. This is linear interpolation. The answer is an estimate, and it is more accurate when the two rates are close together.

NPV and IRR usually agree for a single project. NPV is generally the better measure because it shows the absolute increase in wealth. IRR gives a percentage, which is easy to compare with the cost of capital, but it ignores the size of the project.

Key formulas to remember

IRR interpolation formula: IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L); L is the lower rate (positive NPV). H is the higher rate (negative NPV). Subtracting a negative NPV means you add the two NPVs ignoring signs.
Definition of IRR: IRR = the discount rate where NPV = 0; It is the break-even cost of capital for the project.
Decision rule: Accept if IRR > cost of capital; reject if IRR < cost of capital; This applies to a single project with conventional cash flows (outflow first, then inflows).
Annuity shortcut for level cash flows: Annuity factor = initial investment ÷ annual cash inflow; Find this factor in the annuity table on the row for the project life. The matching rate is the IRR.

How to solve Internal Rate of Return (IRR) questions

Use this method for any IRR question that gives cash flows and a cost of capital.

1List the cash flows by year. Put the initial outlay at time 0 as a negative figure.
2Calculate NPV at the cost of capital if given, or at a sensible first guess. Note whether it is positive or negative.
3If NPV is positive, try a higher rate. If NPV is negative, try a lower rate. Keep going until you have one positive NPV and one negative NPV.
4Label the rate with the positive NPV as L and the rate with the negative NPV as H.
5Apply the formula: IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L). Ignore the minus sign on the negative NPV when adding the two together.
6Check the answer lies between L and H. If it does not, you have made an arithmetic slip.
7Compare the IRR with the cost of capital and state the decision: accept if IRR is higher, reject if lower.

Quickest way: Annuity factor shortcut and sense-check

When to use it: Use when the annual cash inflows are level and you need a fast estimate, or when you want to check an interpolation answer.

Divide the initial investment by the annual inflow to get the annuity factor.
Look along the row for the project life in the annuity table.
Find the two rates whose factors sit either side of your figure.
Choose the nearest rate, or interpolate between the two columns if you need a closer value.
In multiple choice, work out roughly where zero NPV falls. The answer is closer to the rate whose NPV is smaller in size.

Common mistakes in Internal Rate of Return (IRR)

Using two rates that both give positive NPVs, or both negative.
Students stop after two attempts without checking the signs.
Fix: Keep trying rates until the signs are opposite. Then interpolate. Extrapolating outside the two rates is far less reliable.
Subtracting the NPVs wrongly in the denominator.
The negative NPV is treated as a positive number, or its sign is dropped inconsistently.
Fix: Write the denominator as NPV at L minus NPV at H. For example, 1,780 − (−2,760) = 4,540. Add the two sizes together.
Multiplying by the wrong gap, or forgetting to add L.
The formula is memorised as a pattern rather than understood.
Fix: Remember the logic: you start at L and move up a fraction of the gap between H and L. The result must lie between L and H.
Treating IRR as the cost of capital or comparing it with the wrong rate.
Both are percentages and they sound similar.
Fix: IRR is the project's return. Cost of capital is the required return. Accept only if IRR is higher.
Forgetting that the outlay at time 0 is not discounted.
Students apply a factor to every figure out of habit.
Fix: Time 0 has a discount factor of 1.000 at every rate. Use the full outlay.
Claiming the interpolated IRR is exact.
The calculation produces a precise-looking decimal.
Fix: Treat it as an estimate. In the exam, round as the question requires and do not over-interpret the decimals.

Worked examples

Example 1

A project costs $50,000 now and gives cash inflows of $20,000 at the end of each of the next 4 years. The cost of capital is 10%. NPV at 20% is +$1,780 and NPV at 25% is −$2,760. Estimate the IRR and advise whether to accept the project.

Show the solution

Check the signs. NPV is positive at 20% and negative at 25%, so IRR lies between 20% and 25%.
Set L = 20%, H = 25%, NPV at L = 1,780, NPV at H = −2,760.
Denominator: 1,780 − (−2,760) = 4,540.
Fraction: 1,780 ÷ 4,540 = 0.392.
Gap: 25 − 20 = 5 percentage points. 0.392 × 5 = 1.96.
IRR = 20 + 1.96 = 21.96%, or about 22.0%.
Compare with the cost of capital: 22.0% is well above 10%.

Answer: The estimated IRR is about 22.0%. This exceeds the 10% cost of capital, so accept the project.

Example 2

A project has an NPV of +$8,000 at a discount rate of 12% and an NPV of −$4,000 at 20%. The company's cost of capital is 15%. What is the estimated IRR, and what is the decision? Options for the IRR: 14.7%, 16.0%, 17.3%, 20.0%.

Show the solution

L = 12%, H = 20%, NPV at L = 8,000, NPV at H = −4,000.
Denominator: 8,000 − (−4,000) = 12,000.
Fraction: 8,000 ÷ 12,000 = 0.6667.
Gap: 20 − 12 = 8. 0.6667 × 8 = 5.33.
IRR = 12 + 5.33 = 17.33%, so 17.3%.
Compare with the cost of capital: 17.3% is above 15%, so NPV at 15% is positive.

Answer: The IRR is about 17.3% (third option). It exceeds the 15% cost of capital, so accept the project.

Exam tips

In multiple choice, a quick sense-check removes bad options. The IRR must lie between your two rates, and it sits closer to the rate with the smaller NPV (in size).
For number entry questions, read the rounding instruction. Carry full figures through the calculation and round only at the end.
If the question gives you both NPVs and both rates, go straight to the formula. Do not recalculate the NPVs.
Know the decision rule cold: IRR above cost of capital means accept. Many questions are only testing this.
Be ready for short written points in multiple response questions: IRR is a percentage, uses the time value of money, but ignores project size and is only an estimate when interpolated.

Practice questions from Asset budgeting and investment appraisal

Internal Rate of Return (IRR) 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 (IRR): frequently asked questions

What is the IRR interpolation formula in ACCA?

IRR = L + [NPV at L ÷ (NPV at L − NPV at H)] × (H − L). L is the lower rate with a positive NPV and H is the higher rate with a negative NPV. The result is an estimate that lies between L and H.

How do I calculate IRR by hand?

Calculate NPV at one rate, then at a second rate so that one NPV is positive and one is negative. Put these into the interpolation formula. If the cash inflows are level, you can also divide the outlay by the annual inflow and find the matching annuity factor in the tables.

What is the difference between NPV and IRR?

NPV is a dollar figure showing the gain in wealth at the cost of capital. IRR is a percentage showing the discount rate at which NPV is zero. NPV is generally preferred because it shows the size of the gain, while IRR ignores project size.

Is the IRR from interpolation exact?

No. Interpolation assumes NPV falls in a straight line between the two rates, but the true relationship is a curve. The estimate is closer to the true IRR when your two rates are near each other.