Financial Management · Investment appraisal techniques

Internal Rate of Return (IRR) and Interpolation in ACCA FM

Q: What is the IRR interpolation formula in ACCA FM?

IRR = A + [NPV at A ÷ (NPV at A − NPV at B)] × (B − A). A is the lower rate with a positive NPV and B is the higher rate with a negative NPV. The result is an estimate because it assumes a straight line between the two NPVs.

Q: How do I calculate IRR by hand?

Calculate NPV at two discount rates so that one NPV is positive and one is negative. Apply the interpolation formula to find the rate where NPV would be zero. Keep the two rates close together for a better estimate.

Q: What is the difference between NPV and IRR?

NPV is an absolute figure in money that shows the value a project adds at the cost of capital. IRR is a percentage that shows the project's break-even discount rate. NPV considers project size, while IRR does not.

Q: Why do IRR and NPV sometimes give conflicting decisions?

Conflict arises when you rank mutually exclusive projects of different size or with different timing of cash flows. A small project can have a higher IRR but a lower NPV. In that case choose the higher NPV, as it increases shareholder wealth the most.

Updated 11 October 2026 · Fact-checked

The internal rate of return (IRR) is the discount rate at which a project's NPV equals zero. Calculate NPV at two discount rates, one positive and one negative, then interpolate. For a conventional project, accept it if the IRR is above the cost of capital. If it is below, reject it.

Understand Internal Rate of Return (IRR)

Net present value (NPV) tells you how much value a project adds at a given discount rate. The internal rate of return (IRR) asks a different question. It asks what discount rate would make the NPV exactly zero.

Think of the IRR as the project's break-even cost of finance. If the company's cost of capital is lower than the IRR, the project earns more than it costs to fund. NPV is then positive at the cost of capital. If the cost of capital is higher than the IRR, NPV is negative.

You cannot solve for the IRR directly when cash flows are uneven. So in the exam you estimate it. You calculate NPV at one rate, then at a second rate, and assume the NPV moves in a straight line between them. In fact the NPV curve is slightly curved, so the answer is an approximation. The closer your two rates are to the true IRR, the better it is.

IRR is popular because it gives a percentage that managers find easy to compare with a required return. It also has weaknesses. It is a relative measure, so it ignores the size of the project. It assumes cash flows are reinvested at the IRR itself. With non-conventional cash flows (more than one change of sign) there can be several IRRs or none that make sense.

For a single, conventional project, NPV and IRR always give the same accept or reject answer. They can conflict only when you rank mutually exclusive projects. In that case NPV is the better guide, because it measures the absolute increase in shareholder wealth.

Key rules to remember

IRR interpolation formula: IRR = A + [NPV at A ÷ (NPV at A − NPV at B)] × (B − A); A is the lower rate and gives a positive NPV. B is the higher rate and gives a negative NPV. The bottom line is the two NPVs added together, ignoring the minus sign.
Decision rule for one project: Accept if IRR > cost of capital; reject if IRR < cost of capital; Valid for conventional cash flows (an outflow followed by inflows). At equality the NPV is zero and the project is marginal.
Annuity shortcut for a first guess: Annuity factor = initial investment ÷ annual cash inflow; Use this when inflows are level. Find the factor in the annuity table for the project life. The matching rate is a good first estimate of the IRR.
Margin of safety on the discount rate: IRR − cost of capital; Shows how far the cost of capital can rise before the project's NPV turns negative.

How to solve Internal Rate of Return (IRR) questions

Use the same method for any IRR question. Keep the cash flows in a clean table so you can reuse them at the second rate.

1Lay out the cash flows by year, with the outlay at time 0 as a negative figure.
2Calculate NPV at the cost of capital, or at a sensible first guess. Use the annuity shortcut if the inflows are level.
3Check the sign. If NPV is positive, the IRR is higher than your rate, so try a higher rate. If NPV is negative, try a lower rate.
4Calculate NPV at a second rate so that you have one positive and one negative NPV. Keep the two rates reasonably close, ideally within a few percentage points.
5Apply the interpolation formula: IRR = A + [NA ÷ (NA − NB)] × (B − A).
6Compare the IRR with the cost of capital and state the decision. Add that the IRR is an estimate.
7If asked, comment on limitations. Mention project size, reinvestment assumption, and multiple IRRs for non-conventional cash flows.

Quickest way: Two-NPV interpolation with an annuity shortcut

When to use it: Use this when the question asks for the IRR and you have limited time. It works best when the cash flows are given and your two trial rates are close to the IRR.

If inflows are level, divide the outlay by the annual inflow. Find the nearest factor in the annuity table to pick your first rate.
If the inflows are uneven, start at the cost of capital. The NPV you calculate there is often needed for the answer anyway.
Pick the second rate 3 to 5 percentage points away, in the direction that flips the sign of NPV.
Write both NPVs next to their rates. Check one is positive and one is negative before you use the formula.
Do the division first, multiply by the rate gap, then add to the lower rate. Quote the result to one decimal place.

Common mistakes in Internal Rate of Return (IRR)

Using two rates that both give a positive NPV or both give a negative NPV
Students pick the second rate without checking the sign of the first NPV.
Fix: If NPV is positive, go up. If it is negative, go down. Always have one of each before you interpolate. Using two same-sign NPVs is extrapolation and is less reliable.
Subtracting the NPVs incorrectly in the denominator
Students forget that one NPV is negative and so subtract the wrong way, or drop the sign.
Fix: The denominator is the positive NPV minus the negative NPV, which means the two absolute values added together. For example, 3,560 − (−5,520) = 9,080.
Adding the interpolated amount to the wrong rate
The formula starts from rate A. Some students add to the higher rate B.
Fix: Start from the rate that gave the positive NPV (the lower rate) and add the fraction of the gap. The answer must lie between A and B. If it does not, recheck.
Choosing the project with the higher IRR when ranking mutually exclusive projects
Students assume the higher percentage always means the better project.
Fix: IRR ignores scale and timing. Compare NPVs at the cost of capital and pick the higher NPV. Explain that NPV measures the absolute gain in shareholder wealth.
Using rates far apart, then quoting the answer as exact
The NPV curve is convex. A wide gap makes the straight-line estimate less accurate.
Fix: Keep the rates close together. State in your answer that the IRR is an approximation.
Applying the IRR rule to non-conventional cash flows without comment
Students apply the accept rule mechanically when there are outflows later in the project.
Fix: Check for more than one sign change in the cash flows. If there is one, say that multiple IRRs are possible and that NPV is more reliable.

Worked examples

Example 1

A project costs $100,000 now and produces net cash inflows of $40,000 a year for four years, starting at the end of year 1. The cost of capital is 12%. Use the annuity factors 20% (4 years) = 2.589 and 25% (4 years) = 2.362 to estimate the IRR and advise whether to accept the project.

Show the solution

The inflows are level, so first check the annuity factor: 100,000 ÷ 40,000 = 2.5. In the 4-year row this lies between 2.589 (20%) and 2.362 (25%), so the IRR is between 20% and 25%.
NPV at 20% = (40,000 × 2.589) − 100,000 = 103,560 − 100,000 = +3,560.
NPV at 25% = (40,000 × 2.362) − 100,000 = 94,480 − 100,000 = −5,520.
IRR = 20% + [3,560 ÷ (3,560 + 5,520)] × (25 − 20).
3,560 ÷ 9,080 = 0.392, and 0.392 × 5 = 1.96, so IRR ≈ 20 + 1.96 = 21.96%, which is 22.0% to one decimal place.
Compare with the cost of capital: 22.0% is well above 12%.

Answer: The IRR is approximately 22.0% (21.96%). This is above the 12% cost of capital, so the project should be accepted. The IRR is an estimate because it uses linear interpolation.

Example 2

A project needs an outlay of $200,000 now. It returns net cash inflows of $80,000, $90,000 and $100,000 at the end of years 1, 2 and 3. The cost of capital is 14%. Using discount factors for 10%: 0.909, 0.826, 0.751 and for 20%: 0.833, 0.694, 0.579, estimate the IRR and state the decision.

Show the solution

NPV at 10%: (80,000 × 0.909) + (90,000 × 0.826) + (100,000 × 0.751) = 72,720 + 74,340 + 75,100 = 222,160.
NPV at 10% = 222,160 − 200,000 = +22,160.
NPV at 20%: (80,000 × 0.833) + (90,000 × 0.694) + (100,000 × 0.579) = 66,640 + 62,460 + 57,900 = 187,000.
NPV at 20% = 187,000 − 200,000 = −13,000.
IRR = 10% + [22,160 ÷ (22,160 + 13,000)] × (20 − 10).
22,160 ÷ 35,160 = 0.630, and 0.630 × 10 = 6.30, so IRR ≈ 10 + 6.30 = 16.3%.
Compare with the cost of capital: 16.3% is above 14%. Because the two rates are 10 points apart, the straight-line estimate overstates the true IRR, which is about 16.0%. The margin over 14% is therefore a little smaller than the 2.3 points the estimate suggests, at about 2 points. The accept decision still holds.

Answer: The IRR is approximately 16.3%. It exceeds the 14% cost of capital, so accept the project. The two rates are 10 points apart, so the figure is a rough estimate that slightly overstates the true IRR (about 16.0%), and the margin over 14% is a little smaller than 2.3 points. A closer second rate such as 15% or 16% would improve accuracy.

Exam tips

In Section C, show both NPVs and the interpolation line in full. Method marks are available even if you make an arithmetic slip.
In objective test questions, check whether the options are given to one decimal place. Do the interpolation properly rather than guessing from the nearest rate.
For a conflict between NPV and IRR, name the cause (scale, timing or reinvestment assumption) and recommend the higher NPV for mutually exclusive projects.
Remember that the cash flows given may already include inflation and tax. Use them as presented and discount at the rate stated in the question.
Always end with a clear decision and a one-line comment that the IRR is an approximation.

Practice questions from Investment appraisal techniques

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 FM?

IRR = A + [NPV at A ÷ (NPV at A − NPV at B)] × (B − A). A is the lower rate with a positive NPV and B is the higher rate with a negative NPV. The result is an estimate because it assumes a straight line between the two NPVs.

How do I calculate IRR by hand?

Calculate NPV at two discount rates so that one NPV is positive and one is negative. Apply the interpolation formula to find the rate where NPV would be zero. Keep the two rates close together for a better estimate.

What is the difference between NPV and IRR?

NPV is an absolute figure in money that shows the value a project adds at the cost of capital. IRR is a percentage that shows the project's break-even discount rate. NPV considers project size, while IRR does not.

Why do IRR and NPV sometimes give conflicting decisions?

Conflict arises when you rank mutually exclusive projects of different size or with different timing of cash flows. A small project can have a higher IRR but a lower NPV. In that case choose the higher NPV, as it increases shareholder wealth the most.