Internal Rate of Return and NPV versus IRR

Net present value says how many naira a project adds. Managers often prefer a percentage: “this project returns 20%” is easier to compare with a cost of money of 14% than “this project has a net present value (NPV) of ₦1.7m”. The internal rate of return answers that question.

It is the discount rate at which the net present value is exactly zero, in other words the average annual return the project is expected to earn on the money invested, allowing for the timing of the cash flows.

This topic explains the internal rate of return, how to find it by trial and interpolation (and quickly for level cash flows), the decision rule, a worked example, and then the important comparison with NPV: where they agree, where they conflict, and why NPV is preferred. The next topic covers tax and capital allowances in discounted cash flow.

The internal rate of return was tested in May Diet, 2026, Question 3 (20 marks), where three divisional projects with internal rates of return of 16%, 16% and 18% were compared with divisional costs of capital to decide which to accept. Read net present value and the case for discounted cash flow first.

The internal rate of return

The internal rate of return (IRR) is the cost of capital at which the project has an NPV of zero. In the NPV method the rate is the return required by the investor; in the IRR method it is the actual return expected from the project, an average annual percentage over its life, which is compared with the minimum required return.

Decision rule: undertake the project if its IRR exceeds the required return. For several independent projects, accept every project whose IRR exceeds the cost of capital; for mutually exclusive projects the IRR ranking is unreliable (see below).

Finding the IRR by interpolation

1. Calculate the NPV at a convenient rate, a.
2. Calculate the NPV at a second rate, b, chosen so that the NPV changes sign (a positive NPV means the IRR is higher than the rate, a negative NPV means it is lower).
3. Interpolate: IRR = a + [NPV_a ÷ (NPV_a − NPV_b)] × (b − a).

Delta Packaging, from the last topic, had an NPV of +₦1.70m at 12% and +₦0.05m at 20%, so the IRR is a little above 20%. At 24% the present values are 2.6 × 0.806 = 2.096, 3.1 × 0.650 = 2.015, 3.4 × 0.524 = 1.782 and 4.7 × 0.423 = 1.988, a total of 7.880, so the NPV is 7.880 − 8.5 = −₦0.62m. Interpolating between 20% and 24%: IRR = 20 + [0.0512 ÷ (0.0512 + 0.6197)] × 4 = 20 + 0.31 = 20.3%. Compared with a cost of capital of 12% the project should be accepted.

Interpolation assumes the NPV changes in a straight line between the two rates, which is not exactly true, so the answer is an approximation. It is more accurate when the two rates are close together (the earlier choice of 12% and 24% would have given about 20.8%).

Level cash flows: a shortcut

If the project has a single outlay and equal annual inflows, the IRR is the rate whose annuity factor equals outlay ÷ annual flow. A ₦24m outlay returning ₦7.5m a year for 5 years needs a factor of 24 ÷ 7.5 = 3.2. In the 5-year row of the table, 17% gives 3.199, so the IRR is about 17%.

NPV or IRR?

For a single project with an outlay followed by inflows, NPV and IRR give the same decision: the NPV is positive exactly when the IRR exceeds the cost of capital. They can disagree in other cases.

• Mutually exclusive projects. The IRR is a percentage and takes no account of scale or of the size of the naira return, so a smaller project with a higher IRR can add less value than a larger one with a lower IRR; the NPV measures value directly. A timing difference can also reverse the ranking, because the two projects respond differently to a change in the discount rate.
• Multiple IRRs. If the cash flows change sign more than once (for example a large cost at the end of a project, such as site restoration), there can be more than one rate at which the NPV is zero, or none, so the IRR is not meaningful.
• Reinvestment assumption. The NPV implicitly assumes that interim cash flows are reinvested at the cost of capital, which is realistic; the IRR assumes reinvestment at the IRR itself, which is usually optimistic for a high-return project.
• A changing cost of capital. The IRR is compared with one rate; if the discount rate changes over the life, the NPV can use a different rate in each year but the IRR cannot.

Example of a conflict. Project P costs ₦100m and returns ₦70m, ₦50m and ₦30m in years 1 to 3; project Q costs ₦100m and returns ₦20m, ₦50m and ₦90m. The IRR of P is 27.6% and of Q 21.8%, so the IRR prefers P. At a cost of capital of 5% the NPV of P is ₦37.9m and of Q ₦42.1m, so the NPV prefers Q; at 10% the order is reversed (P ₦27.5m, Q ₦27.1m).

Q has its cash later, so its value is more sensitive to the discount rate. Which is right depends on the true cost of capital, and the NPV, which uses it, is the correct basis. The crossing point of the two NPV profiles is about 9%.

Why the IRR is still used

Managers find a percentage easy to understand and compare with a target, it does not need the cost of capital to be known exactly (only whether the IRR is clearly above or below it). It is a measure of the margin of safety: a project with an IRR of 20% against a cost of 12% can tolerate a large error in the cash flows or the rate. These are good reasons for reporting it alongside the NPV.

The ideas in more depth

Why the IRR rule works for one project

The NPV falls as the discount rate rises. At the IRR it is zero; below the IRR it is positive; above it, negative. So if the cost of capital is below the IRR the NPV is positive. This holds only when there is one sign change, an outlay followed by inflows.

Why interpolation is an estimate

The NPV curve is a curve, not a straight line, so joining two points with a line gives an approximate answer. The nearer the two trial rates are to the true IRR, the closer the estimate, and in the exam it is enough to use rates 4 to 5 points apart that bracket the answer.

Sources and references

• No accounting standard governs the method. Discount factors and annuity factors are given in the tables on the formula sheet of the paper.

These points are stated as at 5 November 2026.

Worked examples

1. Share.
   0.0512 ÷ (0.0512 + 0.6197) = 0.076.
2. IRR.
   20 + 0.076 × 4 = 20.3%.

1. Factor.
   24 ÷ 7.5 = 3.2.
2. IRR. The 5-year factor at 17% is 3.199, so about 17%.

1. Compare. 20.3% exceeds 12%, so accept, with a margin of 8.3 percentage points.

How to answer in the exam

• Calculate two NPVs on either side of zero and interpolate; show the formula.
• For level flows, use the factor shortcut and check it in the table.
• State the decision against the required return.
• Compare with NPV, giving at least two of the weaknesses: scale, timing, multiple rates, reinvestment.

Common mistakes in the exam

• Choosing two trial rates that both give a positive NPV.
• Using the wrong sign in the interpolation.
• Ranking mutually exclusive projects by IRR.
• Comparing the IRR with a rate that is not the project’s required return.
• Treating the interpolated figure as exact.
• Using the IRR for projects with several sign changes.

Quick revision

• IRR = discount rate at which NPV = 0; accept if above the cost of capital.
• IRR = a + NPV_a ÷ (NPV_a − NPV_b) × (b − a).
• Level flows: outlay ÷ flow = annuity factor.
• NPV preferred: scale, timing, multiple IRRs, reinvestment.

Practice questions

These are original questions written in the style of the Institute’s exams. Try each one before you open the answer. Section A is a recall check, because the paper itself is written.

Recall check: multiple choice

Section B style: written questions

A further written question

Model answers

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