Two projects both pass the NPV test. One returns 20% on ₦100,000. The other returns 15% on ₦1,000,000. A manager who looks only at the percentage picks the first. The shareholders would be better off with the second, because it adds more money. The internal rate of return (IRR) is popular because a percentage is easy to read, but it can mislead.
This topic shows how to calculate it, when it fails and how the modified internal rate of return (MIRR) repairs part of the problem. It also shows which costs belong in an appraisal at all.
This topic covers the internal rate of return and how to estimate it, the decision rule, the merits of NPV and IRR (including conflicts, multiple IRRs and no IRR), the modified internal rate of return and relevant costs and benefits for materials, existing equipment, working capital, labour and opportunity costs. The previous topic covered NPV, annuities and perpetuities. The next covers tax and capital allowances.
The November 2014 Pathfinder paper asked for NPV and MIRR on a cement project.
In 30 seconds
The IRR is the discount rate at which NPV is zero. Accept if IRR is at or above the required return. Estimate it by linear interpolation between a positive and a negative NPV: IRR = A% + [NPVA ÷ (NPVA − NPVB)] × (B − A)%. For single accept or reject decisions, NPV and IRR agree. For mutually exclusive projects they can conflict, and NPV is the better guide.
IRR can have multiple values or none, and assumes cash flows are reinvested at the IRR. MIRR assumes reinvestment at the cost of capital: MIRR = (terminal value ÷ PV of investment)1/n − 1. Relevant costs are future, cash, incremental flows. Ignore sunk costs, depreciation and unchanged fixed costs. Use opportunity cost for scarce resources.
Key terms in plain English
| Term | What it means |
|---|---|
| Internal rate of return (IRR) | The project’s own percentage return: the rate that gives NPV = 0 |
| Interpolation | Estimating a value between two known points as if they lie on a straight line |
| Mutually exclusive | Projects where choosing one means rejecting the others |
| Reinvestment assumption | The rate at which a method assumes project cash can be reinvested |
| Modified IRR (MIRR) | A return that assumes cash is reinvested at the cost of capital |
| Relevant cost | A future, incremental cash flow caused by the decision |
| Sunk cost | Money already spent, which no decision can recover |
| Opportunity cost | The benefit lost by using a resource for one purpose and not the next best use |
Syllabus check
This topic is in syllabus area B (advanced investment appraisal, 30% of the marks). It covers:
- the internal rate of return, the modified internal rate of return and their relative merits;
- the cash flows that are relevant to an investment decision;
- the treatment of materials, equipment, working capital, labour and opportunity costs.
CAMA 2020 and appraisal
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria. The sections below apply to this topic.
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Projects and decisions should add value to the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should test the assumptions behind a decision |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
The internal rate of return
The IRR is the project’s average yearly return, discounted. It is the rate at which the NPV of the cash flows is zero. The decision rule: if the IRR is equal to or above the minimum required return, accept the project. If it is lower, reject it. For a single project, NPV and IRR give the same decision.
Estimating the IRR
- Calculate the NPV at two discount rates. Ideally one NPV is positive and the other negative, and both are near zero.
- The IRR lies between them. Assume a straight line between the two points.
- Use the formula: IRR = A% + [NPVA ÷ (NPVA − NPVB)] × (B − A)%, where A is the lower rate with the positive NPV and B the higher rate with the negative NPV.
The true NPV curve bends, so interpolation gives an estimate, usually slightly above the actual IRR for a typical project. It is more accurate when both NPVs are close to zero and have opposite signs. Spreadsheet functions find the IRR by repeating the calculation until NPV is almost exactly zero.
ICAN’s example. A project costs ₦80,000 and returns ₦20,000, ₦36,000, ₦30,000 and ₦17,000 in years 1 to 4. The business requires 12%.
- NPV at 10% = +₦2,057.
NPV at 15% = −₦5,920. - IRR = 10% + [2,057 ÷ (2,057 + 5,920)] × 5% = 10% + 1.29% = 11.3%.
- 11.3% is below the 12% required, so reject.
NPV or IRR?
What they share
Both use cash flows, not accounting profit, both allow for the time value of money, and both consider all the cash flows and not just those inside a payback period.
NPV
Advantage: it gives an absolute money value and is consistent with maximising shareholders’ wealth. Criticisms: it is harder to explain to people without finance training, and it needs a cost of capital that may be uncertain. The replies: decision makers are educated and can have it explained, and capital markets are often efficient enough to give a usable rate.
IRR also needs a benchmark, normally the cost of capital, so it faces the same problem.
IRR
Advantage: a percentage is easier to grasp than a money value, and the IRR can be calculated without knowing the cost of capital. Both points are weaker than they look. The percentage is an average that allows for time value, which also needs a financial education to understand, and once calculated the IRR must be compared with the cost of capital.
Disadvantages:
- It is a relative measure. It cannot choose between a 10% return on ₦10,000 and a 10% return on ₦10,000,000. People, not techniques, make decisions, but the figure alone does not show the scale.
- Mutually exclusive projects. NPV and IRR can rank them differently. Resolve the conflict with NPV. NPV assumes cash is reinvested at the cost of capital, which is realistic, while IRR assumes reinvestment at the IRR itself.
- Multiple IRRs. If some cash flows during the project are negative, there may be two or more IRRs. Which one is right?
- No IRR. Some projects have none. NPV solves both problems.
The modified internal rate of return
IRR assumes that all cash earned by the project is reinvested at the IRR. If a project has an NPV of ₦300m at an 8% cost of capital and an IRR of 14%, the IRR assumes reinvestment at 14%, though the company’s cost of capital is 8%. MIRR assumes reinvestment at the cost of capital, which is more realistic.
- Step 1: find the PV of the investment phase: the negative cash flows in the early years, discounted to time 0 at the cost of capital, and added to the time 0 outflow.
- Step 2: compound each positive cash flow of the recovery phase to the end of the project at the cost of capital. Add the cash flow of the final year. This is the terminal value.
- Step 3:
MIRR = (terminal value ÷ PV of investment)1/n − 1, where n is the project life in years.
ICAN’s example. The required return is 8%. Cash flows: year 0 (₦60,000); year 1 (₦20,000); year 2 ₦30,000; year 3 ₦50,000; year 4 ₦40,000; year 5 (₦10,000).
- PV of the investment phase.
60,000 + 20,000 × 0.926 = ₦78,520. - Terminal value.
30,000 × 1.083 + 50,000 × 1.082 + 40,000 × 1.08 − 10,000 = 37,791 + 58,320 + 43,200 − 10,000 = ₦129,311. - MIRR = (129,311 ÷ 78,520)1/5 − 1 = 10.5%.
The IRR is about 11.5%.
MIRR is lower because the reinvestment rate is lower.
Relevant costs in investment decisions
Relevant costs and benefits are future cash flows that arise as a direct consequence of the decision.
- Cash: ignore depreciation and other non-cash items.
- Future: ignore costs already incurred, such as ₦20,000 spent on an earlier investigation (sunk).
- Incremental: ignore costs that will arise whatever is decided.
Materials
If the material must be bought, the relevant cost is the future purchase price. If it is in inventory and in regular use, so that it will be replaced, the relevant cost is the replacement cost. If it is in inventory and will not be replaced, the relevant cost is the higher of its disposal value and the contribution from its next best use.
Existing equipment
If the project uses equipment the firm already owns, the relevant cost is the higher of its current disposal value and the present value of the cash it could earn in an alternative use. ICAN also explains deprival value: the loss to the business if it were deprived of the asset. Its recoverable amount is the higher of net realisable value and value in use.
If the recoverable amount is below the replacement cost, the asset would not be replaced and the relevant cost is the recoverable amount. If it is above, the asset would be replaced, and the relevant cost is the replacement cost.
Working capital
An increase in working capital is treated as a cash outflow, and a reduction as an inflow, because actual cash flow differs from cash profit by the change in working capital. A cash flow early in a year is treated as at the end of the previous year.
Labour
- If labour is variable and not scarce, the relevant cost is the variable cost.
- If labour is a fixed cost and there is spare time, the relevant cost is nil.
- If labour is scarce, the relevant cost is the pay plus the contribution lost by moving the workers from their next best use.
Opportunity costs
The benefit lost by using a resource in one way and not the next best way, often measured as contribution forgone or as a present value in DCF. In ICAN’s example, six scarce IT specialists on salaries of ₦100,000 each cost ₦600,000 in pay plus ₦500,000 of lost contribution, a relevant cost of ₦1,100,000.
The rules in more depth
Why NPV wins a conflict
Shareholders gain money, not percentages. A project with a lower IRR but a bigger NPV leaves them richer. The reinvestment assumption backs this up: cost of capital is the realistic rate for reinvesting cash.
Why a negative cash flow later creates multiple IRRs
The sign of the flows changes more than once, so the NPV curve can cross zero at more than one rate. IRR has no single answer.
Why sunk costs are ignored
They are the same whichever option is chosen. Including them would only bias the decision.
Why opportunity cost matters for scarce resources
Using a scarce resource here means losing what it would have earned there. That loss is a real cost of the decision.
Effect on analysis
List each item, say whether it is relevant and why, give the number, and add up the relevant cash flows by year.
Worked examples
Example 1: a conflict between NPV and IRR. Project X costs ₦100,000 and returns ₦120,000 in one year. Project Y costs ₦1,000,000 and returns ₦1,150,000 in one year. They are mutually exclusive. The cost of capital is 10%.
- IRR of X = 20%. IRR of Y = 15%. IRR prefers X.
- NPV of X.
120,000 ÷ 1.10 − 100,000 = ₦9,091.
NPV of Y = 1,150,000 ÷ 1.10 − 1,000,000 = ₦45,455. - Choose Y: it adds ₦36,364 more to shareholder wealth.
Example 2: multiple IRRs. A project has cash flows (₦100), +₦230 and (₦132) at times 0, 1 and 2.
- At 10%: −100 + 230 ÷ 1.10 − 132 ÷ 1.21 = −100 + 209.09 − 109.09 = 0.
- At 20%: −100 + 230 ÷ 1.20 − 132 ÷ 1.44 = −100 + 191.67 − 91.67 = 0.
- There are two IRRs, 10% and 20%, so the IRR cannot give one answer. Use NPV at the cost of capital.
Example 3: relevant cost of materials. A project needs 5,000 kg of material X a year. The firm holds 4,000 kg bought at ₦400 a kg. The current price is ₦420 and X is in regular use. It also needs material Y, of which 1,500 kg are held at a historic cost of ₦144,000, no longer used, and sellable as scrap for ₦36,000.
- X is in regular use, so the relevant cost is replacement cost: 5,000 × 420 = ₦2,100,000.
The historic ₦400 is irrelevant. - Y in stock is not replaced, so use the scrap value of ₦36,000 for the 1,500 kg, plus the purchase price for any further kilograms needed.
Example 4: deprival value. A machine has a net realisable value of ₦3m and a value in use of ₦5m. It is needed for a new contract.
- Recoverable amount = higher of 3m and 5m = ₦5m.
- If replacement costs ₦4.5m, the machine would be replaced, so the relevant cost is the replacement cost, ₦4.5m.
- If replacement costs ₦6m, it would not be replaced, so the relevant cost is the recoverable amount, ₦5m.
How to score full marks
- Choose two rates that bracket the IRR and show both NPVs.
- Compare the IRR with the required return and state the decision.
- For conflicts, recommend NPV and give the reinvestment reason.
- For MIRR, show the investment PV, the terminal value and the root.
- For relevant cost, state for each item whether it is relevant and why.
- Exclude sunk costs, depreciation, financing and apportioned overheads.
Common mistakes in the exam
- Choosing rates for interpolation that are far apart.
- Choosing the higher IRR between mutually exclusive projects.
- Including interest, apportioned head office costs or sunk survey fees.
- Using historic cost for inventory that will be replaced.
- Forgetting the lost contribution when labour is scarce.
- Confusing the investment phase and the recovery phase in MIRR.
Quick revision
- IRR: NPV = 0. Interpolate. Accept if IRR is at least the required return.
- NPV beats IRR for mutually exclusive projects; IRR can be multiple or absent.
- MIRR = (TV ÷ PV of investment)1/n − 1.
- Relevant: future, cash, incremental. Opportunity cost for scarce resources.
- Materials: purchase price, replacement cost, or the higher of disposal and best use.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. The IRR is the discount rate at which NPV equals:
- Zero
- One
- The cost of capital
- Cash profit
Show the answer
A. It is the project’s own return.
Question 2. If a project’s IRR is above the required return, the project is:
- Rejected
- Accepted, other things equal
- Delayed
- Taxed
Show the answer
B. The decision rule.
Question 3. For mutually exclusive projects with conflicting rankings, use:
- IRR
- NPV
- Payback
- ROCE
Show the answer
B. It has a realistic reinvestment assumption.
Question 4. IRR assumes cash is reinvested at:
- The cost of capital
- The IRR
- Zero
- The inflation rate
Show the answer
B. MIRR assumes the cost of capital.
Question 5. A project with changing signs of cash flow may have:
- Multiple IRRs
- No NPV
- A negative cost of capital
- A perfect IRR
Show the answer
A. NPV avoids the problem.
Question 6. Sunk costs are:
- Relevant
- Irrelevant to the decision
- Always doubled
- Depreciated
Show the answer
B. They cannot be recovered or changed.
Question 7. Material in regular use, held in stock, is costed at:
- Historic cost
- Replacement cost
- Zero
- Scrap value
Show the answer
B. It will be replaced.
Question 8. Labour with spare capacity and a fixed cost has a relevant cost of:
- Full pay
- Nil
- Double pay
- Overtime rate
Show the answer
B. No extra cash is spent.
Question 9. The MIRR terminal value is found by:
- Discounting the inflows
- Compounding the recovery-phase cash flows at the cost of capital
- Adding the outflows
- Taking the IRR
Show the answer
B. To the end of the project.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test an appraisal’s assumptions.
Written questions
Question 11 (10 marks). (a) Explain the IRR and the decision rule. (4 marks) (b) State three weaknesses of the IRR compared with NPV. (6 marks)
Question 12 (20 marks). Kano Plc is considering a project costing ₦120m, returning ₦50m, ₦60m and ₦55m in years 1 to 3. The cost of capital is 10%. Discount factors: 10%: 0.909, 0.826, 0.751. 20%: 0.833, 0.694, 0.579. (a) Calculate the NPV at 10% and 20% and estimate the IRR. (8 marks) (b) Calculate the MIRR.
(6 marks) (c) The project would use 2,000 kg of a material held in stock at a historic cost of ₦300 a kg, which is in regular use, with a replacement cost of ₦340 a kg. State and calculate its relevant cost, and say what is irrelevant. (3 marks) (d) State two other items you would leave out of the cash flows. (3 marks)
Question 13 (10 marks). (a) Project X costs ₦100,000 and returns ₦120,000 in one year. Project Y costs ₦1,000,000 and returns ₦1,150,000 in one year. They are mutually exclusive and the cost of capital is 10%. Calculate the IRR and NPV of each and advise. (6 marks) (b) Explain the reinvestment assumption in IRR and in MIRR. (4 marks)
Model answers
Show the model answer to Question 11
(a) IRR (4 marks). The IRR is the discount rate at which the NPV of the project’s cash flows is zero, the average yearly return allowing for time value. (2) Accept if it is at least the required return and reject if it is lower. (2)
(b) Weaknesses (6 marks). It is a relative measure and does not show the money value added. (2) It can conflict with NPV for mutually exclusive projects, and NPV should then be used. (2) It may give multiple IRRs or none, and assumes reinvestment at the IRR. (2)
Show the model answer to Question 12
(a) IRR (8 marks).
- NPV at 10% = 50 × 0.909 + 60 × 0.826 + 55 × 0.751 − 120 = 45.45 + 49.56 + 41.31 − 120 = +₦16.3m. (3)
- NPV at 20% = 50 × 0.833 + 60 × 0.694 + 55 × 0.579 − 120 = 41.65 + 41.64 + 31.85 − 120 = −₦4.9m. (3)
- IRR = 10% + [16.3 ÷ (16.3 + 4.9)] × 10% = 17.7%. (2)
(b) MIRR (6 marks).
- PV of investment = ₦120m. (1)
- Terminal value.
50 × 1.102 + 60 × 1.10 + 55 = 60.5 + 66 + 55 = ₦181.5m. (3) - MIRR = (181.5 ÷ 120)1/3 − 1 = 14.8%. (2)
(c) Material (3 marks).
- It is in regular use, so the relevant cost is replacement cost: 2,000 × 340 = ₦680,000. (2)
- The historic cost of ₦300 is irrelevant. (1)
(d) Exclusions (3 marks). Any two: depreciation; interest and loan repayments; sunk costs such as earlier survey fees; apportioned head office charges. (1.5 each, up to 3)
Show the model answer to Question 13
(a) Project choice (6 marks).
- IRR of X.
20% and of Y = 15%. (1) - NPV of X.
120,000 ÷ 1.10 − 100,000 = ₦9,091. - NPV of Y.
1,150,000 ÷ 1.10 − 1,000,000 = ₦45,455. (3) - Choose Y, which has the higher NPV and adds more to shareholder wealth, although its IRR is lower. (2)
(b) Reinvestment (4 marks). IRR assumes that cash earned is reinvested at the IRR itself. (2) MIRR assumes reinvestment at the cost of capital, which is more realistic. (2)
Pathfinder on IRR, MIRR and relevant cash flows
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
The compulsory question asked for the NPV (₦404m) and the MIRR of a cement project at 13%, then a recommendation using both. The model answer divides the PV of the return stage (₦2,509m) by the investment at time 0 (₦2,105m), takes the fifth root, multiplies by 1.13 and subtracts 1, giving about 17%, which beats the 13% cost of capital, so the project is accepted. It also marks loan repayments and interest, apportioned overheads and sunk survey costs as irrelevant. Read the sections on the MIRR and on relevant costs.
Revise first: The modified internal rate of return
The same question tests which items are irrelevant and how working capital is treated. See the section on relevant costs.
Revise first: Relevant costs in investment decisions
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: November 2014 question 1(b) and (c).
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