Every cash flow in an appraisal is a forecast, and forecasts are wrong. The sales may be lower, the materials dearer, the machine late, the life shorter. The net present value (NPV) of ₦16m on the page may become a loss in practice. The first question a sensible manager asks is not “what is the NPV?” but “how wrong can we be before the project stops being worth doing?”
Sensitivity analysis answers that question for each input in turn. This topic distinguishes risk from uncertainty, lists the methods used to deal with them, and then explains sensitivity analysis: the principle, how to calculate the sensitivity margin of each variable (price, volume, costs, outlay, cost of capital and project life), a full worked example, and the strengths and weaknesses of the method. The next topic covers expected values and decision trees.
Sensitivity analysis was part of the compulsory question in November Diet, 2025, Question 1: 6 marks asked for a statement showing how sensitive the NPV was to errors in product life, annual sales volume and material cost, and 8 marks for the strengths and weaknesses of the method and how simulation improves on it. Read inflation in discounted cash flow: real and money terms first.
In 30 seconds
Risk exists when the outcomes and their probabilities can be estimated; uncertainty exists when there is not enough information to know the outcomes or probabilities.
Sensitivity analysis takes each uncertain input one at a time and finds the change that would make the NPV zero: sensitivity margin = NPV ÷ present value (PV) of the variable × 100% (for sales volume use the PV of contribution; for the cost of capital use the internal rate of return (IRR); for the project’s life use the discounted payback period). The smaller the margin, the more critical the variable.
Strengths: simple, identifies the critical variables to check and monitor. Weaknesses: changes one variable at a time, ignores probability, and gives no decision rule.
Risk and uncertainty
Investment projects are long-term, so the cash-flow estimates may be wrong for two reasons. Risk exists when the actual outcome could be any of several known possibilities and it is not possible to say in advance which will happen. The probabilities can often be estimated, as with a die (each face has a probability of 1/6), so risk can be measured mathematically.
Uncertainty exists when there is not enough information to be sure what will happen or what the probabilities are: a forecast that sales in three years will be ₦500,000 may be largely guesswork. Uncertainty can be assessed, but with less mathematical precision. Management should take both into account, along with the expected return and NPV.
The main methods are sensitivity analysis (for uncertainty about cash flows), probability analysis (expected values, decision trees and the value of information, for risk), simulation, risk-adjusted discount rates and discounted payback. The next topics cover the other methods.
The method: sensitivity margins
Sensitivity analysis tests how sensitive the NPV is to a change in each item of cost or benefit. For each uncertain factor, calculate the change that would reduce the NPV to zero: it is breakeven analysis in NPV terms. It helps to find the present values of each element of cost and revenue separately. The sensitivity margin is:
Sensitivity = NPV ÷ PV of the variable × 100%
A worked example: Adaobi Ltd
Adaobi plans a 4-year project. The machine costs ₦50m. Annual sales are 100,000 units at ₦800, the variable cost is ₦500 a unit and fixed cash costs are ₦9m a year. The cost of capital is 10% (annual factor 3.170).
| Item | Annual ₦m | PV at 10% ₦m |
|---|---|---|
| Machine (year 0) | (50.00) | |
| Sales 100,000 × ₦800 | 80.0 | 253.60 |
| Variable costs 100,000 × ₦500 | (50.0) | (158.50) |
| Fixed costs | (9.0) | (28.53) |
| NPV | net 21.0 | 16.57 |
| Variable | Working | Margin | Meaning |
|---|---|---|---|
| Machine cost | 16.57 ÷ 50.00 | 33.1% | can rise by 33.1% |
| Selling price | 16.57 ÷ 253.60 | 6.5% | can fall by 6.5% |
| Sales volume | 16.57 ÷ (30 × 3.170 = 95.10) | 17.4% | can fall by 17.4% |
| Unit variable cost | 16.57 ÷ 158.50 | 10.5% | can rise by 10.5% |
| Fixed costs | 16.57 ÷ 28.53 | 58.1% | can rise by 58.1% |
| Cost of capital | IRR 24.5% against 10% | (24.5 − 10) ÷ 10 = 145% | can rise to 24.5% |
| Project life | discounted payback 2.86 years | (4 − 2.86) ÷ 4 = 28.5% | can fall to 2.86 years |
For sales volume, the PV used is that of the contribution (₦30m a year × 3.170), because volume changes both sales and variable costs. For the project’s life, the discounted payback period is found by accumulating the present values of the cash flows: 21 × 0.909 = 19.09, then 17.35, then 15.77, so the cumulative position is −30.91, −13.57, then +2.21 after year 3, and the project has paid back in 2 + 13.57 ÷ 15.77 = 2.86 years.
The selling price is the critical variable here: a fall of only 6.5% would remove the NPV, followed by the unit variable cost (10.5%) and the sales volume (17.4%). These are the figures to check, perhaps by market research, or to protect through fixed-price contracts.
The one-at-a-time approach can mislead. If the price fell by 5% (₦12.7m of present value lost) and the variable cost rose by 5% (₦7.9m), each within its own margin, the NPV would be 16.57 − 12.68 − 7.93 = −₦4.04m: a combination of two modest errors turns the project into a loss.
Strengths and weaknesses
- Strength: information for judgement. The results are presented in a form that lets management judge how likely each change is.
- Strength: identifies critical variables. The areas critical to success can be monitored, and where sales are critical, more market research may help; where the cost of materials is critical, fixed-price contracts, futures or options might limit it. These actions are not free: research costs money, option premiums must be paid and suppliers may ask for payments in advance.
- Strength: simple. It needs no complicated theory.
- Weakness: independence. It changes one variable at a time, but variables move together: if materials rise the firm may raise its price, with little effect on the NPV.
- Weakness: ignores probability. It shows how far a variable must change, not how likely that is.
- Weakness: no clear answer. It is not an optimising technique; it gives information from which a decision can be made, and does not point to a correct decision.
Simulation addresses the weaknesses by letting several variables vary together according to probability distributions, producing a range of NPVs and the probability of a loss, as the next topics explain.
The ideas in more depth
Why the margin is NPV divided by present value
A fall of x% in the selling price reduces the present value of sales by x% of that present value. The NPV reaches zero when x% × PV of sales = NPV, so x = NPV ÷ PV of sales. The same reasoning applies to every cash flow, which is why each variable needs its own present value, and why volume uses contribution (the net effect of a volume change) and not sales.
Why the answer is a screening tool
Sensitivity analysis tells management where to look, not what to decide. A margin of 6.5% on price is worrying only if prices really can fall by that much; a margin of 58% on fixed costs is reassuring because fixed costs are fairly predictable. The judgement about the likelihood is the manager’s.
Sources and references
- No accounting standard governs the technique. Disclosure of sources of estimation uncertainty in the financial statements is required by International Accounting Standard (IAS) 1 Presentation of Financial Statements, a related but separate matter.
These points are stated as at 5 November 2026.
Worked examples
Example 1: the price margin. NPV ₦16.57m; present value of sales ₦253.6m. Required: the sensitivity of price.
- Margin.
16.57 ÷ 253.6 = 6.5%.
Example 2: the volume margin. Annual contribution ₦30m; factor 3.170. Required: the sensitivity of volume.
- PV of contribution.
30 × 3.170 = ₦95.1m. - Margin.
16.57 ÷ 95.1 = 17.4%.
Example 3: the life. Discounted payback 2.86 years of 4. Required: the sensitivity of life.
- Margin.
(4 − 2.86) ÷ 4 = 28.5%.
How to answer in the exam
- Calculate the base NPV first with the present value of each element shown separately.
- Compute each margin as NPV ÷ relevant PV, using contribution for volume and discounted payback for life.
- Rank them and name the critical variable.
- Evaluate the technique: two or three strengths and weaknesses, and state that simulation improves on it.
Common mistakes in the exam
- Using sales instead of contribution for the volume margin.
- Using the NPV of the whole project as the divisor for every variable.
- Forgetting that costs sensitivity measures an increase while sales measures a decrease.
- Calculating the life margin without a discounted payback.
- Presenting margins without saying which is critical.
- Stating weaknesses without noting that variables move together.
Quick revision
- Risk: probabilities known. Uncertainty: not enough information.
- Margin = NPV ÷ PV of the variable × 100%.
- Volume: PV of contribution. Life: discounted payback. Cost of capital: the IRR.
- Strengths: simple, shows critical variables. Weaknesses: independence, no probability, no decision.
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
Question 1. Risk exists when:
- Nothing is known
- There is no information
- The possible outcomes and their probabilities can be estimated
- There is only one outcome
Show the answer
C. For example the six faces of a die.
Question 2. Uncertainty exists when:
- Probabilities are known exactly
- There is insufficient information to know the outcomes or probabilities
- The project is risk-free
- The NPV is positive
Show the answer
B. It is assessed less precisely than risk.
Question 3. Sensitivity analysis finds:
- The change in each variable that makes the NPV zero
- The probability of each outcome
- The best decision
- The average NPV
Show the answer
A. It is breakeven analysis in NPV terms.
Question 4. The sensitivity margin of a variable is:
- NPV × PV
- PV ÷ NPV
- NPV − PV
- NPV ÷ PV of the variable × 100%
Show the answer
D. The percentage change that eliminates the NPV.
Question 5. The divisor for the sales volume margin is the PV of:
- Sales
- Contribution
- Fixed costs
- The outlay
Show the answer
B. Volume changes sales and variable costs together.
Question 6. The most critical variable is the one with the:
- Smallest margin
- Largest margin
- Largest cash flow
- Longest life
Show the answer
A. It takes the smallest error to remove the NPV.
Question 7. A weakness of sensitivity analysis is that it:
- Is too complex
- Gives probabilities
- Changes variables one at a time and ignores probability
- Needs no forecasts
Show the answer
C. It also gives no decision rule.
Question 8. A strength of sensitivity analysis is that it:
- Removes risk
- Gives the optimum
- Predicts the future exactly
- Identifies the critical variables to monitor
Show the answer
D. And it is simple.
Question 9. The margin for the project’s life uses:
- The payback period
- The ARR
- The discounted payback period
- The tax rate
Show the answer
C. The point at which cumulative PV turns positive.
Question 10. The break-even cost of capital is:
- The IRR
- The inflation rate
- The tax rate
- Zero
Show the answer
A. The rate at which the NPV is zero.
Section B style: written questions
Question 11 (10 marks). (a) Distinguish between risk and uncertainty. (4 marks) (b) State the formula for a sensitivity margin and the divisor for volume. (3 marks) (c) State THREE strengths of sensitivity analysis. (3 marks)
Question 12 (15 marks). A project costs ₦10m and returns ₦3.0m a year for 5 years. The cost of capital is 10% (annuity factor 3.791). (a) Calculate the NPV. (4 marks) (b) Calculate the sensitivity of the outlay and of the annual cash flow. (6 marks) (c) The annuity factor needed for a zero NPV is 3.333; given factors of 3.352 at 15% and 3.274 at 16%, estimate the IRR. (5 marks)
A further written question
Question 13 (20 marks). A project costs ₦40m and lasts 5 years at a cost of capital of 8% (annuity factor 3.993; year factors 0.926, 0.857, 0.794, 0.735, 0.681). Annual sales are 20,000 units at ₦2,100; materials ₦700 a unit; other variable costs ₦500 a unit; fixed cash costs ₦6m.
(a) Calculate the NPV. (6 marks) (b) Calculate the sensitivity of sales volume and of material cost. (6 marks) (c) Calculate the sensitivity of the project’s life. (5 marks) (d) State TWO weaknesses of the method. (3 marks)
Model answers
Show the model answer to Question 11
(a) Risk: outcomes and their probabilities can be estimated (2). Uncertainty: insufficient information to know the outcomes or probabilities (2).
(b) NPV ÷ PV of the variable × 100% (2); for volume, the PV of contribution (1).
(c) Simple; identifies critical variables; shows information for managerial judgement (1 each).
Show the model answer to Question 12
(a)
- 3.0 × 3.791 = 11.373
- NPV = 11.373 − 10 = +₦1.37m (4).
(b)
- Outlay.
1.373 ÷ 10 = 13.7% (3). - Annual cash flow.
1.373 ÷ 11.373 = 12.1% (3).
(c) IRR = 15 + (3.352 − 3.333) ÷ (3.352 − 3.274) × 1 = 15.2% (5).
Show the model answer to Question 13
(a) Sales 42m; materials 14m; other variable 10m; fixed 6m; annual cash flow 12m (3); PV 12 × 3.993 = 47.92; NPV = 47.92 − 40 = +₦7.92m (3).
(b)
- Volume.
contribution 42 − 14 − 10 = 18m; PV 71.87; 7.92 ÷ 71.87 = 11.0% (3). - Materials.
PV 14 × 3.993 = 55.90 - 7.92 ÷ 55.90 = 14.2% (3).
(c) Cumulative PVs: −28.89, −18.60, −9.08, −0.26, +7.92; discounted payback 4 + 0.26 ÷ 8.17 = 4.03 years (3); margin (5 − 4.03) ÷ 5 = 19.4% (2).
(d) (1.5 each) Variables are changed one at a time though they move together; probability is ignored; no decision rule is given.
Pathfinder on Risk and uncertainty in appraisal: sensitivity analysis
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.
Part (b) asks for a statement of how sensitive the NPV is to errors in product life, annual sales volume and material cost per unit (6 marks).
Revise first: The method: sensitivity margins
The strengths and weaknesses of sensitivity analysis as a method of evaluating risk and uncertainty, and how simulation improves on it.
Revise first: Strengths and weaknesses
Source: ICAN Pathfinder, Financial Management. Diets covered: Every diet in the Pathfinder, November Diet, 2025 and May Diet, 2026, was reviewed here. The diets listed above test this topic directly..
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