A forecast of ₦5m a year is a guess dressed as a number. Sales may fall short, costs may overrun, the machine may be needed for fewer years. A manager who reports only the single NPV is hiding the part that matters most: how wrong can the forecast be before the decision flips? This topic shows the tools for answering that.
This topic covers risk and uncertainty, sensitivity analysis (how far each input can move before NPV reaches zero), discounted payback as a measure of life sensitivity, expected values of NPV with probabilities, the standard deviation of NPV and the strengths and weaknesses of each. The previous topic covered inflation. The next covers decision trees, risk-adjusted discount rates and simulation.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on this topic.
In 30 seconds
Risk means several possible outcomes with probabilities that can be estimated. Uncertainty means too little information to assign probabilities. Sensitivity analysis asks: by how much can each input change before NPV falls to zero? Sensitivity % = NPV ÷ PV of the item × 100. The smaller the percentage, the more critical the input. For project life, use the discounted payback period.
Expected value (EV) = Σpx. Accept if the EV of NPV is positive, but look at the spread too: the standard deviation measures it, and the chance of a negative NPV shows the downside. An EV is an average that may never happen, and probabilities are often subjective.
Key terms in plain English
| Term | What it means |
|---|---|
| Risk | Several possible outcomes whose probabilities can be estimated |
| Uncertainty | Too little information to know the outcomes or their probabilities |
| Sensitivity analysis | Finding how much each input can change before NPV reaches zero |
| Discounted payback | The time taken for discounted cash flows to repay the investment |
| Expected value (EV) | A probability-weighted average of possible outcomes |
| Standard deviation | A measure of how widely outcomes are spread around the EV |
| Probability distribution | The list of outcomes and their probabilities |
Syllabus check
This topic is in syllabus area B (advanced investment appraisal, 30% of the marks). It covers:
- sensitivity analysis and comments on its results;
- expected values as a way to take account of risk and uncertainty;
- the strengths and limits of each technique.
CAMA 2020 and risk in appraisal
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 | Risk should be weighed against return for the benefit of the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should test the risk and the assumptions in a proposal |
| 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 |
Risk and uncertainty
Cash flow estimates may be wrong because of risk and uncertainty. In a dice roll, the outcomes are 1 to 6, each with probability one-sixth: that is risk, and it can be measured with probabilities. When a business predicts sales in three years’ time of ₦500,000 on little information, that is uncertainty, which can still be assessed but with less precision. Management should consider both, as well as the expected return.
Methods include sensitivity analysis and probability analysis, and also simulation, risk-adjusted discount rates and discounted payback, which the next topic covers.
Sensitivity analysis
Sensitivity analysis takes each uncertain input in turn and finds the change needed to bring NPV to zero. It is break-even analysis in NPV terms. To do it, find the PV of each element of cost and revenue separately.
Sensitivity (%) = NPV ÷ PV of the item × 100, where the item is the one being tested.
- Initial investment: NPV ÷ investment.
- Selling price: NPV ÷ PV of sales revenue.
- Sales volume: NPV ÷ PV of contribution (sales less variable costs).
- Variable cost per unit: NPV ÷ PV of variable costs.
- Fixed costs: NPV ÷ PV of fixed costs.
- Cost of capital: the IRR is the break-even rate. Sensitivity = (IRR − cost of capital) ÷ cost of capital.
- Project life: use the discounted payback. If it is 3.2 years on a 4-year project, the life can fall by about 20% before NPV is zero.
ICAN’s example (Kola Ltd): NPV ₦785,000; investment ₦3,750,000, so it can rise by 20.9%. Sales revenue PV is ₦16,195,000, so price can fall by 4.8%. Volume can fall by 12.1%, variable costs rise by 8.1% and fixed costs rise by 40.4%. The discounted payback is 3.21 years.
Strengths and weaknesses
Strengths: it shows which inputs are critical, so management can monitor them or reduce the risk (more market research on price or volume, fixed-price contracts or hedging on materials, noting that these cost money). It is simple. It gives information in a form that supports judgement.
Weaknesses: it treats inputs as independent when they often move together (higher materials cost may be passed into price). It ignores probability: it says how far a variable must move but not how likely that is. It does not give a decision, only information.
Expected values
An expected value is a weighted average: EV = Σpx, where p is the probability and x the outcome. Applied to a project, x is the NPV for each outcome, and the EV of the NPV is the basic measure. Accept if the EV of the NPV is positive, but a project may still be refused if the risk seems too large for the return.
ICAN’s example. A five-year project costs ₦200,000. NPVs and probabilities: (₦8,000) at 0.10; ₦4,000 at 0.30; ₦12,000 at 0.40; ₦20,000 at 0.20.
- EV = (0.10 × −8,000) + (0.30 × 4,000) + (0.40 × 12,000) + (0.20 × 20,000) = −800 + 1,200 + 4,800 + 4,000 = ₦9,200.
- The EV is positive, so undertake it provided the risk is acceptable.
- There is a 10% chance of a negative NPV of ₦8,000.
The spread of outcomes
To measure risk, compute the standard deviation: take each outcome’s difference from the EV, square it, weight it by its probability, add the results (the variance) and take the square root. A larger standard deviation means more spread, so more risk. The probability of a negative NPV is simply the sum of probabilities of the outcomes below zero.
The coefficient of variation (standard deviation ÷ EV) compares risk per unit of return across projects.
Advantages and disadvantages of EV
Advantages: it uses all the outcomes and gives one figure for the decision.
Disadvantages: probabilities are subjective; the EV may never actually happen; it suits repeated events better than a one-off project; and by itself it does not show the risk.
The rules in more depth
Why the smallest percentage matters most
A small percentage means a small error in that input wipes out the NPV. That is the input to check and control.
Why independence is a problem
If price and volume move together, testing each alone gives a misleading picture of the combined effect.
Why EV alone can mislead
A project with an EV of ₦20m could come from a safe ₦20m or from a gamble between ₦100m and a loss of ₦60m. The spread tells the two apart.
Why discounted payback measures life risk
If cash flows stop early, the project must still repay the investment in PV terms. The payback shows how much life is needed.
Effect on analysis
Give the NPV, rank the inputs by sensitivity, state the EV and the spread, and recommend with reasons.
Worked examples
Example 1: sensitivity. A project has an NPV of ₦500,000, an investment of ₦2,000,000, PV of sales ₦6,000,000, PV of variable costs ₦3,200,000 and PV of fixed costs ₦800,000.
- Investment.
500 ÷ 2,000 = 25.0%. - Price.
500 ÷ 6,000 = 8.3%.
Volume: 500 ÷ (6,000 − 3,200) = 17.9%. - Variable cost.
500 ÷ 3,200 = 15.6%.
Fixed cost: 500 ÷ 800 = 62.5%.
Price is the most critical input.
Example 2: expected value and spread. NPV of ₦80m with probability 0.3, ₦30m with 0.5 and −₦40m with 0.2.
- EV = 24 + 15 − 8 = ₦31m.
- Variance = 0.3(80−31)2 + 0.5(30−31)2 + 0.2(−40−31)2 = 720.3 + 0.5 + 1,008.2 = 1,729.
Standard deviation = ₦41.6m. - The chance of a loss is 20%.
Example 3: ICAN’s two-state case. An investment of ₦1m has a 75% chance of a strong economy (NPV ₦364,500) and 25% of a weak one (NPV −₦476,950).
- EV = 0.75 × 364,500 + 0.25 × (−476,950) = 273,375 − 119,238 = ₦154,138.
- The EV is positive, so the project passes.
- There is a 25% risk that the NPV is negative.
Example 4: discounted payback. A ₦3,000k project returns ₦1,100k a year for 4 years. The cost of capital is 10%.
- Cumulative PV: year 1, 1,000; year 2, 1,909; year 3, 2,735; year 4, 3,487.
- Payback = 3 + (3,000 − 2,735) ÷ 752 = 3.35 years.
- The life could fall by about 16% (0.65 ÷ 4) before NPV reaches zero.
How to score full marks
- Calculate the PV of each input before the sensitivities.
- Present the sensitivities in a table and rank them.
- Comment on the most critical inputs and how to manage them.
- For EV, show the table and state the decision.
- Add the standard deviation or the chance of a loss if risk is asked.
- Give limits of each method.
Common mistakes in the exam
- Dividing by the wrong PV (for volume, use contribution).
- Using undiscounted amounts in sensitivity.
- Calling an EV the most likely outcome.
- Not mentioning the downside risk.
- Saying sensitivity analysis gives the answer.
- Forgetting that inputs may move together.
Quick revision
- Sensitivity % = NPV ÷ PV of the item; volume uses contribution; life uses discounted payback.
- EV = Σpx; accept if positive and the risk is tolerable.
- Standard deviation and chance of loss show the spread.
- Sensitivity ignores probability and assumes independence.
- 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. Risk differs from uncertainty because with risk:
- Probabilities can be estimated
- Nothing is known
- Outcomes are certain
- Tax is ignored
Show the answer
A. Uncertainty lacks them.
Question 2. Sensitivity of selling price is NPV divided by the PV of:
- Sales revenue
- Fixed costs
- The investment
- Contribution
Show the answer
A. Then multiplied by 100.
Question 3. Sensitivity of sales volume uses the PV of:
- Sales revenue
- Contribution
- Fixed costs
- Tax
Show the answer
B. Sales less variable costs.
Question 4. The inputs with the smallest sensitivity percentages are:
- Least critical
- Most critical
- Irrelevant
- Fixed
Show the answer
B. Small changes cancel the NPV.
Question 5. Life sensitivity is found using:
- Discounted payback
- Accounting rate of return
- Tax rate
- Inflation
Show the answer
A. Compared with the project life.
Question 6. EV =
- Σpx
- Σp ÷ x
- x ÷ p
- Σx only
Show the answer
A. A weighted average.
Question 7. A weakness of sensitivity analysis is that it:
- Ignores probability
- Is too complex
- Needs simulation
- Uses profit
Show the answer
A. And treats inputs as independent.
Question 8. A disadvantage of EV is that it:
- Shows the risk fully
- May describe an outcome that never occurs
- Ignores NPV
- Is exact
Show the answer
B. It is only an average.
Question 9. A larger standard deviation of NPV means:
- Less risk
- More spread and more risk
- Higher EV
- Lower tax
Show the answer
B. Outcomes vary more.
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 the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) Distinguish risk and uncertainty. (3 marks) (b) Explain how sensitivity analysis is carried out and state its weaknesses. (7 marks)
Question 12 (20 marks). Kano Ltd has a 4-year project: outlay ₦3,000k; 8,000 units a year at ₦500 with variable cost ₦300 and fixed cash costs ₦500,000. The cost of capital is 10% (annuity factor 3.170; discount factors 0.909, 0.826, 0.751, 0.683). Ignore tax. (a) Calculate the NPV. (5 marks) (b) Calculate the sensitivity of the NPV to the outlay, price, volume, variable cost and fixed cost. (8 marks) (c) Calculate the discounted payback and comment. (4 marks) (d) Advise on the key risks. (3 marks)
Question 13 (10 marks). NPV outcomes are ₦80m (probability 0.3), ₦30m (0.5) and −₦40m (0.2). (a) Calculate the EV. (3 marks) (b) Calculate the standard deviation. (4 marks) (c) State the probability of a loss and two limits of EV. (3 marks)
Model answers
Show the model answer to Question 11
(a) Terms (3 marks). Risk: several possible outcomes whose probabilities can be estimated. (1.5) Uncertainty: too little information to know the outcomes or their probabilities. (1.5)
(b) Sensitivity (7 marks). Take each uncertain input in turn and find the percentage change needed to bring NPV to zero: NPV ÷ PV of the item. (3) Weaknesses: inputs are treated as independent; it ignores probability; it gives information, not a decision. (4)
Show the model answer to Question 12
(a) NPV (5 marks).
- Contribution = 8,000 × 200 = ₦1,600k
- less fixed 500 = ₦1,100k.
- PV = 1,100 × 3.170 = ₦3,487k.
- NPV = 3,487 − 3,000 = ₦487k. (5)
(b) Sensitivities (8 marks).
- PV of sales.
4,000 × 3.170 = 12,680 - PV of variable costs.
2,400 × 3.170 = 7,608 - PV of fixed.
500 × 3.170 = 1,585. - Outlay.
487 ÷ 3,000 = 16.2%. - Price.
487 ÷ 12,680 = 3.8%. - Volume.
487 ÷ (12,680 − 7,608) = 9.6%. - Variable cost.
487 ÷ 7,608 = 6.4%. - Fixed cost.
487 ÷ 1,585 = 30.7%. - (1.5 each, 0.5 working)
(c) Payback (4 marks).
- Cumulative PV.
year 3 = 2,735 - year 4 PV = 752.
- Payback = 3 + 265 ÷ 752 = 3.35 years. (2.5)
- The life could fall by about 16% before NPV reaches zero. (1.5)
(d) Risks (3 marks). Price is the most critical input, since a 3.8% fall removes the NPV, so confirm the price through market research or contracts. (1.5) Variable cost and volume are next at 6.4% and 9.6%, so consider fixed-price supply contracts. (1.5)
Show the model answer to Question 13
(a) EV (3 marks). 0.3 × 80 + 0.5 × 30 + 0.2 × (−40) = 24 + 15 − 8 = ₦31m. (3)
(b) Standard deviation (4 marks).
- Variance = 0.3 × 492 + 0.5 × 12 + 0.2 × 712 = 720.3 + 0.5 + 1,008.2 = 1,729. (3)
- Standard deviation = ₦41.6m. (1)
(c) Loss and limits (3 marks). Probability of a loss = 20%. (1) Limits: probabilities are subjective; the EV may never occur and suits repeated events better than a one-off project. (2)
Pathfinder on Sensitivity analysis, expected NPV and standard deviation
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.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on this topic. Sensitivity and expected values appear inside project questions. Worked examples 1 to 4 show how to apply them.
Revise first: Worked examples
Questions ask for the percentage change in each input that makes NPV zero. See the section on sensitivity analysis.
Revise first: Sensitivity analysis
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
Spotted a mistake? Tell us and we will check it.
