Sensitivity analysis changes one input at a time and probability analysis needs a handful of discrete outcomes. Real projects have many uncertain inputs moving together and each can take a range of values. Two further tools deal with this. Simulation lets a computer (or a table of random numbers) try thousands of combinations and report the spread of results.
Risk-adjusted discount rates take a different route: leave the cash flows as they are and make the discount rate higher for a riskier project. A third approach, discounted payback, asks how long the project takes to recover its cost when the cash flows are discounted, a simple test of exposure to the long-term future.
This topic explains each method, how to use it with a worked example, and its strengths and weaknesses, including how simulation improves on sensitivity analysis. The next topic covers leasing and borrowing to buy.
The comparison of sensitivity analysis with simulation was worth 8 marks in November Diet, 2025, Question 1(c). Read the value of perfect information first.
In 30 seconds
Simulation models the uncertain variables with probability distributions, uses random numbers to pick a value for each variable in each trial, calculates the net present value (NPV) for every trial and builds up a distribution of NPVs (expected value, spread, probability of a loss). It lets variables vary together, but needs data, effort and software, and depends on the distributions assumed.
A risk-adjusted discount rate adds a risk premium for projects riskier than the company’s average. It is simple but the premium is subjective and it penalises distant cash flows heavily. Discounted payback is the time to recover the outlay from discounted cash flows: it allows for the time value of money but ignores the cash flows after payback.
Simulation
Simulation (often called Monte Carlo simulation) is a method of risk modelling. The steps are:
- Identify the uncertain variables (demand, price, costs, life) and the probability distribution of each.
- Allocate a range of random numbers to each possible value in proportion to its probability (for example a probability of 0.2 gets 20 of the 100 two-digit numbers).
- In each trial, draw a random number for each variable, read off the value and calculate the cash flow and the NPV.
- Repeat many times (hundreds or thousands, normally by computer) and analyse the results: the expected NPV, the standard deviation, the probability that the NPV is negative.
Example. A project’s yearly demand is 1,000 units with probability 0.2 (random numbers 00 to 19), 1,500 with 0.5 (20 to 69) and 2,000 with 0.3 (70 to 99). The unit contribution is ₦400 with probability 0.4 (00 to 39) or ₦500 with 0.6 (40 to 99). Five trials use these pairs of random numbers (demand, contribution): (12, 45) gives 1,000 × ₦500 = ₦500,000; (55, 81) gives 1,500 × 500 = ₦750,000; (83, 22) gives 2,000 × 400 = ₦800,000; (07, 63) gives 1,000 × 500 = ₦500,000; (91, 38) gives 2,000 × 400 = ₦800,000. The average is ₦670,000. The exact expected value is (0.2 × 1,000 + 0.5 × 1,500 + 0.3 × 2,000) × (0.4 × 400 + 0.6 × 500) = 1,550 × 460 = ₦713,000, so five trials give only a rough estimate; thousands of trials give a reliable distribution.
How simulation improves on sensitivity analysis
- Several variables vary at the same time, with their interdependence modelled if needed.
- It produces a probability distribution of the NPV, not just a margin for each variable, so the risk of a loss and the likely range are known.
- It uses probabilities for each variable.
- It is a more complete assessment of risk: a manager can compare the spread of two projects.
Its weaknesses are that it needs large amounts of reliable data to estimate distributions, a complex model and software, the results depend on the accuracy of the assumed distributions. It does not itself give a decision rule.
Risk-adjusted discount rates
A risk-adjusted discount rate (RADR) is the cost of capital plus a premium for the extra risk of the project. A company with a cost of capital of 12% might use 10% for a low-risk replacement project, 12% for a project like its existing business and 16% for a new, uncertain venture. (A rate can also be derived from the capital asset pricing model by using the beta of a comparable business, studied later.)
A project with cash flows of ₦8m, ₦9m, ₦9m and ₦6m and an outlay of ₦20m has an NPV of +₦4.54m at 12% (7.14 + 7.17 + 6.41 + 3.82 − 20) but only +₦2.66m at 16%. If the project is high risk, the second figure is the one to use, and it is still positive.
- Advantages: simple, and it is easy to explain; it builds the attitude to risk into one number; it reflects the idea that investors need more return for more risk.
- Disadvantages: the premium is subjective and usually arbitrary; it assumes the risk rises at a constant rate over time (a premium compounded each year penalises later cash flows heavily, whether or not they are more uncertain); it gives no information about the range of outcomes; and it can reject a project that has one-off, early risks.
Discounted payback
The discounted payback period is the time it takes for the cumulative present value of the cash flows to recover the outlay. For the project above at 12%: the present values are 7.14, 7.17, 6.41 and 3.82, so the cumulative position is −12.86 after year 1, −5.68 after year 2 and +0.73 after year 3. The discounted payback is 2 + 5.68 ÷ 6.41 = 2.89 years, against an undiscounted payback of just over two years (8 + 9 = 17 after two years, 26 after three: 2 + 3 ÷ 9 = 2.33 years).
It is a risk measure, because a project that recovers its cost sooner is less exposed to the uncertain future, and it uses the time value of money, unlike simple payback. But it still ignores cash flows after the payback date, so it can reject a profitable long-term project, and the maximum period allowed is arbitrary.
It is best used as a supplement to NPV: the study of the project’s life in sensitivity analysis used it.
The ideas in more depth
Why a risk premium compounds
Adding 4 points to the rate is the same as saying each year’s cash flow is less valuable than the last by an extra 4% a year of uncertainty. A cash flow in year 5 is therefore reduced by about 17% more than one in year 1, though it may be no riskier.
The method is best when the risk really does grow with time, as for a venture whose success is only revealed gradually.
Why simulation does not remove uncertainty
Simulation produces a precise-looking distribution from imprecise inputs. If the probability distributions are guesses, the answer is only as good as the guesses. Its real value is to show which combinations of events cause losses and how likely they are, so that management can decide what to do about them.
Sources and references
- No accounting standard governs these methods. Risk-adjusted rates are linked to the capital asset pricing model, covered later in the course.
These points are stated as at 5 November 2026.
Worked examples
Example 1: a trial. Random numbers (55, 81): demand 1,500 (20 to 69) and contribution ₦500 (40 to 99). Required: the contribution.
- Contribution.
1,500 × ₦500 = ₦750,000.
Example 2: a risk-adjusted NPV. Cash flows ₦8m, ₦9m, ₦9m, ₦6m; outlay ₦20m; rate 16% (factors 0.862, 0.743, 0.641, 0.552). Required: the NPV.
- Present value.
6.90 + 6.69 + 5.77 + 3.31 = ₦22.66m. - NPV.
22.66 − 20 = +₦2.66m.
Example 3: discounted payback. Cumulative present value −₦5.68m after year 2; year 3 present value ₦6.41m. Required: the discounted payback.
- Payback.
2 + 5.68 ÷ 6.41 = 2.89 years.
How to answer in the exam
- Describe the steps of simulation (distributions, random numbers, trials, results) and tie them to the scenario.
- When comparing with sensitivity analysis, stress joint variation, probability and the distribution of the NPV.
- For the discount rate, compute the NPV at both rates and comment on the premium.
- For discounted payback, cumulate the present values and interpolate.
Common mistakes in the exam
- Allocating random numbers that do not match the probabilities.
- Describing simulation without saying what it produces.
- Forgetting its limits: data, cost, assumed distributions.
- Adding the risk premium to the cash flows instead of to the discount rate.
- Using undiscounted flows in discounted payback.
- Treating any of the methods as removing risk.
Quick revision
- Simulation: distributions, random numbers, many trials, distribution of NPVs.
- Risk-adjusted rate: cost of capital plus a premium; simple but subjective.
- Discounted payback: time for cumulative present value to turn positive.
- All three supplement the NPV; none gives certainty.
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. Simulation uses:
- Only the most likely outcome
- One variable at a time
- Random numbers and probability distributions over many trials
- The payback rule
Show the answer
C. It models variables varying together.
Question 2. The result of a simulation is:
- One NPV
- A distribution of NPVs
- A payback period
- A tax figure
Show the answer
B. With its expected value and spread.
Question 3. Random numbers are allocated to values:
- In proportion to their probabilities
- Equally
- At random
- By size
Show the answer
A. A probability of 0.2 gets 20 of 100 numbers.
Question 4. A risk-adjusted discount rate:
- Removes risk
- Is the risk-free rate
- Is lower for riskier projects
- Adds a premium to the cost of capital for riskier projects
Show the answer
D. The higher rate lowers the NPV.
Question 5. A weakness of the risk-adjusted discount rate is that:
- It is hard to calculate
- The premium is subjective and compounds over time
- It ignores the cash flows
- It is always too low
Show the answer
B. It penalises distant cash flows heavily.
Question 6. Discounted payback uses:
- Discounted cash flows
- Accounting profits
- Undiscounted cash flows
- Tax only
Show the answer
A. So it allows for the time value of money.
Question 7. A weakness of discounted payback is that it:
- Ignores time value
- Uses profit
- Ignores cash flows after the payback date
- Is complicated
Show the answer
C. So it may reject a profitable long project.
Question 8. A weakness of simulation is that it:
- Uses one variable
- Ignores probability
- Is too simple
- Needs reliable data, effort and assumed distributions
Show the answer
D. The answer is only as good as the inputs.
Question 9. Simulation improves on sensitivity analysis because it:
- Is simpler
- Needs no data
- Lets several variables vary together and gives probabilities
- Gives a decision
Show the answer
C. Sensitivity analysis changes one at a time.
Question 10. For a project riskier than average, the discount rate used should be:
- Higher than the company’s cost of capital
- Lower
- Zero
- The same always
Show the answer
A. To reflect the extra risk.
Section B style: written questions
Question 11 (10 marks). (a) State the steps in a simulation. (4 marks) (b) State TWO ways in which simulation improves on sensitivity analysis. (3 marks) (c) State TWO limits of simulation. (3 marks)
Question 12 (15 marks). Demand is 800 units (probability 0.3, random numbers 00 to 29), 1,000 (0.5, 30 to 79) or 1,200 (0.2, 80 to 99). The unit margin is ₦50 (0.25, 00 to 24), ₦60 (0.5, 25 to 74) or ₦70 (0.25, 75 to 99). The four trials use the pairs (14, 60), (48, 18), (91, 77) and (33, 92). (a) Calculate the profit in each trial. (8 marks) (b) Calculate the mean of the trials. (2 marks) (c) Calculate the exact expected value and comment. (5 marks)
A further written question
Question 13 (20 marks). A project costs ₦30m and returns ₦11m, ₦12m, ₦12m and ₦8m in years 1 to 4. The cost of capital is 12% (factors 0.893, 0.797, 0.712, 0.636); for a high-risk venture a rate of 17% is proposed (factors 0.855, 0.731, 0.624, 0.534).
(a) Calculate the NPV at 12%. (6 marks) (b) Calculate the NPV at 17% and state the decision. (6 marks) (c) Calculate the discounted payback at 12%. (5 marks) (d) State TWO weaknesses of the risk-adjusted rate. (3 marks)
Model answers
Show the model answer to Question 11
(a) Identify the variables and their distributions; allocate random numbers in proportion to the probabilities; run the trials, drawing a value for each variable and computing the NPV; analyse the distribution of results (1 each).
(b) Variables vary together; it produces a probability distribution of NPVs (1.5 each).
(c) Needs reliable data and complex software; depends on the assumed distributions; no decision rule (any two, 1.5 each).
Show the model answer to Question 12
(a)
- (14, 60): 800 × ₦60 = ₦48,000
- (48, 18): 1,000 × ₦50 = ₦50,000
- (91, 77): 1,200 × ₦70 = ₦84,000
- (33, 92): 1,000 × ₦70 = ₦70,000 (2 each).
(b) (48 + 50 + 84 + 70) ÷ 4 = ₦63,000 (2).
(c) Expected demand 980 and margin ₦60, so ₦58,800 (3). Four trials are too few: the mean is above the expected value, and many more trials are needed for a reliable estimate (2).
Show the model answer to Question 13
(a) 11 × 0.893 = 9.82; 12 × 0.797 = 9.56; 12 × 0.712 = 8.54; 8 × 0.636 = 5.09 (4); total 33.02; NPV = +₦3.02m (2).
(b)
- 11 × 0.855 = 9.41; 12 × 0.731 = 8.77; 12 × 0.624 = 7.49; 8 × 0.534 = 4.27 (3); total 29.94; NPV = −₦0.06m (1).
- At the higher rate the project is marginally negative, so for a high-risk venture reject or review the estimates (2).
(c) Cumulative: −20.18, −10.61, −2.07, +3.02; discounted payback = 3 + 2.07 ÷ 5.09 = 3.41 years (5).
(d) (1.5 each) The premium is subjective; it compounds so penalises distant cash flows heavily; it gives no information on the range of outcomes.
Pathfinder on Simulation, risk-adjusted discount rates and discounted payback
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.
Asks how simulation provides an improvement over sensitivity analysis in evaluating risk and uncertainty in capital project appraisal.
Revise first: Simulation
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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