A company must spend ₦20m before it knows whether the product will work, and then before it knows whether customers will buy. Each stage depends on the last. A single NPV cannot show that. A decision tree can: it lays out the choices and chances in order, works backwards from the end and says what the whole plan is worth.
It can also say how much it would be worth paying for a forecast.
This topic covers decision trees, expected net present value, the value of perfect and imperfect information, certainty equivalents, risk-adjusted discount rates (including the lower rate for risky outflows) and simulation (Monte Carlo). The previous topic covered sensitivity analysis and expected values. The next covers lease or buy decisions. Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on these methods.
In 30 seconds
A decision tree shows staged decisions with decision points (square) and outcome points (circle). Draw it left to right, then solve it right to left, taking expected values at outcome points and the best choice at decision points. The value of perfect information = expected value with it − expected value without it. Imperfect information needs revised (conditional) probabilities.
A certainty equivalent is the certain sum with the same worth as a risky one. Multiply risky flows by certainty equivalent factors and discount at the risk-free rate. A risk-adjusted discount rate is raised for riskier inflows and lowered for riskier outflows. Simulation assigns random numbers to variable ranges, repeats the model many times and builds a distribution of NPVs.
Key terms in plain English
| Term | What it means |
|---|---|
| Decision tree | A diagram of choices and chance events in order |
| Decision point | A point where management chooses |
| Outcome point | A point where chance decides, so an expected value is calculated |
| Perfect information | A forecast that is always right |
| Imperfect information | A forecast that is sometimes wrong |
| Certainty equivalent | The certain amount that is as good as a risky cash flow |
| Risk-adjusted discount rate | A discount rate that reflects the risk of the cash flows |
| Monte Carlo simulation | A model that uses random numbers to test many possible outcomes |
Syllabus check
This topic is in syllabus area B (advanced investment appraisal, 30% of the marks). It covers:
- decision trees and the value of perfect and imperfect information;
- risk-adjusted discount rates and certainty equivalents;
- simulation.
CAMA 2020 and risk decisions
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 | Staged risk should be weighed 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 risk and information before committing funds |
| 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 |
Decision trees
When a decision comes in stages, with the second stage depending on the first outcome, a decision tree gives a systematic way to find the expected value. The tree has two kinds of branching point: decision points (where management chooses) and outcome points (where an expected value is calculated). It is drawn from the root outwards, then analysed from the branches back to the root.
ICAN’s example. A project costs ₦12,000. It has an 80% chance of being developed successfully. If so, the PV of future cash flows is ₦50,000 in good economic conditions (probability 0.6) or −₦20,000 in poor conditions (0.4).
- Expected PV at the economy outcome point.
0.6 × 50,000 + 0.4 × (−20,000) = ₦22,000. - Expected PV at the development point.
0.8 × 22,000 + 0.2 × 0 = ₦17,600. - Expected NPV.
17,600 − 12,000 = ₦5,600.
The choice is to invest.
ICAN also uses a tree to find the expected NPV and standard deviation of a project with different possible NPVs (Kenny Ltd): possible NPVs of ₦8.68m, ₦2.88m and −₦10m with probabilities 0.42, 0.28 and 0.30 give an expected NPV of about ₦1.46m and a standard deviation of about ₦7.86m.
The value of information
Perfect information
A company may ask what it would pay to be certain of an outcome. Value of perfect information = expected value with perfect information − expected value without it. In the example, if the company knew the economy would be poor, it would not proceed, so it avoids the −₦20,000. Expected PV at the economy point = 0.6 × 50,000 = ₦30,000. At the development point, 0.8 × 30,000 = ₦24,000. Expected NPV = 24,000 − 12,000 = ₦12,000. Value = 12,000 − 5,600 = ₦6,400.
Imperfect information
A forecaster may be right only some of the time. Build a table of the joint probabilities, then work out the revised probabilities. ICAN’s forecaster is 85% accurate. The probability of a “good” forecast = 0.6 × 0.85 + 0.4 × 0.15 = 0.57. When the forecast is good, the economy is really good with probability 0.51 ÷ 0.57 = 0.895. The company proceeds only after a good forecast.
Expected NPV with the forecast is about ₦7,448, so the value of the imperfect information is 7,448 − 5,600 = about ₦1,848 (rounded figures). It is lower than the value of perfect information, as it should be.
Certainty equivalents and risk-adjusted rates
There are two ways to allow for the riskiness of cash flows in DCF.
Certainty equivalents
A certainty equivalent is the certain sum that someone would accept in place of a risky one. If a 50% chance of ₦1,000,000 is worth ₦310,000 to someone for sure, that is the certainty equivalent. In DCF, multiply each cash flow by a certainty equivalent factor and discount at the risk-free rate. ICAN’s example: outlay ₦55,000; inflows ₦15,000, ₦35,000, ₦30,000 and ₦20,000; factors 0.94, 0.87, 0.79 and 0.70; risk-free rate 8%. The certain flows are 14,100; 30,450; 23,700 and 14,000, and the NPV is about ₦13,260.
Risk-adjusted discount rates
The discount rate should reflect the risk of the flows being discounted. A company’s weighted average cost of capital (WACC) reflects the risk of the company, so it suits a project only if the project has the same risk. Otherwise use a rate suited to the project, often from the capital asset pricing model (CAPM). A higher rate for riskier inflows gives a lower PV, which is right.
For riskier outflows, use a lower rate: the payment is then worth more as a cost, because a third party would charge more to take on a riskier liability.
Simulation
A risk model, often a spreadsheet, tests scenarios such as delays or combinations of changes in cost and benefit. A Monte Carlo simulation uses random numbers. Each variable (sales volume, price, variable cost, fixed cost) has estimated probabilities for different values. Random number ranges are given to each value in line with its probability.
For example, if volume is 10,000 units with probability 20% (random numbers 01 to 20), 20,000 with 50% (21 to 70) and 30,000 with 30% (71 to 00), a random number of 63 gives 20,000 units. The model is run hundreds or thousands of times, producing a probability distribution of NPVs, often close to normal. From its mean and standard deviation, one can find the probability of a negative NPV or the minimum NPV at the 95% or 99% confidence level.
Advantages: it uses all possible values and their probabilities and suits large, complex projects. Disadvantages: models are complex, slow and costly to build, and probabilities may be subjective and unreliable.
The rules in more depth
Why solve a tree backwards
Later decisions depend on earlier chances. Starting at the end tells you what you would do at each later point, so the earlier choices can be valued correctly.
Why information has a ceiling
No forecast can be worth more than certainty, so the value of imperfect information is never above the value of perfect information.
Why certainty equivalents use the risk-free rate
The risk has been moved into the cash flows, so the discount rate should carry only the time value of money.
Why a risky outflow gets a lower rate
A lower rate makes the PV of the payment larger, which recognises that a risky liability is more expensive.
Effect on analysis
Draw the tree, solve it from the right and state the decision at each point. Say what extra information changes and what it is worth.
Worked examples
Example 1: a two-stage tree. A project costs ₦20m. The chance of successful development is 70%. If successful, PV of cash flows is ₦60m with probability 0.5 or −₦10m with probability 0.5.
- Outcome point.
0.5 × 60 + 0.5 × (−10) = ₦25m. - Development point.
0.7 × 25 + 0.3 × 0 = ₦17.5m. - Expected NPV.
17.5 − 20 = −₦2.5m.
Do not proceed.
Example 2: perfect information. The same project, with a forecast of the economy that is always right and is known before launch.
- Proceed only if demand is good.
0.5 × 60 = ₦30m. - Development point.
0.7 × 30 = ₦21m.
Expected NPV = 21 − 20 = ₦1m. - Without the forecast the choice was to reject (NPV 0), so the value of perfect information is ₦1m.
Example 3: a risky outflow. A payment of ₦100m falls due in a year. The risk-free rate is 6%, and a risk-adjusted rate of 10% would apply to a risky inflow.
- For a risky inflow at 10%, PV = 100 ÷ 1.10 = ₦90.9m.
- For a risky outflow, use a lower rate such as 6%: 100 ÷ 1.06 = ₦94.3m.
- The riskier payment is shown as more costly, as it should be.
Example 4: random numbers. Price ₦14 (probability 15%, numbers 01 to 15), ₦15 (40%, 16 to 55), ₦16 (35%, 56 to 90) and ₦17 (10%, 91 to 00). The random number drawn is 63.
- 63 is in the range 56 to 90.
- The simulated price is ₦16.
- Repeating this for each variable many times gives a distribution of outcomes.
How to score full marks
- Draw the tree with decision and outcome points labelled.
- Solve backwards and show the expected value at each point.
- State the decision, then the expected NPV.
- For information, compare the with and without values.
- For certainty equivalents, discount at the risk-free rate.
- Give the strengths and limits of each method.
Common mistakes in the exam
- Solving the tree from left to right.
- Forgetting the initial cost at the first decision point.
- Valuing information without comparing it with the no-information case.
- Discounting certainty equivalents at the risky rate.
- Using a higher rate for a riskier outflow.
- Assuming simulation removes risk.
Quick revision
- Decision tree: draw forward, solve backward; decision and outcome points.
- Value of perfect information = EV with − EV without.
- Imperfect information needs revised probabilities and is worth less.
- Certainty equivalent flows at the risk-free rate.
- Risk-adjusted rate: higher for inflows, lower for outflows.
- Simulation: random number ranges, many runs, a distribution of NPV.
- 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. A decision tree is solved:
- From the root to the branches
- From the branches back to the root
- Randomly
- Only once
Show the answer
B. Right to left.
Question 2. At an outcome point you calculate:
- A choice
- An expected value
- A tax saving
- A payback
Show the answer
B. Weighted by probabilities.
Question 3. The value of perfect information equals:
- EV with it minus EV without it
- EV without it minus EV with it
- Zero
- The cost of the project
Show the answer
A. The gain from certainty.
Question 4. The value of imperfect information is:
- Greater than perfect information
- No greater than perfect information
- Always zero
- Equal to the NPV
Show the answer
B. Certainty is the ceiling.
Question 5. Certainty-equivalent cash flows are discounted at:
- The risky rate
- The risk-free rate
- Zero
- The WACC
Show the answer
B. The risk is already in the flows.
Question 6. For a riskier cash outflow, the discount rate should be:
- Higher
- Lower
- Unchanged
- Negative
Show the answer
B. To show it as more costly.
Question 7. A WACC suits a project when the project has:
- The same risk as the company
- Any risk
- Higher risk
- No cash flows
Show the answer
A. Otherwise adjust the rate.
Question 8. Monte Carlo simulation uses:
- Random numbers
- Only the mean
- A lease
- Tax tables
Show the answer
A. Assigned to probability ranges.
Question 9. A drawback of simulation is that it:
- Ignores all variables
- Is complex and relies on subjective probabilities
- Gives one answer
- Needs no model
Show the answer
B. And can be costly.
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) Explain decision points and outcome points in a decision tree. (4 marks) (b) Explain what is meant by the value of perfect information and why imperfect information is worth less. (6 marks)
Question 12 (20 marks). Sahel Ltd can spend ₦30m on a product. The chance of successful development is 60%. If successful, demand is high (probability 0.5, PV of cash flows ₦120m) or low (probability 0.5, PV −₦30m). If development fails, the PV is nil. (a) Draw the tree in words and calculate the expected NPV. Advise. (8 marks) (b) A market study would reveal demand perfectly before launch, after successful development.
Calculate its maximum worth. (6 marks) (c) Explain risk-adjusted discount rates and why a riskier outflow has a lower rate. (4 marks) (d) State one advantage and one disadvantage of simulation. (2 marks)
Question 13 (10 marks). A project costs ₦80m now. Expected inflows are ₦50m in year 1 and ₦60m in year 2, with certainty equivalent factors of 0.95 and 0.90. The risk-free rate is 6% (factors 0.943 and 0.890). (a) Calculate the NPV using certainty equivalents. (6 marks) (b) Explain why the risk-free rate is used. (4 marks)
Model answers
Show the model answer to Question 11
(a) Points (4 marks). A decision point is where management chooses between actions. (2) An outcome point is where chance determines the result, so an expected value is calculated from the probabilities. (2)
(b) Information (6 marks). Perfect information is a forecast that is always right. Its value is the expected value with it minus the expected value without it. (3) An imperfect forecast is sometimes wrong, so it avoids fewer bad outcomes and captures fewer good ones, so its value cannot exceed that of perfect information. (3)
Show the model answer to Question 12
(a) Expected NPV (8 marks).
- Demand outcome point.
0.5 × 120 + 0.5 × (−30) = ₦45m. (3) - Development point.
0.6 × 45 + 0.4 × 0 = ₦27m. (2) - Expected NPV.
27 − 30 = −₦3m. (2) - Reject. (1)
(b) Perfect information (6 marks).
- With the study, proceed only if demand is high: 0.5 × 120 = ₦60m. (2)
- Development point.
0.6 × 60 = ₦36m. (2) - Expected NPV.
36 − 30 = ₦6m. - The alternative without the study is to reject (NPV 0), so the maximum worth is ₦6m. (2)
(c) Risk-adjusted rates (4 marks). The rate should reflect the risk of the cash flows, so the company’s WACC is used only when risk is the same as the company’s. (2) For a riskier outflow a lower rate gives a larger PV, showing the payment as more costly, as a third party would charge more to take it on. (2)
(d) Simulation (2 marks). Advantage: it uses all possible values and probabilities and suits complex projects. (1) Disadvantage: complex and costly to build and relies on subjective probabilities. (1)
Show the model answer to Question 13
(a) NPV (6 marks).
- Certain flows.
50 × 0.95 = 47.5 - 60 × 0.90 = 54.0. (2)
- PVs at 6%: 47.5 × 0.943 = 44.79
- 54.0 × 0.890 = 48.06.
- Total = 92.85. (2)
- NPV = 92.85 − 80 = +₦12.9m. (2)
(b) Risk-free rate (4 marks). The risk has already been allowed for by reducing each flow to its certain equivalent. (2) Using a risky rate as well would count the risk twice, so the discount rate carries only the time value of money. (2)
Pathfinder on Decision trees, value of information and certainty equivalents
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. Decision trees and the value of information appear inside project risk questions. Worked examples 1 to 4 show how to apply them.
Revise first: Worked examples
Questions ask what a company would pay for a forecast. See the section on the value of information.
Revise first: The value of information
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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