A forecast of a single net present value (NPV) pretends to know the future. In reality the same project may turn out well, badly or middling, and a manager may be able to put rough probabilities on each. Probability analysis turns those into one number, the expected value, and into a map of the decision when choices follow one another, a decision tree.
This topic explains the expected value of the NPV, how to use it in a decision, its advantages and its limits (it hides risk, and it is an average of outcomes that will never happen), how to measure the spread of outcomes with the standard deviation, and how to build and solve a decision tree with decision points and outcome points, with worked examples. The next topic covers the value of perfect information.
Probabilities are used when a question says that the outcomes can be assigned chances, so this topic applies to risk rather than to uncertainty. Read risk and uncertainty in appraisal: sensitivity analysis first.
In 30 seconds
The expected value (EV) of a project’s NPV is the probability-weighted average of the possible NPVs: EV = Σpx. Rule: undertake the project if the expected NPV is positive, unless the risk seems too great for the return. Advantages: uses all outcomes and gives one figure.
Limits: the probabilities are judgements, the EV is not a real outcome, it suits repeated events better than a once-only project, and it does not measure risk; the standard deviation of the outcomes does.
A decision tree shows decisions (squares) and chance outcomes (circles) in sequence; draw it from left to right, solve it from right to left (folding back): at each outcome point take the expected value, at each decision point take the best option.
Expected value
An expected value is a weighted average calculated from the probabilities of the possible outcomes: EV = Σpx, where p is the probability of each outcome and x its value. In appraisal, x is the NPV of each outcome and the result is the expected net present value (the EV of the NPV).
The basic decision rule is to undertake the project if the expected NPV is positive, though a project with a positive EV may be rejected if the risk looks too great for the return.
Example: a project has four possible results.
| Probability p | NPV x, ₦m | px | p(x − EV)2 |
|---|---|---|---|
| 0.15 | 40 | 6.0 | 143.2 |
| 0.35 | 18 | 6.3 | 27.7 |
| 0.30 | 4 | 1.2 | 7.8 |
| 0.20 | (22) | (4.4) | 193.5 |
| Total | 9.1 | 372.2 |
The expected NPV is +₦9.1m, so on the basic rule the project is accepted. But there is a 20% chance of a loss of ₦22m, and the outcomes are spread widely. The variance is Σp(x − EV)2 = 372.2 and the standard deviation is its square root, ₦19.3m, a measure of the risk: when two projects have the same EV, the one with the smaller standard deviation is less risky.
Advantages and disadvantages
- Advantages: it is a weighted average of all possible outcomes, arguably a better measure than the single most likely outcome, and it gives a single figure for the decision.
- Disadvantages: the probabilities are subjective and may be guesswork; the EV is not a value of any actual outcome (the project will not earn ₦9.1m: it will earn one of the four results); it is much more reliable for events that repeat many times than for a once-only capital project; and above all it gives no analysis of risk.
Decision trees
Some decisions are taken in stages, with the second decision depending on what happens after the first. A decision tree sets out all the possible outcomes and probabilities. It is drawn from the root (the first decision) outwards to the branches and then analysed from the branches back to the root.
There are two kinds of branching point: decision points (drawn as squares), where management chooses, and outcome points (drawn as circles), where chance decides and an expected value is calculated.
The steps are:
- draw the tree
- write the probability on each chance branch (they sum to 1 at each outcome point)
- put the cash flow or value at the end of each path
- fold back, taking the expected value at each outcome point and the best option at each decision point
- state the decision at the start
A worked tree: a new product
A company can spend ₦30m on developing a new product (decision A) or do nothing. Development is successful with probability 0.7. If it succeeds, the company must decide (decision C) whether to launch, at an extra cost of ₦15m, or to sell the rights for ₦35m. If launched, demand is high with probability 0.6 (present value of cash flows ₦100m) or low with probability 0.4 (present value ₦30m).
If the development fails, the project is abandoned and nothing is received.
- Outcome point after launch:
expected present value = 0.6 × 100 + 0.4 × 30 = 72; less the launch cost of 15 gives 57. - Decision point C: launch (57) or sell the rights (35): choose to launch, value 57.
- Outcome point B (development):
0.7 × 57 + 0.3 × 0 = 39.9. - Decision point A:
spend ₦30m to get an expected value of 39.9, an expected NPV of 39.9 − 30 = +₦9.9m, or do nothing for zero.
Develop.
The possible results are: success and high demand, probability 0.7 × 0.6 = 0.42, NPV 100 − 15 − 30 = +55; success and low demand, 0.28, NPV 30 − 15 − 30 = −15; failure, 0.30, NPV −30. The expected value is 0.42 × 55 + 0.28 × (−15) + 0.30 × (−30) = 9.9, the same, and the standard deviation is ₦38.8m, large relative to the expected value. There is a 58% probability of a loss, and a manager unwilling to risk losing ₦30m might reject a project that the expected value would accept.
The ideas in more depth
Why the EV is not a promise
If the same project could be undertaken a thousand times, the average result would approach the EV. A company undertakes it once, and the outcome will be one of the possible results, so the EV is a guide and not a forecast. Where the loss from the worst outcome is large relative to the company’s resources, the spread matters as much as the average.
Why the tree is solved backwards
The best second decision depends on the situation reached, so it must be valued first, then the chance event that leads to it, then the first decision. Working forwards would require guessing what later choices will be.
Sources and references
- No accounting standard governs probability analysis. The standard deviation and expected value are statistical tools described in the course on quantitative methods.
These points are stated as at 5 November 2026.
Worked examples
Example 1: the expected NPV. NPVs of ₦40m, ₦18m, ₦4m and −₦22m with probabilities 0.15, 0.35, 0.30 and 0.20. Required: the expected NPV.
- Sum.
6.0 + 6.3 + 1.2 − 4.4 = ₦9.1m.
Example 2: the standard deviation. The variance is 372.2. Required: the standard deviation.
- Standard deviation. √372.2 = ₦19.3m.
Example 3: an outcome point. High demand 0.6 (PV 100) or low 0.4 (PV 30); launch cost 15. Required: the expected value after the cost.
- Expected PV.
0.6 × 100 + 0.4 × 30 = 72. - Net.
72 − 15 = 57.
How to answer in the exam
- Draw the tree clearly with probabilities on branches and values at the ends.
- Fold back, showing the expected value at each outcome point and the choice at each decision point.
- State the decision and the expected NPV, then comment on the risk and the probability of a loss.
- Mention the limits of the EV for a once-only project.
Common mistakes in the exam
- Probabilities that do not sum to 1 at a point.
- Forgetting to deduct the initial cost at the decision.
- Solving the tree from the start instead of folding back.
- Taking an expected value at a decision point instead of choosing the best option.
- Ignoring the risk when the EV is positive.
- Multiplying probabilities wrongly for a second-stage outcome.
Quick revision
- EV = Σpx; accept if the expected NPV is positive, if the risk is acceptable.
- Standard deviation measures the spread; the EV does not measure risk.
- Tree: squares for decisions, circles for chance; draw forwards, solve backwards.
- Probabilities along a path multiply.
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. The expected value is:
- The most likely outcome
- The highest outcome
- The sum of probability times outcome
- The lowest outcome
Show the answer
C. A probability-weighted average.
Question 2. On the basic rule a project is accepted if its expected NPV is:
- Negative
- Positive
- Zero always
- Equal to the outlay
Show the answer
B. Subject to the risk being acceptable.
Question 3. The expected value:
- Is not the value of any actual outcome
- Will certainly happen
- Is always the best outcome
- Is the standard deviation
Show the answer
A. It is an average of possible outcomes.
Question 4. A decision tree is solved:
- From the root forwards
- In any order
- Only at the root
- From the branches back to the root
Show the answer
D. This is folding back.
Question 5. In a decision tree, a decision point is usually drawn as:
- A circle
- A square
- A triangle
- A line
Show the answer
B. An outcome point is a circle.
Question 6. The probabilities on the branches leaving an outcome point sum to:
- 1
- 0
- 100
- Any number
Show the answer
A. One of the outcomes must happen.
Question 7. A major disadvantage of the EV is that it:
- Uses all outcomes
- Gives one figure
- Provides no analysis of risk
- Is complicated
Show the answer
C. The standard deviation is needed for risk.
Question 8. At a decision point you:
- Average the branches
- Add the probabilities
- Ignore the options
- Choose the option with the best expected value
Show the answer
D. Management chooses, not chance.
Question 9. The standard deviation of NPVs measures:
- The average
- The cost
- The spread of the outcomes
- The tax
Show the answer
C. A larger value means more risk.
Question 10. An advantage of the EV is that it:
- Uses all the possible outcomes and gives a single figure
- Removes risk
- Needs no probabilities
- Is exact
Show the answer
A. It helps a decision to be made.
Section B style: written questions
Question 11 (10 marks). (a) Define expected value and state the decision rule. (4 marks) (b) State TWO advantages and TWO disadvantages. (4 marks) (c) State why the standard deviation is also needed. (2 marks)
Question 12 (15 marks). A project has NPVs of ₦30m, ₦12m, −₦6m and −₦25m with probabilities 0.25, 0.40, 0.25 and 0.10. (a) Calculate the expected NPV. (6 marks) (b) State the probability of a loss. (2 marks) (c) The standard deviation is ₦16.9m; comment on the risk. (4 marks) (d) State the decision. (3 marks)
A further written question
Question 13 (20 marks). A company can spend ₦20m on developing a product. Development succeeds with probability 0.75. If it succeeds, the market is good (probability 0.5, present value ₦70m) or poor (probability 0.5, present value ₦10m, but the rights could be sold for ₦18m). If development fails, nothing is received.
(a) Draw and describe the tree. (4 marks) (b) Calculate the expected value at the outcome point after success, allowing for the option to sell. (6 marks) (c) Calculate the expected NPV of developing. (6 marks) (d) Comment on the risk. (4 marks)
Model answers
Show the model answer to Question 11
(a) A probability-weighted average of the possible outcomes, Σpx (2); undertake the project if the expected NPV is positive (2).
(b) Advantages: uses all outcomes; gives a single figure (2). Disadvantages: subjective probabilities; not a real outcome; no analysis of risk (any two, 2).
(c) The EV does not show how widely the outcomes vary (2).
Show the model answer to Question 12
(a)
- 0.25 × 30 = 7.5
- 0.40 × 12 = 4.8
- 0.25 × (−6) = −1.5
- 0.10 × (−25) = −2.5
- EV = +₦8.3m (6).
(b) 0.25 + 0.10 = 35% (2).
(c) The standard deviation of ₦16.9m is about twice the expected value, so the outcomes vary widely and there is a real chance of loss (4).
(d) Accept on the expected NPV if the company can bear a loss of up to ₦25m (3).
Show the model answer to Question 13
(a) Decision A: develop or not; outcome B: success 0.75 or failure 0.25; after success, an outcome point for the market, with a decision to sell in the poor market (4).
(b) In the poor market, sell for ₦18m, which is better than 10; expected value = 0.5 × 70 + 0.5 × 18 = ₦44m (6).
(c)
- At B.
0.75 × 44 + 0.25 × 0 = 33 - expected NPV.
33 − 20 = +₦13m (6).
(d) The outcomes are +50 (probability 0.375), −2 (0.375) and −20 (0.25); the standard deviation is ₦29.5m, large compared with the expected value of ₦13m, and there is a 25% chance of losing the whole ₦20m (4).
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