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Financial Management · Adjusting for risk and uncertainty in investment appraisal

Expected Values and Probability Analysis in Investment Appraisal

Updated 11 October 2026 · Fact-checked

Expected NPV is the probability-weighted average of the possible NPVs of a project. You list each outcome, multiply its NPV by its probability, and add the results. Accept the project if expected NPV is positive, but also check the spread of outcomes, because the average may never actually happen.

Understand Expected Values and Probability Analysis

A normal NPV uses one forecast for each cash flow. In reality you do not know the future. If you can put probabilities on the possible outcomes, the situation is one of risk in the FM sense, and you can use probability analysis.

The expected value (EV) is the long-run average result if the decision could be repeated many times. You get it by multiplying each outcome by its probability and adding. For investment appraisal, the expected NPV (ENPV) is the EV of the project's NPV. A positive ENPV suggests the project adds value on average.

Often more than one variable is uncertain, such as selling price and sales volume. A joint probability is the probability that two events both happen. If the variables are independent, you multiply their probabilities. Listing every combination gives a set of joint outcomes whose probabilities add to 1.

Expected values have limits. The EV is an average, so it may not be a possible outcome. It ignores the spread of results, so two projects with the same ENPV can have very different risk. A one-off decision is not a long-run average. Probabilities are usually subjective estimates. EV also does not show the investor's attitude to risk. You can improve the picture by also stating the probability of a negative NPV and the range of NPVs. For multi-stage decisions, a decision tree shows choices and chance events in order, and you work backwards from the end using expected values.

Key rules to remember

Expected value
EV = Σ (probability × outcome)
Probabilities must add up to 1 (100%). Use it for cash flows or for NPVs.
Expected NPV
ENPV = Σ (probability × NPV of each outcome)
Gives the same answer as discounting the expected cash flows, provided the cash flow structure is linear and the discount rate is the same in every outcome.
Joint probability (independent events)
P(A and B) = P(A) × P(B)
Only valid if A and B are independent. If the second event depends on the first, use the conditional probability instead: P(A and B) = P(A) × P(B given A).
Probability of a negative NPV
P(NPV < 0) = sum of the probabilities of all outcomes with NPV < 0
A simple way to show risk alongside the ENPV.
Decision tree roll-back
Value at a chance node = Σ (probability × value of branch); at a decision node take the best branch
Work from right to left. Discount future values back to the same date.

How to solve Expected Values and Probability Analysis questions

Use this method for any expected value or probability question in investment appraisal.

  1. 1Identify the uncertain variables (sales volume, price, cost, life) and write down the probability of each possible value. Check that the probabilities for each variable add to 1.
  2. 2If more than one variable is uncertain, decide whether they are independent. If so, build a table of joint outcomes and multiply the probabilities. If outcomes are linked, use the conditional probabilities given.
  3. 3Calculate the cash flow or the NPV for each outcome. Include the initial outlay, the correct discount factors and any tax or working capital given in the question.
  4. 4Multiply each NPV by its probability and add to get the ENPV. As a check, you can discount the expected cash flows instead if the structure is linear.
  5. 5Calculate a measure of risk if asked or useful, such as the probability of a negative NPV, or the best and worst NPV.
  6. 6Give a decision: accept if ENPV is positive (for a single project) or choose the highest ENPV for mutually exclusive projects, then comment on the risk.
  7. 7Add the limitations that fit the case: the average may not occur, one-off decision, subjective probabilities, spread ignored, risk attitude ignored.

Quickest way: Expected cash flow shortcut

When to use it: Use it when only the cash flows are uncertain, the discount rate is the same in all outcomes, and you only need the ENPV, not the probability of a loss.

  1. Work out the expected cash flow for each year: Σ (probability × cash flow).
  2. Discount the expected cash flows once, using an annuity factor if they are level.
  3. Deduct the initial outlay to get the ENPV.
  4. If a separate part asks for the chance of a loss, then calculate the NPV of each outcome. Do this only if needed, to save time.

Common mistakes in Expected Values and Probability Analysis

  • Using probabilities that do not add up to 1, or using percentages and decimals mixed together.

    Students copy figures quickly from the scenario and do not check them.

    Fix: Add the probabilities for each variable before you start. If they do not total 100%, re-read the question.

  • Adding probabilities of joint events instead of multiplying them.

    Students confuse 'and' with 'or'.

    Fix: For 'both happen' with independent events, multiply. Check that all joint probabilities in your table add to 1.

  • Treating a conditional probability as if it were independent.

    Students rush into a joint table without reading how one variable depends on the other.

    Fix: If the question says 'if sales are high, the price is likely to be...', use P(A) × P(B given A) for each branch.

  • Forgetting the initial outlay, or discounting it, when finding the ENPV.

    Students focus on the probability table and lose track of the NPV layout.

    Fix: The outlay at time 0 has a discount factor of 1. Always write PV of inflows minus outlay as a final line.

  • Saying the ENPV is the result that will happen, or that a positive ENPV means no risk.

    Students do not distinguish an average from an actual outcome.

    Fix: State that ENPV is a long-run average, may not be a possible outcome, and hides the spread. Support the comment with the probability of a negative NPV.

  • Making the decision on ENPV alone when asked to comment on risk.

    Students stop once they have a number.

    Fix: Add one or two points: the range of NPVs, the chance of a loss, whether it is a one-off decision and how reliable the probabilities are.

Worked examples

Example 1

A project needs an initial outlay of $150,000 now. It will produce level annual cash inflows for three years, starting in one year. The inflow is $50,000 with probability 0.3, $60,000 with probability 0.5 or $80,000 with probability 0.2. The cost of capital is 10%. The 3-year annuity factor at 10% is 2.487. Calculate the expected NPV and the probability that the NPV is negative.

Show the solution
  1. Present value of inflows for each outcome: $50,000 × 2.487 = $124,350; $60,000 × 2.487 = $149,220; $80,000 × 2.487 = $198,960.
  2. NPV of each outcome: 124,350 − 150,000 = −$25,650; 149,220 − 150,000 = −$780; 198,960 − 150,000 = +$48,960.
  3. Weight by probability: 0.3 × (−25,650) = −7,695; 0.5 × (−780) = −390; 0.2 × 48,960 = +9,792.
  4. ENPV = −7,695 − 390 + 9,792 = $1,707.
  5. Check using expected cash flow: (0.3 × 50,000) + (0.5 × 60,000) + (0.2 × 80,000) = 15,000 + 30,000 + 16,000 = $61,000. Then 61,000 × 2.487 = 151,707, and 151,707 − 150,000 = $1,707.
  6. The NPV is negative in the first two outcomes, so the probability of a negative NPV is 0.3 + 0.5 = 0.8.

Answer: ENPV = $1,707 (positive), but there is an 80% probability of a negative NPV. The project is only marginally acceptable on EV and is risky. The ENPV of $1,707 is not a possible outcome of the project.

Example 2

A company is considering a two-year project with an initial outlay of $170,000 now. Annual net cash inflow = sales volume × contribution per unit, received at the end of each of years 1 and 2 and the same in both years. Contribution per unit is $8 (probability 0.6) or $12 (probability 0.4). Annual volume is 10,000 units (probability 0.7) or 15,000 units (probability 0.3). The two variables are independent. The cost of capital is 10% and the 2-year annuity factor is 1.736. Calculate the ENPV.

Show the solution
  1. Joint outcomes and probabilities (multiply, as the variables are independent): $8 and 10,000: 0.6 × 0.7 = 0.42; $8 and 15,000: 0.6 × 0.3 = 0.18; $12 and 10,000: 0.4 × 0.7 = 0.28; $12 and 15,000: 0.4 × 0.3 = 0.12. Total = 1.00.
  2. Annual cash flows: 8 × 10,000 = $80,000; 8 × 15,000 = $120,000; 12 × 10,000 = $120,000; 12 × 15,000 = $180,000.
  3. Group equal cash flows: $80,000 has probability 0.42; $120,000 has probability 0.18 + 0.28 = 0.46; $180,000 has probability 0.12.
  4. Expected annual cash flow = (0.42 × 80,000) + (0.46 × 120,000) + (0.12 × 180,000) = 33,600 + 55,200 + 21,600 = $110,400.
  5. Expected PV of inflows = 110,400 × 1.736 = $191,654.
  6. ENPV = 191,654 − 170,000 = $21,654 (to the nearest dollar).
  7. Risk check: at $80,000 a year the PV is 80,000 × 1.736 = 138,880, so the NPV is −$31,120. At $120,000 the NPV is +$38,320 and at $180,000 it is +$142,480. Probability of a negative NPV = 0.42.

Answer: ENPV = $21,654, so the project is acceptable on expected value. However, there is a 42% chance of a negative NPV (a loss of $31,120 in that case), so the decision depends on the company's attitude to risk.

Exam tips

  • In Section C, set out a probability table with columns for outcome, probability, cash flow or NPV and weighted value. The marker can then follow your working and give you method marks.
  • In objective test questions, you often only need the expected cash flow. Discount it once instead of finding every NPV. This saves time and is exact when the structure is linear.
  • When asked to comment, use three points: the decision on ENPV, a risk measure such as the probability of a loss, and a limitation of expected values that fits the scenario.
  • Check that probabilities add to 1 before calculating, and that your joint probability table also adds to 1. This catches slips quickly.
  • For decision trees, work backwards from the right, put the expected value at each chance node and cross out the rejected branches at decision nodes.

Practice questions from Adjusting for risk and uncertainty in investment appraisal

Expected Values and Probability Analysis in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Expected Values and Probability Analysis: frequently asked questions

How do I calculate expected NPV in ACCA FM?

Find the NPV for each possible outcome, multiply each by its probability and add the results. If only the cash flows are uncertain, you can instead discount the expected cash flows and subtract the outlay. Both give the same answer.

What are the limitations of expected values in investment appraisal?

The ENPV is an average that may not match any real outcome and ignores the spread of results. It suits repeated decisions better than one-off ones. The probabilities are usually subjective and it does not reflect the investor's attitude to risk.

What is a joint probability in investment appraisal?

It is the probability that two uncertain variables take particular values together, such as a certain price and a certain volume. For independent variables you multiply the individual probabilities. You then calculate the cash flows or NPV for every combination.

When should I use a decision tree?

Use it when there is a sequence of decisions and uncertain events, for example choosing whether to expand now or wait for market results. Start at the end, take expected values at chance nodes and choose the best option at decision nodes. Discount any later cash flows to a common date.