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Strategic Financial Management · Evaluation of Risky Proposals for Investment Decisions

Probability Distribution and Expected NPV in Investment Decisions

Updated 11 October 2026 · Fact-checked

The probability distribution approach assigns probabilities to a project's cash flows or NPVs. Expected NPV is the probability-weighted average. Standard deviation measures spread around it, and coefficient of variation (SD ÷ expected NPV) gives risk per rupee of return. Compute each, then compare projects and recommend.

Understand Probability Distribution and Expected NPV

A normal NPV sum uses one cash flow per year. In real projects, you are not sure of that number. Sales may be weak, normal or strong. The probability distribution approach lists each possible outcome and gives it a probability. The probabilities must add up to 1.

The expected value is the weighted average of the outcomes. If you expect ₹80,000 with probability 0.5 and ₹1,20,000 with probability 0.5, the expected value is ₹1,00,000. Discount the expected cash flows at the risk-free or normal rate to get the expected NPV (ENPV). Because discounting is linear, this equals the probability-weighted average of the possible NPVs.

Expected NPV hides risk. Two projects can have the same ENPV, but one has outcomes close together and the other has wide swings. The standard deviation (σ) captures this. It is the square root of the probability-weighted squared deviations from the mean. A higher σ means a riskier project.

To compare projects of different size, use the coefficient of variation (CV). It is σ ÷ ENPV. A lower CV means less risk for each rupee of expected return. Use it when ENPVs differ.

For several years, you need to know how cash flows in different years relate. If they are independent, variances add after discounting. If they are perfectly correlated, standard deviations add after discounting. If NPV is assumed to be normally distributed, you can convert σ into a probability using the Z value, for example the chance that NPV is negative.

Key rules to remember

Expected value
E(X) = Σ pᵢ × Xᵢ
Probabilities must sum to 1. Applies to cash flows or to NPVs.
Expected NPV
ENPV = Σ [E(CFₜ) ÷ (1 + r)ᵗ] − Initial outlay
Initial outlay is treated as certain unless the question gives its distribution.
Standard deviation of a distribution
σ = √[Σ pᵢ × (Xᵢ − E(X))²]
Use deviations from the mean, weighted by probability.
σ of NPV, independent yearly cash flows
σ(NPV) = √[Σ σₜ² ÷ (1 + r)²ᵗ]
Square the discount factor. Add variances, then take the root.
σ of NPV, perfectly correlated cash flows
σ(NPV) = Σ [σₜ ÷ (1 + r)ᵗ]
Add the discounted standard deviations. This gives the highest σ.
Coefficient of variation
CV = σ ÷ ENPV
Lower CV means lower risk per rupee of expected NPV. Meaningful when ENPV is positive.
Z value for probability under normal distribution
Z = (X − ENPV) ÷ σ
For P(NPV < 0), use X = 0. Read the area from the normal table.

How to solve Probability Distribution and Expected NPV questions

Follow the same order for any question on this topic. Check first whether the question gives distributions of cash flows by year or of whole-project NPVs.

  1. 1Check that each probability set adds to 1. Note the discount rate and whether cash flows are independent or perfectly correlated.
  2. 2Compute the expected cash flow for each year as Σ p × CF.
  3. 3Discount each expected cash flow, add them, and subtract the initial outlay to get ENPV.
  4. 4Compute σ for each year, using deviations from that year's mean.
  5. 5Combine into σ of NPV. For independent flows, discount σ² with (1 + r)²ᵗ, add, then take the root. For perfectly correlated flows, add discounted σ.
  6. 6Compute CV = σ ÷ ENPV if projects are being compared.
  7. 7If asked for a probability, compute Z = (0 − ENPV) ÷ σ and read the normal table.
  8. 8Write a recommendation: higher ENPV with lower CV is preferred. If they conflict, state the trade-off and decide using risk appetite.

Quickest way: Table-first method for exam time

When to use it: Use when NPV outcomes are given directly as a list with probabilities, as in comparing two projects.

  1. Draw one table with columns NPV, p, p × NPV. The column total is ENPV.
  2. Add a column p × (NPV − ENPV)². Its total is the variance.
  3. Take the square root for σ, then divide by ENPV for CV.
  4. For multi-year independent flows, build one row per year: E(CF), σ², discount factor, factor squared. Then sum.
  5. Round discount factors to four decimals and keep working to the nearest rupee. Round Z to two decimals.

Common mistakes in Probability Distribution and Expected NPV

  • Adding yearly standard deviations without discounting them

    Students treat σ like a cash flow total but forget the time value.

    Fix: Discount σₜ by (1 + r)ᵗ for correlated flows, or discount σₜ² by (1 + r)²ᵗ for independent flows.

  • Discounting σ² with (1 + r)ᵗ instead of (1 + r)²ᵗ

    Variance is squared, but the factor is applied as for cash flows.

    Fix: Square the discount factor whenever you discount a variance. Take the root only at the end.

  • Taking deviations from the wrong mean

    Students use the most likely value or the simple average, not the probability-weighted mean.

    Fix: Compute E(X) first, then take every deviation from it.

  • Choosing the project with the lower σ without checking CV

    σ is an absolute figure and ignores the size of expected return.

    Fix: When ENPVs differ, compare CV. Mention both ENPV and CV in the recommendation.

  • Using the wrong sign or tail for P(NPV < 0)

    Z comes out negative and students look up the area on the wrong side.

    Fix: For positive ENPV, Z for zero is negative. The probability of loss is the area to the left, which is 1 minus the table area for the positive Z.

  • Assuming independence or perfect correlation without reading the question

    Students apply one formula by habit.

    Fix: Look for words such as independent, uncorrelated or perfectly correlated. If none is stated, state your assumption in the answer.

Worked examples

Example 1

A project needs an initial outlay of ₹2,00,000. Year 1 cash flow: ₹80,000 (p 0.2), ₹1,00,000 (p 0.6), ₹1,20,000 (p 0.2). Year 2 cash flow: ₹1,00,000 (p 0.3), ₹1,40,000 (p 0.4), ₹1,80,000 (p 0.3). The cash flows are independent and the discount rate is 10%. Find (a) expected NPV, (b) standard deviation of NPV, (c) coefficient of variation, and (d) the probability that NPV is negative, assuming NPV is normally distributed. Discount factors at 10%: 0.9091 and 0.8264.

Show the solution
  1. Year 1 expected cash flow = 0.2 × 80,000 + 0.6 × 1,00,000 + 0.2 × 1,20,000 = 16,000 + 60,000 + 24,000 = ₹1,00,000.
  2. Year 2 expected cash flow = 0.3 × 1,00,000 + 0.4 × 1,40,000 + 0.3 × 1,80,000 = 30,000 + 56,000 + 54,000 = ₹1,40,000.
  3. PV of expected inflows = 1,00,000 × 0.9091 + 1,40,000 × 0.8264 = 90,910 + 1,15,696 = ₹2,06,606. ENPV = 2,06,606 − 2,00,000 = ₹6,606.
  4. Year 1 variance = 0.2 × (20,000)² + 0.6 × 0 + 0.2 × (20,000)² = 1,60,00,000 × 10 = 16,00,00,000 (₹1.6 × 10⁸).
  5. Year 2 variance = 0.3 × (40,000)² + 0.4 × 0 + 0.3 × (40,000)² = 0.6 × 16,00,00,000 × 100 = 9.6 × 10⁸.
  6. Variance of NPV = 1.6 × 10⁸ × (0.9091)² + 9.6 × 10⁸ × (0.8264)² = 1.6 × 10⁸ × 0.82646 + 9.6 × 10⁸ × 0.68294 ≈ 13,22,34,050 + 65,56,19,482 ≈ 7.8785 × 10⁸.
  7. σ(NPV) = √(7.8785 × 10⁸) ≈ ₹28,069.
  8. CV = 28,069 ÷ 6,606 ≈ 4.25.
  9. Z = (0 − 6,606) ÷ 28,069 ≈ −0.24. Area to the left of Z = −0.24 is 1 − 0.5948 = 0.4052.

Answer: ENPV = ₹6,606; σ ≈ ₹28,069; CV ≈ 4.25; probability of negative NPV ≈ 40.5%. The ENPV is positive but small against the risk, so the project is very risky for its return.

Example 2

Two mutually exclusive projects have these NPV distributions. Project X: ₹40,000 (p 0.25), ₹80,000 (p 0.50), ₹1,20,000 (p 0.25). Project Y: −₹20,000 (p 0.20), ₹60,000 (p 0.50), ₹1,40,000 (p 0.30). Compute ENPV, standard deviation and CV of each and recommend one.

Show the solution
  1. Project X: ENPV = 0.25 × 40,000 + 0.50 × 80,000 + 0.25 × 1,20,000 = 10,000 + 40,000 + 30,000 = ₹80,000.
  2. Project X variance = 0.25 × (40,000)² + 0 + 0.25 × (40,000)² = 0.5 × 1,60,00,00,000 = 8 × 10⁸. σ = √(8 × 10⁸) ≈ ₹28,284. CV = 28,284 ÷ 80,000 ≈ 0.354.
  3. Project Y: ENPV = 0.20 × (−20,000) + 0.50 × 60,000 + 0.30 × 1,40,000 = −4,000 + 30,000 + 42,000 = ₹68,000.
  4. Project Y deviations: −88,000, −8,000 and +72,000. Variance = 0.20 × 7.744 × 10⁹ + 0.50 × 6.4 × 10⁷ + 0.30 × 5.184 × 10⁹ = 1.5488 × 10⁹ + 3.2 × 10⁷ + 1.5552 × 10⁹ = 3.136 × 10⁹.
  5. σ(Y) = √(3.136 × 10⁹) = ₹56,000. CV = 56,000 ÷ 68,000 ≈ 0.824.
  6. Compare: X has the higher ENPV (₹80,000 against ₹68,000), the lower σ and the lower CV. Y also has a 20% chance of a negative NPV, while X never goes negative.

Answer: X: ENPV ₹80,000, σ ≈ ₹28,284, CV ≈ 0.35. Y: ENPV ₹68,000, σ = ₹56,000, CV ≈ 0.82. Recommend Project X, since it gives higher return with lower risk on every measure.

Exam tips

  • In the MCQ section, expect a quick expected value, a σ from a short table, or a CV comparison. Practise these until each takes about a minute.
  • In written answers, show a table for every distribution. Marks are given for the method even if the arithmetic slips.
  • Always end with a decision sentence that quotes ENPV and CV. A set of numbers with no recommendation loses marks.
  • Read the correlation wording first. It decides whether you add variances or standard deviations.
  • For normal-distribution parts, show Z, the table area and the final probability clearly. Mention that the normal assumption is being used.

Practice questions from Evaluation of Risky Proposals for Investment Decisions

Probability Distribution and Expected NPV in other exams

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

Probability Distribution and Expected NPV: frequently asked questions

What is the difference between standard deviation and coefficient of variation?

Standard deviation is an absolute measure of spread in rupees. Coefficient of variation divides it by expected NPV, so it shows risk per rupee of return. Use CV to compare projects with different expected NPVs.

How do I calculate standard deviation of NPV for a multi-year project?

Find the standard deviation of each year's cash flow. If the flows are independent, discount each variance by (1 + r)²ᵗ, add them and take the square root. If they are perfectly correlated, discount each standard deviation by (1 + r)ᵗ and add.

How do I find the probability that NPV is negative?

Assume NPV is normally distributed. Compute Z = (0 − ENPV) ÷ σ, then read the area to the left of that Z from the normal table. With positive ENPV, Z is negative and the probability is below 50%.

Which discount rate is used for expected NPV?

Use the rate given in the question. In this approach, risk is shown through the spread of outcomes, not by raising the rate. A risk-adjusted rate is a separate method.

What if the project with higher ENPV also has a higher CV?

Then there is a trade-off. State both figures, say which project gives more return per unit of risk, and decide based on the firm's attitude to risk. Examiners accept a reasoned choice.