Advanced Financial Management · Advanced Capital Budgeting Decisions

Probability, Expected NPV and Decision Trees (CA Final AFM)

Updated 5 October 2026 · Fact-checked

Expected NPV is the probability-weighted average of possible NPVs. Standard deviation and coefficient of variation measure the risk around it. A decision tree lays out sequential choices and chance events. You solve it by rolling back from the right: take expected values at chance nodes and the best option at decision nodes.

Understand Probability, Expected NPV and Decision Trees

A project's cash flows are rarely certain. Instead of one forecast, you assign probabilities to different outcomes. Each outcome gives its own NPV. The expected NPV (ENPV) is the average of these NPVs, weighted by their probabilities. It tells you the likely value of the project if you could repeat it many times.

ENPV alone hides risk. Two projects can have the same ENPV, but one may swing widely. Standard deviation (σ) measures how far outcomes spread around the mean. When projects have different ENPVs, use the coefficient of variation (CV), which is risk per rupee of expected return. A lower CV means less risk for each rupee of ENPV.

Many investment decisions are sequential. You decide now, see what happens, and then decide again. A decision tree draws this. A square is a decision node where you choose. A circle is a chance node where nature decides, with probabilities on each branch. Later decisions depend on earlier outcomes, so the probabilities on later branches are conditional.

The tree is solved by rolling back. Start at the far-right end values. At a chance node, compute the expected value. At a decision node, pick the branch with the highest value and cross out the others. Continue until you reach the first decision. This way each later decision is made optimally, given what you know at that stage.

Key rules to remember

Expected NPV: E(NPV) = Σ pᵢ × NPVᵢ; Probabilities across all outcomes must add up to 1.
Expected cash flow and ENPV: E(CFₜ) = Σ p × CFₜ; ENPV = Σ E(CFₜ) ÷ (1 + k)ᵗ − Initial outlay; Use this when probabilities are given on yearly cash flows, not on NPVs.
Standard deviation: σ = √[Σ pᵢ × (xᵢ − x̄)²]; x̄ is the expected value. Weight each squared deviation by its probability.
Coefficient of variation: CV = σ ÷ Expected NPV; Use to compare projects with different ENPVs. Lower CV means lower relative risk. Meaningful only when ENPV is positive.
σ of NPV, independent yearly cash flows: σ(NPV) = √[Σ σₜ² ÷ (1 + k)²ᵗ]; Use only when cash flows of different years are independent.
σ of NPV, perfectly correlated cash flows: σ(NPV) = Σ σₜ ÷ (1 + k)ᵗ; Use when a good year is always followed by a good year.
Probability of a negative NPV (normal assumption): Z = (0 − ENPV) ÷ σ; Read the area from the normal table. State the normal-distribution assumption.
Joint (path) probability in a tree: P(path) = P(first branch) × P(second branch | first branch); Multiply along the path. Never add.
Rolling back: Chance node value = Σ p × value of branch; Decision node value = highest branch value; Deduct costs incurred on a branch before comparing.

How to solve Probability, Expected NPV and Decision Trees questions

Use this order for any probability or decision tree question. It keeps the working clean and easy to mark.

1Read the case and list the decisions, the chance events, the costs and the probabilities. Check that probabilities at each chance node add up to 1.
2For a probability-distribution question, find the NPV of each outcome. If probabilities are on yearly cash flows, find the expected cash flow of each year first.
3Compute ENPV as the sum of probability × NPV.
4Compute the deviations from ENPV, square them, weight them by probability, add them and take the square root to get σ.
5Compute CV = σ ÷ ENPV for each project, and compare projects on ENPV and CV together.
6For a tree, draw it left to right with squares and circles. Put each cost on its branch and the end value at the right.
7Roll back from the right. Take expected values at circles, the best branch at squares, and deduct costs on the way. Show every node value.
8State the decision in words: the best first action, the later actions that follow, and the expected value. Mention any assumption such as ignoring discounting between stages.

Quickest way: Work in thousands and fill the tree from the right

When to use it: Use when time is short, especially for the 30% case-scenario MCQs and for descriptive tree questions with many branches.

Convert large figures to lakhs or thousands so squares stay small, and convert back in the final answer.
For σ, use the shortcut Σ p x² − (ENPV)² for the variance. Use it if the figures are small and clean. Otherwise use deviations to avoid errors.
For a symmetric distribution, such as outcomes equally spaced around the mean with equal tail probabilities, the deviations repeat. Compute one and reuse it.
In a tree, write the end value, then the node value, directly on the diagram. Do not redraw.
Check the answer: the node value must lie between the lowest and highest branch values, and total probability must be 1.
In MCQs, first eliminate options whose ENPV falls outside the range of the NPV outcomes.

Common mistakes in Probability, Expected NPV and Decision Trees

Taking the expected value of the cash flows and then using it as if it were the NPV, without discounting or deducting the outlay.
Students confuse the expected cash flow with expected NPV.
Fix: Discount each year's expected cash flow, add them, and then deduct the outlay. Alternatively, find each scenario's NPV first and then average.
Adding probabilities along a path in a tree instead of multiplying.
Students treat branches as alternatives rather than a sequence.
Fix: Multiply along a path to get the joint probability. Add only the probabilities of different paths that end in the same outcome.
Using CV to rank a project with a negative or very small ENPV.
Students apply the formula mechanically.
Fix: CV is meaningful for positive ENPV. If ENPV is negative, reject the project or compare using another criterion and say so.
Choosing between projects on lower σ alone when ENPVs differ.
Students forget that σ is an absolute measure.
Fix: Compare CV for risk per rupee of return. Then give your final choice with a reason, including the ENPV.
Forgetting to deduct the cost of a test or survey branch before comparing it with the direct option.
The cost is stated early in the case and gets lost at the end of the rolling back.
Fix: Write the cost beside the branch when you draw the tree. Deduct it at that branch's node before comparing.
Using the wrong formula for σ of NPV across years, applying the independent formula when the question says cash flows are perfectly correlated.
Students skip the correlation sentence in the case.
Fix: Underline the words 'independent' or 'perfectly correlated' first. Independent: add discounted variances, then take the root. Perfectly correlated: add discounted standard deviations.

Worked examples

Example 1

A company is choosing between Project X and Project Y. Each needs a similar outlay. After discounting, the NPV outcomes are: Project X: ₹-2,00,000 (probability 0.2), ₹3,00,000 (0.5), ₹6,00,000 (0.3). Project Y: ₹1,00,000 (0.3), ₹2,50,000 (0.4), ₹4,00,000 (0.3). Compute the expected NPV, standard deviation and coefficient of variation of each, and advise the company.

Show the solution

Project X ENPV = 0.2 × (-2,00,000) + 0.5 × 3,00,000 + 0.3 × 6,00,000 = -40,000 + 1,50,000 + 1,80,000 = ₹2,90,000.
Project X deviations: -4,90,000, +10,000 and +3,10,000. Work in thousands: squares are 240,100; 100; 96,100.
Weighted: 0.2 × 240,100 = 48,020; 0.5 × 100 = 50; 0.3 × 96,100 = 28,830. Variance = 76,900 (in thousand²). σ = √76,900 ≈ 277.3 thousand = about ₹2,77,300.
Project X CV = 2,77,300 ÷ 2,90,000 ≈ 0.96.
Project Y ENPV = 0.3 × 1,00,000 + 0.4 × 2,50,000 + 0.3 × 4,00,000 = 30,000 + 1,00,000 + 1,20,000 = ₹2,50,000.
Project Y deviations: -1,50,000, 0 and +1,50,000. Variance = 0.3 × 150² + 0 + 0.3 × 150² = 13,500 (thousand²). σ = √13,500 ≈ 116.2 thousand = about ₹1,16,200.
Project Y CV = 1,16,200 ÷ 2,50,000 ≈ 0.46.
Interpretation: X has the higher ENPV (₹40,000 more) but about twice the relative risk. It is also the only project that can produce a negative NPV.

Answer: ENPV: X ₹2,90,000; Y ₹2,50,000. σ: X about ₹2,77,300; Y about ₹1,16,200. CV: X about 0.96; Y about 0.46. Y gives a lower risk per rupee of return, and all its outcomes are positive. A risk-averse company should choose Y. A company that accepts higher risk for ₹40,000 more of ENPV may choose X.

Example 2

A firm can launch a new product immediately at an outlay of ₹50 lakh. The present value of inflows is ₹100 lakh if demand is high (probability 0.6) and ₹30 lakh if demand is low (probability 0.4). Alternatively, it can first spend ₹5 lakh on a test market. The test result is favourable with probability 0.5 and unfavourable with probability 0.5. After a favourable result, the probability of high demand is 0.9. After an unfavourable result, it is 0.3. Launch after the test needs the same ₹50 lakh. Ignore discounting between the test and the launch. Should the firm test first?

Show the solution

Direct launch: expected PV of inflows = 0.6 × 100 + 0.4 × 30 = 60 + 12 = ₹72 lakh. NPV = 72 − 50 = ₹22 lakh.
Test, favourable result: expected PV = 0.9 × 100 + 0.1 × 30 = 90 + 3 = ₹93 lakh. Launch NPV = 93 − 50 = ₹43 lakh. This is above 0, so launch.
Test, unfavourable result: expected PV = 0.3 × 100 + 0.7 × 30 = 30 + 21 = ₹51 lakh. Launch NPV = 51 − 50 = ₹1 lakh. This is above 0, so still launch, though marginally. (Not launching gives 0.)
Expected value after the test, before its cost = 0.5 × 43 + 0.5 × 1 = 21.5 + 0.5 = ₹22 lakh.
Deduct the test cost: 22 − 5 = ₹17 lakh.
Compare at the first decision node: launch now = ₹22 lakh; test first = ₹17 lakh.
Check: overall P(high demand) via the test = 0.5 × 0.9 + 0.5 × 0.3 = 0.6, which matches the direct launch estimate. So the test gives no extra expected value, because the best action does not change enough to pay for itself: the firm would launch in both cases.

Answer: Launch immediately. Expected NPV is ₹22 lakh, against ₹17 lakh if the firm tests first. The test costs ₹5 lakh but the firm would launch whatever the result, so the test adds no value.

Exam tips

Draw the tree even in MCQ-type cases. Squares for decisions and circles for chances stop you from missing a later decision.
In descriptive answers, show the node values on the diagram and the calculation below it. Marks are given for the method even if one figure is wrong.
Always end with a recommendation in words and name the criterion used: highest ENPV, or lowest CV for a risk-averse view.
Check whether the question wants probabilities on NPVs or on yearly cash flows. The method changes, and so does the discounting.
Read the line on correlation of cash flows carefully, as it decides which σ formula to use. State the assumption if the question is silent.

Practice questions from Advanced Capital Budgeting Decisions

Probability, Expected NPV and Decision Trees in other exams

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

Probability, Expected NPV and Decision Trees: frequently asked questions

How do I solve a decision tree question in AFM?

Draw the tree from left to right and mark the cost on each branch. Roll back from the right: expected value at chance nodes and the best option at decision nodes. Deduct costs on the way and state the first decision.

When should I use coefficient of variation instead of standard deviation?

Use CV when the projects have different expected NPVs. It shows risk per rupee of expected return, so it gives a fairer comparison. If the ENPVs are equal, standard deviation alone is enough.

Do I need to discount in a decision tree?

Yes, if cash flows come at different times and a discount rate is given. Convert all branch values to present value before taking expected values. If the question says to ignore discounting, do not discount.

What is the difference between expected NPV and the NPV of expected cash flows?

For the NPV formula, which is linear in cash flows, both give the same result when discounted at the same rate. The difference lies in the method. Scenario NPVs let you also compute σ, while expected cash flows alone give only the ENPV.