Strategic Management and Corporate Finance · Project Evaluation

Risk Analysis in Capital Budgeting: Methods and Numericals

Updated 11 October 2026 · Fact-checked

Risk analysis in capital budgeting tests how reliable a project's expected cash flows are. You either raise the discount rate (risk-adjusted discount rate), scale down the cash flows (certainty equivalent), or test outcomes using sensitivity, scenario, simulation or decision trees. Then you compare NPV, standard deviation and coefficient of variation to decide.

Understand Risk Analysis in Capital Budgeting

Every project cash flow is a forecast. Sales, costs, prices and the project life can all turn out different from your estimate. Risk here means the variability of the actual cash flows around the expected ones. Plain NPV uses one set of cash flows and one discount rate. It does not show how shaky those numbers are.

There are two ways to build risk into the NPV. In the risk-adjusted discount rate (RADR) method you keep the cash flows and raise the discount rate for riskier projects. The rate is the risk-free rate plus a risk premium. In the certainty equivalent (CE) method you keep the rate at the risk-free rate and cut each cash flow to the amount you would accept for sure. The factor used for this is the CE coefficient (α), between 0 and 1. Riskier cash flows get a lower α.

The main difference is this. RADR puts all the risk into the denominator and so assumes risk grows steadily with time. CE adjusts each year's cash flow separately, so it can treat each year's risk on its own. That is why CE is often seen as more flexible in theory. RADR is easier to use and is more common in practice.

The other techniques test the forecast itself. Sensitivity analysis changes one variable at a time (sales volume, price, cost) and checks how NPV moves. Scenario analysis changes several variables together into a set of cases, usually worst, expected and best, often with probabilities. Simulation (Monte Carlo) draws random values for the uncertain variables many times and gives a probability distribution of NPV. Decision tree analysis maps a sequence of decisions and chance events. You work backwards from the end to pick the best path.

To measure risk you use the expected NPV, the standard deviation of NPV, and the coefficient of variation (risk per rupee of return). A higher standard deviation means more risk. When projects have different expected returns, the coefficient of variation gives the fairer comparison.

Key rules to remember

Risk-adjusted discount rate: r(adj) = Risk-free rate + Risk premium; The premium is higher for riskier projects. Use r(adj) for every year's cash flow.
NPV under RADR: NPV = Σ [CFt ÷ (1 + r(adj))^t] − Initial outlay; Cash flows stay as expected values. Only the rate changes.
Certainty equivalent coefficient: α = Certain cash flow ÷ Risky (expected) cash flow; α lies between 0 and 1. A lower α means higher risk. CE cash flow = α × expected cash flow.
NPV under certainty equivalent: NPV = Σ [αt × CFt ÷ (1 + Rf)^t] − Initial outlay; Discount at the risk-free rate Rf, not the cost of capital, because risk is already removed.
Expected NPV: E(NPV) = Σ [Pi × NPVi]; Pi is the probability of scenario i. The probabilities must add up to 1.
Standard deviation of NPV: σ = √ Σ [Pi × (NPVi − E(NPV))²]; Measures the absolute spread of outcomes around the expected NPV.
Coefficient of variation: CV = σ ÷ E(NPV); Risk per rupee of expected return. Lower CV means less risk per rupee. Use it when expected NPVs differ.
Sensitivity measure: Sensitivity = % change in NPV ÷ % change in the variable; The variable with the largest value is the most critical one.
Decision tree rollback: EMV at chance node = Σ [P × payoff]; at decision node choose the highest EMV; Start at the right-hand end. Subtract any cost on a branch before comparing.

How to solve Risk Analysis in Capital Budgeting questions

Use this method for any numerical or theory question on project risk. It keeps your answer in the provision, analysis, conclusion format that the written paper expects.

1Read the question and name the technique asked for (RADR, CE, sensitivity, scenario, simulation or decision tree). If it says 'evaluate' or 'advise', also plan a conclusion.
2List the data in a small table: outlay, yearly cash flows, probabilities, rates, α values. Check whether the cash flows are already after tax.
3Pick the discount rate correctly. Use the RADR for the RADR method. Use the risk-free rate for the CE method after applying α.
4Compute present values year by year, using discount factors to four decimals. Show each year's working so you earn step marks.
5For probability-based questions, compute expected NPV first. Then find the deviations, the variance, the standard deviation and the CV.
6For decision trees, draw the tree, put probabilities on chance branches, roll back from the right, and write the EMV at every node. Net off the cost of each decision.
7Compare the results and state the decision: accept if NPV is positive, or pick the option with higher expected NPV and an acceptable CV.
8Close with one line on the limits of the method. For example, RADR assumes risk rises uniformly, and sensitivity changes one variable at a time.

Quickest way: Rollback and rank method for risk numericals

When to use it: Use it when time is short and the question gives probabilities or alternative options. It works for scenario and decision tree questions.

Compute the NPV of each outcome first, net of outlay, so every figure is already an NPV.
Multiply by probabilities and add to get expected NPV for each option.
Compute σ only if the question asks for risk. Use the deviation from expected NPV and square it.
Find CV = σ ÷ E(NPV) for each option and rank options by expected NPV, then by CV.
Write the decision in one line: highest expected NPV, or lowest CV if the question stresses risk.

Common mistakes in Risk Analysis in Capital Budgeting

Discounting certainty equivalent cash flows at the cost of capital or RADR.
Students remember 'discount rate' and reuse the rate from the previous method.
Fix: Once α has removed the risk, discount at the risk-free rate. Using a higher rate counts the risk twice.
Multiplying the cash flow by α but forgetting to deduct the initial outlay, or applying α to the outlay.
Students treat the outlay like any other cash flow.
Fix: The outlay at time 0 is certain, so α is 1 for it. Adjust only the future inflows, then subtract the outlay.
Computing standard deviation around zero or around the wrong mean.
Rushing and skipping the expected NPV step.
Fix: Always find E(NPV) first. Then take each NPV minus E(NPV), square it, multiply by its probability and add.
Comparing options only by standard deviation when their expected NPVs are different.
A lower σ looks safer at first glance.
Fix: Compute the coefficient of variation. It shows the risk per rupee of expected return and gives the fair comparison.
In decision trees, ignoring the cost on a branch or rolling back from the left.
Students read the tree in the order events happen.
Fix: Start from the rightmost payoffs, compute EMV at chance nodes, choose the best branch at decision nodes, and subtract costs before comparing.
Describing sensitivity analysis as testing all variables together.
Confusion with scenario analysis.
Fix: Remember that sensitivity changes one variable at a time with the others held constant, and scenario analysis changes several variables together as one case.

Worked examples

Example 1

Nova Fabrics Ltd is considering a project with an outlay of ₹10,00,000. Expected cash flows are ₹4,00,000, ₹5,00,000 and ₹6,00,000 in years 1 to 3. The RADR is 15%. Under the CE method, the α values are 0.90, 0.80 and 0.70 for years 1 to 3, and the risk-free rate is 6%. Find the NPV under both methods and advise. Use discount factors to four decimals.

Show the solution

RADR method: discount factors at 15% are 0.8696, 0.7561 and 0.6575.
PV of year 1 = 4,00,000 × 0.8696 = ₹3,47,840. PV of year 2 = 5,00,000 × 0.7561 = ₹3,78,050. PV of year 3 = 6,00,000 × 0.6575 = ₹3,94,500.
Total PV = 3,47,840 + 3,78,050 + 3,94,500 = ₹11,20,390. NPV = 11,20,390 − 10,00,000 = ₹1,20,390.
CE method: CE cash flows = 4,00,000 × 0.90 = ₹3,60,000; 5,00,000 × 0.80 = ₹4,00,000; 6,00,000 × 0.70 = ₹4,20,000.
Discount factors at 6% are 0.9434, 0.8900 and 0.8396.
PV of year 1 = 3,60,000 × 0.9434 = ₹3,39,624. PV of year 2 = 4,00,000 × 0.8900 = ₹3,56,000. PV of year 3 = 4,20,000 × 0.8396 = ₹3,52,632.
Total PV = 3,39,624 + 3,56,000 + 3,52,632 = ₹10,48,256. NPV = 10,48,256 − 10,00,000 = ₹48,256.

Answer: NPV is about ₹1,20,390 under RADR and ₹48,256 under CE. Both are positive, so the project should be accepted. The CE result is lower, which shows it treats the risk more strictly in this case.

Example 2

Arya Foods Ltd must choose between Option A (large plant, outlay ₹50 lakh) and Option B (small plant, outlay ₹20 lakh). For both, demand is high with probability 0.6 and low with probability 0.4. Present value of inflows: Option A is ₹90 lakh if demand is high and ₹30 lakh if low. Option B is ₹40 lakh if high and ₹25 lakh if low. Using expected NPV and coefficient of variation, advise which option to choose.

Show the solution

Option A NPV: high = 90 − 50 = ₹40 lakh; low = 30 − 50 = −₹20 lakh.
E(NPV) of A = 0.6 × 40 + 0.4 × (−20) = 24 − 8 = ₹16 lakh.
Option B NPV: high = 40 − 20 = ₹20 lakh; low = 25 − 20 = ₹5 lakh.
E(NPV) of B = 0.6 × 20 + 0.4 × 5 = 12 + 2 = ₹14 lakh.
Variance of A = 0.6 × (40 − 16)² + 0.4 × (−20 − 16)² = 0.6 × 576 + 0.4 × 1,296 = 345.6 + 518.4 = 864. σ(A) = √864 = ₹29.39 lakh.
Variance of B = 0.6 × (20 − 14)² + 0.4 × (5 − 14)² = 0.6 × 36 + 0.4 × 81 = 21.6 + 32.4 = 54. σ(B) = √54 = ₹7.35 lakh.
CV of A = 29.39 ÷ 16 = 1.84. CV of B = 7.35 ÷ 14 = 0.525, which is about 0.53.

Answer: Option A has the higher expected NPV (₹16 lakh against ₹14 lakh) but much higher risk (CV 1.84 against 0.53). It can also make a loss. Option B gives almost the same return with far less risk per rupee and no chance of a loss. A risk-averse board should choose Option B. A board willing to accept risk for a higher expected NPV may choose Option A.

Exam tips

Write a one-line definition of the method before the numerical. Short theory lines earn marks in descriptive papers.
Show every present value separately and state the discount factor used. Step marks depend on visible working.
For 'distinguish between RADR and CE' questions, give three or four points: where risk is adjusted (rate or cash flow), discount rate used, treatment of time, and ease of use.
End each numerical with a clear decision line. Examiners look for the conclusion, not just the figures.
For theory on simulation and decision trees, name the steps and one advantage and one limitation, such as heavy data needs or difficulty in assigning probabilities.

Practice questions from Project Evaluation

Risk Analysis in Capital Budgeting in other exams

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

Risk Analysis in Capital Budgeting: frequently asked questions

What is the difference between risk-adjusted discount rate and certainty equivalent?

RADR keeps the expected cash flows and raises the discount rate to allow for risk. CE keeps the rate at the risk-free level and reduces each cash flow by a factor α to the amount you would accept for certain. RADR is simpler. CE can treat the risk of each year separately.

What is the difference between sensitivity analysis and scenario analysis?

Sensitivity analysis changes one variable at a time, keeping the others fixed, to see how NPV reacts. Scenario analysis changes several variables together to form cases such as worst, expected and best. Scenario analysis is closer to real conditions because variables often move together.

How do you solve a decision tree numerical in capital budgeting?

Draw the tree with decision nodes and chance nodes and put probabilities on the chance branches. Start from the right and compute the expected value at each chance node. At each decision node choose the branch with the highest value after deducting its cost. The value at the first node is the answer.

How do you measure risk in project evaluation?

You measure it through the spread of possible NPVs. Find the expected NPV, then the standard deviation, and then the coefficient of variation. A higher standard deviation means higher risk. CV is better when comparing projects with different expected NPVs.

At what rate do you discount in the certainty equivalent method?

You discount the certainty equivalent cash flows at the risk-free rate. The risk has already been removed by α, so using a higher rate would count the risk twice.