Financial Management and Strategic Management · Investment Decisions

Risk Analysis in Capital Budgeting for CA Inter

Updated 4 October 2026 · Fact-checked

Risk analysis in capital budgeting tests how uncertain cash flows affect a project's NPV. You either adjust the discount rate (RADR) or the cash flows (certainty equivalent), or study the spread of outcomes using sensitivity, scenarios, simulation, decision trees, expected NPV and standard deviation. Then accept only if the return justifies the risk.

Understand Risk Analysis in Capital Budgeting

Every project cash flow is a forecast. Sales, costs and project life can all turn out different. Risk here means the chance that actual cash flows differ from the forecast. If you ignore it, NPV looks more certain than it is.

There are two ways to build risk into the NPV itself. The risk-adjusted discount rate (RADR) raises the discount rate above the risk-free rate for riskier projects. The certainty equivalent (CE) method cuts the cash flows to the amount you would accept for sure, then discounts them at the risk-free rate. RADR puts all the risk in the denominator. CE puts it in the numerator and treats each year separately.

Other techniques do not change the rate. They show how bad or good things can get. Sensitivity analysis changes one variable at a time and sees how NPV moves. Scenario analysis changes several variables together into pessimistic, expected and optimistic cases. Simulation runs many random trials on the variables and builds a distribution of NPV.

When probabilities are known, compute expected NPV and its standard deviation. A higher standard deviation means more risk. Compare projects with different sizes using the coefficient of variation. A decision tree is used when decisions come in stages and each stage has chance outcomes. You work backwards (roll back) from the end.

Key rules to remember

Risk-adjusted discount rate: RADR = Risk-free rate + Risk premium; Discount the normal (uncertain) cash flows at RADR. Higher risk means a higher premium.
Certainty equivalent coefficient: α = Certain cash flow ÷ Risky (expected) cash flow; α lies between 0 and 1. Lower α means the management is more risk averse for that year.
NPV by certainty equivalent: NPV = Σ [α(t) × CFt ÷ (1 + Rf)^t] − Initial outlay; Discount at the risk-free rate Rf, not at the cost of capital.
Expected NPV: E(NPV) = Σ [p × NPV]; Probabilities must add up to 1.
Standard deviation of NPV: σ = √ Σ [p × (NPV − E(NPV))²]; Measures total spread of outcomes around the expected NPV.
Coefficient of variation: CV = σ ÷ E(NPV); Risk per rupee of expected return. Lower CV is better when comparing projects.
SD of project NPV, independent cash flows: σNPV = √ Σ [σt² ÷ (1 + i)^(2t)]; Use when cash flows of different years are independent. i is the risk-free rate.
SD of project NPV, perfectly correlated cash flows: σNPV = Σ [σt ÷ (1 + i)^t]; Use when cash flows in each year move together perfectly.
Decision tree rollback: Value at a chance node = Σ (probability × value of branch); at a decision node = best branch; Start at the right-hand end and move left.

How to solve Risk Analysis in Capital Budgeting questions

Use this order for any question on this topic. First identify which technique the question asks for, then follow the steps.

1Read the question and name the technique: RADR, CE, sensitivity, scenario, probability and SD, or decision tree.
2List the initial outlay and the cash flows for each year. Note whether the rate given is risk-free or risk-adjusted.
3For RADR, discount all cash flows at the risk-adjusted rate. For CE, multiply each cash flow by its α first, then discount at the risk-free rate.
4For probability questions, compute the NPV (or cash flow) for each outcome, then E(NPV) = Σ p × NPV.
5Find each deviation from E(NPV), square it, multiply by p, add up and take the square root to get σ. Compute CV if projects are compared.
6For a decision tree, draw the tree, write the probabilities and payoffs, then roll back from the right. At chance nodes take the expected value. At decision nodes pick the best branch. Deduct any cost at that decision.
7State the decision in one line. Link it to NPV, risk (σ or CV) and the management's attitude to risk.

Quickest way: Fast route for MCQs and step-marked written answers

When to use it: Use under time pressure. MCQs carry no negative marking, so always attempt every one.

For MCQs, first eliminate options that break a rule: CE is discounted at the risk-free rate, and α is never above 1 for a risk-averse investor.
If a question asks which project is riskier, a higher σ (same size) or a higher CV (different sizes) is the answer.
In sums with probabilities, write a small table with columns p, NPV, p × NPV, deviation squared, p × deviation squared. This lays out the working and each line can earn step marks.
Use round PV factors given in the question. Do not recompute them.
End every written answer with a one-line conclusion: accept, reject or choose project X because of its NPV and risk.

Common mistakes in Risk Analysis in Capital Budgeting

Discounting certainty equivalent cash flows at the cost of capital or RADR.
Students remember that discount rate means cost of capital.
Fix: CE flows are already adjusted for risk. Discount them at the risk-free rate only.
Treating sensitivity analysis as if it changes several variables at once.
It gets mixed up with scenario analysis.
Fix: Sensitivity changes one variable at a time, others held constant. Scenario analysis changes a set of variables together.
Taking the square root before adding in the standard deviation, or forgetting to square the deviation.
Rushing through the formula.
Fix: Follow the order: deviation, square, multiply by p, add, then root. Compute variance first.
Comparing projects of different sizes using only standard deviation.
A lower σ looks safer.
Fix: Use the coefficient of variation (σ ÷ E(NPV)) when expected NPVs differ widely.
Not deducting the cost of a decision in decision tree problems, or rolling forward instead of back.
Students read the tree left to right as a story.
Fix: Start at the end. At each decision node, take the best branch net of its cost, and then move left.
Using one risk premium for every year in a CE question.
Confusing CE with RADR.
Fix: CE uses a separate α for each year. Apply the α given for that year to that year's cash flow.

Worked examples

Example 1

A project needs an outlay of ₹10,00,000. Expected cash flows are ₹5,00,000, ₹6,00,000 and ₹6,00,000 in years 1 to 3. The certainty equivalent coefficients are 0.9, 0.8 and 0.7. The risk-free rate is 6%. PV factors at 6%: 0.9434, 0.8900, 0.8396. Find the NPV and advise.

Show the solution

Year 1 certain cash flow = 5,00,000 × 0.9 = ₹4,50,000.
Year 2 certain cash flow = 6,00,000 × 0.8 = ₹4,80,000.
Year 3 certain cash flow = 6,00,000 × 0.7 = ₹4,20,000.
Discount at the risk-free rate: 4,50,000 × 0.9434 = ₹4,24,530.
4,80,000 × 0.8900 = ₹4,27,200.
4,20,000 × 0.8396 = ₹3,52,632.
Total PV = 4,24,530 + 4,27,200 + 3,52,632 = ₹12,04,362.
NPV = 12,04,362 − 10,00,000 = ₹2,04,362.

Answer: NPV by the certainty equivalent method is ₹2,04,362. It is positive, so accept the project.

Example 2

A project has three possible outcomes. NPV of −₹2,00,000 with probability 0.2, NPV of ₹4,00,000 with probability 0.5 and NPV of ₹8,00,000 with probability 0.3. Compute the expected NPV, standard deviation and coefficient of variation.

Show the solution

E(NPV) = 0.2 × (−2,00,000) + 0.5 × 4,00,000 + 0.3 × 8,00,000 = −40,000 + 2,00,000 + 2,40,000 = ₹4,00,000.
Deviations from 4,00,000 are −6,00,000, 0 and +4,00,000.
Variance = 0.2 × (6,00,000)² + 0.5 × 0 + 0.3 × (4,00,000)².
= 0.2 × 36 × 10^10 + 0.3 × 16 × 10^10 = 7.2 × 10^10 + 4.8 × 10^10 = 12 × 10^10.
σ = √(12 × 10^10) = √12 × 10^5 = approximately ₹3,46,410.
CV = 3,46,410 ÷ 4,00,000 = approximately 0.87.

Answer: Expected NPV is ₹4,00,000, standard deviation is about ₹3,46,410 and CV is about 0.87. The project has a positive expected NPV, but the spread is large, so management should weigh its risk appetite.

Exam tips

Read which rate is given. If it is the risk-free rate, the question wants a CE answer. If it is a higher rate for risk, it wants RADR.
Write the definition difference in theory answers: sensitivity is one variable at a time, scenario is a combination of variables, simulation is many random trials.
In decision tree questions, draw the tree neatly with squares for decisions and circles for chance events. The diagram itself earns marks.
In theory answers, mention a limitation. RADR assumes risk grows at a constant rate over time. CE needs subjective α values. Sensitivity ignores links between variables.
Always attempt the MCQs, since there is no negative marking. Look for answers using σ, CV or the rollback rule.

Practice questions from Investment Decisions

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 certainty equivalent and risk-adjusted discount rate?

RADR adds a premium to the discount rate and applies it to the expected cash flows. CE reduces each cash flow with a coefficient and discounts at the risk-free rate. CE can treat risk differently each year, while RADR assumes risk rises steadily with time.

What is the difference between sensitivity analysis and scenario analysis?

Sensitivity analysis changes one input at a time to see its effect on NPV. Scenario analysis changes a group of inputs together to create cases such as best, expected and worst. Scenario analysis is closer to real life because variables usually move together.

When do I use the coefficient of variation?

Use it when you compare projects whose expected NPVs differ. It is standard deviation divided by expected NPV. The project with the lower CV has less risk per rupee of expected return.

How do I solve a decision tree problem?

Draw the tree with decision and chance nodes and write probabilities and payoffs on the branches. Work backwards. At a chance node take the expected value, and at a decision node choose the best branch after deducting its cost. The value at the first node is the answer.