CFA Level I Exam · Capital Investments and Capital Allocation

Capital Rationing and Project Risk Analysis for CFA Level I

Updated 7 October 2026 · Fact-checked

Capital rationing means a firm has less capital than the positive-NPV projects it could fund, so it picks the combination with the highest total NPV. Project risk analysis tests how reliable an NPV is, using sensitivity, scenario or simulation analysis, or a higher discount rate for riskier projects.

Understand Capital Rationing and Project Risk Analysis

Normally you accept every project with a positive NPV. That rule assumes the firm can raise all the capital it needs. Capital rationing is the case where it cannot, or chooses not to. Now you must choose among positive-NPV projects, and the goal changes: pick the set of projects that gives the highest total NPV within the budget.

There are two types. Hard capital rationing is an external limit. The firm truly cannot raise more funds, for example because lenders or markets will not supply them. Soft capital rationing is an internal limit set by management, such as a divisional budget. It can be relaxed if the projects justify it. Under either type, the firm may have to reject projects with positive NPV, so value is lost compared with having no limit.

How you choose depends on whether projects are divisible. If you can take part of a project, rank by profitability index (PI) and fund from the top until the money runs out. If projects are all-or-nothing, PI ranking can mislead. Test the feasible combinations and pick the one with the largest sum of NPVs. This guide assumes one budget period and independent projects.

Next is risk. An NPV is built from forecasts, and forecasts are uncertain. Sensitivity analysis changes one input at a time (sales, margin, discount rate) and shows how NPV moves. It tells you which input matters most. Scenario analysis changes several inputs together to build cases such as worst, base and best, and can attach probabilities to get an expected NPV. Simulation analysis (Monte Carlo) draws inputs at random from assumed probability distributions many times and produces a distribution of NPV. It is the most informative, but it depends on the distributions and correlations you assume.

A different approach handles risk inside the discount rate. A risk-adjusted discount rate adds a premium for projects riskier than the firm's average, so a risky project must earn more to show a positive NPV. None of these methods makes the decision for you. They show the range of outcomes so you can judge the risk.

Key formulas to remember

Profitability index: PI = PV of future cash flows ÷ initial investment = 1 + NPV ÷ initial investment; Use PI to rank divisible projects under a single-period budget limit. PI above 1 means NPV above 0.
Capital rationing objective: Maximize Σ NPV of the chosen projects, subject to Σ investment ≤ budget; For indivisible projects, test feasible combinations. Do not simply fund the highest PI first.
Expected NPV from scenarios: Expected NPV = Σ (probability of scenario × NPV in scenario); Probabilities must sum to 1. Expected NPV is a probability-weighted average, not the base-case NPV.
Risk-adjusted discount rate: Project rate = base required return + risk premium for the project's extra risk; A higher rate lowers NPV. Do not also cut the cash flows for the same risk, or you double count.
Hard vs soft rationing: Hard = external, binding limit. Soft = internal, flexible limit.; The test usually asks which one the firm faces and whether it can exceed the budget.

How to solve Capital Rationing and Project Risk Analysis questions

Work out first whether the question is about choosing projects under a budget or about measuring project risk. Then apply the matching steps.

1Read the stem and decide: allocation under a budget (rationing) or risk measurement (sensitivity, scenario, simulation, risk-adjusted rate).
2For rationing, check whether the limit is external (hard) or internal (soft), and whether projects are divisible or all-or-nothing.
3Drop any project with NPV below zero. Compute PI if it is not given: 1 + NPV ÷ investment.
4If divisible, rank by PI and fund in order, taking a fraction of the last project if needed. If indivisible, list every combination within the budget and add the NPVs.
5Choose the combination with the highest total NPV. Check that the total investment does not exceed the budget.
6For risk questions, match the method to the description: one input at a time is sensitivity, several inputs together is scenario, random draws from distributions is simulation, a higher rate for risk is the risk-adjusted rate.
7For scenario questions, multiply each NPV by its probability and add. Confirm the probabilities sum to 1.
8Eliminate the two wrong options by checking direction: higher risk means a higher rate and a lower NPV, and a binding budget means some positive-NPV projects are rejected.

Quickest way: Budget table and keyword match

When to use it: Use it when a question gives several projects and a budget, or asks which risk method fits a description.

Write each project as investment and NPV in one line, then list the combinations that fit the budget.
Add NPVs for each combination and pick the largest. Use PI only as a first screen, and only trust it fully for divisible projects.
Match risk keywords: 'one variable at a time' means sensitivity, 'worst, base, best' means scenario, 'probability distributions, many trials' means simulation.
For expected NPV, enter each product on your calculator and add. For example, press 0.25 × 4 +/- = then add the others, or write the products on the sheet.
If you must compute an NPV on the BA II Plus: press CF, enter CF0 then ENTER, enter each C01 and frequency F01, then press NPV, enter I, ENTER, and CPT.

Common mistakes in Capital Rationing and Project Risk Analysis

Funding projects in order of highest PI when projects are indivisible.
PI ranking feels like the efficient rule, and it is correct for divisible projects.
Fix: For all-or-nothing projects, test the combinations that fit the budget and choose the highest total NPV. Leftover budget can make a lower-PI combination better.
Rejecting a positive-NPV project because the firm is capital rationed, without checking the combination.
Students forget that rationing forces a choice among positive-NPV projects.
Fix: Rank or combine the positive-NPV projects, and accept the set that maximizes total NPV within the budget.
Mixing up hard and soft rationing.
Both words describe a limit, and the names are not obvious.
Fix: Hard means the limit is external and cannot be exceeded. Soft means management sets it and can relax it.
Calling a one-at-a-time change a scenario analysis.
Both change inputs, so they look alike.
Fix: Sensitivity changes one input with all others fixed. Scenario changes several inputs together to form a coherent case.
Treating simulation as a decision rule that gives a single answer, or assuming it removes the need for good inputs.
Output looks precise because it is a full distribution.
Fix: Simulation shows a distribution of NPV, but it relies on the assumed distributions and correlations. Poor inputs give poor output.
Raising the discount rate and also cutting the cash flows for the same risk.
Students try to be extra careful about risk.
Fix: Reflect each risk once, either in the cash flows or in the rate, not both.

Worked examples

Example 1

A firm has a hard budget of $10 million this year. Four independent, indivisible projects each have positive NPV. A: investment $5m, NPV $2.0m. B: investment $4m, NPV $1.8m. C: investment $3m, NPV $1.1m. D: investment $6m, NPV $2.4m. What is the maximum total NPV the firm can obtain? A) $3.8 million B) $4.2 million C) $4.6 million

Show the solution

Compute PI as 1 + NPV ÷ investment: A = 1.40, B = 1.45, C = about 1.37, D = 1.40.
PI ranking would fund B ($4m) first. A and D then tie at a PI of 1.40, so PI ranking alone is ambiguous here. If you break the tie in favour of A, you get B + A = $9m and NPV $3.8m, with $1m unused. If you break it in favour of D, you get B + D = $10m and NPV $4.2m. PI cannot tell you which is right, so check the combinations.
Because the projects are indivisible, test every combination within $10m.
A + B = $9m, NPV 3.8. A + C = $8m, NPV 3.1. B + C = $7m, NPV 2.9. C + D = $9m, NPV 3.5. B + D = $10m, NPV 4.2. A + D = $11m is over budget.
Any three projects cost at least $12m (the cheapest three are C, B and A: 3 + 4 + 5), so no three-project group fits.
The best feasible combination is B + D with total NPV $4.2m. The $4.6m option is not achievable, because no feasible combination gives that total and the maximum feasible total is $4.2m.

Answer: B) $4.2 million, from funding projects B and D.

Example 2

A project is analysed with three scenarios: pessimistic (probability 25%) with NPV of -$4 million, base case (probability 50%) with NPV of $6 million, and optimistic (probability 25%) with NPV of $14 million. What is the expected NPV? A) $4.0 million B) $5.5 million C) $6.0 million

Show the solution

Expected NPV = Σ (probability × NPV).
Pessimistic: 0.25 × (-4) = -1.0.
Base: 0.50 × 6 = 3.0.
Optimistic: 0.25 × 14 = 3.5.
Sum: -1.0 + 3.0 + 3.5 = 5.5.
The base-case NPV of $6.0m is a trap. The two outer scenarios have equal 25% probabilities, but the pessimistic NPV is $10m below the base case while the optimistic NPV is only $8m above it. That pulls the expected NPV below the base case.

Answer: B) $5.5 million. It is below the base-case NPV because the pessimistic NPV is $10m below the base case, while the optimistic NPV is $8m above it, with equal 25% probabilities.

Exam tips

Read for the word 'indivisible' or 'all-or-nothing'. It tells you to test combinations, not just rank by PI.
In a rationing question, the answer is usually the highest total NPV within the budget, not the highest single NPV or the highest PI.
Match the risk method by its description. Keywords like 'one variable', 'several variables together' and 'probability distributions' map directly to the three methods.
Direction checks eliminate options fast: a higher risk-adjusted rate lowers NPV, and a probability-weighted NPV lies between the worst and best scenario NPVs.
With 90 seconds per question and no penalty for wrong answers, answer every item. If unsure, remove the option that violates the direction or budget logic, then pick between the other two.

Practice questions from Capital Investments and Capital Allocation

Capital Rationing and Project Risk Analysis in other exams

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

Capital Rationing and Project Risk Analysis: frequently asked questions

What is the difference between hard and soft capital rationing?

Hard capital rationing is an external limit, so the firm cannot raise more funds. Soft capital rationing is an internal limit set by management, which can be relaxed if the projects justify it. Both can force the firm to pass on positive-NPV projects.

How do you choose projects under capital rationing?

Drop projects with negative NPV. If projects are divisible, rank by profitability index and fund down the list. If they are indivisible, list the combinations within the budget and choose the one with the highest total NPV.

What is the difference between sensitivity analysis and scenario analysis in capital budgeting?

Sensitivity analysis changes one input at a time and holds the rest fixed, showing which input drives NPV most. Scenario analysis changes several inputs together to build cases like worst, base and best, and can assign probabilities to find an expected NPV.

What does Monte Carlo simulation add to capital budgeting?

It draws inputs at random from assumed probability distributions many times and produces a distribution of possible NPVs. This shows the range and likelihood of outcomes. Its quality depends on the distributions and correlations you assume.