Capital budgeting and cost of capital - Risk analysis sensit...

Learning Outcomes

This article explains capital budgeting risk analysis techniques for CFA Level 1, including:

• Distinguishing the purposes and mechanics of sensitivity, scenario, and break-even analysis in evaluating stand-alone project risk
• Identifying, selecting, and interpreting key value drivers of project NPV and IRR, and judging when a variable should be treated as “critical”
• Setting up one‑variable sensitivity tables and spider or tornado diagrams, and drawing exam‑relevant conclusions from the patterns of NPV changes
• Building coherent optimistic, base‑case, and pessimistic scenarios by varying multiple inputs together, and using scenario NPVs and probabilities to compute and interpret expected NPV
• Calculating NPV-based break-even points for sales volume, price, costs, or the discount rate (IRR), and converting these into margins of safety expressed in units and percentages
• Linking the results of risk analysis back to the accept‑or‑reject decision, including assessing downside exposure, commenting on risk tolerance, and recognizing common exam traps such as treating sensitivity results as probabilities or ignoring the base‑case benchmark
• Integrating these tools into a disciplined, exam-ready approach for discussing how robust a project’s valuation is to plausible errors in forecasts and external conditions

CFA Level 1 Syllabus

For the CFA Level 1 exam, you are required to understand capital budgeting risk analysis techniques, with a focus on the following syllabus points:

• Explain the significance of risk analysis in capital budgeting
• Apply sensitivity analysis to determine key NPV drivers
• Describe how scenario analysis is used to assess project outcomes under different conditions
• Evaluate the use of break-even analysis and margin of safety in project assessment
• Identify common errors in interpreting risk analysis results and link risk implications back to NPV/IRR decisions

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. What does one-variable sensitivity analysis show in capital budgeting?
   1. The probability distribution of a project’s NPV
   2. The effect on NPV of changing a single input while holding others constant
   3. The combined effect of changing several correlated inputs
   4. The impact of changing the project’s capital structure
2. In scenario analysis, what mainly distinguishes an optimistic case from a pessimistic case?
   1. The discount rate used is higher in the optimistic case
   2. Only fixed costs differ between the two cases
   3. Several key inputs move favorably together in the optimistic case and unfavorably in the pessimistic case
   4. The project life is longer in the pessimistic case
3. How does NPV-based break-even analysis help assess project risk?
   1. It identifies the payback period at which initial investment is recovered
   2. It finds the discount rate that sets NPV to zero
   3. It shows how sensitive NPV is to a 1% change in each input
   4. It identifies the input level at which project NPV changes sign from positive to negative
4. Why is it important not to treat sensitivity analysis results as probabilities?
   1. Because sensitivity analysis only applies to IRR, not NPV
   2. Because the results assume that changes in different variables are perfectly correlated
   3. Because sensitivity analysis varies one input at a time and does not indicate how likely any particular change is
   4. Because sensitivity analysis always overstates project risk

Introduction

Risk analysis is essential in capital budgeting, as project outcomes depend on variables that are uncertain. Forecasts for sales, prices, costs, and economic conditions will often prove inaccurate, exposing investment projects to risk. Analysts use risk analysis tools to evaluate how deviations from forecasts affect investment decisions, especially where assumptions about sales, costs, or external factors may be wrong.

Key Term: project risk analysis
Project risk analysis is the process of examining how uncertainty in key inputs (such as sales, prices, and costs) affects a project’s NPV or IRR, to assess the range and drivers of potential outcomes.

Sensitivity, scenario, and break-even analysis provide a structured way to understand project risk, so that investment decisions are better informed and more robust. These techniques do not replace NPV or IRR; instead, they complement discounted cash flow analysis by showing how fragile or robust the base‑case evaluation is.

Key Term: sensitivity analysis
Sensitivity analysis examines how a project’s NPV or IRR responds when one input is altered, holding all other variables constant.

Key Term: scenario analysis
Scenario analysis examines project value by changing several variables simultaneously to reflect defined future scenarios (for example, best, base, and worst cases).

Key Term: break-even analysis
Break-even analysis determines the value of an input variable at which a project’s NPV (or profit) becomes zero, helping to assess the risk associated with that input.

Key Term: base‑case forecast
The base‑case forecast is the analyst’s best estimate of project inputs and resulting cash flows, used as the starting point for risk analysis.

These tools help answer questions such as:

• Which assumptions matter most for project value?
• How bad can things get before the project destroys value?
• How different could outcomes be under favorable or unfavorable conditions?

Understanding how to construct and interpret these analyses is specifically tested at Level 1, often in the context of short numerical examples.

Analyzing Project Risk: Sensitivity, Scenario, and Break-Even Analysis

Forecasts for sales, costs, and market conditions will often prove inaccurate, exposing investment projects to risk. Risk analysis examines project exposure and guides management in identifying areas warranting further caution or strategic planning.

Sensitivity Analysis

Sensitivity analysis isolates the effect on project NPV (or IRR) when a single input—such as sales volume, price, variable cost, or discount rate—is changed, with all others held constant. Analysts can identify key value drivers, visualize which variables a project’s value is most sensitive to, and assess the risk from potential input errors.

Key Term: key value driver
A key value driver is an input variable whose changes have a large impact on project NPV or IRR and therefore require especially careful estimation and monitoring.

Typical inputs varied in sensitivity analysis include:

• Unit sales volume
• Selling price per unit
• Variable cost per unit
• Fixed operating costs
• Project life
• Salvage value
• Discount rate (cost of capital)

The basic steps are:

• Specify a base‑case NPV (or IRR) using your best estimates.
• Select one input variable and define a plausible range around the base‑case (for example, ±10%, ±20%).
• Recalculate NPV for several values of that variable, holding all other inputs constant.
• Record and plot the resulting NPVs in a table or graph.

Variables that cause large swings in NPV for relatively small changes are the ones the project is most sensitive to.

Worked Example 1.1

A project requires an initial investment of $2 million and is projected to have an NPV of $110,000 under the base‑case forecast. If unit sales decrease by 8%, NPV drops to –$220,000.

Answer:
The project’s value is highly sensitive to sales volume. An 8% decrease in sales causes NPV to fall by $330,000 (from +$110,000 to –$220,000), turning the project from value‑creating to value‑destroying. Small errors in sales forecasting may result in an overall project loss; sales are therefore a key value driver. Management should focus on improving the sales forecast and mitigating downside risk (for example, through marketing commitments, contracts, or staged investment).

In practice, sensitivity analysis is often summarized in a table or graph (sometimes called a spider or tornado diagram), showing the effect on NPV as input variables are adjusted one at a time.

For example, an analyst might build a table like:

• Sales volume: –20%, –10%, 0%, +10%, +20%
• Sales price: –5%, 0%, +5%
• Variable cost per unit: –5%, 0%, +5%

and compute the NPV for each change, always changing only one variable at a time.

From this, you can calculate a simple measure of sensitivity, such as:

A larger absolute value indicates greater sensitivity.

Strengths and limitations of sensitivity analysis

• Strengths:

  • Identifies which assumptions are most critical for project value.
  • Simple to compute and interpret for exam questions.
  • Highlights where better data collection or more conservative assumptions are needed.

• Limitations:

  • Changes only one variable at a time, ignoring correlations (for example, prices and volumes may fall together in a recession).
  • Often assumes linearity over the range analyzed, which may be unrealistic for large changes.
  • Does not indicate how likely any particular change is.

Exam note: Sensitivity analysis is a “what-if” tool. It shows how NPV would change if an input changed, not how likely that change is.

Worked Example 1.2

An analyst performs one-variable sensitivity analysis on a project’s sales price. The base‑case price is $40 per unit with NPV = $200,000. If price falls to $38, NPV falls to $80,000. If price rises to $42, NPV increases to $320,000.

Answer:
A $2 (5%) change in price leads to a $120,000 change in NPV. The sensitivity ratio to price is $120,000 / 5% = $24,000 per 1% price change. If the same analysis showed that a 5% change in variable cost shifted NPV by only $40,000, the project would be more sensitive to price than to variable cost. Price is therefore a more important value driver than variable cost and should receive more attention in forecasting and risk management.

Scenario Analysis

While sensitivity analysis varies one variable at a time, scenario analysis changes multiple variables simultaneously according to pre‑defined scenarios: optimistic, base, and pessimistic cases. This allows analysts to evaluate how combined changes (for example, a recession causing both sales and price to drop while costs rise) impact overall project profitability.

Key Term: optimistic scenario
An optimistic scenario is a set of input assumptions that are collectively favorable to the project (for example, strong demand, higher prices, lower costs).

Key Term: pessimistic scenario
A pessimistic scenario is a set of input assumptions that are collectively unfavorable to the project (for example, weak demand, lower prices, higher costs).

Scenario analysis helps decision makers:

• Assess the range of possible project outcomes (spread between best and worst NPVs)
• Understand how correlated changes in variables affect project value
• Prepare for tail (extreme) risks and design contingency plans
• Judge whether the downside risk is acceptable given the firm’s risk tolerance

Worked Example 1.3

A capital project has the following NPVs under three scenarios:

• Optimistic: NPV = $800,000 (higher sales, lower costs)
• Base case: NPV = $250,000 (expected inputs)
• Pessimistic: NPV = –$650,000 (lower sales, higher variable costs)

Answer:
The project’s value could vary dramatically. The gap between optimistic and pessimistic NPVs is $1,450,000 (from –$650,000 to +$800,000), indicating high outcome variability. There is a substantial risk of losses if conditions are unfavorable. Management should consider:

• Whether the firm can tolerate a loss of around $650,000
• Actions to reduce downside risk (for example, flexible capacity, cost reduction plans, or contracts that guarantee minimum sales volumes)
• Whether a risk‑adjusted discount rate or additional analysis (such as real options) is warranted

Scenario analysis is particularly relevant when variables may be correlated, such as declines in price and sales volume during an economic downturn, or rising input costs when output prices cannot be increased.

Scenario probabilities and expected NPV

Sometimes analysts assign probabilities to scenarios and compute an expected NPV as a probability‑weighted average. This uses the same logic as the total probability rule in probability theory.

Key Term: expected NPV
Expected NPV is the probability‑weighted average of NPVs across mutually exclusive scenarios.

If scenarios are mutually exclusive and exhaustive, then:

where $S_i$ is scenario i, $P(S_i)$ is its probability, and $\text{NPV}\_i$ is the NPV under that scenario.

Worked Example 1.4

Suppose the project in Worked Example 1.3 has the following scenario probabilities:

• Optimistic (NPV = $800,000): probability 0.3
• Base case (NPV = $250,000): probability 0.5
• Pessimistic (NPV = –$650,000): probability 0.2

Calculate the expected NPV.

Answer:
The expected NPV is:

$$E(\text{NPV}) = (0.3\times800{,}000) + (0.5\times250{,}000) + (0.2\times(-650{,}000))$$

The expected NPV is positive at $235,000, suggesting that on average the project adds value. However, the presence of a sizable negative NPV in the pessimistic scenario means the project still carries significant downside risk. A risk‑averse firm may compare this expected NPV to the potential losses and its risk tolerance before deciding.

Exam tip: You may be asked either to compute expected NPV from scenario NPVs and probabilities or to interpret whether a project is attractive, given a positive expected NPV but a large negative NPV in a pessimistic scenario.

Qualitative and quantitative scenarios

Not all scenarios need to be fully quantitative. In practice:

• Qualitative scenarios describe different states of the world (for example, “severe recession,” “moderate growth”) and their likely impact on key drivers.
• Quantitative scenarios specify numerical values for inputs (for example, sales volumes, prices, costs) and yield specific NPVs.

At Level 1, questions will typically give quantitative assumptions directly, but you should understand that scenario analysis is a structured way of thinking about alternative futures, not just a mechanical calculation.

Break-Even Analysis

Break-even analysis determines the value of an input variable (for example, sales level, selling price) at which project NPV (or accounting profit) equals zero. This clarifies the “margin of safety,” the extent to which actual results can deviate before the project becomes unviable.

Key Term: NPV break-even point
The NPV break-even point for a given input is the value of that input at which project NPV equals zero, holding all other inputs constant.

Key Term: margin of safety
The margin of safety is the difference between the forecast value of an input and its NPV break-even value, often expressed as a percentage of the forecast. It measures how much actual results can worsen before NPV becomes negative.

For example, if:

• Forecast sales volume = 30,000 units per year
• Break-even sales volume (NPV = 0) = 24,800 units

then the margin of safety in units is 5,200, and as a percentage:

The larger the margin of safety, the more robust the project is to unfavorable deviations in that input.

Worked Example 1.5

A company forecasts a project will sell 30,000 units per year for five years, generating a base‑case NPV of $270,000 at the firm’s required rate of return. By solving for sales volume where NPV = 0 (holding price, costs, and discount rate constant), the analyst finds that the break-even sales volume is 24,800 units per year.

Answer:
The project’s NPV becomes zero if sales fall to 24,800 units per year. The margin of safety is:

• In units: 30,000 – 24,800 = 5,200 units
• As a percentage: $5,200 / 30,000 \approx 17.3\%$

If a sales shortfall of 17.3% (or more) is plausible, the project is fairly risky on the sales dimension. If historical sales forecasts for similar products have rarely been off by more than 5%, the margin of safety appears comfortable. Break-even analysis thus helps gauge whether project returns are robust to reasonable estimation errors.

Break-even analysis can also be applied to other variables:

• Price break-even: What minimum price per unit keeps NPV = 0?
• Cost break-even: How high can variable cost per unit rise before NPV becomes zero?
• Discount rate break-even: Setting NPV = 0 and solving for the discount rate yields the IRR.

At Level 1, the emphasis is on interpreting break-even points and margins of safety rather than on complex algebra. You may be given the break-even value and asked to comment on risk.

Worked Example 1.6

A project’s NPV is +$150,000 at a discount rate of 10%. The project’s IRR is found to be 13%. Interpret the “break-even” discount rate.

Answer:
The IRR of 13% is the discount rate at which NPV = 0. It can be interpreted as a break-even required return:

• If the firm’s actual cost of capital is below 13%, the project has a positive NPV.
• If the cost of capital rises above 13%, NPV becomes negative.
  The gap between the IRR (13%) and the current required return (10%) provides a buffer against increases in the cost of capital. A narrow gap would indicate greater risk that modest changes in financing conditions could make the project unattractive.

Interpreting Results and Decision-Making

Risk analysis informs management by:

• Highlighting highly sensitive variables that may need better data, more conservative assumptions, or contractual hedging
• Revealing the consequences if multiple adverse events occur together (through scenario analysis)
• Quantifying the buffer before a project fails to create value (via break-even analysis and margins of safety)

If a project’s NPV is very sensitive to a single input, or if small, plausible deviations cause NPV to turn negative, further investigation or a risk premium may be warranted. Risk analysis findings may lead to:

• Revising input assumptions (for example, lowering forecast sales or raising cost estimates)
• Redesigning the project to reduce fixed costs or increase flexibility
• Renegotiating supplier or customer contracts to share risk
• Delaying or rejecting projects whose downside scenarios conflict with the firm’s risk tolerance

Key Term: stand‑alone risk
Stand‑alone risk is the risk of a project considered in isolation, usually measured by the variability of its NPV or IRR before considering diversification within the firm.

Capital budgeting risk analysis at Level 1 usually treats projects on a stand‑alone basis. In practice, firms also consider how a project affects overall firm risk (for example, whether its cash flows are correlated with existing operations).

Exam Warning

Do not treat sensitivity analysis results as providing explicit probabilities of outcomes. Sensitivity analysis shows the effect of changing one variable at a time; it does not indicate how likely that change is, nor does it combine changes across multiple variables. Also, a project can have a positive expected NPV across scenarios but still be unattractive if downside risk is too large relative to the firm’s risk tolerance.

Common Exam Pitfalls

When answering exam questions on risk analysis in capital budgeting, watch out for these common issues:

• Confusing sensitivity and scenario analysis:
  • Sensitivity: one variable changes at a time
  • Scenario: several variables change together
• Ignoring the base‑case:
  • Always relate sensitivity or scenario results back to the base‑case NPV or IRR.
• Treating scenario NPVs as equally likely when probabilities are given (or implied to be different).
• Forgetting that risk analysis complements, but does not replace, NPV/IRR as the primary decision criteria.
• Misinterpreting break-even points:
  • Break-even means NPV = 0 (or profit = 0), not that the project is risk‑free.
  • A large margin of safety does not guarantee success; it only indicates robustness with respect to that particular input.

Revision Tip

When describing risk analysis, specify both the input changed and the resulting effect on project NPV—for example, “NPV falls by $600,000 if unit price drops 3%,” or “NPV is zero if annual sales fall to 24,800 units.”

Summary

Risk analysis using sensitivity, scenario, and break-even tools is essential for evaluating capital projects under uncertainty. Sensitivity analysis identifies key value drivers by varying one input at a time. Scenario analysis explores combined changes across multiple inputs under optimistic, base, and pessimistic cases and can be extended to probability‑weighted expected NPV. Break-even analysis focuses on the input values at which NPV becomes zero and computes margins of safety.

Together, these approaches help analysts and decision makers:

• Identify the main risks and critical assumptions behind NPV and IRR estimates
• Quantify how far actual outcomes can deviate before the project loses value
• Judge whether downside outcomes are compatible with the firm’s risk tolerance
• Communicate risk clearly to stakeholders and support robust capital allocation decisions

Key Point Checklist

This article has covered the following key knowledge points:

• Explain why risk analysis is important in capital budgeting and how it relates to NPV/IRR
• Use sensitivity analysis to identify and quantify the impact of key value drivers on project NPV or IRR
• Apply scenario analysis to assess project outcomes under simultaneous changes in several inputs
• Compute and interpret expected NPV using scenario NPVs and probabilities
• Perform break-even analysis to determine input values that lead to zero NPV and to measure the project’s margin of safety
• Recognize that sensitivity and scenario analyses provide “what‑if” outcomes, not probabilities by themselves
• Avoid common pitfalls, such as confusing sensitivity with scenario analysis or misinterpreting break-even points

Key Terms and Concepts

• project risk analysis
• sensitivity analysis
• scenario analysis
• break-even analysis
• base‑case forecast
• key value driver
• optimistic scenario
• pessimistic scenario
• expected NPV
• NPV break-even point
• margin of safety
• stand‑alone risk