BFC2140 · Corporate Finance
Capital Budgeting III: Risk Analysis & Decision Trees
A single-point NPV hides the uncertainty in its inputs, so this chapter adds risk analysis to capital budgeting: sensitivity analysis (vary one input), scenario analysis (vary several together) and break-even analysis (find the input level where NPV = 0), then decision-tree (real-option) analysis for sequential decisions under uncertainty. Decision-tree roll-back NPV is a signature learning outcome of the unit and a likely Section C exam item. It is examined through both numerical break-even/decision-tree calculations and written reasoning about how each technique exposes a project's risk.
What this chapter covers
- 01Sensitivity analysis: change one input at a time and observe the change in NPV
- 02Scenario analysis: vary a consistent set of inputs together (base, worst and best cases)
- 03Break-even analysis: the input level (e.g. units sold) that drives NPV to zero
- 04Accounting/EBIT break-even versus financial (NPV = 0) break-even
- 05Operating leverage and why NPV is far more sensitive than the input change that drives it
- 06Decision trees: decision nodes and chance nodes with branch probabilities and payoffs
- 07Roll-back analysis: compute expected values from the right, take the max at each decision node
- 08Real options as the link between decision trees and project flexibility (expand, abandon, delay)
Decision-tree (real-option) roll-back NPV
- 1 markValue the success branch at the chance node (the t = 1 viewpoint perpetuity): PV if success = 180,000/0.12 = $1,500,000.
- 1 markValue the failure branch: it produces nothing, so its payoff is $0.
- 1 markRoll back the chance node to an expected payoff at t = 1: E[payoff] = 0.40 × 1,500,000 + 0.60 × 0 = $600,000.
- 1 markDiscount the expected payoff one period to today: 600,000/1.12 = $535,714.29.
- 1 markNet off the upfront R&D outlay at t = 0: NPV = −600,000 + 535,714.29 = −$64,285.71.
- 1 markApply the decision rule at the decision node: NPV < 0, so do NOT proceed with the R&D.
Key terms
- Sensitivity analysis
- Changing one input (e.g. price, unit sales or the discount rate) at a time while holding the rest fixed, to see how much the NPV moves. It identifies which variables the project's value is most exposed to.
- Scenario analysis
- Varying a self-consistent set of inputs together to build whole scenarios — typically a base, worst and best case. Unlike sensitivity analysis it captures interactions between variables, but it examines only a few discrete combinations.
- Break-even analysis
- Finding the level of a key input at which the project just breaks even. Accounting/EBIT break-even uses (fixed costs + depreciation)/(price − variable cost); financial break-even finds the input level that makes NPV = 0.
- Operating leverage
- The amplification of NPV (or profit) changes relative to the change in a driver such as sales, caused by fixed costs. High operating leverage means a small percentage fall in sales can cause a much larger percentage fall in NPV.
- Decision tree
- A diagram of a sequential decision under uncertainty, with decision nodes (the firm chooses) and chance nodes (nature decides, with probabilities) leading to payoffs. It makes the structure of staged decisions explicit for valuation.
- Roll-back NPV
- The method for valuing a decision tree: work from the right-hand payoffs back to t = 0, taking probability-weighted expected values at chance nodes and the best (max-NPV) choice at decision nodes, discounting at each step.
Capital Budgeting III: Risk Analysis & Decision Trees FAQ
What is the difference between sensitivity and scenario analysis?
Sensitivity analysis changes one input at a time, holding everything else constant, to isolate which single variable the NPV is most sensitive to. Scenario analysis changes a coherent group of inputs together — for example a recession scenario with lower sales, lower price and higher costs all at once — to value plausible joint outcomes. Sensitivity tells you where the risk is concentrated; scenario analysis tells you what a realistic combination of bad (or good) news does to the project. Both are descriptive: they show the spread of NPV outcomes but do not by themselves change the accept/reject rule.
How do I value a decision tree?
Use roll-back analysis, working from the right (the final payoffs) back to today. At each chance node, replace the branches with their probability-weighted expected value; at each decision node, choose the branch with the highest NPV (because the firm controls that choice). Discount cash flows back one step at a time as you move left, and read off the NPV at the initial decision node. Accept only if that NPV is positive. The key discipline is keeping the timing straight as you discount each stage.
What is the link between decision trees and real options?
A real option is the flexibility to change a project as uncertainty resolves — to expand if things go well, abandon if they go badly, or delay until more is known. Decision trees are the natural way to value that flexibility, because the decision nodes capture exactly those future choices. By taking the maximum at each decision node, the roll-back rule automatically values the option to act on good news and avoid bad outcomes, which is why a staged project can be worth more than a now-or-never one.
Why is NPV so much more sensitive than the input I changed?
Operating leverage. Because a project has fixed costs (and a fixed initial outlay), a given percentage change in sales or price flows through to a much larger percentage change in profit and hence NPV. In the unit's worked sensitivity example, a 10% fall in sales produces a roughly 33% fall in NPV. This is why sensitivity analysis is valuable — it reveals that the project's value can be far more fragile than the modest variation in a single input would suggest.
How is risk analysis examined in BFC2140?
Decision-tree analysis is an explicit learning outcome (LO2: 'apply decision-tree analysis in investment decision making') and a likely Section C question, where you roll back a tree to a t = 0 NPV and recommend. Expect also Section B numerical break-even questions and Section C explanations of how sensitivity and scenario analysis expose project risk. Decision-tree roll-back NPV is flagged among the high-yield recurring exam skills.
Exam move
Treat this chapter as 'NPV under uncertainty' and keep the three descriptive tools straight: sensitivity (one input), scenario (a consistent set of inputs), and break-even (the input level for NPV = 0). Practise the financial break-even calculation and be ready to explain, using operating leverage, why NPV moves by a larger percentage than its driver. For decision trees, master the roll-back discipline — expected values at chance nodes, the maximum at decision nodes, discounting one stage at a time — and always draw the tree before computing, labelling each branch with its probability, payoff and timing. Watch the timing of perpetuities and annuities inside a tree, the single most common error. Finally, connect roll-back to real options so you can explain in Section C why staging a decision (the option to expand, abandon or delay) adds value over a now-or-never commitment.
