Three tools, one question.

If the NPV depends on twenty assumptions, which one should I actually worry about?

Break-even analysis.

The break-even point is the level of a parameter at which the project's economic result is neither positive nor negative.

Break-even is the threshold below which a project loses money and above which it makes money. It answers "how good do things have to be?" — a useful framing for decisions with uncertain demand.

Break-even volume.

Software example: An AI feature has $200K of fixed annual cost (engineering, observability) and $0.05 marginal cost per request. Price per request: $0.25.

If you can confidently project > 1M requests per year, the unit economics work. If you cannot, no amount of cost discipline will save the project.

Discounted break-even (time).

A more rigorous time-based break-even discounts each period's cash flow before summing — answering "in what year does cumulative PV first turn positive?"

Discounted payback ≈ 3.02 years. The simple (undiscounted) payback would be 2.5 years — too optimistic by half a year. The discount rate has real teeth.

Single-variable sensitivity analysis.

Hold all inputs at their base-case values. Vary one input by ±X%. Record the resulting NPV. Repeat for every important input. The slope of NPV vs. input is the sensitivity.

User count drives NPV more than 4× as strongly as discount rate. That is where the team's research effort should go.

Multi-variable sensitivity.

Many inputs are correlated: if user growth slows, token spend falls too. Single-variable sensitivity overstates the role of inputs whose movement is naturally hedged.

Multi-variable sensitivity captures these joint movements. Common techniques:

• Correlated scenarios: define groups of inputs that move together (e.g., "low user adoption" implies low revenue AND low token cost).
• Spider plots: plot NPV vs each input, one line per input.
• Two-way tables: NPV as a function of two key inputs varying simultaneously.

The Tornado diagram.

A horizontal bar chart, one bar per input, length proportional to the NPV range when that input varies through its plausible range. Sorted longest-bar-at-top → the silhouette resembles a tornado.

In one glance, leadership knows where the project's uncertainty really lives. Spend research budget on the top of the tornado — never on the bottom.

Optimistic · realistic · pessimistic.

A board sees ONE number: $78K NPV. Showing the three scenarios reframes the decision honestly: positive expected outcome with downside risk of $45K.

AI feature — fully stress-tested.

Hypothetical AI sales-assistant: $250K dev cost, $0.04 / request marginal cost, monetised at $0.20 / qualified lead, 5-year horizon, MARR 12%.

Lead-conversion rate dominates the tornado. Before scaling, the team must run a measurement experiment to nail down this single input. Token cost matters far less than the team would have guessed.

Tomorrow we put probabilities on the assumptions.

Sensitivity says which inputs matter. Tomorrow we say how likely each input is — decision trees, expected value, Monte Carlo simulation.

Build your tornado.

1. Pick a project you have NPV-ed (yours or HW5's mini-case). Build a one-variable sensitivity table for at least five inputs.
2. Construct a Tornado diagram. Identify the top three NPV drivers.
3. Run three scenarios (pess / real / opt). Decide whether the project would still be approved in the pessimistic case.
4. One paragraph: what experiment would you propose to reduce the uncertainty of the top driver?

What today bought you.

1. Break-even tells you how good things need to be. Useful when demand is uncertain.
2. Sensitivity tells you which assumptions deserve scrutiny. A Tornado diagram is the standard format.
3. Scenarios turn a single NPV into a decision narrative leadership can actually act on.