Every appraisal is a forecast, and forecasts are uncertain. A single-number NPV computed from best-guess inputs hides the real question — how confident should we be, and what could change the answer? This module covers handling risk and uncertainty in CBA, and the subtle but powerful idea of option value, which changes how we appraise irreversible decisions.
Risk versus uncertainty
Frank Knight's distinction: risk is when outcomes are unknown but their probabilities can be quantified (you can put numbers on the odds); uncertainty is when even the probabilities are unknown. Much of project appraisal is genuinely uncertain (we don't know the probability distribution of a new technology's costs), but we often proceed as if it were risk, attaching subjective probabilities — usefully, as long as we remember they are subjective. The tools differ: risk can be handled with expected values and distributions; deep uncertainty calls for robustness, scenarios, and option value.
Expected NPV and sensitivity analysis
Two essential tools
Expected NPV — where probabilities can be assigned, compute the NPV under each outcome and weight by probability: E[NPV] = Σ pᵢ · NPVᵢ. This collapses a distribution of possible NPVs into one expected figure (with the caveat that it ignores the spread/risk, addressed below). Sensitivity analysis — vary each key input and see how much the NPV moves; identify the assumptions the answer actually depends on. The most useful form is the switching value: the value an input would have to take for the NPV to fall to zero (the decision to flip). If a project's NPV stays positive unless, say, the benefit is overestimated by 60%, that is reassuring; if a 5% cost overrun flips it, the project is fragile. Switching values turn a fragile-looking single NPV into a statement about how much margin for error there is — which is far more decision-useful than the point estimate alone.
Scenarios and Monte Carlo
Beyond one-at-a-time sensitivity, scenario analysis bundles inputs into coherent stories (optimistic, central, pessimistic) and computes NPV for each. Monte Carlo simulation goes further: assign probability distributions to the uncertain inputs, draw thousands of random combinations, and generate a full probability distribution of the NPV — showing not just the expected value but the probability the project loses money. Monte Carlo is powerful where the input distributions are genuinely knowable, and spurious where they are not (it can dress deep uncertainty in a false precision — a confident-looking distribution built on guessed inputs). Use it when you have real information about the input distributions; otherwise prefer transparent scenarios and switching values.
Risk belongs in the cash flows, usually not the rate
A common, costly error
A frequent mistake is to handle a project's riskiness by adding a 'risk premium' to the discount rate. This is usually wrong for public CBA. Bumping the discount rate penalises ALL future cash flows more, and at a compounding rate that grows with time — so it conflates risk with time and arbitrarily punishes long-dated benefits regardless of their actual riskiness. The correct treatment is to deal with risk in the cash flows themselves — use expected values, and if society is risk-averse about the project's outcome, apply an explicit certainty-equivalent or risk adjustment to the relevant flows — while discounting at the social rate that reflects time, not risk. (For a government able to pool many independent project risks across the whole population, the Arrow-Lind theorem argues the cost of risk per project is small anyway.) Keep risk and time separate.
Option value and irreversibility
The value of waiting and of preserving flexibility
Standard NPV implicitly assumes now-or-never: do it now or not at all. But many decisions can be delayed, and many are irreversible — and that combination creates option value. Dixit and Pindyck (investment under uncertainty): when an investment is irreversible and there is uncertainty that time will resolve, there is a value to waiting — keeping the option open until you learn more — so the NPV rule 'invest if NPV > 0' is too eager; you should invest only if NPV exceeds the value of the option to wait. Arrow and Fisher's quasi-option value applies the same logic to irreversible harms, especially environmental: because an irreversible action (flooding a forest, building over a wetland) forecloses future choices and you will learn more about the lost asset's value over time, there is a value in preserving flexibility — delaying the irreversible action until the uncertainty resolves. The practical upshot: for irreversible projects under resolvable uncertainty, conventional CBA understates the case for waiting, and the appraisal should explicitly value the option to delay and learn.
Exercise
A government considers building a large dam that would flood a forest valley of uncertain but possibly high ecological and tourism value; the flooding is irreversible. A standard CBA shows a small positive NPV at central estimates. (1) Identify the key uncertainties and use a switching value to express the project's fragility. (2) The appraiser proposes adding 3% to the discount rate 'to account for risk' — critique this. (3) Apply quasi-option value: why might the small positive NPV still not justify proceeding now? (4) Recommend how to handle the decision given irreversibility and resolvable uncertainty.