Methods for assessing risk and uncertainty

Risk and uncertainty arise because cash flows, lives and interest rates used in an analysis are estimates of the future. The common methods to assess their effect are:

1. Break-even analysis

Finds the value of one variable (for example annual output, price or life) at which PW or AW becomes zero, i.e. income equals cost. If the expected value is well above the break-even point, the risk is small. It is simple but treats one variable at a time.

2. Sensitivity analysis

One factor (investment, annual revenue, life, MARR) is changed over a range (for example ±30%) while others stay fixed, and the change in PW, AW or IRR is recorded.

• Results are shown in a sensitivity diagram: the steeper the line, the more sensitive the decision is to that factor.
• It shows which estimate must be made most carefully, but gives no probabilities and ignores the link between factors.

3. Scenario analysis

Three sets of estimates (optimistic, most likely, pessimistic) are assumed and the project value is computed for each. It shows the range of outcomes and the downside.

4. Expected value analysis

When probabilities are known, the expected value is E(X) = Σ pᵢXᵢ. For PW, E(PW) = Σ p × PW(outcome). The alternative with the highest expected PW (or lowest expected cost) is chosen. The variance or standard deviation, σ² = Σ pᵢ(Xᵢ − E)², measures the risk: a larger spread means higher risk.

5. Decision tree analysis

Used for sequential decisions with chance events. The tree is rolled back from the end: expected values at chance nodes and best choice at decision nodes.

6. Monte Carlo simulation

Probability distributions are given to the uncertain variables; many random trials are run by computer and the distribution of PW or IRR is built. It gives the probability that PW < 0 and is suitable for complex projects.

7. Other approaches

• Risk-adjusted discount rate: a higher MARR for riskier projects.
• Reduction of useful life / payback cut-off: accept only projects that recover investment quickly.
• Certainty equivalent: replace uncertain cash flows by sure equivalents.

Summary: break-even and sensitivity analyses are the simplest and are most used in practice; expected value, decision trees and simulation need probability estimates.