Expected value and sensitivity analysis

Expected value converts uncertain outcomes into a probability-weighted planning estimate. Sensitivity analysis then tests how strongly the decision depends on assumptions that may prove wrong. Used together, they replace a single confident forecast with a more transparent view of uncertainty.

When to use it

• Use expected value for discrete, consequential outcomes such as winning a major contract, obtaining approval or renewing a large lease. It is especially useful when a small number of “yes or no” events drive the result.
• Use sensitivity analysis in every material investment appraisal, whether or not expected values are part of the base case.

Origins

Expected value emerged from the development of probability theory by Blaise Pascal, Pierre de Fermat and Christiaan Huygens, including work on how to divide the stakes of an unfinished game. Decision theory later connected probability-weighted outcomes with choices under uncertainty. Sensitivity analysis developed across applied mathematics, economics and operations research as a complementary discipline: vary an important assumption and observe whether the result or preferred decision changes.

What it is

For a set of mutually exclusive outcomes, expected value is the sum of each outcome multiplied by its probability. In a contract pipeline, that means weighting every opportunity by its estimated chance of success and adding the weighted amounts.

The result is an average over repeated comparable situations, not a prediction that the portfolio will produce that exact amount. A single large contract will still be won or lost. Expected value is therefore most useful for planning a portfolio of uncertain events and understanding the central case across them.

Sensitivity analysis asks “what if?” It changes one or more uncertain inputs—probability, price, volume, cost or timing—and recalculates the outputs. The analyst learns which assumptions drive the model, where downside threatens cash or capacity, and whether a decision remains attractive across a credible range.

The two methods work together: expected value supplies a probability-weighted base case, and sensitivity analysis describes the range and fragility around it.

How to use it

Suppose the firm has confirmed orders of 1000 units and 10 prospective orders, each worth between 30 and 300 units, across a three-year planning horizon.

A common scenario approach includes a downside case containing opportunities above a 50 per cent probability and an upside case containing every opportunity. In Year 3, those assumptions produce 1490 units and 2000 units respectively; simply taking their midpoint would produce a base case of 1745.

Expected value: an example

The midpoint approach overstates the central case because the largest opportunity, worth 300 units, has a stated probability of only 10 per cent. The downside excludes it, while the upside includes it fully, so their midpoint effectively gives it a 50 per cent weight rather than 10 per cent. The second-largest opportunity, worth 180 units, appears fully in both scenarios and therefore receives an implied 100 per cent weight instead of its stated 60 per cent.

Weighting each opportunity separately produces an expected-value total of 1457 units in Year 3. That figure is a more coherent base case, provided the opportunity values and probabilities are independently estimated, mutually compatible and kept current.

Now test the assumptions. Vary uncertain inputs by 5 to 10 per cent where that range is credible, and test discrete contract outcomes separately. In this example:

• With Order E, expected-value orders in Year 3 become 1529 units. The increase is 72 because the model replaces 60 per cent with 100 per cent of 180.
• Without Order E, the result becomes 1349 units.
• With Order J, the result becomes 1727 units.
• Without Order J, the result becomes 1427 units.
• With Orders E and J, the result becomes 1799 units.
• Without Orders E and J, the result becomes 1319 units.

Translate each case into revenue, profit and loss, cash flow, staffing and capacity. Build operational flexibility around the exposures that matter. Typical parameter tests might include:

• sales volume up or down 5–10 per cent;
• unit price up or down 2–3 per cent;
• labour cost up 5 per cent;
• capital expenditure up 10 per cent.

Record the owner and evidence for every probability. Review the model as new information arrives, and use scenario analysis when outcomes interact or when a common shock could affect several opportunities at once.

Top practical tip

Separate the base-case calculation from the management response. First make probabilities explicit and calibrated; then identify the assumptions whose movement changes cash, capacity or the decision. Assign a trigger and contingency to each material exposure so the analysis leads to action.

Top pitfall

Probability estimates are vulnerable to optimism, double counting and false precision. Expected value also hides the shape of the distribution: two choices can have the same average but radically different downside. Challenge estimates with evidence, model dependencies and consider risk tolerance before committing.

Further reading

Howard, R. A. “Decision Analysis: Applied Decision Theory.”

von Neumann, J., and Morgenstern, O. Theory of Games and Economic Behavior.