Average price times average volume can misstate expected revenue

A revenue forecast may average possible prices and possible volumes before multiplying them together. That shortcut can be wrong when price and volume move together. A high-volume outcome might require a discount, while a lower-volume outcome retains the full price.

OpenStax defines the expected value of a discrete outcome as its probability-weighted average. For revenue, the outcome being averaged is the revenue produced in each complete commercial case. Averaging the inputs first is a separate operation that needs its own justification.

State whether the scenarios have defensible probabilities. If they are simply low, central and high management cases, keep their financial results visible without calling an unweighted average expected revenue. The arithmetic below uses explicitly assumed equal probabilities only to illustrate the issue.

Suppose an invented period has two possible commercial outcomes. In a discounted high-volume outcome, the company sells 1,500 units at $80, generating $120,000. In a full-price low-volume outcome, it sells 500 units at $100, generating $50,000. Assign each outcome an illustrative 50% probability.

Expected revenue is half of $120,000 plus half of $50,000, or $85,000. The scenario-average price is $90, and expected quantity is 1,000 units. Multiplying those two averages gives $90,000, which overstates the correct weighted revenue by $5,000.

The missing feature is the pairing. The lower price occurs with more units, while the higher price occurs with fewer. Replacing those pairs with an average price and an average quantity creates a synthetic combination that does not preserve their relationship.

Mathematically, expected price times quantity equals expected price multiplied by expected quantity plus the covariance between price and quantity, when these expectations exist. The covariance term captures whether above-average prices tend to occur with above-average or below-average quantities.

In this example, the covariance is negative $5,000 in revenue units. It explains the difference between $90,000 from multiplying averages and the correct $85,000 expectation. If the high-volume case instead carried the higher price, the covariance could be positive and the shortcut could understate revenue.

Independence is a sufficient reason for the product of expectations to equal the expected product. Zero covariance also makes that particular equality hold. Do not assume either merely because price and volume occupy separate input cells in the financial model.

The $90 scenario-average price gives equal weight to the two possible prices because the scenarios have equal probability. It is not weighted by how many units are sold in each outcome. That distinction matters whenever commercial reports use average selling price without naming the denominator.

Dividing expected revenue of $85,000 by expected quantity of 1,000 gives $85 per expected unit. Multiplying that volume-weighted measure by expected quantity does reproduce $85,000. But it is a different average, derived from the joint outcomes rather than from the separate price probabilities.

The expected realized price across these two scenarios remains $90 because each scenario has a single realized price and equal weight. A ratio of expected totals need not equal the expectation of a scenario-specific ratio. Label the chosen measure so readers know which question it answers.

A quantity-linked price agreement may create the pairing directly. Customer mix can also produce it if larger orders come from lower-priced accounts. In other cases, demand pressure can increase both price and volume. Identify the mechanism before choosing a correlation or a scenario pairing.

Use the same approach for connected variable costs. If higher volume requires overtime or triggers a supplier discount, calculate that cost within each scenario. A revenue expectation that preserves price-volume dependence can still misstate contribution if cost is calculated from incompatible average inputs.

Avoid adding a generic correlation adjustment after a contractual formula already connects the variables. If the price automatically changes with quantity in each scenario, the relationship is already represented. Another adjustment may count the same effect twice.

If management fixes the same selling price for every outcome of a proposed policy, that price is no longer an uncertain input within that policy. Expected revenue can then be calculated as the fixed price multiplied by expected quantity, subject to the actual discounts and mix represented in the model.

Do not average different discretionary price proposals as though the business will randomly select one. Evaluate each proposal under its relevant demand outcomes. The worked example instead assumes two uncertain commercial outcomes with different realized prices and quantities.

Identify which inputs management controls before assigning probabilities. This keeps a decision comparison from becoming a forecast of an artificial blended policy that nobody intends to implement.

When generating simulated outcomes, preserve supported relationships between price, quantity and cost. Independently drawing each input from its marginal distribution can create combinations that the commercial terms never permit, such as discounted pricing without the required volume.

Inspect a few generated outcomes in ordinary business language. Ask whether the customer mix, delivery capacity and contract terms could all occur together. This can reveal a modeling inconsistency that a smooth distribution of total revenue does not make obvious.

Adding expected revenues from different products does not require those product revenues to be independent. Dependence matters for the distribution of their total and the chance of missing a target. The product-of-inputs problem inside each revenue calculation is different from summing already valid expectations.

The example's $85,000 expectation is useful for a defined planning purpose, but the actual modeled outcomes are $120,000 and $50,000. A fixed commitment that requires more than $50,000 of receipts still needs an assessment of the lower outcome and its timing.

Retain the scenario table, probabilities and commercial evidence with the forecast. Reconcile every weighted total to its underlying outcome calculations. That gives management an expected revenue measure that preserves the business relationships, along with a direct view of the cases the average cannot fund by itself.