Adding project cost percentiles does not give the same percentile for the total

If two projects each have a P80 cost allowance, their combined allowance is not automatically the P80 of the portfolio. The total depends on which costs can occur together. Adding two dollar amounts does not preserve their separate percentile labels.

The Government Accountability Office's cost guide explains the use of cumulative cost distributions and warns that budgeting individual programs at a given percentile can produce a much higher percentile for the portfolio. Its guidance also discusses correlation between cost elements. Those are reasons to model the combined distribution explicitly.

For the examples below, a percentile means the smallest cost at which the cumulative probability reaches or exceeds the stated level. That definition matters for discrete outcomes, where a distribution jumps between a few possible values instead of moving smoothly through every dollar amount.

Consider two invented, independent cost risks. Each creates no additional cost with 60% probability or a $100,000 additional cost with 40% probability. The P80 of each individual risk is $100,000 because the zero-cost outcome covers only 60% of cases.

Adding the individual P80 figures gives $200,000. But the combined risk has three outcomes. Neither cost occurs with probability 0.60 times 0.60, or 36%. Exactly one occurs with probability 48%. Both occur with probability 16%.

The cumulative probability at $100,000 is 36% plus 48%, or 84%. The total's P80 is therefore $100,000. In this illustration, adding the two individual P80 amounts produces enough allowance for every modeled outcome, not an 80th-percentile total.

Now suppose both risks are triggered by the same event and always occur together. The individual distributions remain unchanged: each still has a 60% chance of zero and a 40% chance of $100,000. Their individual P80 amounts are still $100,000.

The combined distribution is now zero with 60% probability and $200,000 with 40% probability. Its P80 is $200,000. The same two individual cost distributions therefore support a different total allowance when their joint behavior changes.

Ask what produces the costs. A shared supplier disruption, common price index or single technical failure may connect several rows. Independence should be an evidence-based modeling assumption, not a default chosen because the spreadsheet stores costs in separate cells.

It is tempting to conclude that adding percentiles always overstates the portfolio amount. Discrete tail risks provide a counterexample. Suppose two different independent risks each cost $100,000 with 9% probability and zero with 91% probability.

Each individual P90 is zero under the stated percentile definition. Adding them gives zero. Yet the probability that neither occurs is 0.91 squared, or 82.81%. The total's P90 is $100,000, because the cumulative probability including exactly one event is 99.19%.

This counterexample is why the direction of the error cannot be assumed from the word percentile alone. The relevant inputs are the full marginal distributions and their dependence. Means add directly across costs; a chosen percentile generally requires the distribution of their sum.

For example, expected total cost in the first model is $80,000: two risks each with $40,000 of expected cost. That remains true when they always occur together, even though the total P80 changes from $100,000 to $200,000. An expected-cost budget and an allowance chosen to cover a stated share of outcomes therefore serve different purposes.

For a small set of discrete risks, enumerate the feasible joint outcomes and their probabilities, then add costs within each outcome. Sort the resulting totals to obtain the cumulative distribution. The first example can be solved in a four-row table before collapsing the two identical $100,000 outcomes.

For a larger model, add all relevant cost elements within each complete simulation run and calculate the desired percentile across those total costs. Preserve shared drivers within the run so a supplier shock affects all exposed elements consistently.

Do not independently sort each cost column and then add matching percentile rows. That operation can create artificial combinations that never occurred in the modeled scenarios. Similarly, adding separately calculated P80 outputs from several models requires an additional model of how their outcomes relate.

The first example's $100,000 portfolio P80 assumes the amount can cover either project when needed. A company might instead have two restricted funding accounts that cannot transfer money. A dollar held for project A would then be unavailable when only project B incurs the cost.

The total-cost distribution is still mathematically useful, but the funding decision must reflect those restrictions. Model which account pays each obligation and when transfers can occur. A pooled allowance and two independently controlled project budgets solve different operational problems.

Timing can create another constraint. Costs in different years may compete for the same nominal reserve but require separate annual approvals. Check the liquidity and authorization schedule instead of treating a lifetime cost percentile as proof that every payment date is funded.

Choosing an 80th-percentile allowance leaves modeled outcomes above it. Management still needs to know their severity and the available response. The percentile answers how often the modeled total is at or below an amount; it does not describe every consequence beyond that amount.

Avoid counting the same buffer twice. If individual cost estimates already include specific risk allowances, reconcile those allowances before adding a portfolio reserve. Conversely, removing project buffers without modeling their underlying risks can make the aggregate estimate incomplete.

State the cost scope, horizon, chosen percentile, dependence assumptions and restrictions on using the allowance. For the first illustration, a defensible description is a $100,000 allowance covering at least 80% of the modeled combined additional costs under independence.

Review that conclusion when a shared risk emerges or the funding rules change. The calculation should help management choose an affordable response to uncertainty, while making clear that two P80 labels do not by themselves establish the probability attached to their sum.