Explainer · Decision analysis
Expected Monetary Value (EMV)
The formula, a worked risk register and a decision tree you can check by hand
What is expected monetary value (EMV)?
Expected monetary value (EMV) is the average financial result of an uncertain event: each possible outcome in money, weighted by its probability. A 30% chance of a €130,000 delay has an EMV of −€39,000. Threats carry negative values and opportunities positive ones, so EMVs can be added across a risk register or compared between the options in a decision tree.
“Expected” is meant in its statistical sense. EMV is what the event would cost on average if you faced it many times. On a single project the delay either happens or it does not, so the result is €0 or −€130,000, never −€39,000. That is the most useful thing to know about EMV, and the easiest to forget: it is a good way to rank and compare risks, and a poor way to decide how much money to hold.
EMV turns up in two places. In quantitative risk analysis it ranks the risks in a register and gives a first figure for the exposure. In decision analysis it is the number a decision tree rolls back to choose between options. This page works one example of each, with figures small enough to check by hand.
How do you calculate EMV?
Multiply each possible outcome by its probability and add the results. For a risk event with one impact, that reduces to probability × impact, because the other outcome, nothing happening, is worth €0. Give threats a negative sign and opportunities a positive one, and write probabilities as decimals: 30% is 0.30.
EMV = p1 × v1 + p2 × v2 + … + pn × vn
Here p is the probability of each outcome and v its value in money. For the delay above:
EMV = 0.30 × (−€130,000) + 0.70 × €0 = −€39,000
Three conventions keep the arithmetic straight:
| Convention | Why it matters |
|---|---|
| Threats negative, opportunities positive | EMVs can then be added, and an opportunity offsets a threat. Some textbooks write costs as positive numbers and look for the lowest EMV instead. Either works if you keep to one convention. |
| The probabilities of an event’s outcomes add up to 1 | A 30% chance of a delay means a 70% chance of no delay. That outcome drops out of the sum only because it is worth €0. |
| The impact is the full money consequence | A 20-day delay is converted at the cost of a day, including knock-on costs such as standing plant and penalties, so that every risk is measured in the same unit. |
How is EMV used in a risk register?
Each risk in a quantitative register gets a probability and an impact in money, and its EMV is their product. Sorting by EMV ranks risks by their average cost, and adding the EMVs gives the expected net exposure of the whole register. Opportunities reduce the total, because their EMVs are positive.
Take a building project with three threats and two opportunities:
| Risk | Probability | Impact | EMV |
|---|---|---|---|
| Poor ground conditions under the foundations | 20% | −€400,000 | −€80,000 |
| Late delivery of structural steel | 40% | −€150,000 | −€60,000 |
| A planning objection delays the start | 30% | −€130,000 | −€39,000 |
| The steel price falls before the order is placed | 25% | +€80,000 | +€20,000 |
| The neighbouring plot is released early for storage | 50% | +€40,000 | +€20,000 |
| Expected net exposure | −€139,000 |
Ranked by probability, late steel would come first. Ranked by EMV, poor ground does, because a one-in-five chance of a €400,000 problem costs more on average than a two-in-five chance of a €150,000 one. That ordering is the main use of EMV in a register: it shows which risks deserve attention and money first.
EMV also prices a response. Suppose reserving fabrication capacity with the steel supplier costs €20,000 and cuts the chance of late delivery from 40% to 10%. The risk then costs €20,000 for certain plus an EMV of 0.10 × (−€150,000):
−€20,000 + 0.10 × (−€150,000) = −€35,000, against −€60,000 without it
On average the response saves €25,000, so it is worth doing. A response that costs more than the EMV it removes is not, unless the size of the possible loss is the real worry, which is where the next section comes in.
Is the EMV total the contingency you should hold?
No. The EMV total is an average over every combination of risks happening or not, and no single project will see it. In this register the risks cost more than €139,000 in 38% of cases. A contingency set at the EMV total would be exceeded more than one time in three, before any other uncertainty is counted.
Because the five risks in the example are independent, the exact distribution of the total can be worked out from the 32 combinations of risks happening or not:
| What can happen | Chance |
|---|---|
| No threat happens, so the result is €0 or a gain | 34% |
| The risks cost more than the EMV total of €139,000 | 38% |
| The risks cost €400,000 or more | 15% |
| Every threat happens and neither opportunity does: −€680,000 | 0.9% |
The median, or P50, of the net cost is €110,000. The P80, the value with an 80% chance of not being exceeded, is €280,000, twice the EMV total. The gap comes from the large, unlikely risks that EMV flattens: poor ground adds €80,000 to the EMV total, but €400,000 to the bill when it happens.
Key point
EMV ranks risks well and sizes budgets badly. Use EMV to decide which risks to treat first, and the distribution of the total to decide how much to hold.
Our guide to calculating cost contingency shows how to choose between the P50, P80 and P90 once you have that distribution.
How do you use EMV in a decision tree?
Work from right to left. At each chance node, replace the branches with their EMV, the probability-weighted sum of their values. At each decision node, keep the branch with the best EMV. The value left at the root is the EMV of the best strategy, and the branches you kept are the recommendation. This is called rolling back the tree.
A contractor is deciding whether to bid for a contract. Preparing the bid costs €50,000 whatever happens. The chance of winning is 40%. If it wins, there is a 60% chance the contract makes €500,000 and a 40% chance that design and ground problems turn it into a €200,000 loss. Not bidding costs nothing and earns nothing.
The bid decision, rolled back
The square is the decision and the circles are chance events. Results are net of the €50,000 bid cost. Blue marks the branch the rollback keeps.
Bidding is worth +€38,000 on average and not bidding €0, so the rollback keeps Bid.
See the numbers
| Path | Chance | Net result | Chance × result |
|---|---|---|---|
| Bid, win, good contract | 0.40 × 0.60 = 24% | +€450,000 | +€108,000 |
| Bid, win, poor contract | 0.40 × 0.40 = 16% | −€250,000 | −€40,000 |
| Bid, lose | 60% | −€50,000 | −€30,000 |
| Bid: EMV | 100% | +€38,000 | |
| No bid | 100% | €0 | €0 |
Rolling it back takes three steps. First the contract node, using results net of the bid cost:
EMV(contract) = 0.60 × €450,000 + 0.40 × (−€250,000) = +€170,000
Then the node that decides whether the bid wins:
EMV(bid) = 0.40 × €170,000 + 0.60 × (−€50,000) = +€38,000
Finally the decision: bidding is worth +€38,000 and not bidding €0, so on EMV the contractor bids.
The register’s caveat applies here too. If the contractor bids, it makes €450,000 with a 24% chance (0.40 × 0.60), loses €250,000 with a 16% chance and loses the €50,000 bid cost with a 60% chance. That is a 76% chance of losing money on an option with a positive EMV. For a firm that bids often, EMV is the right guide, because the results average out over many bids. For a one-off bid where a €250,000 loss would really hurt, the risk profile matters as much as the average. ModelChoice draws the risk profile from the same tree as the rollback.
What does the PMP exam ask about EMV?
In the PMBOK Guide’s sixth edition, EMV sits in the Perform Quantitative Risk Analysis process, where decision tree analysis uses it to compare options. Practice questions are usually short calculations: the EMV of one risk, the net EMV of a few threats and opportunities, or the better of two options in a small tree. The usual traps are signs and forgotten costs.
For example, a threat with a 25% chance of costing €40,000 and an opportunity with a 10% chance of saving €20,000 give:
EMV = 0.25 × (−€40,000) + 0.10 × €20,000 = −€10,000 + €2,000 = −€8,000
| Common mistake | What to do instead |
|---|---|
| Treating every impact as a positive number | Make threats negative and opportunities positive, or treat everything as a cost and look for the lowest EMV. |
| Leaving an option’s own cost out of the tree | Subtract it on the branch, or from every end value of that option, as with the bid cost above. |
| Chance branches that do not add up to 100% | Add the missing outcome, usually “nothing happens” at €0. |
| Picking the option most likely to succeed | Pick the option with the best EMV. The bid above is more likely to lose money than to make it, and still has the higher EMV. |
Monte Carlo simulation sits in the same PMBOK process. Our guide to Monte Carlo simulation in project management covers where it fits in the exam.
When does EMV give the wrong answer?
EMV assumes you are neutral to risk: that a gamble worth +€38,000 on average is as good as €38,000 for certain. It also treats each probability and impact as exact and each risk as independent. When a loss would hurt badly, when risks share a cause, or when impacts are really ranges, EMV understates what is at stake.
Risk attitude
A firm that cannot absorb a €250,000 loss should not value the bid like one that can. Decision analysts handle this with a utility function, which scores each result by how much it matters to the decision-maker, and compare expected utilities instead.
Impacts are ranges
“Late steel costs €150,000” is shorthand for something between perhaps €90,000 and €250,000. A three-point estimate keeps the range. EMV keeps only the average.
Risks share causes
Poor ground and a late start often come together, because both depend on the same site. Correlation of this kind leaves the EMV total unchanged and makes large totals more likely.
Probabilities are estimates
A 20% from one workshop could easily be 10% or 35%. If a small change flips the decision, a sensitivity analysis shows it, and the value of information tells you whether finding out is worth paying for.
Monte Carlo simulation deals with the first three together. Each risk becomes an event that happens or not in each iteration, with its impact drawn from a range and correlations kept, and the result is the full distribution of the total: the P50, the P80 and the chance of exceeding any budget. Monte Carlo simulation explained covers how it works.
How do you calculate EMV in Excel?
Put the probabilities in one column and the signed impacts in the next, then multiply them row by row with a formula such as =C2*D2. The register total is one formula, =SUMPRODUCT(C2:C6,D2:D6), which returns −139,000 for the example above. Excel treats 20% and 0.2 as the same number, so either format works.
With the five risks in rows 2 to 6, probabilities in column C and impacts in column D:
| Cell | Formula | Result |
|---|---|---|
| E2, copied down to E6 | =C2*D2 | −80,000 for poor ground |
| E7 | =SUM(E2:E6) | −139,000 |
| Or in one step | =SUMPRODUCT(C2:C6,D2:D6) | −139,000 |
Decision trees are harder to keep straight in plain formulas, because every rollback step depends on the structure of the tree. ModelChoice builds the tree on the worksheet and rolls it back as you type, so the EMV of every node updates when a probability or payoff changes. The risk profile, sensitivity analysis and value of information come from the same tree, and utility functions are there for when the average is not enough.
To see the spread of a register rather than its average, ModelRisk turns each risk into a single cell. Written as a cost, late steel becomes =VoseRiskEvent(0.4,VoseModPERTObject(90000,150000,250000,4)): zero in about 60% of iterations, and in the rest a cost between €90,000 and €250,000, most likely €150,000. Simulating the register then gives the distribution of the total, like the table above, with the ranges included, and correlations too if you add them.
Frequently asked questions
What is the difference between EMV and expected value?
None in the method. Expected value is the general statistical term for a probability-weighted average, and EMV is the same calculation when the values are amounts of money. Project management and decision analysis texts tend to say EMV; statistics texts say expected value.
Can EMV be negative?
Yes. A threat has a negative EMV because its impact is a loss, and an option has a negative EMV when its expected losses outweigh its expected gains. An option with a negative EMV can still be the right choice when every alternative is worse.
What is the difference between EMV and EVPI?
EMV is the average value of a decision as things stand. EVPI, the expected value of perfect information, is how much that EMV would rise if the uncertainty were resolved before you decide. It is calculated from EMVs and caps what any study is worth. Our value of information example works one through.
How is EMV different from a probability-impact score?
A probability-impact score multiplies ratings, such as 3 × 4 on five-point scales, and the result has no unit. EMV multiplies a probability by an amount of money, so EMVs can be added, compared with the cost of a response and used in a budget. Two risks with the same score can have very different EMVs.
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