Expected value: the formula, and what it actually tells you

Why a $1 binary reduces to two terms

Expected value normally means summing every outcome weighted by its probability. A binary event contract has two outcomes and one of them pays nothing, so the sum shortens:

Multiply out (p − pc − c + pc) and the pc terms cancel, leaving p − c. The full weighted sum and the short form give the same number.

Prices are quoted in cents on a $1 contract, so a quote of 41¢ corresponds to an implied probability of roughly 41%. That correspondence is approximate rather than exact — spread, fees and multi-outcome normalization all move it — and odds vs probability works through why.

Worked example

The numbers below are illustrative, chosen to show the arithmetic rather than to describe any real market.

Suppose a contract asks whether one party holds Senate control after the next election, and yes is quoted at 41¢. Suppose further that a reader has worked through the seat map and the relevant base rates and arrived at 48% as their own estimate.

Substituting:

Seven cents per contract, before any costs.

That figure is the arithmetic consequence of the two inputs above, not a forecast, a return, or a projection. Change either input and the output changes with it, which the next two sections work through.

The other outcome

The same illustration, worked the other way. A yes position taken at 41¢ has two possible ends:

If the event doesn't happen, the full 41¢ paid is lost, on every contract held. A 41¢ quote is conventionally read as an implied probability of roughly 59% that this is the outcome.

A positive expected value therefore doesn't mean a positive outcome. It means that, given the two inputs supplied (that estimate and that price), the average result across many repetitions of the same situation would be positive. An individual contract resolves once, and on that single resolution the two rows above are the only possibilities.

The formula also can't tell you when an input is wrong. Substituting 38% instead of 48%, at the same 41¢ quote, gives:

The same arithmetic, the opposite sign. Nothing about the +7¢ result flagged that it depended on the 48% being right.

Lumina Markets Inc. is a CFTC-registered Introducing Broker and Member NFA (NFA ID: 0576536). Trading event contracts involves substantial risk of loss and is not appropriate for all.

Fees and the effective price

c in the formula is the amount actually paid, which is generally not the quote on screen. Two things typically sit between them:

• The spread. A buy order is usually filled at the ask rather than at the midpoint, and on a wide market the ask can sit several cents above the mid a reader was looking at.
• Transaction costs. Execution fees are charged per contract and vary by venue and product, so the current published schedule for the market in question is the figure to substitute here.

Call the adjusted figure the effective cost, c_eff. The formula is unchanged:

Returning to the illustration: a 41¢ mid with a 42¢ ask, plus Lumina's 1¢ exchange fee and 0.35% commission on the 42¢ (0.147¢), gives c_eff = 43.147¢, and:

The +7¢ becomes +4.853¢. About two cents matters here because the figure it is subtracted from is single-digit to begin with, which is generally why small estimated differences do not survive costs.

Run your own numbers in the EV calculator →

Break-even is the same equation, rearranged

Set EV = 0 and solve for p:

break-even probability ≈ c_eff

The effective cost, read as a percentage, is the break-even probability. At c_eff = 43.147¢, the event has to occur more than 43.147% of the time for the position to come out ahead of not taking it.

Break-even probability calculator →

What EV can't tell you

The formula evaluates the estimate it is given. It contains no test of whether that estimate is any good.

• A positive result describes the reader's number, not the market's. The quote is a price participants transacted at; the 48% is one person's judgment.
• It says nothing about a single outcome. Expected value is an average across repetition, and an individual contract resolves once.
• It excludes position size, concentration, and everything else held alongside it. Two positions with identical EV per contract can carry very different consequences.
• It degrades as a market thins out. Where volume is low, the quote carries less information and the fill is less predictable, so both p and c_eff are less reliable than they appear. Market efficiency covers the conditions involved.

Checking whether an estimate is any good

An estimate is assessed against outcomes across a run of forecasts, not by how the most recent one turned out. Two standard measures:

• Brier score: a single figure for how far stated probabilities landed from outcomes; lower is better.
• Calibration: whether things stated at 70% resolve yes about 70% of the time.

Both require the probabilities to have been recorded, dated, before resolution, which is what makes an estimate checkable rather than just stated.

The next question people usually ask — how much a disagreement between an estimate and a price is worth committing — has a standard published answer in the Kelly criterion, along with a set of assumptions that don't transfer cleanly to event contracts.