The Kelly criterion, and why it's hard to apply to event contracts
The Kelly criterion for event contracts: the formula, where it comes from, the four assumptions behind it, and why they rarely hold in practice.
Updated November 10, 2026 · Lumina Markets
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Expected value answers whether an estimate and a price disagree. The Kelly criterion is the formula usually reached for next, when the question becomes how much: what fraction of the money available a stake would represent.
Its one-line form makes it look like a settled answer. The assumptions behind that line are where most of the difficulty sits, and on event contracts several of them hold poorly.
What the formula says
For a contract that pays $1 if an event happens and nothing if it doesn't, bought at price c, against a probability estimate p:
f* p c1 c
f* is a fraction of a total bankroll. Where p equals c the formula returns zero; where p is below c it returns a negative number, which in the original framing means the position isn't taken.
What it's a solution to
Kelly answers a specific question: across a long run of repeated opportunities, what constant fraction of your money produces the fastest growth without eventually wiping you out?
The "without wiping you out" half is what makes the question non-trivial. If the goal were simply the largest average profit, the answer on any favorable opportunity would be to commit everything every time. That maximizes the average, but across a run of such positions a single loss takes you to zero, and there is no recovering from zero. So the useful version of the question has to trade some average return for survival.
Kelly's answer is to maximize the expected logarithm of wealth rather than expected wealth itself. Two properties of the log do that work: it grows more slowly than its input, so each additional unit of gain counts for less than the last, which discourages concentration; and it is undefined at zero, so any strategy with a real chance of total loss is excluded automatically rather than by an added rule. Maximizing expected log wealth is also equivalent to maximizing the long-run compound growth rate, which is the interpretation most commonly attached to it.
The objective itself is an assumption. Kelly is optimal for one thing — compound growth over a long horizon — and someone who wants a steadier path, or who has a fixed end date, is asking a question the formula doesn't answer.
The general form is usually written:
f* bp qb
where b is net odds and:
q 1 p
A $1 contract bought at c returns (1 − c) net on a stake of c, so:
b 1 cc
and substituting gives the binary form above.
Working the arithmetic
The figures below are illustrative: the same invented contract, invented quote and invented estimate used on the expected value page, carried over so the two calculations can be compared. A quote of 41¢, and an estimate of 48%.
f* 0.48 0.411 0.41 0.070.59 0.119
The formula returns roughly 11.9%. With Lumina's 1¢ fee and 0.35% commission the all-in cost is 42.1435¢, and the same estimate returns about 10.1%; the Kelly criterion glossary entry works through that version.
That figure is not a recommended position size, and nothing here recommends one. It's what the arithmetic returns given those two inputs and the four assumptions set out below, assumptions that hold poorly on event contracts.
- Quoted price (c)
- 41¢
- Probability estimate (p)
- 48%
- What the formula returns (f*)
- 11.9%
- A fraction of bankroll, not a recommended size
The four assumptions
- The probability is known, not estimated
In the derivation, p is the actual frequency with which the event occurs, the kind of number you have for a fair coin. Substituting a judgment for that changes what the output means, and the arithmetic gives no sign that the substitution happened: it returns a figure to three decimal places either way.
- The opportunity repeats, independently, many times
The result is asymptotic, meaning it describes what happens to wealth in the limit, across a long sequence of similar bets, rather than what happens on any one of them. "Independently" means the outcome of one tells you nothing about the next, the way successive coin flips do. Neither condition describes a set of election contracts: there is one election, and contracts on the same race or the same macro release resolve together.
- The stake is infinitely divisible
The formula returns a continuous fraction: 11.9% of whatever is available. That assumes you can commit exactly 11.9%. Contracts trade in whole units, on Lumina at prices between 2¢ and 99¢, so the achievable fractions are a coarse grid, and the smaller the amount available the coarser it gets: at $100, one 41¢ contract moves the committed fraction by about 0.4 percentage points at a time, so 11.9% rounds to whatever whole number of contracts lands nearest. The formula's third decimal place has nothing behind it.
- The objective is maximum long-run growth
Kelly is optimal for one goal: growing money at the fastest sustainable compound rate over a long horizon, accepting whatever swings that involves along the way. Those swings are substantial: a strategy that commits double-digit percentages repeatedly will see large drawdowns as an ordinary feature, not a malfunction. Someone whose actual objective is a smoother path, or who has a fixed horizon rather than an indefinite one, is optimizing for something Kelly does not solve, and the formula has no term for that difference.
Where these hold up worst on event contracts
Estimate error is generally the binding one of the four.
Estimate error dominates. Kelly's output is highly sensitive to the input a reader is least certain about. Keeping the 41¢ quote and moving the estimate from 48% to 44%, four percentage points, well inside the range a careful person could be wrong by:
f* 0.44 0.410.59 0.051
The fraction falls from about 11.9% to about 5.1%. A four-point error in p roughly halves it, and at 41% it goes to zero. It looks exact to a tenth of a percent, but the estimate behind it isn't.
- Formula output at a 48% estimate
- 11.9%
- Formula output at a 44% estimate
- 5.1%
- Four points lower, roughly half
- Formula output at a 41% estimate
- 0%
- The estimate equals the price
So the prerequisite for using the formula is a record: whether the probabilities someone states have historically matched what happened. That is measurable: it's what Brier score and calibration measure. Absent that record, p is a number someone chose rather than a frequency.
Contracts on the same election, the same data release or the same broad macro question tend to resolve together. Applying the formula to each one separately, as though they were independent, will generally understate total exposure, and the formula has no term for correlation to correct with.
The result describes the limit of a long sequence; most readers encounter a given contract once. An asymptotic optimality property says little about one resolution, for the same reason a positive expected value says little about one outcome. Both are statements about averages that a single event doesn't sample.
Where a contract is thinly traded, the quote is a less reliable reading of anything and the fill is less predictable, so c is least well defined at the same time p is least checkable, and the output inherits both problems. Market efficiency covers those conditions.
Fractional Kelly
Because the full figure is so sensitive to estimate error, published treatments commonly discuss taking some fraction of f*, a half or a quarter, instead. The trade being made is a lower theoretical growth rate in exchange for less sensitivity to p being wrong, and smaller swings along the way.
What the adjustment doesn't do is improve the input. A quarter of a figure derived from an unreliable estimate is still derived from an unreliable estimate. No fraction is recommended here — not full Kelly, not fractional Kelly, not any other.
What this doesn't answer
Nothing above establishes what to size, whether to take a position, or how to allocate money. Those depend on a person's own circumstances, and none of this is tailored to them.
A binary contract that resolves against you pays nothing, and the full amount paid is lost. A sizing formula does not reduce that outcome on any individual contract. It is a statement about a long run of repeated, independent opportunities, which is not the situation most readers are in.
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.
FAQ
What is the Kelly criterion formula?
For a binary contract paying $1, bought at price c against a probability estimate p, it is f* = (p − c) / (1 − c), expressed as a fraction of total bankroll. The general form is f* = (bp − q) / b for net odds b.
What does the Kelly criterion maximize?
The expected logarithm of wealth, which is equivalent to the long-run compound growth rate, not expected profit. The distinction matters: maximizing expected profit alone would commit everything to any favorable opportunity, which produces the highest average and eventual ruin, since one total loss ends the sequence. The log has two properties that fix that: each additional unit of gain counts for less than the last, which discourages concentration, and it is undefined at zero, so any strategy carrying a real chance of total loss is excluded automatically.
Why is Kelly hard to use on event contracts?
It rests on four assumptions, and an event contract strains each of them. It assumes the probability is known rather than estimated: on an election or a data release nobody has the true frequency, only a judgment. It assumes opportunities repeat independently, where in practice one election happens once and contracts on related questions resolve together. It assumes stakes divide continuously, where contracts trade in whole units, on Lumina at 2¢–99¢, so the achievable fractions are a coarse grid. And it assumes the objective is maximum long-run compound growth, accepting large swings along the way, which is a specific goal rather than a default one. The probability input drives the answer: at a 41¢ price, an estimate of 48% returns about 11.9% while 44% returns about 5.1%.
Does a smaller fraction of Kelly fix the estimate problem?
It reduces sensitivity to an error in p at the cost of theoretical growth rate. It doesn't make the estimate better, which is why forecast-accuracy measures matter more than the sizing algebra.