Kelly criterion
Updated November 10, 2026
The Kelly criterion gives the fraction of a bankroll that maximizes long-run compound growth. For a $1 binary at price c against probability p:
f* p c1 c
The derivation treats p as a known frequency; on an event contract it is an estimate. Sizing does not reduce the loss on a contract that resolves at $0.
Illustrative figures. At 41¢, Lumina's 1¢ exchange fee and 0.1435¢ commission make the all-in cost 42.1435¢. With an estimate of 48%, the Kelly fraction is:
0.48 0.4214350.578565 0.101
About 10.1% of the bankroll. If the contract settles no, that stake and its fees are lost.
- Kelly fraction at a 48% estimate
- 10.1%
- Kelly fraction at a 44% estimate
- 3.2%
- Four points lower, about a third of the fraction
Change the estimate by four points, to 44%, and the output falls to about 3.2%, roughly a third, from an input revision most people would consider small. At 42% the estimate sits below the all-in cost, and the trade loses after fees. Because p is estimated rather than measured, that sensitivity is usually the binding constraint on using the formula, and it makes a calibration record (whether stated probabilities have historically matched outcomes) the prerequisite the algebra assumes.
Where the binary form comes from
Kelly's general form is:
f* bp qb
where b is the net odds received on a win and:
q 1 p
is the probability of losing. A $1 contract bought at c returns 1 − c net on a stake of c, so:
b 1 cc
Substituting that into the general form and simplifying gives (p − c) / (1 − c).
At 41¢:
b 0.590.41 1.44
Each dollar staked returns about $1.44 net if the contract settles yes. The general form with p = 0.48 and q = 0.52 gives:
1.44 0.48 0.521.44 0.119
the same 11.9% the binary form gives at 41¢ before fees.
The formula's edges follow from the same arithmetic. Where p equals c, the output is zero. Where p is below c, it is negative, which in the original framing means the position isn't taken.
What it maximizes
Kelly maximizes the expected logarithm of wealth, which is the same as maximizing the long-run compound growth rate. That is a different target from expected profit. Committing everything to every favorable opportunity gives the highest average result, and a single loss ends the sequence at zero. The logarithm rules that out, because it falls without limit as wealth approaches zero.
Growth is the only objective the formula serves. Someone who wants a steadier path, or who has a fixed end date, is asking a question it doesn't answer. Published treatments often discuss taking a fraction of f*, such as half, which lowers the theoretical growth rate in exchange for less sensitivity to an error in p and smaller swings. It doesn't improve the estimate itself.
The derivation also assumes the opportunity repeats independently. Contracts on the same election or the same data release tend to resolve together, and applying the formula to each one separately treats them as unrelated. The full Kelly criterion guide works through each assumption.
FAQ
What is the Kelly criterion formula for a binary contract?
For a contract paying $1, bought at price c against a probability estimate p, it is f* = (p − c) / (1 − c), a fraction of total bankroll. The general form is f* = (bp − q) / b for net odds b.
What does a negative Kelly fraction mean?
That your estimate is below the price, so the formula finds no advantage in buying at that price. In the original framing, the position isn't taken.
What is fractional Kelly?
Using a set fraction of the Kelly output, such as half. It reduces sensitivity to estimate error and the size of swings, at the cost of a lower theoretical growth rate.