Kelly Criterion for Binary Bets & Prediction Markets: Formula

DefinitionThe fraction of wealth to stake that maximises the expected logarithm of wealth, and so the long-run growth rate. For a $1 event contract costing c with true probability p, it is (p − c)/(1 − c).

Formula for event contracts#

f^{*} = \frac{p - c}{1 - c}

where $c$ is the all-in cost per contract (price plus fee). Introduced by Kelly (1956) in the context of information theory.

Worked example#

Probability 0.60, price 50¢, no fee: $f^{*} = 0.1/0.5 = 0.2$ , or 20% of wealth, with expected log growth of about 2.0% per bet. Staking 40% gives roughly zero growth; more than that loses money in the long run.

Why fractional Kelly#

Kelly assumes $p$ is known. In our 4,000-path simulation, three percentage points of estimation error made half Kelly beat full Kelly, and six points left quarter Kelly as the only sizing that still grew in the median (full results).

Questions#

What is the Kelly criterion?

A formula for the fraction of wealth to bet that maximises long-run growth: for an event contract, (p − c)/(1 − c).

Why do traders use half Kelly?

Because probability estimates are uncertain. Half Kelly keeps about three quarters of the growth rate with far smaller drawdowns, and is more robust to estimation error.