Kelly Criterion Calculator: Betting, Investing, Poker

The Kelly Criterion answers a precise question: given a sequence of bets with a positive expected value, what fraction of your bankroll should you stake on each one to maximise the long-run growth rate of that bankroll?

This is a different question from "what maximises expected profit?" Expected profit is maximised by betting everything on the highest-EV opportunity available. Long-run growth is maximised by Kelly - because growth is multiplicative across bets, and a 100% loss takes you out of the game forever regardless of your edge.

The intuition is straightforward. Each bet multiplies your bankroll by some random factor. Over many bets, your final bankroll is the product of all those factors. Maximising that product is equivalent to maximising the sum of their logarithms - and the Kelly fraction is exactly the stake size that maximises expected log-growth per bet.

If this is your first exposure to the framework, the full Kelly Criterion explainer covers the derivation and the philosophical case for it. This page is the practical calculator.

The discrete Kelly formula - for a single bet that either wins or loses - has several equivalent forms, depending on whether you express the odds as decimal, fractional, or as a payout ratio. They all reduce to the same number.

The most general form, using a payout ratio b (net profit per unit staked if you win):

f* = (b × p − q) / b

Where:

If f* comes out negative, the bet has negative expected value and Kelly says do not bet at all. If it comes out above 1, the formula is telling you to stake more than your bankroll - meaning the edge is so large that capital constraints, not Kelly, are the binding limit.

For continuous outcomes (investing in an asset with a distribution of returns rather than a binary win/lose), the formula generalises to f* = (μ − r) / σ², where μ is the asset's expected return, r is the risk-free rate, and σ² is the return variance. This is the version most relevant for portfolio sizing.

Bookmakers quote decimal odds: 2.20 means a £1 stake returns £2.20 (your stake plus £1.20 profit). To plug decimal odds d into the Kelly formula, set b = d − 1. The simplified form is:

f* = (p × (d − 1) − (1 − p)) / (d − 1)

This is the calculator most matched-betting and value-betting communities use. The hard part is estimating p honestly - see the overconfidence bias piece for why most punters overstate their edge.

Two extensions are worth knowing. First, if you are betting into a market where the bookmaker takes a commission (a betting exchange like Smarkets or Betfair takes 2–5% on winnings), reduce b accordingly: b = (d − 1) × (1 − commission). Second, for multi-outcome markets such as football match results (home / draw / away), the standard Kelly does not apply directly - use the simultaneous-Kelly formulation, which adjusts each stake for the correlation between outcomes.

For investments where returns are continuously distributed rather than binary, the discrete Kelly formula does not apply. The continuous version uses expected excess return and variance:

f* = (μ − r) / σ²

Where μ is the expected annual return, r is the risk-free rate (UK gilts, US Treasuries, your savings rate), and σ is the standard deviation of returns. σ² is variance - square the volatility figure.

This is mathematically equivalent to f* = Sharpe / σ, which is how some portfolio managers prefer to think about it: position size scales with the Sharpe ratio and inversely with volatility.

Two cautions on this version. The continuous Kelly assumes returns are roughly log-normal and the inputs are stable. Both assumptions break in crises - see ergodicity for why long-run averages can lie about your actual experience. And the formula assumes you know μ and σ². You do not. Both are estimated from historical data, with sampling error that swamps any apparent precision in the calculator output.

For deeper treatment of the investing application, see position sizing with Kelly, which works through stock-specific and portfolio-level versions.

Poker is a multi-outcome game with very high variance relative to expected return, which is why most poker bankroll advice ends up well below full Kelly without explicitly invoking the formula.

For a cash-game session with winrate w in big blinds per 100 hands and standard deviation σ (also in bb/100), the per-hand variance is σ²/100 and per-hand EV is w/100. The Kelly fraction per hand is:

f* = w / σ²

This gives the fraction of bankroll to risk on each hand. To convert to buy-ins, divide a typical 100bb buy-in by the bankroll at which one buy-in equals that Kelly fraction.

Tournament poker is harder still because the variance is dominated by rare large outcomes (top-three finishes) rather than per-hand swings. Most professional tournament players target 100+ buy-ins for the level they play - closer to one-tenth Kelly. The general principle: as variance rises relative to edge, the fraction of Kelly you should use falls. For the expected-value calculations behind poker decisions themselves, see expected value in poker.

Full Kelly maximises long-run growth in theory. In practice almost every professional uses some fraction of it - typically half ("half Kelly") or quarter ("quarter Kelly"). There are two reasons.

The first is estimation error. Kelly assumes you know p exactly. You do not - you have an estimate of p with some uncertainty around it. Because the cost of overestimating p is dramatically worse than the cost of underestimating it (the formula is asymmetric around the true value), the rational response to estimation uncertainty is to bet less than the formula suggests.

The second is variance tolerance. Full Kelly produces extraordinary drawdowns. Edward Thorp's analysis showed that under full Kelly, the probability of seeing your bankroll halve at some point before doubling is roughly 50%. Most people, even professional investors and bettors, find that level of volatility psychologically untenable - and the worst time to abandon a strategy is during a drawdown, which converts paper losses into permanent ones.

Half Kelly is the working compromise. It captures around 75% of full Kelly's growth rate with roughly half the variance. Quarter Kelly drops growth to about 44% of full but cuts variance another half. For most practical applications - sports betting, portfolio management, poker bankroll - half or quarter is where you should set the calculator output.

• The Kelly Criterion explainer
• Position sizing with Kelly
• Expected value calculator
• Expected value vs expected utility
• Probability calibration training