Kelly Criterion — Sizing Positive EV Bets
Kelly is the bankroll-optimal staking rule. Full Kelly maximises long-run growth rate; fractional Kelly accepts a smaller growth rate to massively reduce variance.
Updated · 7 min read
Key takeaways
- Kelly fraction = (decimal_odds × p_true − 1) / (decimal_odds − 1).
- Full Kelly maximises geometric growth rate but is extremely volatile.
- Quarter-Kelly is the practical default — gives ~94% of full Kelly's growth at ~1/16th the variance.
- Kelly assumes you know p_true exactly. You don't. Overestimating edge by 2× while full-Kelly betting will bankrupt you.
- Never bet more than 5% of bankroll on a single signal regardless of what Kelly says.
The Kelly formula
For a binary bet at decimal odds:
f* = (b × p − q) / b b = decimal_odds − 1 (net odds) p = true probability of winning q = 1 − p (true probability of losing) Stake = bankroll × f*
A worked example
Bankroll = $10,000. Sharp baseline says Cowboys win 56%. AU corporate has them at $1.95.
b = 1.95 − 1 = 0.95 p = 0.56, q = 0.44 f* = (0.95 × 0.56 − 0.44) / 0.95 f* = (0.532 − 0.44) / 0.95 f* = 0.0968 → 9.68% of bankroll Full Kelly stake = $968 Quarter-Kelly stake = $242
Full Kelly vs fractional Kelly
Full Kelly maximises the long-run geometric growth rate of your bankroll. The catch is volatility: a 50% drawdown is roughly the median outcome over 100 bets even with positive EV. Most professional bettors use quarter-Kelly or half-Kelly.
Quarter-Kelly captures about 94% of the growth rate while reducing variance to roughly 1/16 of full Kelly. The math is symmetric — fractional Kelly is the practical sweet spot.
Practical Kelly fractions
- Full Kelly — only if p_true is known with very high confidence (rare in sports betting).
- Half Kelly — for tightly modelled markets with clean closing-line value data.
- Quarter Kelly — sensible default for AU +EV bettors.
- Tenth Kelly — early in a strategy, before you trust your model.
Why edge estimation matters more than the formula
Kelly assumes you know p_true. You don't. If your model says p_true = 0.56 but the real number is 0.52, full Kelly bets are heavily over-sized and you will steadily lose bankroll.
Rule of thumb: discount your estimated p_true by 2–5 percentage points before applying Kelly. The asymmetry is brutal — over-betting wipes you out, under-betting only slows you down.
When Kelly says zero or negative
If the formula returns f* ≤ 0, do not bet. A negative Kelly means the bet has negative expected value and the formula is telling you to lay it instead.
Most retail +EV signals at AU corporates yield Kelly fractions between 1% and 5%. Anything over 10% should be treated with suspicion — either the odds error is genuine and the book will void it, or your p_true is wrong.
FAQ
Why don't professional bettors use full Kelly?
Variance. Full Kelly's 50%-drawdown median outcome over a season is psychologically and operationally unacceptable. Quarter-Kelly is the standard practical compromise.
Does Kelly work for parlays?
Yes mathematically — replace decimal_odds with parlay_odds and p_true with the product of leg probabilities. In practice the joint probability is hard to estimate and bet limits on parlays are tight, so most pros stake parlays at flat fractional amounts.
How do I estimate p_true without modelling?
Use the devigged probability from the sharpest available market as your p_true proxy. Krok Odds publishes this sharp-line probability for every AU market we cover.
Should I use Kelly for arbitrage?
No. Arbitrage has no probability of loss (subject to execution), so Kelly is undefined. Size arbs by stake-cap policy — typically 10–20% of bankroll per arb across both legs.