What Kelly answers, and what it doesn't
Whether to bet is an expected-value question: is this price in your favour? Kelly starts only after that answer is yes. It solves a different problem — how much — and its objective is specific: maximise the growth rate of the bankroll over a long sequence of bets. Not the profit on one bet, not the chance of a good week, but the compound growth rate over time.
That objective is what makes Kelly self-correcting. Stake too little and you leave growth on the table; stake too much and volatility eats the compounding. Kelly finds the single fraction that balances them, and the answer is a percentage of the bankroll you hold right now.
# Kelly fraction for a simple win/lose bet # b = profit per unit at decimal odds = D − 1 # p = your probability of winning # q = 1 − p f = (b × p − q) / b # stake = f × current_bankroll # if f is zero or negative, the bet has no edge — do not bet
A worked fraction
Take an illustrative bet at decimal 2.10 where you estimate the win probability at 52 percent. Note this is the same setup as a positive-EV bet — Kelly turns that edge into a stake.
# D = 2.10 to b = 1.10 ; p = 0.52 ; q = 0.48 f = (1.10 × 0.52 − 0.48) / 1.10 = (0.572 − 0.48) / 1.10 = 0.092 / 1.10 = 0.0836 # full Kelly ≈ 8.4% of the current bankroll
Full Kelly says stake 8.4 percent of the roll. For most bettors that is a large, nerve-testing bet on a single edge — and it is correct only if 0.52 is exactly right. Shave your true probability to 0.50 and the edge, and the correct stake, collapse to zero. The fraction is exquisitely sensitive to the input, which is the entire case for betting less than it says.
Why fractional Kelly is the practical default
Kelly's promised growth assumes your probability is exact. In reality every edge is estimated with error, and errors are not symmetric in their consequences. The growth rate as a function of stake is a hump: it climbs to a peak at the full-Kelly fraction, then falls away — and it falls faster than it rose.
| Stake vs full Kelly | Share of max growth kept | Relative volatility |
|---|---|---|
| Quarter Kelly | high — roughly three-quarters | much lower |
| Half Kelly | high — about three-quarters | lower |
| Full Kelly | maximum | high |
| Double Kelly | zero | very high |
The lesson lives in the two extremes. Half-Kelly keeps most of the growth for a fraction of the volatility. Double-Kelly — a plausible result of overestimating your edge by a factor most bettors would not notice — drives long-run growth back to zero on a bet that genuinely had an edge. Because overestimation is common and its cost is severe, staying left of the peak is the disciplined error, and quarter-Kelly is a widely used default. The Kelly stake calculator exposes these fractional presets directly.
Simultaneous and correlated positions
The clean formula covers one bet against one bankroll. Real betting rarely looks like that, and two complications break the simple version.
- Shared bankroll. Three simultaneous bets cannot each claim their full Kelly fraction of the same roll — the exposures add up against one balance. Either scale each down so the total exposure stays sane, or impose a hard cap on combined stake regardless of the individual fractions.
- Correlation. When legs move together — same game, same driver, correlated outcomes — treating them as independent double-counts the edge and overbets the combined position. Correlated bets have to be sized down, or solved jointly rather than one at a time.
A practical desk therefore runs Kelly with guardrails: a per-bet fractional multiplier, a ceiling on any single stake, and a ceiling on total open exposure. The formula sets the shape; the caps keep a single bad estimate from being catastrophic.
The recalculation cadence
Kelly is a fraction of the bankroll you hold now, not the one you started with, and this is where its compounding comes from. Size every bet off the current balance and the stake grows after wins and shrinks after losses on its own — the mechanism that maximises long-run growth and stops a cold streak from ending the account.
- Re-anchor to the live bankroll before every bet, not to a fixed starting figure or a round number you are comfortable with.
- Re-estimate the edge as prices move; a stale probability against a moved line is a stale stake. Line movement is a signal your inputs may need refreshing — see the line movement guide.
- Respect the caps even when the fraction tempts you past them; the caps exist for the bets where your estimate is most wrong.