Fractional Kelly is staking a set fraction of the Kelly stake rather than all of it, most often a half (half Kelly) or a quarter (quarter Kelly). The Kelly stake is only as good as the edge you feed it. Staking too much costs far more than staking too little by the same factor: at twice the true Kelly stake a bankroll's long-run growth falls to about zero, while at half it keeps about 75%.
With illustrative numbers, a bet at $2.50 that you rate a 6% edge has a full Kelly stake of 4% of the bankroll, so half Kelly is 2% and quarter Kelly is 1%. If your estimate is right, half Kelly gives up a quarter of the growth for half the swings; if the true edge is half what you thought, half Kelly is exactly the stake that grows the bankroll fastest.
How fractional Kelly works
The Kelly criterion sizes a bet as a share of the bankroll from the decimal price and your estimate of the chance. Fractional Kelly adds one multiplier at the end:
- edge = your chance x decimal odds - 1
- full Kelly share = edge / (decimal odds - 1)
- fractional Kelly stake = bankroll x full Kelly share x fraction
Example: Illustrative numbers, not a real market. A runner is $2.50, you rate its chance at 42.4%, and your bankroll is $2,000. Edge = 0.424 x 2.50 - 1 = 0.06, or 6%. Full Kelly share = 0.06 / (2.50 - 1) = 0.04, or 4%.
| Fraction | Share of bankroll | Stake on a $2,000 bankroll |
|---|---|---|
| Full Kelly | 4% | $80 |
| Half Kelly | 2% | $40 |
| Quarter Kelly | 1% | $20 |
The share applies to the bankroll at the time of each bet, so stakes fall after losses on their own: if the bankroll drops to $1,800, half Kelly on the same bet is $1,800 x 0.02 = $36. The Kelly criterion calculator works out the stake at any fraction you choose. Kelly only sizes bets; finding them is the rest of a betting strategy.
Why use a fraction of Kelly: noisy edge estimates
Your edge comes from the gap between your chance and the chance the price implies, 1 / odds. A small error in your chance becomes a much larger error in the edge, because each percentage point you misjudge moves the edge by as many points as the price: 2.5 points at $2.50, and 8 points at $8.00.
Illustrative numbers: a 6% edge at three prices, and what it becomes if the true chance is one point lower than yours:
| Price | Chance the price implies | Your chance | Your edge | True chance | True edge |
|---|---|---|---|---|---|
| $2.50 | 40.0% | 42.4% | 6.0% | 41.4% | 3.5% |
| $4.00 | 25.0% | 26.5% | 6.0% | 25.5% | 2.0% |
| $8.00 | 12.5% | 13.25% | 6.0% | 12.25% | -2.0% |
Each true edge is the true chance x the price - 1, so at $8.00 it is 0.1225 x 8.00 - 1 = -0.02. There the whole 6% edge is three quarters of a point of win chance (13.25% against 12.5%), and a rating that is slightly too kind turns a value bet into a losing one. Errors that size come from small samples behind ratings, from the way the margin is stripped from a market to get a fair price, and from prices that move before your bet is accepted.
Results cannot settle it quickly either: telling a 6% edge from a 3% one takes thousands of bets, as sample size in betting shows.
Half Kelly vs quarter Kelly: growth and drawdown
Two measures compare the fractions: long-run growth, how fast the bankroll compounds over many bets (the average log of the bankroll), and drawdown, the fall from a high point to a later low. Write c for your stake divided by the true Kelly stake, and two approximations cover both:
- growth kept, as a share of the best possible = 2 x c - c x c
- chance the bankroll ever falls to a share x of its starting size = x to the power (2 / c - 1)
The second comes from a continuous model with the stake re-sized after every bet and no end to the betting, so treat it as a guide to scale.
When your edge estimate is right
On the $2.50 bet with a true 6% edge, c is the fraction itself:
| Stake | c | Growth kept | Size of the swings | Bankroll ever falls to 70% of its start | Bankroll ever halves |
|---|---|---|---|---|---|
| Full Kelly, 4% | 1 | 100% | Full | 70% | 50% |
| Half Kelly, 2% | 0.5 | 75% | Half | 34% | 12.5% |
| Quarter Kelly, 1% | 0.25 | 44% | A quarter | 8% | 0.8% |
The working for half Kelly: growth kept = 2 x 0.5 - 0.5 x 0.5 = 0.75, and the chance of halving = 0.5 to the power (2 / 0.5 - 1) = 0.5 to the power 3 = 12.5%. For quarter Kelly: 2 x 0.25 - 0.25 x 0.25 = 0.4375, about 44%, and 0.5 to the power 7 = 0.78%. Worked exactly for this bet from the log of the bankroll, half Kelly keeps 75.1% and quarter Kelly 43.9%.
When the true edge is half your estimate
Now say your 42.4% is optimistic and the runner's true chance is 41.2%. The true edge is 0.412 x 2.50 - 1 = 0.03, or 3%, and the true Kelly share is 0.03 / 1.50 = 2%. Your stakes are the same as before, but c has doubled:
| Your stake | c | Growth kept | Chance the bankroll ever halves |
|---|---|---|---|
| Full Kelly on your estimate, 4% | 2 | About 0% | Certain, given enough bets |
| Half Kelly, 2% | 1 | 100% | 50% |
| Quarter Kelly, 1% | 0.5 | 75% | 12.5% |
Half Kelly on an estimate twice too high is exactly full Kelly on the truth. Size for size, erring low costs less than erring high: 1.5 times the true Kelly stake keeps 2 x 1.5 - 1.5 x 1.5 = 75% of the best growth, and two thirds of it about 89%. Erring low still costs growth. If the true edge were 9%, the true Kelly share would be 6%, and your full, half and quarter stakes (c = 0.67, 0.33 and 0.17) would keep about 89%, 56% and 31% of the best growth. Staking too little gives up growth; staking too much gives up growth and puts the bankroll at risk.
Risk: Betting involves risk. These figures assume independent bets that settle at the prices shown, and any growth depends on the edge being real: with no real edge, any stake loses over time at bookmaker prices, and a smaller one only loses more slowly. There is no guarantee of profit whatever fraction you choose. See responsible gambling for limits and support.
Kelly across several bets at once
Kelly is worked out for one bet in isolation. Bets open together need a second look, depending on how their results are linked.
Independent bets settled together
Illustrative numbers: take 10 bets on 10 different games on one afternoon, each like the $2.50 bet with a true 6% edge. Alone, each gets 4%. Solving for the best growth across all 1,024 ways the 10 can land gives 3.92% each, 39.2% of the bankroll in total. The share per bet barely moves, but almost 40% of the bankroll now rides on estimates made the same way, and if your method overrates its bets, all 10 are wrong together. Quarter Kelly puts about 1% on each and about 10% at risk at once.
Runners in the same race
Only one runner can win, so two bets in one race partly cover each other. With illustrative prices, Runner A is $4.00 and you rate it 26.5%, and Runner B is $8.00 and you rate it 13.25%, a 6% edge on each. Sized one at a time, A gets 0.06 / 3.00 = 2.00% of the bankroll and B gets 0.06 / 7.00 = 0.86%.
The joint answer, from the multi-runner form of Kelly, starts from a reserve figure, R:
- R = (1 - total chance of the runners you back) / (1 - total of 1 / odds for those runners) = (1 - 0.265 - 0.1325) / (1 - 0.25 - 0.125) = 0.6025 / 0.625 = 0.964
- share on each runner = its chance - R / its odds
- Runner A: 0.265 - 0.964 / 4.00 = 0.024, or 2.40%
- Runner B: 0.1325 - 0.964 / 8.00 = 0.012, or 1.20%
A runner belongs in the set only if its chance x odds is above R, and here both are 1.06. The joint shares total 3.60% against 2.86% for the two singles, because a win by either covers part of the other's stake. They also lean on two estimates at once, so the same fraction applies: half Kelly is 1.20% and 0.60%.
Bets on the same game
A head to head bet on a team and a line bet on the same team mostly win and lose together, so staking each at its own Kelly share is close to staking one opinion twice. Size strongly linked bets as one position.
When to scale stakes down
Use a smaller fraction, or no bet at all, in these cases:
| Situation | Why it calls for a smaller fraction |
|---|---|
| A method with only a few hundred bets behind it | Its edge estimate has no record yet, and results take thousands of bets to test it |
| Long prices | Each point of win chance you misjudge moves the edge by as many points as the price |
| Prices that no longer beat the closing price | Negative closing line value suggests the edge is smaller than you think, or gone |
| Many bets open at once from one method | Their errors arrive together |
| Swings you cannot sit through | Quarter Kelly's swings are a quarter the size of full Kelly's |
Kelly already cuts the stake after losses. Do not top the bankroll up to bring the stakes back after a losing run: that is chasing losses, and the stake is meant to follow the bankroll down.
Risk: Betting involves risk. A smaller fraction makes the swings smaller, not impossible, and an overestimated edge loses money at any fraction of Kelly. See responsible gambling for limits and support.
Mistakes with fractional Kelly
With illustrative numbers:
| Mistake | What it costs |
|---|---|
| Shading the edge and then the fraction | Rating the 6% edge down to 3% and then staking half Kelly on it puts 1% on the $2.50 bet: a quarter of the original Kelly stake, not the half you meant |
| Rounding stakes up | Quarter Kelly on $1,100 is $11; rounded up to $20 it is 1.82% of the bankroll, close to half Kelly |
| Raising the fraction after a winning run | A hot streak does not make your estimates any sharper, and it usually cools, which is regression to the mean at work |
| Staking a negative Kelly share at a small fraction | A fraction of a negative number is still negative: the formula is saying not to bet |