Staking
Kelly criterion: the staking formula and why most bet a fraction
The Kelly criterion tells you what share of your bankroll to stake so that the bankroll grows fastest over the long run: f = (p × odds − 1) / (odds − 1). For odds of 2.50 and an estimate of p = 0.45 that is 8.33% of the bankroll. The formula only holds if p is right. With an overestimated p the suggested stake is too big, and a stake twice the Kelly size already cancels the growth, so in practice people often bet half Kelly or less.
Contents
Key terms
- Kelly criterion (Kelly formula)
- A staking rule that gives the fastest bankroll growth over the long run, when the probability estimate is correct.
- Bankroll
- The money set aside for betting. A Kelly stake is always a percentage of the current bankroll, so it changes after every bet.
- Fractional Kelly (half Kelly)
- A stake that is a fraction of the Kelly stake, for example half. It grows more slowly, but swings far less and copes better with an error in the estimate.
- Growth rate
- The average logarithmic growth of the bankroll per bet. The typical result of a long run depends on it.
The Kelly criterion formula
f = (p × odds − 1) / (odds − 1)The numerator is the expected value per unit and the denominator is the net win per unit when the bet wins. When EV is zero or negative, f is zero or less: Kelly then says do not bet.
Odds 2.50, estimate p = 0.45, bankroll 10,000.
- EV = 0.45 × 2.50 − 1 = 0.125;
- f = 0.125 / 1.50 = 0.0833, or 8.33% of the bankroll;
- stake: 833.33, and half Kelly 416.67.
After a win the bankroll is bigger, so the next stake is bigger too. After a loss the stake shrinks.
How does the stake depend on the estimate and the odds?
| Estimate p | EV | Kelly stake |
|---|---|---|
| 0.40 | 0.000 | 0.00% |
| 0.42 | +0.050 | 3.33% |
| 0.45 | +0.125 | 8.33% |
| 0.50 | +0.250 | 16.67% |
The Kelly stake rises fast with the estimate. A difference of 5 percentage points in p (0.45 or 0.50) doubles the stake.
| Odds | Estimate p | EV | Kelly stake |
|---|---|---|---|
| 1.50 | 0.700 | +0.05 | 10.00% |
| 2.00 | 0.525 | +0.05 | 5.00% |
| 3.00 | 0.350 | +0.05 | 2.50% |
| 5.00 | 0.210 | +0.05 | 1.25% |
For the same EV, Kelly gives a smaller stake at bigger odds, because that bet loses more often.
Why does hardly anyone bet full Kelly?
Kelly maximises growth only when p is right. A stake bigger than Kelly does not grow faster but slower, and double Kelly brings growth back to zero.
Odds 2.10, estimate p = 0.50 (value +5%), so the Kelly stake is 4.55%. The typical result (median) after 1,000 such bets with a starting bankroll of 10,000, if p really is 0.50:
| Stake per bet | Share of current bankroll | Median after 1,000 bets |
|---|---|---|
| 1% | 1.00% | 15,602 |
| 2% | 2.00% | 21,801 |
| Half Kelly | 2.27% | 23,433 |
| Kelly | 4.55% | 31,114 |
| Double Kelly | 9.09% | 10,000 |
| Triple Kelly | 13.64% | 327 |
Half Kelly keeps 75% of the full Kelly growth rate with half the stake. Double Kelly ends a typical run of 1,000 bets exactly where it started, and triple Kelly loses almost everything. The median is exact because the median number of wins is exactly 500.
The estimate p is always an estimate. If the true probability is lower than you think, the Kelly stake is too big and growth quickly turns into a loss, which is the main topic of the guide Flat vs Kelly.
Kelly for several bets and for a ticket
- For an accumulator or a system, p is the probability of the whole ticket and the odds are the ticket odds, so errors in the estimate are multiplied too.
- When several bets run at the same time, the sum of the Kelly stakes can get too big. The single-bet stakes should then be scaled down.
- Use the odds as they are when you place the bet, not later changes.
Limitations
- Kelly assumes p is known. The true probability is never seen, and an overestimated p gives a stake that is too big.
- The fastest long-run growth does not mean small swings: even with the right p, a bankroll on full Kelly goes through deep drawdowns.
- The examples use made-up estimates, with no tax on winnings and no stake limits.
- The median after 1,000 bets holds only if every bet really had p = 0.50. Over a short run the result can be anywhere.
Related guides
Sources
- Kelly, J. L. (1956). A New Interpretation of Information Rate. Bell System Technical Journal, 35(4), 917–926.
- Thorp, E. O. (2006). The Kelly criterion in blackjack, sports betting and the stock market. Handbook of Asset and Liability Management, Vol. 1, 385–428.