Every investor eventually faces the same question: once you have found an opportunity you believe in, how much money should you put behind it? Too little, and a good idea barely moves your portfolio. Too much, and a single bad outcome can set you back years. In 1956, a scientist at Bell Labs published a formula that gave a precise mathematical answer. Today, the Kelly criterion is used by hedge funds, professional poker players and sports bettors alike. Here is how it works, why the professionals rarely use it in full, and what it teaches anyone who puts money at risk.
A formula born in a telephone laboratory
John L. Kelly Jr. was a researcher at Bell Labs working on information theory, the science of how signals are transmitted over noisy channels. In 1956 he published a paper in the Bell System Technical Journal that applied those ideas to a gambler receiving information about the outcome of a race. His conclusion was a simple rule for how much of a bankroll to stake in order to grow it as fast as possible over the long run.
The idea did not stay in the laboratory for long. The mathematician Edward Thorp used it in the early 1960s to size his bets at the blackjack table, and later applied the same thinking to the stock market as a pioneering quantitative investor. Since then, Kelly’s formula has become a standard reference in discussions of position sizing, from options trading to sports betting.
How the Kelly formula works
At its core, the Kelly criterion compares two things: the probability that you will win and the odds you are being offered. In its simplest form for a bet with decimal odds, the formula reads: stake fraction = (b × p − q) / b, where b is the decimal odds minus one, p is your estimated probability of winning, and q is the probability of losing, which is 1 − p.
Take a football match where a team is offered at decimal odds of 2.50. Those odds imply a 40 percent chance of winning. Now suppose your own analysis suggests the true probability is 45 percent. Plug the numbers in, and Kelly says you should stake 8.3 percent of your bankroll. If you believed the chance was 50 percent, the formula would recommend 16.7 percent. If you agreed with the market at 40 percent, it would recommend nothing at all, because there is no edge to exploit.
Side by side, the three scenarios look like this:
- You estimate 40 percent, the same as the odds imply: expected return £0.00 per £1 staked, and Kelly recommends a stake of 0 percent.
- You estimate 45 percent: expected return £0.125 per £1 staked, and Kelly recommends 8.3 percent of your bankroll.
- You estimate 50 percent: expected return £0.25 per £1 staked, and Kelly recommends 16.7 percent of your bankroll.
No edge, no bet: the link to value
The most important lesson of Kelly is hidden in the first of those scenarios. The formula only recommends a stake when your estimated probability is higher than the probability implied by the price. In investing, that is the difference between buying a stock you think is undervalued and simply buying the market. In betting, it is known as value: odds that are higher than they should be, given the real chance of the outcome. As the value betting guide on FootyPulse.org explains, a bet is only worth considering when the price is better than the probability behind it.
Without value, Kelly tells you to stay out. That is a powerful discipline, because most bets offered by bookmakers include a margin that makes the implied probabilities add up to more than 100 percent. In other words, the default position for most bets is negative expected value. Kelly does not create an edge. It only tells you how to size one once you genuinely have it.
Why professionals use fractional Kelly
If Kelly maximizes long-term growth, why do so few professionals bet the full amount? The answer lies in uncertainty. The formula assumes you know the true probability exactly, but in practice every estimate comes with error. Overestimate your edge, and full Kelly quickly turns into overbetting, with large swings in your bankroll and a real risk of deep losses.
The maths is striking. In the example above, betting half the Kelly stake still captures around 75 percent of the long-term growth rate, while sharply reducing volatility. Betting double the Kelly stake pushes long-term growth to almost zero, and betting three times Kelly turns it negative, meaning the bankroll is expected to shrink despite a genuine edge. That asymmetry is why many investors and bettors use half Kelly or less. Being too cautious costs a little growth. Being too aggressive can cost everything.
Kelly in the investment world
In portfolio management, the Kelly criterion appears in discussions of how much capital to allocate to a single position or strategy. The principle is the same: size your exposure according to your edge and your confidence in it, not according to how exciting the opportunity feels. Concentrated bets on high-conviction ideas and smaller positions on uncertain ones follow the same intuition.
Critics point out that markets are rarely as clean as a coin flip or a football match. Returns are not binary, probabilities shift constantly and correlations between positions matter. For that reason, Kelly is more often used as a guiding principle than as a precise instruction. Its real value is the mindset it encourages: an honest estimate of your edge, and discipline about how much you risk on any single idea.
Putting it into practice as a football fan
For sports fans, the first step is the hardest: forming an independent estimate of probability that is better than the bookmaker’s. That requires data, from form and home and away records to expected goals and head-to-head trends. The good news is that much of that information is now freely available, along with tools that convert odds into implied probabilities and calculate expected value.
The free football statistics site FootyPulse.org also publishes data-driven predictions and calculators that make the arithmetic behind Kelly easy to check. Even if you never place a bet, running the numbers on a few matches is a useful exercise. It shows how often the market’s price is close to fair, and how rare genuine value really is.
Frequently asked questions
What is the Kelly criterion in simple terms?
The Kelly criterion is a formula that tells you what fraction of your bankroll to stake on an opportunity, based on your estimated chance of winning and the odds or return on offer. It aims to maximize long-term growth. If you have no edge, it recommends staking nothing. The bigger your edge, the larger the recommended stake, although most professionals bet only a fraction of it.
Why is half Kelly so popular?
Because probability estimates are never perfect. If you overestimate your edge, full Kelly leads to overbetting and large swings. Betting half the Kelly amount captures around three quarters of the long-term growth while cutting volatility substantially. Many investors and bettors see that trade-off as worth it, since the downside of betting too much is far more severe than the downside of betting too little.
Can the Kelly criterion make betting profitable?
No formula can create an edge. Kelly only tells you how to size a bet once you genuinely have one, and bookmakers build a margin into their odds that makes most bets negative in expected value. If your probability estimates are no better than the market’s, Kelly will recommend staking nothing. Free tools such as the EV calculator on FootyPulse.org make it easy to check whether a price offers any value at all. Betting should be treated as entertainment, not as an investment strategy.
Gambling should be entertainment, never a way to make money. You must be 18 or over to gamble in the UK. If you need support, contact the National Gambling Helpline on 0808 8020 133 or visit BeGambleAware.org.
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