The Lab

Where it comes from

• John L. Kelly Jr., a scientist at Bell Labs, published the formula in 1956, in a paper called "A New Interpretation of Information Rate". He framed it as a gambler who gets tips over a noisy private wire, and showed that the best rate at which the gambler's money can grow equals the rate at which the wire carries information. It was built on Claude Shannon's information theory. Shannon was a colleague at Bell Labs.
• The idea underneath is older. In 1738 Daniel Bernoulli argued that people value money by its logarithm: each doubling is worth the same. Kelly's rule is what you get if you aim to make the logarithm of your money grow as fast as possible.
• Others took it further. Henry Latané (1959) proposed the same aim for investing. Leo Breiman (1961) proved that, over the long run, Kelly betting beats any essentially different strategy.
• Edward Thorp used it in blackjack, in Beat the Dealer (1962), and later to size bets in his investment fund.

The maths, step by step

1. A simple bet. A coin lands your way 60% of the time, and pays even money. Kelly's stake is the chance of winning minus the chance of losing:

0.6 − 0.4 = 20% of the pot on each flip.

For other odds, the rule is p − q ÷ b, where b is what a win pays per 1 staked. At 2 to 1 on a 40% chance: 0.4 − 0.6 ÷ 2 = 10%.

2. Why growth, not the average. A win multiplies the pot by 1.2 and a loss by 0.8. What matters over many bets is the typical multiplier per bet:

1.2^0.6 × 0.8^0.4 = 1.0203, about +2.03% a bet.

3. Betting different amounts on the same coin:

• At double Kelly the same winning coin loses money over time.
• Half Kelly keeps 75% of the growth, for much less risk.
• Betting more than Kelly is never better: more risk and less growth.

4. The same idea for a coin you hold. In the usual approximation, the growth a year from holding a share f of your money in a coin is about

f × edge − f² × swings² ÷ 2,

and Kelly's share is edge ÷ swings².

• Example (a made-up coin): it beats cash by 20% a year on average and swings about 60% a year. Kelly's share: 0.2 ÷ 0.6² = 56% of the money. That is less than holding the coin outright, with no leverage at all.
• Holding it 1×: 0.2 − 0.36 ÷ 2 = +0.02 a year (log growth).
• At 3× leverage: 0.6 − 9 × 0.36 ÷ 2 = −1.02 a year. The same winning coin wipes you out.

5. When you guess the edge wrong. Say you think the edge is 20% and hold 56%, but the real edge is 10%. Then

0.556 × 0.1 − 0.556² × 0.36 ÷ 2 = 0.

You took the risk and got no growth. Guessing your edge twice too high puts you at double Kelly.

6. The ride is rough even at Kelly. Under the same approximation, the chance your money ever falls to half its starting value is:

• 50% at full Kelly;
• 12.5% at half Kelly.

What others found

This section is other people's work, not ours.

• The long-run result is proven. Breiman (1961) showed Kelly betting grows fastest in the long run, and reaches a far-off goal soonest.
• It isn't right for everyone. Paul Samuelson (1979) argued that growing as fast as possible isn't the right aim for every investor, even over long horizons. He wrote the paper in words of one syllable to make his point.
• Practitioners bet a fraction of Kelly. The edge is never known, only estimated, and step 5 shows that overestimating it is costly. That is why half Kelly or less is the common practice. Leonard MacLean, Edward Thorp and William Ziemba collected the main papers in one book (2011).

What we tested and found

This section is ours. We have not tested Kelly sizing itself. Our tests bear on the part of it that matters most to us: what too much size does.

• Our rule is no leverage. The maths in step 4 is one reason: for a coin swinging 60% a year, Kelly's share only passes 1× if the coin beats cash by more than 36% a year (0.6² = 0.36), every year, on average.
• Leverage on our buying ladders. We ran our dip-buying ladder at 2×, 3×, 5× and 10×. In a falling market the best settings picked on the first half were ruined in the second half on 45% to 83% of coins and draws, and no setting at 2× or more avoided ruin. On plain spot, 50 of 240 settings did.
• More leverage, more lopsided returns. In the same study, in a falling market, the share of runs with rare large losses rose from 11% on spot to 86% at 10×. Those returns look good right up to the big loss.
• High leverage on both sides. In Bitcoin's drop and bounce after one US inflation report, the liquidation prices of trades at 50× and 100× were reached on both sides within hours. Shorts at 25× or less, opened at the low, were not.

These agree with what the maths says about overbetting. They don't test Kelly's formula itself.

Credits

• John L. Kelly Jr., "A New Interpretation of Information Rate", Bell System Technical Journal, 1956.
• Daniel Bernoulli, "Specimen theoriae novae de mensura sortis", 1738, translated as "Exposition of a New Theory on the Measurement of Risk", Econometrica, 1954.
• Henry A. Latané, "Criteria for Choice Among Risky Ventures", Journal of Political Economy, 1959.
• Leo Breiman, "Optimal Gambling Systems for Favorable Games", Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961.
• Edward O. Thorp, Beat the Dealer, 1962.
• Paul A. Samuelson, "Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long", Journal of Banking and Finance, 1979.
• Leonard C. MacLean, Edward O. Thorp and William T. Ziemba (editors), The Kelly Capital Growth Investment Criterion: Theory and Practice, World Scientific, 2011.
• Claude E. Shannon, "A Mathematical Theory of Communication", Bell System Technical Journal, 1948.
• Our tests: "Buying the dip in steps: three tests of our DCA ladder", "How far to trust a Sharpe ratio", and "Bitcoin after the PPI flush: where the liquidations sit".
• PerpGremlin and Claude, The Kelly formula: how much to bet when you have an edge, TerminalGremlin, 2026: the tests, the code and the words.