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Bet size should come from a formula, not a feeling. Given an honest probability of winning and the real payoff if you do, there is a specific fraction of your capital that grows it fastest over the long run — and betting more than that doesn't just add risk, it can make you grow slower, or go broke, even with a genuine edge.
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Given a win probability (p) and the payoff if you're right (b), the Kelly fraction f* = p − q/b tells you what share of your capital grows fastest in the long run. Bet less than that and you leave growth on the table; bet more and the extra volatility can cost you more than it earns — a positive edge is not a license to bet big.
Important: the relationship between stake size and long-run growth isn't a straight line. It rises, peaks exactly at the Kelly fraction, and then falls — past roughly twice the Kelly fraction, growth can turn negative even though the underlying bet still has a positive edge.
Most decision tools stop at "is this worth doing." This model picks up right after that, whenever the decision is repeatable and divisible: a position size, a budget split, a resourcing commitment, a traffic allocation.
Formulated by John L. Kelly Jr. at Bell Labs in 1956. The formula answers one specific question well: given a quantified edge, what fraction of capital should be staked to grow it fastest over many repeated bets.
Your best honest estimate of the probability the bet pays off. Everything downstream depends on this number being realistic, not optimistic.
The payoff ratio: how much you gain per unit staked if you win, relative to what you lose if you don't. Even-money odds make b equal to 1.
bp − q, the expected value per unit staked. A bet with zero or negative edge gets a 0% allocation from this model, however exciting it feels.
The fraction of current capital to stake, derived directly from edge and odds: f* = p − q/b. It's a growth-rate optimizer, not a confidence score.
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When you have a quantifiable probability and payoff estimate for a trade, size the position so a realistic string of losses can't knock you out of the game while still compounding the real edge over time.
Treat each channel's estimated ROI and conversion odds like a bet's edge and odds. A channel with no real edge gets the 0% Kelly answer, however appealing the pitch for it sounds.
Use an honest estimated probability of success and payoff size to size commitment to an experimental initiative, instead of going all-in on conviction or starving it on caution alone.
Translate a concentrated opportunity (a side venture, a concentrated position) into explicit probability and payoff terms before deciding what fraction of savings is reasonable to put behind it.
Frame a new hire or new team as a bet with an estimated success probability and payoff, then size the resourcing commitment instead of over-investing on excitement or under-investing on caution.
Where you have a probability-weighted estimate of a change's impact, scale exposure to match the actual edge rather than defaulting to an arbitrary fixed split.
This is an illustrative example designed to teach the method; the numbers are for demonstration only, not an actual backtest.
Expand for the model's origin, its precise formal claim, and how it reached markets.
John L. Kelly Jr., a scientist at Bell Labs, published the formula in 1956 in the Bell System Technical Journal under the title "A New Interpretation of Information Rate" (an earlier draft was reportedly titled "Information Theory and Gambling," which AT&T's executives declined to use). Kelly built directly on Claude Shannon's information theory: he showed that the optimal long-run growth rate of a bankroll in a repeated betting scenario corresponds mathematically to the capacity of a noisy information channel carrying the bettor's edge over the odds. Kelly died in 1965 at age 41, well before the formula found its later fame.
For a simple win/lose bet, f* = p − q/b (equivalently (bp − q)/b), where p is the true win probability, q = 1 − p, and b is the net odds — the profit per unit staked if you win. f* is the fraction of current capital to stake. The formula maximizes the expected logarithm of wealth across a long sequence of repeated, compounding bets. It is not a formula for the expected value of a single bet, and it assumes the probability estimate fed into it is accurate and that the stake is resized each round based on current capital.
Edward Thorp read Kelly's paper and used it to size blackjack card-counting bets, detailed in his 1962 book Beat the Dealer. He later applied the same growth-rate logic to sizing positions in securities markets through his hedge fund, Princeton-Newport Partners, which reportedly returned about 15.1% annualized net over 19 years. The jump from casino games to markets is the main thing that made the formula famous outside of information theory, even though the underlying math is unchanged.
The formula is only as good as p. A probability estimate that's even slightly too optimistic leads it to recommend a stake that's too large, and the error compounds every round.
The mathematically optimal fraction has very high real-world volatility. Most practitioners deliberately stake a fraction of it (half-Kelly, quarter-Kelly) to trade some growth rate for a far smoother ride.
Kelly answers the fraction to compound across many similar repeated bets, not how confident to feel about a single decision. For a genuinely one-off, non-repeating bet, the growth-rate framing doesn't cleanly apply.
The math implicitly assumes outcomes don't move in lockstep. A set of positions that all lose together in the same shock isn't the diversified sequence the formula is modeling.
A positive edge only means f* is greater than zero, not that f* is large. Small or noisy edges produce small Kelly fractions; the formula is a sizing brake as often as it's a green light.
Transaction costs, taxes, leverage limits, and liquidity constraints all change the practical answer. A clean formula can create false confidence in a number that ignores these costs.
there's no reliable way to estimate a probability at all — deep, Knightian uncertainty rather than quantifiable risk. Margin of Safety or The Precautionary Principle fit a situation with no honest p better than a formula that requires one.
this is a genuinely single, non-repeating, high-stakes choice rather than one of many similar bets made over time. Pre-Mortem Analysis or Second-Order Thinking suit a one-off decision better than a growth-rate optimizer built for repetition.
correlated, fat-tail events could wipe out several "independent" bets at once. Antifragility or Black Swan Events are a better starting lens than a formula that assumes well-behaved, roughly independent repetition.
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The long-run growth rate is a concave function of stake size: it rises to a peak exactly at f*, then falls, and crosses back to zero around twice f*. Past that point, the extra variance from oversized bets outweighs the edge, so the bankroll's expected long-run growth rate actually turns negative even though each individual bet still has a positive expected value. More variance, not bad luck alone, is what erodes compounded growth.
Full Kelly maximizes long-run growth rate but comes with large swings in capital along the way, and it assumes the probability and odds inputs are exactly correct. Betting a fraction of full Kelly gives up some theoretical growth rate in exchange for a much smoother path and more room for the inevitable error in the probability estimate — a trade most practitioners consider worth making.
The math was derived from a betting and information-theory context, but the underlying logic — size a repeated, divisible commitment to match a quantified edge, not to match confidence — transfers to any situation with an estimable probability, a real payoff, and the ability to repeat or scale the decision, such as marketing budgets, R&D bets, and resource allocation across a portfolio of initiatives.