In 1956, John L. Kelly Jr., a physicist at Bell Labs working on information theory, published a paper deriving the optimal fraction of a bankroll to bet on a series of favourable wagers. The formula — later refined and popularised by others — provides a specific answer to the position-sizing question, given specific assumptions. Understanding both the formula and the assumptions is essential to using it, and even more essential to knowing when not to.
The formula
For a bet with a probability p of a gain of amount b, and probability (1-p) of losing 1 unit, the Kelly criterion says the optimal bet fraction is:
f = (bp - (1-p)) / b
Simpler in words: bet size equals your edge divided by the odds. If a bet offers 2-to-1 odds (b=2) and you estimate a 60% chance of winning (p=0.6), the Kelly formula says to bet 40% of your bankroll. If you estimate a 55% chance of winning at the same odds, it says to bet 32.5%. If you estimate a 50% chance, it says to bet zero.
The property that makes this mathematically interesting is that Kelly-sized bets, played over an infinite sequence of independent bets with correctly-estimated probabilities, produce the maximum expected long-term growth rate of the bankroll. No other bet size does better in expectation. Larger bets have a higher chance of ruin; smaller bets grow slower.
Why almost every practitioner runs half-Kelly or less
Three well-documented reasons.
The volatility is enormous. A Kelly-sized bankroll following the formula exactly experiences drawdowns of 50% or more with high frequency. In gambling contexts these drawdowns are theoretically recoverable. In investing contexts, they are practically catastrophic — either because clients withdraw, careers end, or the emotional cost of the drawdown alters decision-making in ways the formula does not model.
The probabilities are almost never known. Kelly assumes you know p with certainty. In real markets, p is estimated from a small sample and carries substantial estimation error. If your estimate of p is off by even a small amount, the optimal bet size collapses much faster than the estimate of expected return declines. This is called "estimation error asymmetry" and it is the single strongest argument for running Kelly at a fraction.
The bets are correlated. Kelly's derivation assumes each bet is independent of the others. In an investment portfolio with multiple positions, this is nearly never true — correlations spike during stress, and the effective number of independent bets shrinks precisely when it matters most. A Kelly-sized portfolio built on assumed independence is over-levered whenever the assumption breaks.
The half-Kelly convention
Most quantitative practitioners who use Kelly-style sizing run one-half Kelly or less. The reason is not conservatism for its own sake; it is that half-Kelly delivers roughly three-quarters of the growth rate of full Kelly with roughly one-quarter of the volatility. The trade-off is favourable in almost every real-world portfolio context.
At one-quarter Kelly — the fraction some hedge funds use — the growth rate is about half of full Kelly, but the drawdown properties become manageable enough for institutional risk management.
What Kelly is useful for even if you don't use it directly
The formula's greatest value in real investment work is not as a bet-sizing rule but as a discipline for making the position-sizing question quantitative at all. Most investors size positions by intuition — "I'll put 5% in, that feels right." Kelly forces the underlying question: what is my estimated edge, and how confident am I in the estimate?
If you cannot articulate the edge, Kelly says the bet should be zero. If your edge is small — say a 52% probability of the good outcome at 1-to-1 odds — Kelly says 4%, not 20%. If your edge is large but your confidence in the estimate is low, running half-Kelly on the formula's output is still likely too aggressive.
The formula's most useful diagnostic property is that it collapses to zero at the point where your estimated edge falls below breakeven. If you cannot articulate a positive edge with defensible reasoning, no position size is optimal — the correct action is not to take the position.
Why the formula fails silently in equity portfolios
In a single-bet gambling context, Kelly's failure modes are visible — you have one number for edge, one for odds, and both are usually knowable from the game's rules. In a multi-position equity portfolio, the failure modes are subtle. Correlations shift over time. Edges are estimated, not known. The bet doesn't resolve on a fixed schedule; it continues, with the "probability" and "payoff" both changing continuously.
The rigorous application of Kelly to real portfolios is a specialist quantitative task. The intuitive application — "size positions in proportion to conviction, divided by the volatility of the outcome" — captures the useful spirit without pretending to the precision the formula suggests.
The rule to internalise
Kelly answers a specific question: what is the mathematically optimal bet size for an infinite sequence of independent bets with known probabilities? Almost no real investment situation satisfies those conditions. The formula's practical value is not the number it produces but the questions it forces you to ask: what is my edge, how confident am I in the estimate, and what fraction of my capital does the answer justify? A discipline of running well below what Kelly says is not a failure to apply the formula; it is the correct response to the fact that the formula's assumptions are almost never satisfied.
Educational content only. Not investment advice.