The formula
For a simple bet with two outcomes, the Kelly fraction (f*) is: f* = (bp − q) ÷ b, where b is the net odds received (profit per £1 bet on a win), p is the probability of winning, and q = 1 − p is the probability of losing. In investment terms, this becomes f* = (expected return) ÷ (variance of returns), or approximately the Sharpe ratio divided by the volatility of returns. The formula tells you the fraction of wealth to invest to maximise the compound growth rate.
A worked example
You have a bet that wins £2 for every £1 risked, with a 55% win probability. Here b = 2, p = 0.55, q = 0.45. Full Kelly = (2 × 0.55 − 0.45) ÷ 2 = (1.10 − 0.45) ÷ 2 = 0.65 ÷ 2 = 32.5% of capital. This is the fraction that maximises the expected log of wealth (i.e., the compound growth rate). Bet more than this, and your long-run growth rate actually decreases — the drawdowns become so severe that compound growth suffers even with positive expected value.
Why full Kelly is dangerous in practice
Full Kelly produces maximum long-run growth only if: the edge is known with certainty; the probability distribution of outcomes is fully known; and you can sustain large drawdowns psychologically. In practice, all three conditions fail. Edge estimation error — overstating your win probability — causes Kelly to systematically overbetting. A 50% Kelly drawdown (which occurs regularly even in favourable situations) would likely cause most investors to liquidate, abandoning the strategy at the worst moment.
The Kelly framework in portfolio management
Berkshire Hathaway's concentrated portfolio structure is sometimes described as approximately "Kelly-optimal" — Buffett and Munger's willingness to hold large positions reflects their high confidence in edge and long time horizon. Most institutional investors implicitly use a fractional Kelly approach: diversifying risk across many positions (reducing effective bet size per position), while maintaining meaningful sizing in their highest-conviction ideas. The insight is the same: position size should be proportional to edge, not arbitrary.
"Bet too little and you leave compounding on the table. Bet too much and the variance will bankrupt you. Kelly tells you where the optimum is — and the answer is usually: smaller than you think." — An edge in portfolio management
What this means for you
The Kelly Criterion's most important message is not the formula itself but the principle: the optimal bet size is a function of your edge and your uncertainty. Sizing positions based on conviction alone, without considering the variance of outcomes or the probability of being wrong, leads to systematically poor risk-adjusted returns. Whether you use full Kelly, half-Kelly, or a simpler volatility-based sizing approach, the discipline of explicit position sizing based on edge estimation is a hallmark of sophisticated portfolio management.