The Kelly Criterion: Optimal Bet Sizing and Why Overbetting Destroys Growth

Growth Rate at Different Bet Sizes

The expected logarithmic growth per bet is g(f) = p ln(1 + bf) + q ln(1 − f). For p = 0.55 and b = 1: at f = 10%, g = 0.50% per bet; at f = 5%, g = 0.375%; at f = 20%, g ≈ −0.01% (essentially zero); at f = 30%, g = −1.6% per bet, a long-run loss even though each bet has a positive expected value. Betting twice the Kelly fraction yields no long-term growth, and betting more than twice Kelly produces a decline toward zero. Underbetting is much less costly than overbetting: half-Kelly (5%) captures 75% of the maximum growth with a much lower variance.

The reason is the asymmetry of percentage changes: a 30% loss requires a 43% gain to recover.

Other Examples

With a 60% chance of winning at even money: f* = (0.6 − 0.4) = 20%. With a 55% win probability on a bet that pays 1.5-to-1: f* = (1.5 × 0.55 − 0.45) ÷ 1.5 = 25%. With a 40% chance on a bet paying 2-to-1: f* = (2 × 0.4 − 0.6) ÷ 2 = 10%. If the edge is zero or negative (bp ≤ q), Kelly says to bet nothing. For an investment with an expected excess return μ and variance σ², the continuous version is f* = μ ÷ σ²: for a stock with a 6% expected excess return and 20% volatility, f* = 0.06 ÷ 0.04 = 1.5, or 150% of your wealth in the stock (leveraged), which most people would find too aggressive.

Practical Use and Limits

Kelly assumes you know the true probabilities and odds. In real markets and games, estimates are uncertain, and overestimating your edge leads to overbetting. Most practitioners use a fraction of Kelly, such as half or a quarter, to reduce volatility and protect against estimation error. Kelly bets can lead to large drawdowns even when it is optimal: a full-Kelly bettor can expect to lose half of the bankroll at some point with a decent probability. The criterion also ignores consumption needs, taxes, and utility. For most people, the main takeaway is qualitative: bet size must be proportional to edge, never bet more than twice Kelly, and with no edge, do not bet.

Connection to Investing

The Kelly logic underlies advice on position sizing and leverage. It explains why diversified portfolios of many small positions compound better than a few big concentrated bets, and why high-volatility leveraged strategies can have negative long-run growth even when average returns are positive. This article is educational and does not recommend gambling or any investment strategy.