Where Kelly Comes From
John Kelly developed his criterion in 1956 while working at Bell Labs on information theory problems. The original question was about optimizing data transmission over noisy channels, but the math translates directly to gambling and investing. The Kelly Criterion answers a specific question: given a bet with known probability and payoff, what fraction of your bankroll should you wager to maximize the long-term growth rate of your capital?
The formula for a simple binary bet is: f* = (bp - q) / b, where f* is the optimal fraction of your capital to bet, b is the net odds received on the bet (profit per dollar risked), p is the probability of winning, and q is the probability of losing (1 - p).
A Concrete Example
Suppose you have a trading strategy that wins 55% of the time and, when it wins, you make $1.50 for every $1 risked. When it loses, you lose the full $1. Here, p = 0.55, q = 0.45, and b = 1.5.
Kelly fraction = (1.5 * 0.55 - 0.45) / 1.5 = (0.825 - 0.45) / 1.5 = 0.375 / 1.5 = 0.25
The Kelly Criterion says you should risk 25% of your capital on each trade. This seems aggressive, and it is. Full Kelly sizing maximizes the long-run geometric growth rate of your portfolio, but it comes with substantial drawdowns along the way. In practice, most professional traders and fund managers use fractional Kelly, typically half-Kelly or quarter-Kelly, to reduce the volatility of the equity curve at the cost of slightly lower long-run growth.
Why Geometric Growth Matters
Kelly optimizes for the geometric mean return, not the arithmetic mean. This distinction matters because investment returns compound multiplicatively, not additively. If you gain 50% and then lose 50%, you are not break-even, you are down 25%. The geometric mean captures this compounding reality.
Overbetting relative to Kelly, risking a larger fraction than the formula suggests, actually reduces your long-run growth rate even though it increases your expected return on any single bet. This is counterintuitive but mathematical. Betting too large introduces so much variance that the compounding penalty from bad sequences of trades overwhelms the higher expected return. At exactly 2x Kelly, your expected long-run geometric growth rate drops to zero, meaning you will almost certainly go broke given enough time.
Estimating Inputs for Trading
The hard part of applying Kelly to trading is estimating the inputs. For a coin flip, p and b are known. For a trading strategy, you need to estimate your win rate and your payoff ratio, and those estimates carry uncertainty.
You can estimate these from backtest results, but backtests are subject to the overfitting and data issues discussed elsewhere. A more conservative approach is to estimate your edge (the expected return per trade) and then apply Kelly to that estimate with a discount for uncertainty. If your backtest suggests a 60% win rate with a 1.2:1 payoff ratio, but you think the live performance will be worse, you might use 52% and 1.1:1 as more conservative inputs.
For trading strategies with variable position sizes and non-binary outcomes (you might win 2%, 5%, or 15% on different trades), the generalized Kelly Criterion accounts for the full distribution of outcomes. The formula becomes more complex, but the principle is the same: find the fraction that maximizes E[log(1 + f*r)], where r is the random return of your strategy.
Kelly for Multiple Simultaneous Positions
When you have multiple simultaneous trades, Kelly extends to a portfolio context. The optimal allocation depends not just on each individual trade's edge but also on the correlations between trades. Two highly correlated positions should be treated almost as a single position, while uncorrelated positions allow for larger aggregate sizing because the diversification benefit reduces portfolio variance.
In practice, most crypto traders deal with highly correlated assets. BTC and ETH have a correlation above 0.8 most of the time. Running full Kelly on a BTC long and simultaneously full Kelly on an ETH long is effectively doubling your bet on the same underlying factor, which puts you well above the Kelly-optimal allocation for that factor.
Practical Implementation
A reasonable implementation for most traders: estimate your strategy's edge conservatively, calculate the full Kelly fraction, and then use 25-50% of that number as your actual position size. This fractional Kelly approach sacrifices some theoretical growth in exchange for a much smoother equity curve and more margin for error in your edge estimates.
Track your actual win rate and payoff ratio in live trading and update your Kelly sizing periodically. If your live results differ significantly from your estimates, adjust. Kelly sizing is only as good as the inputs you feed it, but even an approximate application is better than the arbitrary position sizing (1% risk per trade, or whatever round number feels right) that most traders default to.