Worked example
Win rate 55 %, win/loss ratio 2: full Kelly 32.5 %, half Kelly 16.25 %, quarter Kelly 8.125 %. Expected log growth per trade: 9.86 % at full Kelly, 7.50 % at half Kelly. Continuous form with 8 % expected return, 3 % risk-free rate and 20 % volatility: 125 %, which means leverage.
Formula
- Binary outcome, general form (Kelly 1956): f = p / l - q / g, where p is the win probability, q = 1 - p, g is the fraction of the stake gained on a win and l the fraction lost on a loss.
- Trading form with a full loss of the amount risked (l = 1) and payoff ratio R = average win / average loss: f = p - q / R.
- Expected log growth per bet at fraction f: G(f) = p ln(1 + g f) + q ln(1 - l f). Betting twice the Kelly fraction brings growth back to about zero.
- Continuous approximation for an investment: f = (mu - r) / sigma^2, with mu the expected return, r the risk-free rate and sigma the volatility, all annual. A result above 100 % implies leverage.
Limits
- p and R are estimates. A small error in the win rate moves the Kelly fraction a lot, and an overestimated edge leads to over-betting, which lowers growth and deepens drawdowns.
- The formula assumes independent repeated bets with known odds. Trades are often correlated, fat tailed and affected by costs and slippage.
- Fractional Kelly (half or quarter) reduces volatility and the chance of ruin at the cost of slower growth.
- The continuous form assumes a symmetric return distribution.
Questions
What is a good Kelly fraction?
There is no universal value. Full Kelly maximises long-run log growth only if the inputs are exact. Many traders use half or quarter Kelly to absorb estimation error.
What if the Kelly result is negative?
A zero or negative result means the inputs show no edge: the criterion says not to take the bet.
Is Kelly the amount to risk or the position size?
In the trading form it is the fraction of capital put at risk on a trade that loses the full amount risked at the stop. Convert it into units with the position size calculator.