Kelly Criterion Calculator

What the formula says

The Kelly criterion answers one question: given a bet you can repeat, what fraction of your capital should you stake each time so that your money compounds as fast as possible? For a bet that either wins b times the stake or loses the stake, the answer is f* = p - (1 - p) / b, where p is the probability of winning and b is the payoff ratio. The numerator of that second term is the chance of losing, so the formula subtracts your loss frequency scaled by how little you win when you win.

Kelly maximizes the expected logarithm of wealth. That matters because compounding is multiplicative: a 50% loss needs a 100% gain to undo, so the arithmetic average return of a strategy tells you less about its long run outcome than the average log return does. The drawdown recovery calculator shows that asymmetry directly.

A worked example with the default numbers

Take the defaults: a 55% win rate and a payoff ratio of 1.6, which is the same as an average win of $400 against an average loss of $250.

• f* = 0.55 - 0.45 / 1.6 = 0.55 - 0.28125 = 0.26875, or 26.88% of capital.
• On a $25,000 account, full Kelly stakes $6,718.75 on every trade.
• Half Kelly is 13.44% of capital, or $3,359.38. Quarter Kelly is 6.72%, or $1,679.69.
• Expected value per dollar risked is 0.55 x 1.6 - 0.45 = $0.43.

The growth rate follows from g = p ln(1 + fb) + (1 - p) ln(1 - f). At full Kelly that gives 0.55 ln(1.43) + 0.45 ln(0.73125) = 0.0559, about 5.59% of log growth per trade. At half Kelly it gives 4.22%. Halving the stake costs you a quarter of the growth rate and removes half of the volatility, which is the trade every professional sizing framework is built around.

Why traders use a fraction of Kelly

A 26.88% stake sounds absurd next to the 1% rule taught in risk management for traders, and the two numbers measure different things. Kelly stakes the amount you can lose entirely, which in a stock trade is the distance from your entry to your stop rather than the full position value. A 27% Kelly fraction on a trade where the stop is 8% below entry is a position worth about three times the account, which no cash account can hold.

The deeper problem is estimation. Kelly is sharply asymmetric around its peak: betting half the optimal amount costs you a quarter of your growth, while betting twice the optimal amount costs you all of it. Since a win rate estimated from a few dozen trades carries a wide error band, and since traders tend to remember their good trades, the fraction you calculate is usually too big. Betting a quarter or a half of it moves you to the safe side of the curve where the cost of being wrong is small.

How to use the output

Read the quarter Kelly line as an upper bound rather than a target. If your log of real trades says quarter Kelly is 6.7% of equity and your written plan risks 1%, the plan is conservative and you can hold it with confidence. If the plan risks more than quarter Kelly, you are betting past the point where your own numbers support the size. The derivation and the portfolio version are covered in the Kelly criterion and position sizing, and the risk measures that tell you whether the edge is real live in measuring portfolio risk.

Before you raise size, check the quality of the return stream with the Sharpe ratio calculator and understand what a bad stretch does to the account. Sizing rules fail at the moment they are least comfortable to follow, which is the subject of trading psychology and discipline.