The binary Kelly formula and its units
Let p be win probability, q = 1 - p, and b the net winning profit per dollar risked. A win multiplies wealth by 1 + fb; a loss multiplies it by 1 - f. For this model, maximizing expected log growth gives:
G(f) = p ln(1 + fb) + (1 - p) ln(1 - f)
f* = p - (1 - p) / b
Risk fraction = max(0, f*)
Fractional risk = chosen multiplier × max(0, f*)
The feasible choice here is to place the specified bet or hold cash, without borrowing. A negative raw Kelly value therefore produces zero risk. The break-even probability is 1 / (1 + b): at b = 1.5 it is 40%. Positive historical expectancy alone does not establish a future edge.
The binary solution and its cash alternative follow the two-outcome derivation in Busseti, Ryu and Boyd, Risk-Constrained Kelly Gambling, section 2.3.
A calculation you can reproduce
Enter a 55% win rate, a 1.5 win/loss ratio and a $10,000 bankroll. The arithmetic is 0.55 - 0.45 / 1.5 = 0.25. Full Kelly risks 25%, or $2,500; a 50% Kelly multiplier risks 12.5%, or $1,250. These are illustrations under the stated model, not proposed account risk limits.
| Multiplier | Account at risk | Risk dollars | Expected log growth G |
|---|---|---|---|
| 25% Kelly | 6.25% | $625 | 0.02024 |
| 50% Kelly | 12.50% | $1,250 | 0.03443 |
| 100% Kelly | 25.00% | $2,500 | 0.04569 |
| 200% Kelly | 50.00% | $5,000 | -0.00413 |
Each growth value uses the formula above, with natural logarithms, per bet. Double Kelly is not universally zero growth: in this example it is negative. Half Kelly is about 75.3% of full Kelly log growth here, not an exact 75% rule. Ten consecutive modeled losses leave 0.75^10 = 5.63% of starting capital at full Kelly, versus 0.875^10 = 26.31% at half Kelly. Fractional sizing still permits severe losses.
Estimation error changes the sizing decision
Keep b = 1.5 and lower the estimated win rate from 55% to 45%: full Kelly falls from 25% to 8.33%. At 40% it is zero; at 35% raw Kelly is -8.33%, so the feasible amount remains zero. A 10 percentage point probability error changes full Kelly by 16.67 percentage points because its slope with respect to p is 1 + 1/b.
If the true win probability were 45%, risking the earlier 25% estimate would give G = 0.45 ln(1.375) + 0.55 ln(0.75) = -0.01492 per bet. A seemingly favorable input can therefore produce a losing growth rate when the estimate is wrong. Reducing the fraction limits exposure but does not prove the edge exists.
- Use trades from a consistent rule set and period. Include commissions, spreads, financing and slippage in the payoff estimates.
- Check lower win-rate and payoff scenarios, then compare on held-out trades without retuning to that test set.
- Record gaps and unusually large losses. Do not treat average loss as a guaranteed maximum loss.
- Review overlapping positions together. Separate Kelly calculations do not account for trades that lose at the same time.
Variable returns need a different model
Advanced mode still compresses trades into two outcomes using average win divided by average loss. Real returns may have many outcomes or a continuous distribution. The general objective is to maximize E[ln(1 + fR)] over feasible f, where R is the net return on the chosen exposure. Replacing outcomes by their averages generally changes that objective.
For example, returns of +10% or -5% with equal probability have a 2:1 payoff ratio. Binary risk Kelly is 25%. Converting that risk into exposure gives 0.25 / 0.05 = 5 times capital, which requires leverage. An unlevered allocation must remain within its own constraints. This calculator does not solve those constraints or a correlated portfolio.
Risk-Constrained Kelly Gambling, sections 2 and 3 formulates the broader return-distribution problem and distinguishes growth maximization from drawdown control. Fractional Kelly scales exposure; it does not impose a specified maximum drawdown.
Read the calculator output within its limits
Basic mode requires a positive payoff ratio. Advanced mode requires positive average win and loss amounts. Bankroll must be positive, win rate must be between 0% and 100%, and the Kelly multiplier must be between 0% and 100%. Empty, nonfinite or out-of-range inputs produce an error instead of a result.
At 0% win probability the model chooses zero. At 100% it chooses all capital for a positive payoff, because it assumes losses are impossible. A streak of wins cannot establish that assumption. The Kelly curve excludes a 100% stake because any nonzero loss probability makes expected log growth negative infinite there.
The growth tab shows a deterministic log-growth illustration, W₀ exp(nG), with constant inputs. It is not expected dollar wealth, a backtest or a sampled equity path. It cannot display actual drawdowns. The doubling figure is ln(2) / G, not an expected waiting time or deadline.
For the next calculation, use the position size calculator to convert a separately chosen risk budget and stop distance into units. A stop order does not guarantee its execution price. For methodology, return to TradingView backtesting guides.
Source checked October 17, 2026. Numerical examples are direct calculations under the stated binary model. Educational information only, not investment advice. Historical results and model outputs do not guarantee future returns.