A positive expectancy does not specify how much to risk. The same payoff distribution can produce steady participation, intolerable drawdown, or practical ruin depending on exposure. Position sizing is not a second edge; it is the constraint that keeps the first edge available.
Arithmetic expectation ignores capital path
Return to the binary model: a 30% chance of +3R and a 70% chance of −1R. Expected R is +0.20 per trade. If one R is always a fixed $100, expected dollars are +$20 per trade.
Fixed-dollar risk makes outcomes additive. It also causes risk as a percentage of equity to rise during a drawdown and fall as the account grows. Fixed-fraction risk instead recalculates one R as a percentage f of current equity. Outcomes then multiply:
Losing equity multiplier = 1 − f
Compounding makes logarithmic growth relevant
For repeated multiplicative outcomes, expected log growth is:
Logarithms turn the product of sequential wealth multipliers into a sum. Maximizing expected log wealth balances using the positive edge against the increasing damage from large percentage losses.
| Fraction f | Expected log growth / trade | After 10 straight losses | Recovery required |
|---|---|---|---|
| 0.5% | about 0.096% | −4.9% | +5.3% |
| 1.0% | about 0.183% | −9.6% | +10.6% |
| 2.0% | about 0.334% | −18.3% | +22.4% |
| 3.33% | about 0.487% | −28.8% | +40.4% |
| 6.67% | about 0.640% | −49.8% | +99.2% |
| 10.0% | about 0.496% | −65.1% | +186.8% |
Values assume exact independent +3R / −1R outcomes and no costs, gaps, position limits, or parameter error.
The clean model's Kelly fraction
For win W, loss L, and probabilities p and q = 1 − p, differentiate expected log growth and solve:
For p = 0.30, q = 0.70, W = 3, and L = 1:
In the perfectly known binary model, 6.67% maximizes asymptotic expected log growth. It also loses nearly half of equity during ten consecutive losses, a run that has roughly a 58% chance of appearing somewhere in 100 independent trades at this hit rate.
Why fractional Kelly is discussed
Half Kelly in the clean model is about 3.33%; quarter Kelly is about 1.67%. Reducing the fraction gives up some theoretical growth in exchange for materially lower path volatility. Around the mathematical optimum, expected growth falls more slowly than drawdown discomfort does.
Fractional Kelly also offers a limited buffer against estimation error. Limited is the important word. If the true win rate is 25% rather than 30% while W remains 3R, the edge is zero and every positive fraction has negative expected log growth after costs. A fraction of a nonexistent edge is still a bad bet.
Sizing must reflect uncertainty in the inputs
The Kelly fraction reacts sharply to small changes near breakeven:
| Win rate | Average win | Average loss | Expectancy | Model Kelly f* |
|---|---|---|---|---|
| 32% | 3R | 1R | +0.28R | 9.33% |
| 30% | 3R | 1R | +0.20R | 6.67% |
| 28% | 3R | 1R | +0.12R | 4.00% |
| 26% | 3R | 1R | +0.04R | 1.33% |
| 25% | 3R | 1R | 0.00R | 0.00% |
A win-rate estimate from a noisy, selected backtest does not justify treating 6.67% as known. Conservative work uses lower-bound assumptions, portfolio constraints, and stress losses beyond the ordinary stop.
One percent on ten trades may be one ten-percent bet
Per-trade sizing ignores portfolio dependence. Ten long equity positions triggered by the same market regime can all fail together. Their names differ; their underlying risk factor may not.
Portfolio sizing should account for:
- Open risk: the sum of plausible losses across current positions.
- Correlation: how exposures behave during stress, not only in calm samples.
- Gap scenarios: losses beyond stops after overnight or event moves.
- Liquidity: whether intended size can exit without moving the market.
- Leverage and margin: forced liquidation can occur before economic ruin.
- Concentration: shared sector, direction, volatility, and macro factors.
Mathematical nonzero is not practical survival
Under ideal fixed-fraction sizing with losses strictly bounded below 100%, equity approaches zero without literally reaching it. That observation is not useful comfort. Practical ruin occurs when capital falls below minimum position or margin requirements, when costs dominate smaller positions, or when the process is abandoned.
Risk capacity therefore includes a behavioral and operational boundary. If a 25% drawdown will force the strategy to stop, the relevant question is not whether 25% is survivable in theory. It is whether the chosen fraction makes that drawdown sufficiently remote under conservative assumptions.
A sizing framework without false precision
- Estimate net R outcomes with costs, slippage, and missed trades included.
- Use out-of-sample and nearby-parameter results to create conservative inputs.
- Stress win rate, average win, average loss, correlation, and gap size together.
- Calculate drawdowns from ordinary streaks and clustered portfolio losses.
- Set hard portfolio-level exposure and leverage limits.
- Choose a fraction that remains executable beyond the modeled bad path.
- Reduce exposure when uncertainty rises; never increase it to recover a loss.
The objective is not to locate a perfect percentage. It is to prevent a plausible sequence or estimation mistake from removing the ability to keep taking valid trades. The tail can pay only if the path leaves you present.