How Fractional Kelly Sizer works

Scope

The tool answers one question: given a known win rate and a known win/loss ratio, what fraction of bankroll maximizes long-run log-growth, and how does that strategy behave over many paths?

It does not estimate win rate or win/loss ratio from data. It does not infer distributional assumptions. It does not account for serial correlation, regime change, or fat tails beyond the Gaussian noise floor described below.

Formulas

Kelly fraction

For a bet with known win probability p, loss probability q = 1 − p, and win/loss ratio b = avg_win / avg_loss, the Kelly-optimal bet fraction of bankroll per trade is:

f* = (b·p − q) / b

Edges where f* ≤ 0 imply no positive-expectancy bet exists — the tool clamps these to zero.

Fractional Kelly

The tool scales f* by a user-selected multiplier:

full    = 1.000 × f*
half    = 0.500 × f*
quarter = 0.250 × f*
eighth  = 0.125 × f*

Fractional Kelly is standard practice when win rate and win/loss ratio are estimated rather than known. Ed Thorp and William Poundstone recommend half-Kelly or smaller for real-world bankroll management.

Monte Carlo simulation

For each of S independent paths across T trades:

• Sample win/loss via Bernoulli with parameter p.
• Apply outcome magnitude as b (win) or −1 (loss), scaled by a Gaussian noise factor (1 + 0.1·N(0,1)) floored at 0.1 to prevent degenerate zero-change paths.
• Compound bankroll: B_t+1 = B_t · (1 + f · outcome).
• Track peak bankroll and running max drawdown: DD_t = 1 − B_t / peak_t.
• Mark a path "ruined" if B_t ≤ ruin_threshold at any point.

Bet fraction f is clamped by the user's absolute single-trade cap before each trade.

Reported statistics

Across S paths:

• Median / 5th / 95th pct ending bankroll — distribution of final compound result.
• Median / 95th pct max drawdown — typical and tail drawdown experience.
• Ruin rate — fraction of paths that touched the ruin threshold.

Quantiles computed by linear interpolation on the sorted sample.

Assumptions

1. Stationary edge. p and b are constant across all trades. Real markets exhibit regime change.
2. Independent trades. Each trade's outcome is independent. Real strategies often exhibit serial correlation.
3. Symmetric noise. Outcome magnitudes have Gaussian multiplicative noise (σ = 0.1). Real returns frequently have fat tails.
4. No slippage or costs. The simulator ignores commissions, spreads, and market impact. For rough sizing decisions this is acceptable; for live trading, adjust b downward by expected cost.
5. Kelly is a growth-maximizing rule, not a risk-management rule. Full Kelly can tolerate severe drawdowns in exchange for maximum long-run growth. Fractional Kelly reduces drawdowns at the cost of growth.

Limitations

• The tool cannot tell you what your edge actually is. Bad inputs → bad sizing.
• Heavy tails are under-represented. For strategies with rare large losses (e.g. option selling, illiquid products), consider using a smaller Kelly fraction than the tool suggests.
• Monte Carlo results vary run-to-run. Increase the simulation count for tighter intervals.
• The "ruin" definition is a fraction-of-bankroll threshold, not a capital requirement. Regulatory minimums may be more restrictive.

Reproducibility

The simulation runs client-side in the browser, so every step is inspectable through DevTools (the Sources panel will surface the bundled component as KellySizer). Pseudo-random draws use the browser's Math.random() via Box-Muller for the Gaussian component; different runs of the same inputs will yield different specific paths, but distributional statistics converge as S grows.

References

• Kelly, J. L. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal 35.4.
• Thorp, E. O. (2006). "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." Handbook of Asset and Liability Management.
• MacLean, L. C.; Thorp, E. O.; Ziemba, W. T. (2011). The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific.
• Poundstone, W. (2005). Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. Hill and Wang.

Changelog

• 2026-04-20 — Initial release.