Kelly Criterion — Optimal Bet Sizing

The Kelly criterion is the mathematically optimal bet size that maximizes long-term geometric growth of capital. Developed by John Kelly at Bell Labs in 1956, it answers a precise question: given a known edge, what fraction of your bankroll should you risk to maximize the compounding rate?

Core insight: Betting too small leaves growth on the table. Betting too large increases ruin risk and actually reduces long-term growth. Kelly finds the exact optimum between these extremes.

Practical insight: You should almost never use full Kelly. Estimation error in your edge means full Kelly will overbets in practice. Use fractional Kelly (0.25x to 0.5x) for real trading.

The Kelly Formula

For a binary outcome (win or lose):

f* = (p * b - q) / b

Where:

• f* = optimal fraction of bankroll to bet
• p = probability of winning
• q = probability of losing (1 - p)
• b = payoff ratio (average win / average loss)

Equivalent forms:

f* = p - q / b
f* = p - (1 - p) / b
f* = (p * b - (1 - p)) / b

Edge = p * b - q = expected value per unit risked. Kelly only makes sense when edge > 0. If edge is zero or negative, the optimal bet is zero — do not trade.

Quick Reference

Values are full Kelly fraction. In practice, use 0.25x to 0.5x of these numbers.

Why Use Fractional Kelly

Full Kelly assumes you know p and b exactly. You never do. Here is why fractional Kelly is essential:

1. Estimation Error

Your win rate estimate from 100 trades has a standard error of roughly ±5%. If your true win rate is 55% but you estimate 60%, full Kelly will overbets by ~50%, which reduces long-term growth below what half Kelly would achieve.

2. Variance and Drawdowns

Full Kelly has extremely high variance. Expected maximum drawdown for full Kelly is roughly 50-80% of account. This is psychologically devastating and practically dangerous (margin calls, inability to continue trading).

3. Asymmetry of Over vs. Under Betting

Overbetting by 2x (betting at 2f) produces zero long-term growth — the same as not trading at all. Underbetting by 2x (betting at 0.5f) still captures ~75% of the optimal growth rate. The penalty for overbetting is catastrophically worse than for underbetting.

Recommended Fractions

Estimating Your Edge

Kelly requires two inputs: win rate (p) and payoff ratio (b). Both must be estimated from data.

Minimum Data Requirements

• 50 trades minimum for any Kelly calculation. Below this, estimation error dominates.
• 100+ trades preferred for half Kelly sizing.
• 200+ trades before considering aggressive fractions.

Calculation from Trade History

wins = [t for t in trades if t > 0]
losses = [t for t in trades if t < 0]

win_rate = len(wins) / len(trades)               # p
payoff_ratio = mean(wins) / abs(mean(losses))     # b
edge = win_rate * payoff_ratio - (1 - win_rate)   # should be > 0

kelly_full = (win_rate * payoff_ratio - (1 - win_rate)) / payoff_ratio

Conservative Estimation

Use the lower bound of a Wilson confidence interval for win rate rather than the point estimate:

import math

def wilson_lower(wins: int, total: int, z: float = 1.96) -> float:
    """Lower bound of Wilson score interval (95% confidence)."""
    p = wins / total
    denominator = 1 + z**2 / total
    centre = p + z**2 / (2 * total)
    spread = z * math.sqrt((p * (1 - p) + z**2 / (4 * total)) / total)
    return (centre - spread) / denominator

Using the lower bound of the confidence interval for win rate automatically builds in conservatism, reducing the risk of overbetting due to sampling luck.

Edge Strength Classification

Multi-Bet Kelly (Simultaneous Positions)

When holding multiple positions simultaneously:

Independent Bets

If bets are uncorrelated, each can be sized at its individual Kelly fraction. However, the sum of all Kelly fractions should not exceed 1.0 (total portfolio). If it does, scale each proportionally:

kelly_fractions = [0.15, 0.10, 0.12, 0.08]  # individual Kelly fractions
total = sum(kelly_fractions)  # 0.45
if total > 1.0:
    scale = 1.0 / total
    kelly_fractions = [f * scale for f in kelly_fractions]

Correlated Bets

Correlated positions (e.g., multiple SOL memecoins) are effectively one larger bet. Reduce each position proportionally to the correlation:

# Simple correlation adjustment
def adjust_for_correlation(kelly_fractions: list, avg_correlation: float) -> list:
    """Reduce Kelly fractions based on average inter-position correlation."""
    n = len(kelly_fractions)
    # Effective number of independent bets
    n_eff = n / (1 + (n - 1) * avg_correlation)
    scale = n_eff / n
    return [f * scale for f in kelly_fractions]

In crypto, meme token positions often have correlations of 0.5-0.8 with each other (they all dump together in risk-off). Treat them as partially one bet.

Portfolio Kelly Cap

Regardless of individual calculations, enforce a hard cap: total Kelly allocation should never exceed 1.0 (100% of portfolio). A practical maximum is 0.6-0.8 to leave cash buffer for drawdowns and new opportunities.

PumpFun / Meme Token Kelly

Meme token trading presents specific challenges for Kelly:

1. Edge is hard to estimate: Win rates and payoff ratios shift rapidly with market regime.
2. Fat tails dominate: A few large winners and many small losers. Standard Kelly assumes thin tails.
3. Correlation spikes in drawdowns: All meme tokens can dump simultaneously.

Practical Adjustments

• Use 0.1x to 0.25x Kelly maximum for meme tokens.
• Cap absolute position size at 2-5% of portfolio regardless of Kelly output.
• Recalculate edge weekly — stale estimates are dangerous.
• If Kelly suggests > 30%, your edge estimate is almost certainly wrong. Use 5% maximum.

def meme_kelly(win_rate: float, payoff_ratio: float, account: float) -> float:
    """Conservative Kelly for high-uncertainty meme token trades."""
    kelly_full = (win_rate * payoff_ratio - (1 - win_rate)) / payoff_ratio
    kelly_conservative = kelly_full * 0.15  # 0.15x fractional
    max_fraction = 0.05                     # hard cap at 5%
    return min(max(kelly_conservative, 0), max_fraction) * account

When Kelly Does Not Work

Kelly optimality relies on assumptions that are often violated:

Mitigations

1. Use fractional Kelly (addresses estimation error)
2. Adjust for correlation (addresses dependence)
3. Use continuous Kelly for non-binary returns (see references/kelly_derivation.md)
4. Recalculate regularly (addresses non-stationarity)
5. Subtract estimated costs from edge before calculating Kelly

Continuous Kelly (For Portfolio Returns)

When returns are continuous rather than binary win/lose:

f* = (μ - r) / σ²

Where:

• μ = expected return of the strategy
• r = risk-free rate (often 0 for crypto)
• σ² = variance of returns

This is equivalent to Sharpe² / (2 * σ) when the Sharpe ratio is computed as (μ - r) / σ.

Use this form when you have a return stream rather than discrete win/loss trades. See references/kelly_derivation.md for the full derivation.

Integration with Other Skills

• position-sizing: Kelly provides the optimal fraction; position-sizing translates that into units. Use Kelly as one input, then apply liquidity and volatility constraints from position-sizing.
• risk-management: Kelly sizing must respect portfolio-level risk limits. If Kelly suggests 10% per trade but your risk policy caps at 5%, the cap wins.
• strategy-framework: Document your Kelly parameters (fraction used, sample size, recalculation frequency) as part of strategy specification.
• regime-detection: Recalculate Kelly when regime changes. Edge in a trending market differs from edge in a ranging market.

Files

References

• references/kelly_derivation.md — Full mathematical derivation of Kelly criterion, fractional Kelly growth rates, continuous Kelly, and multi-outcome Kelly
• references/practical_kelly.md — Edge estimation from trading data, confidence intervals, worked examples, common pitfalls, and danger zones

Scripts

• scripts/kelly_calculator.py — Kelly calculator from win rate, payoff ratio, and account size. Prints fractional Kelly recommendations and sensitivity analysis. Dependencies: none.
• scripts/kelly_from_trades.py — Estimate Kelly from a list of trade P&L values. Computes confidence intervals, rolling stability analysis, and recommended fraction. Dependencies: numpy.