Kelly Criterion

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.