Fractional Kelly: Balancing Growth and Safety

## The Problem With Full Kelly Full Kelly betting produces the maximum long-run growth rate — but the variance is enormous. In the short term (hundreds of bets), you can experience 30–50% drawdowns even with well-calibrated estimates. Most bettors cannot psychologically sustain these drawdowns without abandoning the strategy. ## Fractional Kelly Fractional Kelly means betting a fixed fraction of the full Kelly stake. Common fractions: - **Quarter Kelly (25%):** Very safe, captures ~75% of Kelly growth rate - **Half Kelly (50%):** Moderate, captures ~87.5% of Kelly growth rate - **Full Kelly (100%):** Maximum theoretical growth, extreme variance ## The Mathematical Trade-Off At f fraction of Kelly, the growth rate is proportional to f − f²/2 (relative to full Kelly at 1 − 1/2 = 0.5). At 50% Kelly: growth rate = 2(0.5) − (0.5)² × (full Kelly growth) ≈ 75% of full Kelly growth, with ~25% of the variance. At 25% Kelly: growth rate ≈ 44% of full Kelly growth, with ~6% of the variance. The sharp reduction in variance for modest reduction in expected growth makes fractional Kelly significantly more practical. ## The Professional Standard Most professional bettors use 25–33% Kelly. This: - Produces meaningful compounding over a season - Limits drawdowns to roughly 15–20% of peak (manageable psychologically) - Accounts for the certainty that probability estimates are not perfectly calibrated ## Implementing Fractional Kelly Full Kelly calculation: 8.33% (from previous lesson) At 25% Kelly: stake = 0.25 × 8.33% = 2.08% of bankroll At £2,000 bankroll: stake = £41.60 This is a concrete, mathematically grounded stake — not a guess.