kelly-criterion-reference

Kelly Criterion — Position Sizing Reference

> Mathematical framework for optimal bet sizing. Applied by Rohan (MKT-ROHAN-001) for prediction market position sizing.

The Formula

Full Kelly

f* = (p × b - q) / b

Where:

• f* = optimal fraction of bankroll to bet
• p = probability of winning
• q = 1 - p (probability of losing)
• b = net odds (decimal odds - 1, or reward/risk ratio)

Fractional Kelly

f_adj = fraction × f*

Where fraction is typically 0.25 (quarter-Kelly) or 0.50 (half-Kelly).

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Why Fractional Kelly?

Full Kelly is mathematically optimal for maximizing long-term geometric growth rate. However:

1. Extreme Volatility: Full Kelly produces massive bankroll swings
2. Estimation Error: If your probability estimates are even slightly off, full Kelly over-bets
3. Psychological Stress: Large drawdowns are emotionally difficult to sustain
4. Model Uncertainty: No model is perfectly calibrated

Professional practice: Quarter-Kelly (0.25x) or Half-Kelly (0.50x).

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Worked Example

Scenario: $10,000 bankroll, 70% win probability, 2:1 reward/risk

Calculation:

• p = 0.70, q = 0.30, b = 2.0
• f* = (0.70 × 2.0 - 0.30) / 2.0 = (1.40 - 0.30) / 2.0 = 1.10 / 2.0 = 0.55

Full Kelly: 55% = $5,500 per trade (extremely aggressive)

Comparison:

| Fraction | f_adj | Position Size | Character | |----------|-------|---------------|-----------| | Full Kelly (1.0x) | 55.0% | $5,500 | Mathematically optimal, psychologically brutal | | Half Kelly (0.5x) | 27.5% | $2,750 | Aggressive but manageable | | Quarter Kelly (0.25x) | 13.75% | $1,375 | Conservative, consistent |

Over 100+ trades, quarter-Kelly generates more consistent returns with substantially reduced ruin risk.

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Kelly with Edge in Prediction Markets

In prediction markets, the odds are implied by the market price:

b = (1 / market_price) - 1

Example: Market price = 0.40 (40¢)

• b = (1 / 0.40) - 1 = 2.5 - 1 = 1.5
• If your model says probability = 0.55:
• f* = (0.55 × 1.5 - 0.45) / 1.5 = (0.825 - 0.45) / 1.5 = 0.25
• Quarter-Kelly: 0.25 × 0.25 = 6.25% of bankroll

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Position Size Constraints

Even with Kelly, apply hard limits:

• Max single position: 5% of bankroll (regardless of Kelly output)
• Max concurrent: 15 positions
• Minimum edge: 4% (don't trade small edges even if Kelly says to)

If Kelly recommends > 5%, cap at 5%. Bankroll protection overrides optimization.

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When Kelly Says "Don't Trade"

If f* ≤ 0, the edge is negative or zero. Do not trade.

• This means the market is fairly priced or mispriced against you
• No position is the correct position
• Log as "insufficient edge" and move on

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Bankroll Management

• Starting bankroll: Define clearly before trading
• High-water mark: Track the highest bankroll ever achieved
• Drawdown: Current bankroll / high-water mark - 1
• Kill switch: If drawdown exceeds 8%, stop trading until review

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Common Mistakes

1. Using full Kelly: Almost always over-bets. Use fractional.
2. Ignoring estimation error: If your 70% is actually 55%, full Kelly is catastrophic
3. Betting on tiny edges: Kelly says bet small, but commission and slippage eat the edge
4. Not tracking calibration: Brier Score tells you if your probability estimates are reliable
5. Compounding Kelly with correlated bets: If positions are correlated, total exposure is larger than it looks