Kelly Criterion
Definition
The Kelly Criterion is a bankroll sizing rule that maximizes long-run growth when you have a known edge; can be scaled (e.g., half‑Kelly).
The Formula
For a binary bet with win probability p and odds b:1:
Kelly % = (bp - q) / b
Where:
- p = probability of winning
- q = probability of losing (1 - p)
- b = odds received (payout ÷ stake)
Prediction Market Version
For prediction markets where contracts pay $1:
Kelly % = (p - price) / (1 - price)
Where:
- p = your estimated probability
- price = current Yes contract price
Example
Market: "Will it rain tomorrow?"
- Market price: $0.40 (40%)
- Your estimate: 60% (p = 0.60)
- Edge: 0.60 - 0.40 = 0.20 (20%)
Kelly calculation:
Kelly % = (0.60 - 0.40) / (1 - 0.40)
Kelly % = 0.20 / 0.60
Kelly % = 0.333 (33.3%)
Action: Bet 33.3% of your bankroll
Why Kelly Works
Maximizes Growth
- Optimal between too aggressive and too conservative
- Grows bankroll fastest in long run
- Mathematically proven optimal
Prevents Ruin
- Never bets entire bankroll
- Scales with edge and confidence
- Built-in risk management
Compound Growth
- Winners compound
- Losers don't destroy bankroll
- Long-term wealth maximization
Kelly Fractions
Full Kelly (1.0x)
- Most aggressive
- Maximum growth rate
- Highest volatility
- Can feel scary
Half Kelly (0.5x)
- Most popular
- 75% of full Kelly growth
- Much lower volatility
- Better sleep at night
Quarter Kelly (0.25x)
- Conservative
- ~50% of full Kelly growth
- Very stable
- Good for beginners
Example Comparison
Scenario: 60% edge, $10,000 bankroll
| Strategy | Bet Size | Growth Rate | Volatility | |----------|----------|-------------|------------| | Full Kelly | $3,330 | 100% | High | | Half Kelly | $1,665 | 75% | Medium | | Quarter Kelly | $833 | 50% | Low | | Fixed $1,000 | $1,000 | Varies | Fixed |
Practical Application
Step 1: Estimate Probability
Your independent analysis:
True probability = 65%
Step 2: Check Market Price
Current Yes price:
Market price = $0.45 (45%)
Step 3: Calculate Edge
Edge = 0.65 - 0.45 = 0.20 (20% edge)
Step 4: Calculate Kelly
Kelly % = (0.65 - 0.45) / (1 - 0.45)
Kelly % = 0.20 / 0.55
Kelly % = 0.364 (36.4%)
Step 5: Apply Fraction
Using Half Kelly:
Half Kelly = 0.364 / 2 = 0.182 (18.2%)
Step 6: Size Position
With $10,000 bankroll:
Bet = $10,000 × 0.182 = $1,820
Common Mistakes
Overestimating Edge
- Thinking you have 20% edge when really 5%
- Results in overbetting
- Solution: Be conservative with estimates
Ignoring Correlation
- Kelly assumes independent bets
- Correlated markets need adjustment
- Solution: Reduce size for correlated positions
Using Full Kelly
- Too aggressive for most
- High drawdowns
- Solution: Use fractional Kelly
Not Updating
- Edge changes as prices move
- Need to recalculate
- Solution: Dynamic position sizing
When Kelly Doesn't Apply
Unknown Probabilities
- Can't calculate edge accurately
- Garbage in, garbage out
- Alternative: Conservative sizing
Few Bets
- Kelly optimizes long-run
- Short-term variance high
- Alternative: Fixed sizing for one-offs
Liquidity Constraints
- Can't easily exit
- Position limits
- Alternative: Reduce to fit constraints
Emotional Tolerance
- Can't handle volatility
- Stress affects decisions
- Alternative: Use smaller fraction
Kelly for Multiple Markets
Portfolio Kelly
When trading multiple markets:
- Calculate edge for each
- Consider correlations
- Allocate fraction of bankroll
- Rebalance as edges change
Correlation Adjustment
- Reduce Kelly fraction for correlated bets
- Sum of positions should be less than Kelly
- More diversification = closer to full Kelly
Advanced: Growth vs Risk
Growth Rate Formula
Growth Rate = r × (1 - σ²/2r²)
Where:
- r = edge × bet fraction
- σ = standard deviation
Optimal Trade-off
- Full Kelly maximizes growth
- Half Kelly = 75% growth, 50% variance
- Sweet spot for most traders
Historical Performance
Simulations Show
- Kelly outperforms fixed sizing
- Half-Kelly beats Full Kelly risk-adjusted
- Quarter-Kelly stable but slower
- Over-Kelly leads to ruin eventually
Related Terms
- Expected Value
- Edge
- Position
- [Bankroll Management]