12 KiB
Kelly Criterion Deep Dive
Mathematical foundation for optimal bet sizing under uncertainty.
Table of Contents
- Mathematical Derivation
- Formula Variations
- Fractional Kelly
- Extensions
- Common Mistakes
- Practical Implementation
- Historical Examples
- Comparison to Other Methods
1. Mathematical Derivation
The Core Question
Problem: What fraction of your bankroll maximizes long-term growth?
Why it matters: Bet too little → Leave money on the table. Bet too much → Risk ruin, high variance.
Logarithmic Utility Framework
Key insight: Maximize expected logarithm of wealth, not expected wealth.
Why log utility?
- Captures diminishing marginal utility ($1 matters more when you have $100 vs $1M)
- Makes repeated multiplicative bets additive: log(AB) = log(A) + log(B)
- Geometric mean emerges naturally (what matters for repeated bets)
- Prevents betting 100% (avoids ruin)
Derivation for Binary Bet
Setup:
- Current bankroll: W
- Bet fraction: f
- Win probability: p, Loss probability: q = 1 - p
- Net odds: b (bet $1, win $b net)
Outcomes:
- Win (probability p): New wealth = W(1 + fb)
- Lose (probability q): New wealth = W(1 - f)
Expected log utility:
E[log(W_new)] = p × log(1 + fb) + q × log(1 - f) + log(W)
Objective: Maximize g(f) = p × log(1 + fb) + q × log(1 - f)
Finding the Optimum
Take derivative:
dg/df = pb/(1 + fb) - q/(1 - f)
Set equal to zero and solve:
pb/(1 + fb) = q/(1 - f)
pb(1 - f) = q(1 + fb)
pb - pbf = q + qfb
pb - q = f(pb + qb) = fb(p + q) = fb
f* = (pb - q) / b = (bp - q) / b
The Kelly Criterion:
f* = (bp - q) / b
Where:
f* = Optimal fraction to bet
b = Net odds received
p = Win probability
q = 1 - p
Alternative Form
Edge = Expected return per dollar bet = bp - q
Kelly formula: f* = Edge / Odds = (bp - q) / b
Example: p = 60%, b = 1.0 (even money)
- Edge = 0.6 × 1 - 0.4 = 0.2
- f* = 0.2 / 1 = 20%
Optimality
Second derivative: d²g/df² < 0 at f = f* → Maximum confirmed
Growth rate: G(f*) maximizes long-run geometric growth
Comparison:
- f < f*: Lower growth (too conservative)
- f > f*: Lower growth (too aggressive, variance dominates)
- f > 2f*: Negative growth (eventual ruin)
2. Formula Variations
Converting Market Odds
Decimal odds (e.g., 2.50): b = Decimal - 1 = 1.50
American odds:
- Positive (+150): b = 150/100 = 1.50
- Negative (-150): b = 100/150 = 0.667
Fractional odds (3/1): b = 3.0
Implied probability: Market p = 1/(b + 1)
Multi-Outcome Bet
Horse race: Multiple options, bet on any with positive Kelly
Formula for outcome i:
f_i* = (p_i(b_i + 1) - 1) / b_i
If f_i* > 0: Bet f_i* on outcome i
If f_i* ≤ 0: Don't bet
Continuous Outcomes (Merton's Formula)
Stock market application:
f* = μ / σ²
Where:
μ = Expected return (drift)
σ² = Variance of returns
Example: μ = 8%, σ = 20% → f* = 0.08/0.04 = 2.0 (200%, use leverage)
Reality: Too aggressive, use fractional Kelly → 50-100% more reasonable
3. Fractional Kelly
Why Fractional Kelly?
Problems with full Kelly:
- Extreme volatility: Wild swings, can lose 50%+ in bad runs
- Model error: If probability estimate wrong, full Kelly overbets dramatically
- Practical ruin: 20% chance of 50% drawdown before doubling
- Non-ergodic: Most can't bet infinitely many times
Formula
Fractional Kelly = f* × Fraction
Common choices:
- Half Kelly: f*/2
- Quarter Kelly: f*/4 (recommended)
- Third Kelly: f*/3
Growth vs. Variance Trade-off
| Strategy | Growth Rate | Volatility | Max Drawdown |
|---|---|---|---|
| Full Kelly | 100% | 100% | -50% |
| Half Kelly | ~75% | 50% | -25% |
| Quarter Kelly | ~50% | 25% | -12% |
Key: Half Kelly gives 75% of growth with 25% of variance → Better risk-adjusted return
Robustness to Error
Example: You think p = 0.60, true p = 0.55, even money bet
Full Kelly (f = 20%):
- Growth rate = 0.55×log(1.20) + 0.45×log(0.80) ≈ 0 (breakeven!)
Half Kelly (f = 10%):
- Growth rate = 0.55×log(1.10) + 0.45×log(0.90) ≈ 0.005 (still positive)
Lesson: Overbetting much worse than underbetting. Fractional Kelly provides buffer.
Recommended Fractions
| Situation | Fraction | Reasoning |
|---|---|---|
| Professional gambler | 1/4 to 1/3 | Reduces career risk |
| High model uncertainty | 1/4 or less | Error buffer crucial |
| High confidence | 1/2 to 2/3 | Can use more aggression |
| Institutional | 1/4 to 1/3 | Drawdown = career risk |
Default: Quarter Kelly (1/4) for most real-world situations
4. Extensions
Multiple Simultaneous Bets
Matrix form (N assets):
f* = Σ⁻¹ × μ
Where:
Σ = Covariance matrix
μ = Expected returns vector
Key insight: Correlated bets reduce optimal sizing
Heuristic: Adjusted Kelly = Individual Kelly × (1 - ρ/2), where ρ = correlation
Example: ρ = 0.6, Individual Kelly = 15%
- Adjusted: 15% × (1 - 0.3) = 10.5%
Correlated Outcomes
Common correlations:
- Political: Presidential + Senate races
- Sports: Team championship + Player MVP
- Markets: Tech stock A + Tech stock B
Extreme cases:
- ρ = 1 (perfect correlation): Only bet on one
- ρ = -1 (negative correlation): Bets hedge, can bet more
- ρ = 0 (independent): No adjustment
Dynamic Kelly
Problem: Probability changes over time (new information)
Process:
- Start with p₀, bet f₀*
- New information → Update to p₁ (Bayesian)
- Recalculate f₁*
- Rebalance (adjust bet size)
Consideration: Transaction costs limit rebalancing frequency
5. Common Mistakes
Mistake 1: Full Kelly Overbet
The error: Using full Kelly in practice
Why wrong: Assumes perfect probability estimate (never true)
Impact: Bet 2×f* → Negative growth rate
Fix: Always use fractional Kelly (1/4 to 1/2)
Mistake 2: Ignoring Model Error
The error: Treating probability as certain
Adjustment:
Uncertain Kelly = f* × Confidence
Example: f* = 20%, 80% confident → Bet 16%
Better: Use fractional Kelly (implicitly adjusts)
Mistake 3: Neglecting Bankruptcy
Reality: Finite games + estimation error → real ruin risk
Drawdown stats (full Kelly, p=0.55):
- 25% chance of -40% before recovery
- 10% chance of -50% before recovery
Practical bankruptcy: Client fires you, forced liquidation, can't maintain discipline
Fix: Use fractional Kelly, set stop-loss (if down 25%, pause)
Mistake 4: Ignoring Correlation
Example disaster:
- 10 bets, each Kelly 10%
- All highly correlated (same theme)
- Bet 100% total → Single adverse event → Large loss
Fix: Measure correlations, use portfolio Kelly, diversify themes
Mistake 5: Misestimating Odds
Common confusion:
- Decimal 2.0: b = 1.0 (not 2.0)
- "3-to-1": b = 3.0 ✓
- American +200: b = 2.0 (not 200)
Fix: Always convert to NET payout (b = total return - 1)
Mistake 6: Static Bankroll
Problem: Calculate once, never update
Fix: Recalculate before each bet using current bankroll
6. Practical Implementation
Step-by-Step Process
1. Convert odds to decimal:
# Decimal odds: b = decimal - 1
# American +150: b = 150/100 = 1.50
# American -150: b = 100/150 = 0.667
# Fractional 3/1: b = 3.0
2. Determine probability: Use forecasting process (base rates, Bayesian updating, etc.)
3. Calculate edge:
edge = net_odds * probability - (1 - probability)
4. Calculate Kelly:
kelly_fraction = edge / net_odds
5. Apply fractional Kelly:
fraction = 0.25 # Quarter Kelly recommended
adjusted_kelly = kelly_fraction * fraction
6. Calculate bet size:
bet_size = current_bankroll * adjusted_kelly
7. Execute and track:
- Record: Date, event, probability, odds, edge, Kelly%, bet
- Set reminder for resolution
- Note new information
Position Tracking Template
Date: 2024-01-15
Event: Candidate A wins
Your probability: 65%
Market odds: 2.20 (implied 45%)
Net odds (b): 1.20
Edge: 0.43 (43%)
Full Kelly: 35.8%
Fractional (1/4): 8.9%
Bankroll: $10,000
Bet size: $890
Resolution: 2024-11-05
7. Historical Examples
Ed Thorp - Blackjack (1960s)
Application: Card counting edge varies with count → Dynamic Kelly
Implementation:
- True count +1: Edge ~0.5%, bet ~0.5% of bankroll
- True count +5: Edge ~2.5%, bet ~2.5% of bankroll
Results: Turned $10k into $100k+, proved Kelly works in practice
Lessons: Used fractional Kelly (~1/2), dynamic sizing, managed "heat" (detection risk)
Princeton-Newport Partners (1970s-1980s)
Strategy: Statistical arbitrage, convertible bonds
Kelly application: 1-3% per position, 50-100 positions (diversification)
Results: 19.1% annual (1969-1988), only 4 down months in 19 years, <5% max drawdown
Lessons: Fractional Kelly + diversification = low volatility, dominant strategy
Renaissance Technologies / Medallion Fund
Strategy: Thousands of small edges, high frequency
Kelly application:
- Each signal: 0.1-0.5% (tiny fractional Kelly)
- Portfolio: 10,000+ positions
- Leverage: 2-4× (portfolio Kelly supports with diversification)
Results: 66% annual (gross) over 30+ years, never down year
Lessons: Kelly optimal for repeated bets with edge. Diversification enables leverage. Discipline crucial.
Warren Buffett (Implicit Kelly)
Concentrated bets: American Express (40%), Coca-Cola (25%), Apple (40%)
Why Kelly-like: High conviction → High p → Large Kelly → Large position
Quote: "Diversification is protection against ignorance."
Lessons: Kelly justifies concentration with edge. Still uses fractional (~40% max, not 100%).
8. Comparison to Other Methods
Fixed Fraction
Method: Always bet same percentage
Pros: Simple, prevents ruin
Cons: Ignores edge, suboptimal growth
When to use: Don't trust probability estimates, want simplicity
Martingale (Double After Loss)
Method: Double bet after each loss
Fatal flaws:
- Requires infinite bankroll
- Exponential growth (10 losses → need $10,240)
- Negative edge → lose faster
- Betting limits prevent recovery
Conclusion: NEVER use. Mathematically certain to fail.
Fixed Amount
Method: Always bet same dollar amount
Cons: As bankroll changes, fraction changes inappropriately
When to use: Very small recreational betting
Constant Proportion
Method: Fixed percentage, not optimized for edge
Difference from Kelly: Doesn't adjust for edge/odds
Conclusion: Better than fixed dollar, worse than Kelly
Risk Parity
Method: Allocate to equalize risk contribution
Difference from Kelly: Doesn't use expected returns (ignores edge)
When better: Don't have reliable return estimates, defensive portfolio
When Kelly better: Have edge estimates, goal is growth
Summary Comparison
| Method | Growth | Ruin Risk | When to Use |
|---|---|---|---|
| Kelly | Highest | None* | Active betting with edge |
| Fractional Kelly | High | Very low | Real-world (recommended) |
| Fixed Fraction | Medium | Low | Simple discipline |
| Fixed Amount | Low | Medium | Recreational only |
| Martingale | Negative | Certain | NEVER |
| Risk Parity | Low-Med | Low | Defensive portfolios |
*Kelly theoretically no ruin risk, but model error creates practical risk → use fractional Kelly
Final Recommendation: Quarter Kelly (f/4)* for nearly all real-world scenarios.
Return to Main Skill
← Back to Market Mechanics & Betting
Related resources: