Kelly Criterion / Kelly Formula
ES: Kelly Criterion PT: Critério de Kelly
La fórmula matemática óptima de position sizing — John Kelly (1956), popularizada por Ed Thorp (Beat the Dealer, 1962) y Claude Shannon. Maximiza crecimiento geométrico a largo plazo. Warren Buffett y Charlie Munger usan variaciones. Pero full Kelly es demasiado agresivo — profesionales usan Half-Kelly o Quarter-Kelly. Comprender Kelly es entender el balance óptimo entre risk y return.
Qué es el Kelly Criterion
El Kelly Criterion (Criterio de Kelly, en portugués Critério de Kelly) es la fórmula matemática que maximiza el crecimiento geométrico a largo plazo de una serie de apuestas con probabilidades conocidas. Desarrollado por John L. Kelly Jr. (Bell Labs, 1956) originalmente para telecommunications problem, aplicada luego a gambling y trading por Ed Thorp (MIT professor, blackjack team pionero) y Claude Shannon (information theory). Fórmula básica: Kelly % = (W × R - L) / R. Donde: W = probability de winning. L = probability de losing (1 - W). R = ratio of win/loss (avg win / avg loss). Kelly % = fraction of capital a bet. Alternative formulation: Kelly = edge / odds. Edge = expected value of bet. Odds = payout ratio. Ejemplo numérico: strategy with 60% win rate, average win $200, average loss $100. W = 0.6, L = 0.4, R = 200/100 = 2. Kelly % = (0.6 × 2 - 0.4) / 2 = 0.8 / 2 = 0.40 = 40%. Full Kelly says bet 40% of capital per trade. Por qué Kelly maximiza growth: Mathematical proof: Kelly maximizes expected value de log(capital). Over many trials, log-optimal strategies outperform all others. Intuition: betting less than Kelly = suboptimal growth. Betting more than Kelly = volatility hurts compounding (volatility drag). Kelly is precise optimum. Ed Thorp's application: applied Kelly to blackjack counting en 1960s. Then to options trading (Princeton Newport Partners, 1969-1988) — 20% annualized returns over 19 years, Sharpe >4. Then Warren Buffett, according to Thorp's books, uses Kelly-style thinking intuitively. Kelly en real investing: full Kelly rarely used directly. Reasons: (1) Estimates of W and R are uncertain. (2) Real returns are continuous, not binary. (3) Psychological difficulty of losing 40%+ during inevitable drawdowns. Most practitioners use Half-Kelly (Kelly% / 2) o Quarter-Kelly (Kelly% / 4).
Cálculo para Diferentes Escenarios
El Kelly adaptado a different situations requires careful formulation. Escenario 1: Coin flip con odds: W = 0.52, R = 1 (even money). Kelly = (0.52 × 1 - 0.48) / 1 = 0.04 = 4%. Slight edge = small Kelly. Correct intuition. Escenario 2: High win rate, low R/R: W = 0.80, R = 0.5 (win $1, lose $2). Kelly = (0.80 × 0.5 - 0.20) / 0.5 = 0.20 / 0.5 = 0.40 = 40%. Even with bad R/R, high win rate justifies big bet. Escenario 3: Low win rate, high R/R: W = 0.30, R = 5 (win $5, lose $1). Kelly = (0.30 × 5 - 0.70) / 5 = 0.80 / 5 = 0.16 = 16%. Big winners justify position sizing despite low hit rate. Escenario 4: Negative expectancy: W = 0.40, R = 1. Kelly = (0.40 × 1 - 0.60) / 1 = -0.20 = -20%. Negative Kelly = don't bet, o bet opposite. Useful for identifying unprofitable strategies. Kelly for continuous returns (investments): formula more complex. Mean-variance approximation: Kelly = μ / σ². Where μ = expected return, σ² = variance. Example: investment with 10% expected return, 20% standard deviation: Kelly = 0.10 / 0.04 = 2.5 = 250%. Ridiculous, because formula assumes single-period. For continuous compounding: Kelly = μ / σ² × (assumption adjustments). Real application: for equity portfolios, Kelly suggests high concentrations, which is impractical. Multiple adjustments needed. Kelly con correlación múltiple: for multiple correlated bets/investments, full Kelly fórmula uses covariance matrix. Optimal allocation across positions adjusting for correlations. Complex but tractable with software. Kelly en options trading: specialized application. Long calls/puts: max loss bounded (premium), max gain variable. Kelly simpler. Credit spreads: high win rate, low R/R. Kelly often suggests moderate positions. Naked short options: unbounded risk. Kelly formulation problematic — infinite variance scenarios.
Full Kelly vs Fractional Kelly
El debate fundamental en práctica: usar full Kelly o fractional Kelly? Full Kelly (100%): matemáticamente óptima para maximum long-term growth IF estimates of W y R are perfect. Characteristics: large drawdowns (40-60% temporarily). Highest long-term returns. Highly volatile. Problems: estimates never perfect. Overconfidence in edge = betting more than true Kelly = catastrophic losses. Psychological stress of 40%+ drawdowns difficult. Half-Kelly (50%): most common practitioner choice. Mathematical property: Half-Kelly captures ~75% of Full Kelly's growth rate with ~50% less volatility. Drawdown ~25% vs 50% for Full Kelly. Massively better risk-adjusted. Used by: many professional investors, quant funds, option traders. Quarter-Kelly (25%): conservative practitioner choice. Returns: ~55% of Full Kelly's growth rate. Volatility: much smaller. Drawdown: ~12%. Used by: risk-averse investors, retirement planning. Ed Thorp's recommendation: Half-Kelly for most. Adjust based on confidence in edge estimates. If very uncertain, use Quarter-Kelly. Buffett/Munger implicit approach: appears to use fractional Kelly intuitively. Concentrated positions (top 10 holdings 50-70% of Berkshire portfolio) pero never over 30-40% in single name. Quarter to Half Kelly equivalent. Charlie Munger: "Intelligent people bet heavily when odds are extreme. Most of the time, odds are not extreme, so they don't bet." Kelly-style. Why not Full Kelly: (1) Estimate error: if W = 0.55 estimated but actually 0.52, Full Kelly will over-bet significantly. Ruin possible. (2) Uncertainty bands: real edges are estimates with confidence intervals. Full Kelly assumes point estimates. (3) Volatility drag: even correct Kelly produces large drawdowns. Psychological challenges. (4) Sequential decisions: Kelly optimal for isolated bets. Continuous investments require adjustment. (5) Black swans: fat tails not in Kelly model. Full Kelly underestimates true downside. Practical rule: Maximum edge confidence → Half-Kelly. Moderate confidence → Quarter-Kelly. Low confidence → Eighth-Kelly or lower. No confidence (random strategy) → don't trade.
Kelly en Trading y Portfolio Management
Aplicar Kelly en real investing requires careful calibration. Application 1: Single trade sizing: if expecting 60% win rate con R/R 1:2, Kelly = 40%, Half-Kelly = 20%, Quarter-Kelly = 10%. Size each trade 5-20% of capital. Reality: most retail traders over-size at 20-50% per trade. Kelly suggests 5-10% more appropriate given estimate uncertainty. Application 2: Portfolio concentration: how much of portfolio en single asset? Warren Buffett's biggest positions (Apple 40%+ of equity portfolio) represent Kelly-style thinking applied. Kelly framework: very high confidence position = higher allocation. Calculated edge high = larger fraction. Diversification vs Kelly: Kelly can suggest concentration if single asset has large edge. Traditional diversification suggests spreading risk. Tension. Resolution: use fractional Kelly to balance concentration and diversification. Top position 10-30% (below Kelly suggestion), diversify remaining 70-90%. Application 3: Hedge fund sizing: sophisticated hedge funds use Kelly-variant sizing across strategies. Each strategy: Kelly calculates ideal fraction. Portfolio: sum of Kelly fractions across all strategies. If exceeds available capital, leverage or sizing reduction. Complex optimization. Application 4: Options trading: Long options: max loss = premium. Kelly straightforward. Credit spreads: risk defined by spread width. Kelly uses narrower spread, higher win rate. Often suggests 5-15% sizing. Straddles/strangles: complex Kelly for multi-leg. Monte Carlo simulation often used instead of closed-form. Application 5: Long-term investing: Kelly for asset allocation. Treats stocks, bonds, etc. as "bets" with expected returns and volatilities. Generally suggests higher equity allocations than traditional 60/40. Example: stocks 8% expected return, 15% vol → Kelly = 8% / 2.25% = 356%. Clearly unrealistic, but suggests maximize equity within safety constraints. Risk management alongside Kelly: (1) Maximum position limit: regardless of Kelly, never >25% single position. (2) Total leverage limit: even if Kelly suggests 150%, cap at 100% (no leverage for most retail). (3) Correlation considerations: adjust Kelly down for correlated positions. (4) Black swan reserve: keep 10-20% in "safe" assets regardless. (5) Regular re-estimation: Kelly fractions change as edge estimates change. Reassess quarterly.
Limitaciones y Consideraciones Psicológicas
El Kelly Criterion tiene limitaciones significativas. Limitación #1: Estimate Uncertainty. Kelly assumes we know W and R exactly. En reality, these are estimates from historical data. Overestimate edge: bet more than true Kelly = over-betting = potential ruin. Underestimate: bet less than true Kelly = suboptimal growth. Solution: use fractional Kelly as margin of safety. Limitación #2: Normal Distribution Assumption. Kelly math assumes well-behaved returns. Real returns have fat tails (black swans). Full Kelly ignores tail risk. Fix: fractional Kelly partially addresses. Stress-testing Kelly assumptions with extreme scenarios useful. Limitación #3: Time Horizons. Kelly maximizes long-term geometric growth. Requires many trials. Single trades: Kelly irrelevant. Institutional time horizons: Kelly appropriate. Retail with retirement 20 years out: Kelly useful. Limitación #4: Strategy Capacity. Large Kelly suggests big positions, but market capacity may limit. Can't bet 40% of portfolio if position would move markets significantly. Limitación #5: Transaction Costs y Taxes. Kelly ignores costs. Real implementation reduces effective edge. Adjust Kelly down for costs. Limitación #6: Psychological. Even "optimal" drawdowns are painful. Humans will capitulate at 20-30% drawdowns despite math saying hold. Kelly ignores behavioral component. Consequence: practical Kelly sizing is 50-75% of "mathematical" Kelly to allow psychological sustainability. Limitación #7: Changing Edge. Markets evolve. Yesterday's edge may not persist. Kelly based on historical estimates may be wrong for future. Solution: frequent re-estimation, adaptive sizing. Limitación #8: Drawdown Tolerance. Full Kelly mathematically "recovers" from any drawdown over time. But practical investors can't wait 20 years for recovery. Shorter horizons punish over-betting asymmetrically. Psychological framing: Most retail over-bet: conviction + FOMO = 50%+ positions = full Kelly or more. Usually leads to ruin over time. Most retail under-bet on high-edge opportunities: fear-based = 1-2% positions on obvious winners. Suboptimal. Kelly teaches both: some bets should be bigger than comfortable. Some should be much smaller than tempting. Quote of Ed Thorp: "Bet your best ideas biggest, but never so big that a bad luck streak could ruin you."
Kelly Variations y Trade-offs
Most practitioners use Half or Quarter Kelly; Full Kelly too aggressive usually.
| Version | % of Full Kelly | Growth Captured | Drawdown Typical |
|---|---|---|---|
| Full Kelly | 100% | 100% | 40-60% |
| Half Kelly | 50% | ~75% | 20-25% |
| Quarter Kelly | 25% | ~55% | 10-15% |
| Eighth Kelly | 12.5% | ~30% | 5-10% |
| 1% fixed rule | ~5-10% equivalent | ~15-25% | <5% typical |