Kelly Criterion Formula
Quick Reference
Formula matematica per calcolare leverage ottimale che massimizza crescita geometrica del capitale.
Formula Base Kelly
f* = (r - b) / s²
Variabili
- f*: Optimal leverage factor
- r: Expected annual return (senza leverage)
- b: Borrowing rate (risk-free rate)
- s: Annual standard deviation of returns
Optimal Volatility Target
Versione piu utile per trading:
Optimal Vol Target = (r - b) / s = SR
Dove SR = Sharpe Ratio
In parole: Optimal volatility target = Expected Sharpe Ratio
Half-Kelly (Raccomandato)
Half-Kelly Target = SR / 2
Con aggiustamenti realistici:
Recommended Target = (Backtest SR × Discount Factor) / 2
Discount Factors
- Out-of-sample bootstrap: 0.75
- Rolling window: 0.60
- In-sample only: 0.25
Esempio Numerico Completo
Step 1: Backtest Results
Sistema trend-following: - Backtest SR = 1.0 - Out-of-sample bootstrap - Zero skew
Step 2: Apply Discount
Realistic SR = 1.0 × 0.75 = 0.75
Step 3: Half-Kelly
Target Vol = 0.75 / 2 = 0.375 = 37.5%
Raccomandazione finale: 35-40% target volatility
Formula Derivation (Semplificata)
Massimizzare geometric mean growth rate:
Obiettivo:
Maximize: E[log(1 + return)]
Con leverage f:
Return = f × (r - b) - f × volatility shocks
Soluzione:
f_optimal = (r - b) / s²
Optimal risk:
Risk_optimal = f × s = (r - b) / s = SR
Negative Skew Adjustment
Per strategie con negative skew (~-2):
Target Vol = (SR / 2) / 2 = SR / 4
Esempio: - Realistic SR = 0.50 - Negative skew strategy - Target = 0.50 / 4 = 12.5%
Expected Return Calculation
Date leverage/volatility target, expected return:
Expected Return = SR × Vol Target
Esempi:
| SR | Vol Target | Expected Return |
|---|---|---|
| 0.25 | 12.5% | 3.1% |
| 0.50 | 25% | 12.5% |
| 0.75 | 37.5% | 28% |
| 1.0 | 50% | 50% |
Questi sono geometric means attesi.
Overconfidence Example
Trader overconfident: - Crede backtest SR = 2.0 (nessun discount!) - Full Kelly: Target = 200% - Expected return = 400% annuo
Realta: - True SR = 0.5 (delusione!) - Con 200% vol e SR 0.5: - Molto probabile rovina
Lesson: SEMPRE discount backtest e usa Half-Kelly.
Kelly per Diversi Trader Types
Staunch Systems Trader
Target = (Realistic SR × 0.75) / 2 [zero skew]
Target = (Realistic SR × 0.75) / 4 [negative skew]
Max realistic SR assumed: 1.0
Semi-Automatic Trader
Target = (Expected SR) / 2 [positive/zero skew]
Target = (Expected SR) / 4 [negative skew]
Max safe expected SR: 0.5
Asset Allocator
Target = Expected SR / 2
Max realistic SR: 0.4 (large diversified portfolio)
Probability of Ruin
Con optimal Half-Kelly e SR 0.5, 10 anni:
| Vol Target | Prob Lose 50% | Prob Lose 90% |
|---|---|---|
| 12.5% (1/4 Kelly) | <1% | <1% |
| 25% (Half Kelly) | 10% | 1.1% |
| 50% (Full Kelly) | 40% | 22% |
| 100% | 93% | 88% |
Nota: Anche Half-Kelly ha 10% prob di perdere meta capitale!
Continuous Rebalancing
Kelly richiede continuous position adjustment:
New Position = Current Capital × (SR / Price × σ%)
Questo è rolling up profits and losses.
Non rebalancing → rischio di rovina aumenta drammaticamente.
Kelly con Transaction Costs
Costi riducono optimal Kelly:
Effective SR = Raw SR - (Annual Costs / Vol Target)
Usa effective SR per calcolo target.
Esempio: - Raw SR = 0.60 - Costs = 0.10 SR units - Effective SR = 0.50 - Half-Kelly target = 25% (non 30%)
Fractional Kelly
General formula per frazione f del Kelly:
Target Vol = (Fraction) × SR
- f = 0.25: Quarter-Kelly
- f = 0.50: Half-Kelly
- f = 1.00: Full Kelly
Raccomandazioni: - Retail: 0.25-0.50 - Professional: 0.25-0.50 (!!) - Overconfident: 1.0+ (bankruptcy soon)
Verifica Formula
Sanity check dopo calcolo:
Implied Leverage = Target Vol / Natural Vol
Red flags: - Implied leverage > 50: Strumento troppo low-vol - Target vol > 50%: Troppo aggressivo per most traders - Expected return > 100%: Sei overconfident
Alternative: Fixed Fraction
Se non hai SR estimate, usa fixed conservative fraction:
Ultra-conservative: 10-12% vol target Conservative: 15-20% vol target Moderate: 20-25% vol target
Errors Common
- Using full Kelly: Recipe for ruin
- No discount to backtest: Overconfidence
- Ignoring skew: Negative skew needs even lower target
- Not rebalancing: Kelly assumes continuous adjustment
- Wrong SR units: Confondere daily con annual SR
- Forgetting costs: Use net SR after costs
Concetti Correlati
- [[Sharpe Ratio]] - input chiave per Kelly
- [[Volatility Targeting]] - implementazione Kelly in practice
- [[Geometric Mean]] - Kelly massimizza questo
- [[Position Sizing]] - output del Kelly criterion
- [[Skew]] - richiede adjustment al Kelly