Comment calculer le Kelly Criterion fraction optimale pari

Calculez fraction optimale Kelly Criterion (1956) pour bet sizing paris et investissements. Half-Kelly Thorp, edge expected value, log-growth, doubling time, risque bankroll.

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Comment calculer le Kelly Criterion: formule de John L. Kelly Jr. (1956) determine fraction optimale du capital (bankroll) a parier/investir maximisant croissance logarithmique attendue long terme. Formule classique: f* = (p*(b+1) - 1) / b = p - q/b. Formule generalisee: f* = (p*(b+L) - L) / (b*L). Edge = p*b - q*L > 0 condition nécessaire. Kelly maximise E[log(W_n)]. Half-Kelly Thorp recommande en pratique reduit volatilite 75%. Essentiel sports betting, blackjack, hedge fund position sizing, poker bankroll management.

Formules Kelly

Classique: f* = p - q/b. p=0,55, b=1: f* = 0,10 = 10% bankroll.

Generalisee: f* = (p*(b+L) - L) / (b*L). Asymmetric payoffs.

Expected log-growth: G(f) = p*log(1+f*b) + q*log(1-f*L). G(half-Kelly)/G(full-Kelly) = 0,75.

Probabilite ruine: full-Kelly 50% bankroll a 50%; half-Kelly 25%; quarter-Kelly 12,5%.

Connexion utility theory: Kelly equivaut max logarithmic utility E[log(W)]. Bernoulli 1738 St. Petersburg paradox.

Histoire et contexte

Kelly Criterion publie par John L. Kelly Jr. (1956) en "A New Interprétation of Information Rate" (Bell System Technical Journal). Edward O. Thorp applique au blackjack 1960-62 "Beat the Dealer". Thorp fonda Princeton-Newport Partners (1969-1988) premier statistical arbitrage. Warren Buffett (rejete nominalement, concentrations coherentes). Bill Gross PIMCO. Critique Paul Samuelson 1971 "fallacy maximizing geometric mean". MacLean-Thorp-Ziemba 2011 World Scientific. Renaissance Technologies, Citadel, Two Sigma. Kelly conference Hong Kong 2011. AMF France regulation traders professionnels. ARJEL France jeux en ligne autorisation.

Table Kelly fraction

  • p=0,5, b=1: f* = 0 (NO bet)
  • p=0,52, b=1: f* = 4%
  • p=0,55, b=1: f* = 10%
  • p=0,60, b=1: f* = 20% (blackjack strong)
  • p=0,55, b=2: f* = 32,5%
  • p=0,40, b=3: f* = 20% (longshot)
  • CAC 40: ~Kelly 100%
  • Half-Kelly: 0,5x volatilite -75%
  • Quarter-Kelly: 0,25x conservateur
  • p=0,4, b=2: f* = 10% (longshot edge)

Exemples numériques

Exemple 1 - poker France. p=0,55, b=1. Kelly = 10% bankroll/session. 50.000€ = 5.000€ max. Half-Kelly 2.500€. ANJ Autorite Nationale Jeux France.

Exemple 2 - parie sportif rugby Top 14. p=0,53, b=0,909. Kelly = 1,25%. 20.000€ = 250€. Quarter-Kelly 63€.

Exemple 3 - hedge fund equity. Sharpe 2,0, return 20%, vol 10%. f* = 20 (cap 2-3x). Renaissance Medallion.

Exemple 4 - blackjack Hi-Lo. True count +5 = +2,5%. Kelly = 2,5% bankroll. 100k€ -> 2.500€ max. MIT Blackjack Team.

Exemple 5 - venture capital startup. Prob 20%, payoff 10x. Kelly = 12% bankroll. 1M€ -> 120k€ single deal. France Bpifrance, EIC UE.

Limites

  1. Suppose p et b connus exactement.
  2. Long-run convergence 100-1000 trades.
  3. Full-Kelly 50% prob 50% drawdown.
  4. Discrete bets approximation continue.
  5. Tax: 30% PFU France modifie Kelly.
  6. Couts transaction reduisent edge.
  7. Correlation entre trades.
  8. Max drawdown 20% institutionnel.
  9. Biais comportementaux paper loss.
  10. Kelly vs Markowitz objectifs differents.

Erreurs courantes

  1. p surestimee.
  2. Confondre American/decimal/fractional odds.
  3. Sans fractional cushion.
  4. Pas correlation paralleles.
  5. Bankroll initial vs current.
  6. Ignorer commissions/taxes.
  7. Stop-loss arbitraires.
  8. Kelly dans jeux edge negatif.
  9. Sous-estimer drawdown tolerance.
  10. Sample petit sans IC.

Applications pratiques

  1. Paris sportifs: NFL/NBA, football handicap asiatique.
  2. Card counting: blackjack Hi-Lo, MIT Blackjack Team.
  3. Poker: WPT, EPT Paris.
  4. Quantitative trading: hedge fund position.
  5. Stock investing: Berkshire Hathaway concentration.
  6. Options trading: covered calls, iron condors.
  7. Venture capital: France Digitale, Bpifrance.
  8. Crypto: Bitcoin spot, perpetual futures.
  9. Assurance: reinsurance AXA, Scor.
  10. Game theory research: HEC Paris, INSEAD.

Support

Questions: info@calcoi.com. Applications professionnelles: Kelly 1956, Thorp "Beat the Dealer" 1962, Poundstone "Fortune's Formula" 2005, MacLean-Thorp-Ziemba 2011 World Scientific.

Mis à jour en juillet 2026

Questions fréquentes

Que mesure le Kelly Criterion?

Mesure fraction optimale bankroll a parier/investir maximisant croissance logarithmique attendue capital (E[log(W_n)]) long terme. Different de maximize expected value (peut mener a ruine). Kelly 1956 derivation information theory (Shannon Bell Labs). Edward Thorp blackjack/Princeton-Newport Partners, Warren Buffett implicite.

Difference Kelly et expected value?

EV = p*b - q*L mesure seulement gain moyen. Maximiser EV peut suggerer bet excessifs. Kelly maximise E[log(W)] = croissance géométrique, bounded -infini gauche. Kelly toujours <= 100% bankroll, survie garantie long terme. p=0,55 b=1: EV = 0,10, Kelly = 0,10.

Half-Kelly: pourquoi recommande?

Half-Kelly (0,5*f*) sacrifie seulement 25% croissance mais reduit volatilite 75%. Prob 50% bankroll loss passe de 50% (full) a 25% (half) a 12,5% (quarter). Thorp empirical advice: full-Kelly emotionnellement difficile. Bookmakers professionnels, hedge funds adoptent 1/4 a 1/2 Kelly.

Kelly fonctionne si pas sur p et b?

Sensibilite a estimation. +/- 5% erreur en p change Kelly drastiquement. Pratique fractional Kelly cushion. Bayesian update prior + evidence empirique. Insufficient data: Quarter-Kelly conservatif. Confidence interval p lower bound IC. Renaissance Bayesian estimators.

Kelly marches financiers continus?

Asset continus: f* = mu/sigma^2. Exemple CAC40 mu=9%, sigma=19%: f* = 2,5 (250% leverage). Pratique cap 1-2x risk parity. Growth rate G = mu^2/(2*sigma^2). Equivaut max Sharpe single asset. Multi-asset f* = Sigma^-1 * mu (Kelly-Markowitz hybrid).

Quand NE PAS appliquer Kelly?

Edge negatif (slot, loterie): f=0. Bankroll petit indivisible. Tail risk fat-tailed log-normal breaks. Time horizon court. Non-fungible bankroll. LTCM 1998 deviation Kelly catastrophique.

Kelly portefeuille multi-asset?

f* = Sigma^-1 * mu vectoriel. Equivaut Markowitz mean-variance log utility. Kelly diversifie naturellement. Short-selling constraint f<0. CFA portfolio management: Kelly long terme, Markowitz single-period. Industry hybrid blend.

Formule

f* = (p*(b+1) - 1) / b = p - q/b
f* = (p*(b+L) - L) / (b*L)
E[X] = p*b - q*L > 0 (edge necessario)
G(f) = p*log(1+f*b) + q*log(1-f*L)
f* = mu / sigma^2 (continuous-time)
0.5 * f* riduce volatility 75%, growth -25%
prob bankroll a 50% iniziale = 50% full-Kelly, 25% half-Kelly
Kelly 1956 Bell System Tech J; Thorp Beat the Dealer 1962; MacLean-Thorp-Ziemba 2011

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