How to compute the Kelly Criterion optimal bet sizing

Compute optimal Kelly Criterion fraction (1956) for bet sizing in betting and investing. Half-Kelly Thorp, edge expected value, log-growth, doubling time, bankroll risk.

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How to compute the Kelly Criterion: John L. Kelly Jr. (1956) formula determines optimal capital fraction (bankroll) to bet/invest to maximize expected logarithmic growth long-term. Classic formula: f* = (p*(b+1) - 1) / b = p - q/b, where p is win probability, q = 1-p, b net odds payoff (e.g., 1:1 = 1, 2:1 = 2). Generalized formula: f* = (p*(b+L) - L) / (b*L), where L = loss multiplier. Edge = E[X] = p*b - q*L > 0 necessary condition. Kelly maximizes E[log(W_n)] (geometric mean), avoiding "gambler's ruin". Half-Kelly (Thorp, Ed Thorp 1962 "Beat the Dealer") recommended in practice to reduce volatility 75% sacrificing only 25% growth. Essential tool for sports betting, blackjack card counting, hedge fund position sizing, professional poker bankroll management.

Kelly Criterion formulas

Classic formula: f* = p - q/b = (p*(b+1) - 1) / b. Example: 55% win probability (p=0.55, q=0.45), even-money payoff (b=1): f* = 0.55 - 0.45/1 = 0.10 = 10% of bankroll. On $10,000 = $1,000 bet. Doubling time E[log(W)] ~ 0.005 = 0.5% per bet, doubles bankroll in ~140 bets.

Generalized formula (asymmetric payoff): f* = (p*(b+L) - L) / (b*L), where L loss multiplier (1 = total loss, 0.5 = 50% stop-loss). Example investment 60% prob, +50% gain b=0.5, -30% loss L=0.3: f* = (0.6*(0.5+0.3) - 0.3) / (0.5*0.3) = (0.48 - 0.3)/0.15 = 0.18/0.15 = 1.20 (>100%, leverage). In practice leverage cap 1.0.

Expected log-growth: G(f) = p*log(1 + f*b) + q*log(1 - f*L). Maximized at f*. G(f*) is geometric growth rate per bet. Ratio G(half-Kelly) / G(full-Kelly) ~ 0.75 but volatility 0.5x: half-Kelly Pareto-superior in many practical scenarios. Ed Thorp empirical: most managers adopt 1/4 to 1/2 Kelly.

Probability of ruin: with full-Kelly, probability that bankroll drops to 50% initial = 50% (long-run). With half-Kelly, probability = 25%. Quarter-Kelly = 12.5%. Professional sports bettors often use 0.1-0.25 Kelly for drawdown management. CFA Institute curriculum coverage: comparison Kelly vs Markowitz mean-variance.

Connection with utility theory: Kelly is equivalent to maximizing logarithmic utility E[log(W)]. Bernoulli (1738 St. Petersburg paradox) already proposed log utility. Daniel Kahneman-Amos Tversky (1979 prospect theory) showed different utilities for real investors (loss aversion). Real-world bet sizing additional factors: regret aversion, transaction costs, bid-ask spread, liquidity constraints.

History and context

The Kelly Criterion was published by John L. Kelly Jr. (1923-1965) in 1956 in the paper "A New Interpretation of Information Rate" (Bell System Technical Journal vol 35), originally as an information transmission problem (Shannon Bell Labs colleague): binary transmission with noise via "horseracing" channel - what maximizes information rate? Kelly showed = log-optimal bet sizing. Edward O. Thorp (MIT/UC Irvine mathematics) applied to blackjack 1960-62 in "Beat the Dealer" (Vintage 1962) with card counting and Kelly bet sizing, generating millions in Las Vegas/Reno. Subsequently Thorp founded Princeton-Newport Partners (1969-1988) first statistical arbitrage hedge fund, applying Kelly-derived position sizing. Kelly was also adopted by Warren Buffett (although Buffett nominally rejects it, his portfolio concentrations are Kelly-consistent), Charlie Munger, William Poundstone "Fortune's Formula" (Hill 2005). Bill Gross (PIMCO) used explicitly. Criticisms: Paul Samuelson 1971 "The 'fallacy' of maximizing the geometric mean" (Proc Nat Acad Sci) argued max geometric != classic utility maximization. Discussion Hakansson-Ziemba "Capital Growth Theory" 1995 (Handbooks in Operations Research). MIT, Stanford, Princeton continue research. Kelly Capital Growth Investment Criterion 2011 conference Hong Kong. Industry: Renaissance Technologies, Citadel, Two Sigma use Kelly-related sizing for quantitative strategies. SEC US, FINRA regulation professional traders.

Kelly fraction table for typical scenarios

  • Fair coin flip p=0.5, b=1, L=1: edge 0, f* = 0 (NO bet)
  • Small advantage p=0.52, b=1: f* = 0.04 = 4% bankroll
  • p=0.55, b=1: f* = 0.10 = 10% (typical sports betting)
  • p=0.60, b=1: f* = 0.20 = 20% (strong blackjack card counting)
  • p=0.55, b=2: f* = 0.325 = 32.5% (favorite high payoff)
  • p=0.40, b=2: f* = 0.10 = 10% (longshot bet with edge)
  • p=0.40, b=3: f* = 0.20 (longshot edge 20%)
  • Asset 60% prob +20%, -10%: f* = (0.6*0.3 - 0.1)/(0.2*0.1) = 4.0 (capped 1.0)
  • S&P 500 stock: assuming +10% annual mean, vol 20%: ~Kelly 100% (CAPM logical)
  • Half-Kelly: 0.5x above values = -75% volatility, -25% growth
  • Quarter-Kelly: 0.25x = -94% volatility, -44% growth (conservative)

Worked numerical examples

Example 1 - professional Hold'em poker. Texas Hold'em professional player average win rate 55% (showdown), even-money payoff. p=0.55, b=1, L=1. Kelly = 0.10 = 10% bankroll/session. Bankroll $50,000, max bet session $5,000. Half-Kelly recommended Phil Ivey/Dan Smith: $2,500/session. Reduces ruin probability 25% maintaining growth 75%. WSOP World Series of Poker Las Vegas.

Example 2 - NFL spread sports betting. Sharp bettor identifies +EV 53% edge on spread (close to coin flip but -110 odds). p=0.53, b=0.909 (american -110 = 100/110), q=0.47, L=1. Kelly = (0.53*1.909 - 1)/0.909 = 0.0114/0.909 = 1.25%. On bankroll $20,000 bet $250. Quarter-Kelly $63 recommended professional bookmaker. Compliance Nevada Gaming Control Board, NJ DGE.

Example 3 - hedge fund equity strategy. Quant strategy with predicted Sharpe 2.0, expected return 20%, volatility 10%. Continuous-time Kelly: f* = mu/sigma^2 = 0.20/0.01 = 20 (10x leveraged max not realistic). In practice leverage cap 2-3x with risk parity. Quarter-Kelly = 5x leverage, half-Kelly 10x. Renaissance Medallion Fund history achieves Sharpe 2-3 supports aggressive Kelly.

Example 4 - blackjack Hi-Lo card counting. True count +5 = +2.5% player advantage, 1:1 payoff (3:2 blackjack). p=0.5125, b=1. Kelly = 0.025 = 2.5% bankroll. Bankroll $100,000 Vegas player, max bet $2,500. Edward Thorp original, MIT Blackjack Team applied. Casino countermeasure: shuffle, bet spread limits, banning known counters.

Example 5 - venture capital startup investing. Angel investor evaluates startup with success probability 20%, expected payoff 10x ($100k -> $1M), failure prob 80%, total loss L=1. Kelly = (0.2*11 - 1)/(10*1) = 1.2/10 = 0.12 = 12% bankroll. Investor $1M should limit single deal commitment to $120k. Diversification 8-10 deals recommended. US: NVCA National Venture Capital Association, SEC Reg D 506(b)(c) accredited investors.

Limits and considerations

  1. Assumes probability p and payoff b known exactly. Real estimate is uncertain: +/- 10% error in p drastically alters Kelly.
  2. Long-run optimization: convergence in 100-1000 trades. Small sample large variance.
  3. Volatility: full-Kelly 50% probability of 50% drawdown at any moment.
  4. Discrete bets: assumes continuous reinvestment, in practice integer indivisible bets.
  5. Tax efficiency: US capital gains 0-37% reduces net return, modify Kelly to after-tax.
  6. Transaction costs and bid-ask spread reduce effective edge: net Kelly lower.
  7. Correlation between trades: Kelly for portfolio requires covariance considered (Kelly-Markowitz hybrid).
  8. Maximum drawdown tolerance: institutional investors 20% max DD, reduces Kelly fraction.
  9. Behavioral biases: full-Kelly is emotionally hard, 50% paper losses create panic selling.
  10. Kelly-vs-Markowitz: Markowitz max single-period mean-variance; Kelly max long-run geometric. Different objectives.

Common errors

  1. Estimating p too optimistically: nonexistent edge, Kelly recommends not betting.
  2. Confusing American odds (-110) with decimals (1.909) or fractional (10/11).
  3. Applying Kelly without fractional cushion (excessive volatility).
  4. Not considering correlation between parallel bets (concentration risk).
  5. Re-betting initial bankroll instead of current bankroll (Kelly is compounding).
  6. Ignoring commissions, spread, taxes (reduce effective edge).
  7. Mixing Kelly with arbitrary stop-losses (sub-optimal).
  8. Using Kelly in negative-edge games (slots, lottery): formula recommends f=0.
  9. Underestimating personal drawdown tolerance (full-Kelly emotionally devastating for many).
  10. Computing Kelly on small sample without confidence interval (high standard error).

Practical applications

  1. Sports betting: NFL/NBA spreads, soccer Asian handicap, MLB run lines.
  2. Card counting: blackjack Hi-Lo, MIT Blackjack Team, original Edward Thorp.
  3. Poker: cash game/tournament bankroll management, Phil Ivey Dan Smith.
  4. Quantitative trading: hedge fund position sizing, Renaissance Medallion.
  5. Stock investing: portfolio concentration, Buffett-Munger Berkshire Hathaway.
  6. Options trading: covered calls, iron condors, volatility position sizing.
  7. Venture capital: startup angel investing, NVCA, SEC Reg D accredited.
  8. Crypto trading: Bitcoin spot, perpetual futures Binance, dYdX.
  9. Insurance: reinsurance treaty pricing, Lloyd's of London syndication.
  10. Game theory research: MIT, Stanford, Princeton academic Kelly Capital Growth.

Support

Questions: info@calcoi.com. Educational tool for bet sizing. For professional applications consult Kelly 1956 Bell System Tech J, Thorp "Beat the Dealer" 1962, Thorp "The Mathematics of Gambling" 1984, Poundstone "Fortune's Formula" 2005, MacLean-Thorp-Ziemba "The Kelly Capital Growth Investment Criterion" World Scientific 2011.

Updated July 2026

Frequently Asked Questions

What does the Kelly Criterion measure?

Measures optimal bankroll fraction to bet/invest maximizing expected logarithmic growth of capital (E[log(W_n)]) long-term. Different from maximize expected value (can lead to ruin). Kelly guarantees survival + optimal growth. Kelly 1956 original derivation from information theory (Shannon Bell Labs colleague). Adopted by Edward Thorp blackjack/Princeton-Newport Partners, Warren Buffett implicitly.

Difference between Kelly and expected value?

Expected value E[X] = p*b - q*L measures only mean gain. Maximizing EV can suggest excessive bets (even 100% bankroll), causing ruin with probability >0. Kelly maximizes E[log(W)] = geometric growth, naturally bounded by -infinity from left (preventing 100% bet). Kelly is always <= 100% bankroll so survival guaranteed long-run. For p=0.55 b=1: EV = 0.10 but Kelly = 0.10 conservative, keeps bankroll positive.

Half-Kelly: why recommended?

Half-Kelly (0.5*f*) sacrifices only 25% growth but reduces volatility by 75% (1/4 variance). Probability of 50% bankroll loss goes from 50% (full-Kelly) to 25% (half) to 12.5% (quarter). Edward Thorp empirical advice: full-Kelly emotionally hard, predictable -50% drawdowns, panic selling. Professional bookmakers, hedge funds, conservative investors adopt 1/4 to 1/2 Kelly. CFA curriculum covers, MacLean-Thorp-Ziemba 2011 World Scientific.

Does Kelly work if you're not sure of p and b?

Sensitivity to parameter estimation. +/- 5% error in p drastically changes Kelly (e.g., p=0.55 vs p=0.50: Kelly 10% vs 0%). In practice use fractional Kelly (1/4 or 1/2) for cushion. Bayesian update: prior + empirical evidence progressively refine estimate. Insufficient data: bet conservatively Quarter-Kelly. Confidence interval p incorporated: use lower bound CI rather than point estimate. Renaissance, MIT Blackjack Team use Bayesian estimators.

Kelly in continuous financial markets?

For continuous assets (stocks, ETFs) with expected return mu and volatility sigma: continuous-time Kelly f* = mu/sigma^2. Example S&P500 mu=10%, sigma=20%: f* = 0.1/0.04 = 2.5 (250% leverage). Practical: capping leverage 1-2x with risk parity. Growth rate G = mu^2/(2*sigma^2). Equivalent to maximize Sharpe ratio for single asset. Multi-asset with covariance: f* = Sigma^-1 * mu vector (Kelly-Markowitz hybrid). Renaissance Medallion Fund evidence Sharpe 2-3 long-run.

When NOT to apply Kelly?

(1) Negative edge (slots, lottery, casinos house edge): formula recommends f=0, do NOT bet. (2) Small bankroll: indivisible discrete bets make Kelly imprecise. (3) Tail risk (black swan, fat-tailed distributions): log-normal assumption breaks; use Cumulant ESS expansion. (4) Short time horizon: long-run convergence not guaranteed. (5) Non-fungible bankroll (capital needed for living expenses): cannot be lost. Sample LTCM 1998 Kelly deviation catastrophic (Black-Scholes assumption breakdown).

Kelly for multi-asset portfolio?

Vector extension: f* = Sigma^-1 * mu, where mu vector of expected returns, Sigma covariance matrix. Equivalent to Markowitz mean-variance portfolio when log utility approximation. Kelly naturally diversifies: low covariance assets reduce concentration risk. Short-selling constraint prevents f<0. For CFA Institute portfolio management curriculum: Kelly long-run optimal, Markowitz single-period optimal. Differences: Kelly ignores intermediate volatility, Markowitz penalizes. Industry practice: hybrid blend.

Formula

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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