The Kelly Criterion (or Kelly formula, Kelly strategy, Kelly bet) is a mathematical rule that tells you the optimal fraction of your bankroll to stake on a single bet or investment to maximize long‑term (geometric) growth of wealth. First published by John L. Kelly Jr. in 1956 at Bell Labs, it has been used by gamblers and investors to size positions based on the probability of winning and the average payoff when winning versus losing. (Source: Investopedia, Kelly Criterion.)
How the Kelly Criterion Works — Intuition
– Objective: maximize the long‑run growth rate of capital (maximize expected log wealth).
– Inputs: an estimate of the probability of a favorable outcome and an estimate of how large wins are relative to losses.
– Output: the fraction of current capital to allocate to the bet or trade.
– Key tradeoff: Kelly balances reward versus the risk of ruin and frequency of wins/losses. It is often aggressive; many practitioners use a fractional Kelly (e.g., half‑Kelly) to reduce volatility and drawdowns.
Core formulas (discrete, gambling-style)
1) Standard form for a bet that pays net odds b (i.e., you gain b dollars for each $1 staked if you win), with win probability p and loss probability q = 1 − p:
f* = (b p − q) / b
– f* is the optimal fraction of bankroll to wager. If f* ≤ 0, don’t bet.
2) Equivalent form often used by traders (from Investopedia):
K = W − (1 − W) / R
– W = probability of a favorable return (p).
– R = average win size ÷ average loss size (this corresponds to b in the other formulation if wins/losses are measured similarly).
– K is the fraction of the bankroll to commit.
Continuous (lognormal) case for an investment with expected excess return μ (over risk‑free r) and volatility σ:
f* = (μ − r) / σ^2
– This is the Kelly fraction when returns follow geometric Brownian motion and you maximize expected log wealth. If r = 0, f* = μ / σ^2.
Practical examples
Example 1 — Simple betting:
– Coin‑type bet: p = 0.6 (60% to win), on a win you gain equal amount (b = 1), on loss you lose your stake.
– f* = (1*0.6 − 0.4)/1 = 0.2 → stake 20% of bankroll.
Example 2 — Trader’s W and R:
– Suppose from recent trades you estimate W = 0.55 and average win = $200, average loss = $100 ⇒ R = 200/100 = 2.
– K = 0.55 − (0.45)/2 = 0.55 − 0.225 = 0.325 → 32.5% of bankroll (again, aggressive in practice).
Example 3 — Continuous stock case:
– Excess expected return μ − r = 4% (0.04), volatility σ = 20% (0.20).
– f* = 0.04 / 0.20^2 = 0.04 / 0.04 = 1 → 100% of portfolio in the risky asset (this points out how sensitive Kelly is to input estimates).
How to estimate the inputs (practical steps)
1) Define your bankroll clearly (liquid capital available for the strategy).
2) Estimate win probability (W or p):
• Historical approach: count fraction of past trades that were profitable. Many traders use last 50–100 trades to get a rolling estimate, but beware small-sample noise.
• Probabilistic/statistical approach: build a predictive model for p (signal-based probability), or use Bayesian updating to incorporate prior beliefs and uncertainty.
• Be conservative: shrink raw historical p toward 0.5 or use confidence intervals to reflect estimation uncertainty.
3) Estimate win/loss size (R or b):
• Compute average winning trade P&L and average losing trade P&L (use absolute values). R = avg win / avg loss.
• Use median or trimmed mean if returns have outliers. Consider expected profit per dollar risked, not only average dollar amounts.
4) If you model continuous returns, estimate expected excess return μ − r and variance σ^2 from historical returns; use robust estimators and account for serial correlation.
5) Run sensitivity analysis: check how f* changes for plausible ranges of p, R, μ, σ.
Applying the Kelly recommendation — practical rules
– Use fractional Kelly: many practitioners scale f* down (commonly half‑Kelly) to reduce extreme volatility and drawdowns while retaining much of Kelly’s long‑term advantage.
– Cap allocations: impose limits per asset (e.g., no single position > X% of portfolio) and absolute leverage caps.
– Diversify: Kelly can recommend concentrated or leveraged positions — diversify across uncorrelated bets where practical.
– Transaction costs & constraints: include costs, slippage, taxes, and margin constraints—Kelly ignores these but they matter.
– Rebalance periodically: as bankroll and estimates change, recompute and tilt positions; avoid over‑trading on small fluctuations.
– Stress test: simulate with adverse sequences and realistic estimation errors to see potential drawdowns.
– Record keeping: maintain trade history and update probability estimates with new data.
Multiple bets and portfolio Kelly
– For independent bets with vector expectations and covariances, the general Kelly solution (quadratic optimization) is:
f* = Σ^−1 μ
where μ is a vector of expected excess returns and Σ is the covariance matrix of returns (this comes from maximizing expected log return under multivariate normal assumptions). In practice this is highly sensitive to estimation error.
– When bets are correlated or inputs uncertain, regularize estimates (shrinkage, Bayesian priors) and use fractional Kelly.
Limitations, caveats and warnings
– Input sensitivity: small errors in p, R, μ or σ produce large changes in f*. Biased estimates can produce ruinous recommendations.
– Drawdowns and volatility: full Kelly can imply large drawdowns even while maximizing asymptotic growth; many investors cannot tolerate these psychologically or operationally.
– Single-asset risk: Kelly may recommend concentrating in a single “edge” — diversification and nonfinancial constraints often make this undesirable.
– Real-world frictions: transaction costs, discrete trade sizes, borrowing costs, taxes, margin calls, and liquidity limits are not captured in the simple formula.
– Utility mismatch: Kelly maximizes logarithmic utility (long‑term geometric growth). If your utility function or horizon differs (e.g., you need capital soon), Kelly may be inappropriate.
– Theoretical critiques: some economists stress that practical constraints and different utility considerations can make expected-utility frameworks or other sizing rules preferable.
Fractional Kelly and rule-of-thumb adjustments
– Half‑Kelly: stake 0.5 × f* — common compromise to reduce volatility and estimation sensitivity.
– Conservative floors: never exceed a pre-specified percentage per trade, e.g., 2–5% of bankroll.
– Use ensemble estimates: combine model-based p with subjective adjustments and stress scenarios.
Kelly vs Black‑Scholes (relationship and differences)
– Both are mathematical frameworks applied to investments and often assume lognormal returns in continuous time.
– Black‑Scholes: builds a risk‑neutral option pricing model that values options by hedging and arbitrage; it does not tell you how much to invest to maximize growth.
– Kelly: gives position sizing to maximize long‑term growth under real‑world return probabilities (and can be applied in continuous‑time models where it reduces to f* = (μ − r)/σ^2).
– Key difference: Black‑Scholes uses a risk‑neutral measure to price derivatives; Kelly uses real-world parameters to size positions for growth. They intersect when investors ask how to size an options/hedge position given real expected returns and volatility assumptions.
Practical implementation checklist
1) Define objective and horizon (Kelly is for maximizing long‑term capital growth).
2) Choose a stable bankroll and tracking system.
3) Estimate inputs (p and R, or μ and σ) with robust methods and regular updates.
4) Compute Kelly fraction(s).
5) Adjust: apply fractional Kelly, caps, and limits; include transaction costs.
6) Diversify and compute portfolio-level Kelly if multiple bets.
7) Rebalance and reestimate periodically; track realized performance vs expectations.
8) Stress‑test and document assumptions; be prepared to reduce sizing when model confidence is low.
Quick rules of thumb
– If your win probability and payoff edge are large and well‑estimated, Kelly can yield aggressive sizing.
– If your estimates are noisy or you cannot tolerate large drawdowns, use fractional Kelly and caps.
– Never use raw historical win rate from tiny samples without adjustment.
References and further reading
– Investopedia — “Kelly Criterion” (source URL:
– J. L. Kelly, Jr., “A New Interpretation of Information Rate,” Bell System Technical Journal, 1956 (original Kelly paper)
– Texts on portfolio theory and utility maximization (for the multivariate and continuous Kelly results)
Final note
The Kelly Criterion is a powerful theoretical tool for sizing positions to maximize long‑run growth, but it depends critically on accurate input estimates and a tolerance for volatility. Treat its output as a starting point for position sizing, not a prescription. Always incorporate diversification, transaction costs, personal constraints, and conservative scaling (fractional Kelly) when applying it to real portfolios.
(Not financial advice.)
(Continuing from the earlier overview)
Practical implementation of the Kelly Criterion requires a structured approach, an understanding of its variants, realistic estimation of inputs, and sensible risk controls. Below I expand the explanation, give examples, show how to apply it step‑by‑step, discuss multi‑asset/continuous variants, and summarize practical warnings and best practices.
What the Kelly Criterion maximizes
– The Kelly Criterion chooses bet sizes (or portfolio weights) that maximize the long‑term growth rate of wealth by maximizing the expected logarithm of wealth (expected log utility).
– That makes Kelly “growth‑optimal” over long horizons, but it does not necessarily minimize short‑term losses or maximize other objectives (e.g., probability of having more wealth than a benchmark at a fixed horizon).
Key Kelly formulas (common variants)
1. Binary bet with net odds b and win probability p:
f* = (b p − q) / b where q = 1 − p.
• For even odds (b = 1), f* = 2p − 1.
2. Win/loss ratio form (used by many traders):
K = W − (1 − W) / R
• W = probability of a favorable outcome (win probability).
• R = average win / average loss (win size divided by loss size).
• K is the fraction of bankroll to risk on that bet.
3. Continuous/multi‑asset (Gaussian approximation / growth‑optimal portfolio):
For one risky asset (expected excess return μ, variance σ^2),
f* = μ / σ^2 (fraction of wealth to invest in the risky asset)
For multiple assets, the Kelly vector (f*) under multivariate normal returns solves:
f* = Σ^−1 μ
where μ is the vector of expected excess returns and Σ is the covariance matrix.
• These come from maximizing expected log return under normal-return assumptions.
Worked examples
Example 1 — simple biased coin (binary, even odds)
– Suppose an even‑payout bet (if you win you get $1 for $1 staked) and win probability p = 0.6.
– Use f* = 2p − 1 = 2(0.6) − 1 = 0.2.
– Kelly says bet 20% of current bankroll each round.
Example 2 — trading wins and losses (use R form)
– Suppose in your strategy:
• Probability of a winning trade W = 0.55 (55%),
• Average win = +3% (0.03), average loss = −2% (0.02).
• R = 0.03 / 0.02 = 1.5.
– K = 0.55 − (0.45)/1.5 = 0.55 − 0.30 = 0.25 → bet 25% of bankroll.
– Many traders use a fractional Kelly (e.g., half‑Kelly) to reduce volatility: half‑Kelly = 12.5% here.
Example 3 — single risky asset with mean/variance
– Suppose expected excess return μ = 5% = 0.05 and annual variance σ^2 = 0.20^2 = 0.04.
– f* = μ / σ^2 = 0.05 / 0.04 = 1.25 → implies 125% of bankroll (leverage).
– This highlights Kelly can recommend leverage when the edge is high relative to variance. Real investors may cap leverage.
Step‑by‑step practical procedure to use Kelly
1. Define the bet/investment precisely (one trade, repeated bet, position in an asset).
2. Estimate the edge:
• For binary trades: estimate win probability p (W).
• For trading strategies: estimate average win size, average loss size to compute R, and estimate W.
• For continuous assets: estimate expected excess return μ and covariance Σ (or σ^2 for a single asset).
Use quality data, not hand‑waving guesses. Use large samples if possible.
3. Compute the Kelly fraction using the appropriate formula.
4. Adjust the result for uncertainty:
• Use fractional Kelly (common choices: 1/2 or 1/3 Kelly) to reduce volatility and the impact of estimation error.
• Impose practical portfolio constraints (max position size, no more than X% in one trade).
5. Incorporate transaction costs, taxes, margin requirements, and liquidity limits (these all reduce effective edge).
6. Backtest and stress test: simulate Kelly sizing with historical or bootstrap data to see drawdown behavior.
7. Reestimate inputs periodically and rebalance; avoid over‑reacting to a few trades.
8. Use risk controls: stop losses, maximum drawdown limits, diversification across uncorrelated bets.
Dealing with estimation error and small samples
– Kelly is highly sensitive to input estimates. Small samples can greatly overestimate the true edge.
– Practical mitigations:
• Use shrinkage or Bayesian estimates (pull extreme estimates toward zero).
• Use fractional Kelly (reduces sensitivity to mistakes).
• Increase sample size and use bootstrapping to estimate confidence intervals.
• Impose a maximum allocation cap relative to portfolio wealth.
Multi‑bet and portfolio situations
– When bets are independent or weakly correlated, Kelly can allocate across multiple opportunities.
– The multivariate Kelly solution (f* = Σ^−1 μ) picks weights that account for correlations; if assets are highly correlated, Kelly will reduce combined exposure.
– For many real portfolios, estimating full Σ reliably is difficult; use factor models, shrinkage estimators, or simpler diversification rules.
Kelly and Black‑Scholes — how they relate
– Both are mathematical systems dealing with probabilities and returns, but they address different problems:
• Black‑Scholes prices options in a risk‑neutral world and produces a hedge/repl ication strategy under continuous-time assumptions.
• Kelly concerns position sizing to maximize long‑run growth based on real‑world (physical) probabilities and expected returns.
– In continuous-time portfolio optimization, maximizing expected log‑wealth (Kelly) yields a dynamic allocation rule (growth‑optimal portfolio), while Black‑Scholes is about valuation and hedging of contingent claims.
– Both rely on model assumptions; mismatch between model and reality (wrong probability measure, misestimated volatility) will change practical outcomes.
Limitations and warnings (expanded)
– Estimation risk: small sample bias can produce dangerously large Kelly fractions.
– Drawdowns: full Kelly often creates large interim drawdowns; many investors prefer fractional Kelly to reduce psychological and financial stress.
– Single‑asset concentration: Kelly may recommend large bets on a single perceived edge; this conflicts with diversification preferences and practical constraints.
– Finite horizon: Kelly optimizes asymptotic growth; for finite horizons, different risk preferences or utility functions (e.g., mean–variance or CRRA utility) might be more appropriate.
– Leverage and constraints: margin calls and borrowing costs can destroy the theoretical Kelly advantage.
– Nonstationary edges: if your edge disappears (regime change), Kelly sizes based on past data can be harmful.
– Correlated bets: naive application to correlated bets can lead to unintended concentrations; use multivariate Kelly or conservative caps.
Practical rules of thumb
– Use fractional Kelly (1/2 or 1/3 Kelly) unless you have very high confidence in your estimates.
– Always shrink extreme parameter estimates toward zero.
– Limit any single position to a reasonable percentage (e.g., 5–25%) depending on risk tolerance.
– Reestimate inputs regularly and stress test for worst‑case scenarios.
– Combine Kelly sizing with portfolio diversification rather than betting a single asset.
Key insights to keep in mind
– Kelly maximizes long‑term geometric growth—not short‑term utility or safety.
– It is a disciplined, quantitative approach, but its effectiveness depends critically on accurate probability and return estimates.
– Many successful investors have used Kelly‑like ideas but tempered them with risk limits, fractional Kelly, and practical constraints.
Concluding summary
The Kelly Criterion is a powerful framework for sizing bets or portfolio weights to maximize long‑term growth by maximizing expected log wealth. In simple binary cases, it produces a clear, interpretable fraction of wealth to risk (e.g., 2p − 1 for even‑odds bets). In multi‑asset or continuous settings it links to mean/variance inputs (f* = Σ^−1 μ). However, practical use requires careful parameter estimation, consideration of leverage, controls against drawdowns, and often conservative modifications like fractional Kelly. Use Kelly as a disciplined guide to position sizing, not as an automatic all‑or‑nothing rule: combine it with diversification, robust estimation techniques, and real‑world constraints.
References and further reading
– John L. Kelly Jr., “A New Interpretation of Information Rate,” Bell System Technical Journal, 1956.
– Investopedia, “Kelly Criterion,”