This article explains the full Kelly Criterion from first principles to practical implementation. You will learn the canonical single-bet formula, why maximizing expected log wealth matters for repeated bets, how Kelly extends to multiple assets, and which conservative adjustments professionals use in practice. The intended audience includes sports analysts, quantitative bettors, and anyone interested in disciplined bet sizing and portfolio allocation. The discussion balances theoretical footing with actionable steps and worked examples you can try with a virtual bankroll.

The full Kelly Criterion gives a closed form bet fraction for single binary wagers that maximizes expected log wealth.

In multiasset settings, Kelly generalizes to an objective that often approximates to Sigma inverse mu under small returns.

Practitioners commonly use fractional Kelly to reduce drawdowns and compensate for estimation risk.

What the full Kelly Criterion is

The full Kelly Criterion is a prescription for how much of a bankroll to allocate to a favorable bet or investment if your goal is to maximize long run geometric growth of wealth. The rule follows from choosing the fraction that maximizes expected log wealth and was first derived in the information theory literature as a growth optimal solution, which the original work framed in terms of information rate and betting odds Bell System Technical Journal. See the Wikipedia article Kelly criterion.

What is the full Kelly Criterion and when should I use it?

The full Kelly Criterion is the allocation that maximizes expected log wealth; use it when you have high confidence in probability and return estimates and a long horizon, but prefer fractional Kelly when uncertainty, drawdown limits, or operational constraints are significant.

In plain terms, the Kelly approach picks the stake that, under modeled probabilities and outcomes, gives the highest expected rate of compounded growth. The formal statement and motivating examples were expanded in later treatments that apply the same objective to markets, blackjack, and sports betting, showing how the criterion connects betting with information theory and portfolio choice Thorp, arXiv. See our blog.

Short history and original motivation

John L. Kelly proposed the idea in the 1950s while working on problems of communication and information. His contribution was to relate an information rate objective to a bet sizing rule that maximizes expected logarithmic growth of wealth. That framing made it possible to move from questions about data transmission to concrete betting recommendations under known odds and probabilities Bell System Technical Journal. A practitioner overview is on the CFA blog CFA Institute.

Where Kelly fits in betting, trading, and information theory

Because the Kelly objective targets long run compounding, it is relevant wherever repeated bets or repeated investments occur and the participant cares about geometric growth. The same math underlies certain portfolio selections and trading rules, and later researchers used Kelly style objectives in continuous time and multiasset settings to connect the method with classic portfolio theory Econometrica. See a Berkeley note Berkeley note.

The single-bet Kelly formula - exact result and interpretation

Consider a single binary wager that pays net odds b to 1 on a win and loses the stake on a loss. Let p be the probability of winning and 1 minus p the probability of losing. Maximizing expected log wealth in that simple setting yields the closed form Kelly fraction f star equals (b p minus (1 minus p)) divided by b, so f* = (b p - (1 - p)) / b. This expression is the canonical single-bet result from the original derivation Bell System Technical Journal.

Setup: binary wager with probability p and net odds b

Define the variables clearly. Bankroll refers to the total capital used for repeated wagers. If you stake a fraction f of bankroll on the binary bet, a win multiplies the staked amount by 1 plus b while a loss subtracts the stake. The net odds b is the amount won per unit staked, net of returning the stake on a win.

Derivation sketch and final closed form

The algebra is straightforward. Write the expected logarithm of wealth after one bet as p times log(1 + f b) plus (1 minus p) times log(1 minus f). Differentiate that expression with respect to f, set the derivative to zero, and solve for f to obtain f* = (b p - (1 - p)) / b. When f star is positive you have a positive edge and should bet that fraction; when it is zero you are at breakeven; when it is negative the correct action is to avoid the bet or effectively short it if allowed Thorp, arXiv.

$kelly criterion single bet diagram showing probability p odds b bankroll and resulting fraction in a clean minimal Funded Plays infographic$

Interpretation is direct: the numerator captures the net expected gain per unit of stake, scaled by the odds, and dividing by b converts that net advantage into an optimal fraction of bankroll.

Why maximize expected log wealth - intuition and asymptotic optimality

Maximizing expected log wealth prioritizes geometric growth because repeated multiplications of returns compound, and the long run growth rate is governed by average log returns. That focus makes the Kelly approach natural when the objective is to maximize wealth over many independent opportunities rather than to maximize a single period expected value.

At a high level, log utility penalizes volatile strategies that reduce compounded growth even if they have attractive short term expected returns. This is why a strategy chosen to maximize expected log wealth tends to balance edge and risk in a way that favors consistent compounding rather than large, infrequent gains.

Practice bet sizing in a structured challenge

Try the worked examples in this article using a small virtual bankroll to see how the allocations change when you alter probabilities and odds. Treat the exercises as a learning tool for risk management, not as a promise of outcomes.

View FundedPlays Challenges

The formal mathematical reinforcement of that intuition comes from results proving that, under the Kelly assumptions, the Kelly policy almost surely achieves the highest exponential growth rate and asymptotically dominates any fixed fraction strategy. That theorem, established in the literature on favorable games, gives the long run dominance claim that makes Kelly distinctive among sizing rules Breiman, Annals.

Intuitive explanation: compounding and geometric growth

When wealth is multiplied repeatedly by returns, the median or typical outcome is governed by geometric factors, not arithmetic means. Maximizing expected log return aligns decisions with that geometric objective and therefore aims to maximize the typical compounded outcome rather than occasional large wins that can be offset by deep losses.

Formal result: Breiman's almost-sure exponential growth dominance

Breiman proved that if the assumptions hold, the Kelly strategy attains an almost-sure exponential growth rate that exceeds that of any fixed fraction betting scheme in the long run. The result is asymptotic and relies on stationarity and correctly specified probabilities, so its power is rigorous but conditioned on those model assumptions

What is the full Kelly Criterion in one sentence?

The full Kelly Criterion is the stake fraction that maximizes expected log wealth, giving a closed form for single binary bets and a portfolio objective for multiple assets.

When should I use fractional Kelly instead of full Kelly?

Use fractional Kelly when inputs are uncertain, when you need lower drawdowns, or when operational limits and psychological tolerance require smaller stakes.

Does Kelly guarantee long term profits?

No. Kelly optimizes long run geometric growth under its assumptions, but it does not guarantee profits when probabilities or stationarity assumptions are wrong.

If you want to learn by doing, run the examples here with modest stakes or simulated bankrolls and compare full Kelly with fractional variants to see how paths and drawdowns differ. Keep conservative controls in place and treat Kelly as a disciplined input to a broader risk management process.