Kelly Criterion for Betting: Formula, Checks and Model Risk

Derive the binary formula

Let decimal odds be d, net odds be b = d - 1, estimated win probability be p, and loss probability be q = 1 - p. For a fraction f of bankroll:

• wealth multiplier after a win = 1 + f * b;
• wealth multiplier after a loss = 1 - f.

Maximizing expected log wealth under this model gives the following fraction (Kelly, 1956):

f* = (b * p - q) / b = (p * d - 1) / (d - 1)

Kelly's original paper provides the log-growth basis for proportional staking. The formula does not estimate p; that separate input is usually the main source of risk.

Worked example

Assume an illustrative decimal price of 2.20 and model probability p = 0.50.

f* = (0.50 * 2.20 - 1) / (2.20 - 1)

f* = 0.10 / 1.20 = 0.083333, or 8.333% of the defined bankroll

In this illustrative example, a GBP 600 model bankroll produces the mathematical full-Kelly output:

GBP 600 * 0.083333 = approximately GBP 50

This is a formula check, not a recommended or affordable stake.

Probability sensitivity

Keep the price at 2.20 and change only p:

At p = 0.46, (0.46 * 2.20 - 1) / 1.20 = 0.01. At p = 0.45, the unconstrained expression is negative; for a back-only decision the action is no bet, not a negative back stake.

The small probability change produces a large fraction change. Peer-reviewed sports-wagering work explicitly treating p as unknown found materially different, smaller modified fractions in its studied cases (Chu, Wu and Swartz, 2018).

Preconditions for using the output

Record:

1. probability model, version, and information cutoff;
2. calibration and later-sample evaluation;
3. accepted odds and settlement payoff;
4. commission, tax, limits, and rejected stake treatment;
5. bankroll definition and already-open exposure;
6. dependence with other positions;
7. fraction or risk constraint applied after full Kelly.

Do not use a confidence label, tipster win rate, or closing price as p without a documented conversion and validation.

Growth objective is not drawdown control

Full Kelly optimizes a long-run model objective and can still generate substantial drawdowns. Risk-constrained Kelly research treats drawdown probability as an additional constraint rather than assuming growth optimization is enough.

Personal affordability is outside both models. A mathematically positive fraction remains invalid if the money is needed elsewhere or gambling is causing harm.

Verify boundary cases and portfolio use

Check three binary back-bet boundaries from the Kelly formula. At p = 0, the unconstrained fraction is negative. At p = 1 / d, f must equal zero. At p = 1, f equals one under the idealized model. These checks catch use of gross odds in the wrong term, percentage-versus-decimal mistakes, and missing no-bet constraints.

For simultaneous positions, do not calculate each stake against the full bankroll and add them without a joint model. Build outcome scenarios or a covariance-aware portfolio calculation, then apply cash, event, and drawdown limits. Store the unconstrained model output and every operational reduction separately. That preserves the mathematics while making clear which stake was actually proposed and why.

Next step

Use Kelly Criterion for the next part of this topic.

Continue learning

• Next guide: Martingale and Fibonacci Betting Systems
• Related guide: Percentage Staking
• Definition: Kelly Criterion

Assumptions and limitations

The derivation is for one binary, fully settled back bet with known decimal payoff and no simultaneous positions. Real football markets can have pushes, partial wins, exchange commission, dependent bets, price movement, and uncertain probabilities. This page verifies the formula; it does not validate a model or recommend a stake.