Kelly Criterion for Value Bettors: Formula and Model Risk

Formula

For a binary one-unit back bet:

• p = estimated win probability.
• q = 1 - p.
• b = decimal odds - 1, the net profit per unit won.
• Full Kelly fraction f = (b * p - q) / b.

This is equivalent to f = (p * d - 1) / (d - 1). Kelly's original paper develops the logarithmic-growth objective (Kelly, 1956). The formula does not validate p or guarantee smooth growth.

Checked example

At decimal odds d = 2.20 and p = 0.50:

• b = 2.20 - 1 = 1.20.
• q = 1 - 0.50 = 0.50.
• f = ((1.20 * 0.50) - 0.50) / 1.20 = 0.08333.

Full Kelly is 8.333% of the bankroll recorded at calculation time under those exact assumptions. Half Kelly = 0.5 * 0.08333 = 0.04167, or 4.167%. Quarter Kelly = 0.25 * 0.08333 = 0.02083, or 2.083%. The objective and assumptions come from Kelly's original paper.

Probability error changes the stake quickly

Keep d = 2.20:

At p = 0.46, f = ((1.20 * 0.46) - 0.54) / 1.20 = 0.01. A four-point change from 50% to 46% reduces the fraction from 8.333% to 1%.

Fractional Kelly does not repair calibration

Fractional Kelly deliberately reduces the full fraction. It can reduce growth volatility and sensitivity to error, but an overconfident or stale probability can still prescribe a harmful positive stake. Check probability calibration on later events; scikit-learn's calibration guide explains why stated probabilities need frequency validation.

Cases where the simple formula is insufficient

• Quarter handicaps, pushes and partial settlement.
• Correlated simultaneous bets.
• Exchange commission or tiered costs.
• Limits and partial acceptance.
• Uncertain or changing bankroll values.
• Multiple mutually exclusive opportunities in one market.

Model every net payoff state or use a tested optimization appropriate to the portfolio. Never add individual full-Kelly fractions while ignoring dependence; the simple binary derivation in Kelly's original paper does not establish that independently calculated fractions remain optimal for correlated bets.

Implementation checklist

1. Recalculate EV before stake size.
2. Use the accepted price, not the earlier displayed price.
3. Cap exposure independently of the formula and personal affordability.
4. Store the probability version and stake decision before the event.
5. Recompute from the current ring-fenced bankroll.
6. Stop using the rule when calibration or data quality fails.

Write a staking policy before using the output

A reproducible policy states the bankroll definition, fraction of Kelly, maximum per-event exposure, treatment of correlated bets, minimum probability evidence, price-change rule, and action when a model or data check fails. It also records whether a negative Kelly output means no bet rather than a negative stake under the Kelly objective.

Run a shadow period in which fractions are calculated but no staking change is made. Compare the proposed exposure with flat staking, inspect concentration by team and market, and test deliberately perturbed probabilities. If a small input change produces an unacceptable stake change, reduce exposure or stop. This is a governance decision around the Kelly objective, not proof that one fraction is universally suitable.

Next step

Use Kelly Criterion Betting for the next part of this topic.

Continue learning

• Next guide: Luck vs Edge in Betting
• Related guide: Overround and Value

Assumptions and limitations

The example is illustrative and assumes a repeatable binary payoff with reliable probabilities. Kelly maximizes a mathematical objective, not personal welfare, and it can produce substantial drawdowns. Betting should never use money needed for essentials; the NHS gambling-harm guide provides safeguards and support routes.