Kelly Criterion: Optimal Bet Sizing

The Kelly Criterion gives the theoretically optimal fraction of bankroll to wager on a +EV bet to maximize long-term logarithmic growth while avoiding ruin.

Kelly optimizes growth rate, not comfort. Full Kelly often creates large equity swings that many bettors cannot psychologically tolerate. Overbetting relative to true edge drastically increases risk of deep drawdowns or ruin.

Formula (Decimal Odds)

Let p be your true probability and o the decimal odds.

Profit per unit staked if win = o - 1; loss if lose = 1.

Kelly Fraction f* = (p * (o - 1) - (1 - p)) / (o - 1)
                 = (o * p - 1) / (o - 1)

Stake = f* * Bankroll.

Example: p = 0.55, o = 2.00:

f* = (2 * 0.55 - 1) / (1) = 0.10 (10% of bankroll)

If your real probability was only 0.52 (estimation error):

f* = (2 * 0.52 - 1)/1 = 0.04 (4%)

Betting 10% instead of 4% magnifies variance and regret.

Why Full Kelly Is Often Too Aggressive

Inputs (your probabilities) contain error. Overbetting compounds estimation error and variance. Many pros use ½ Kelly or ¼ Kelly to smooth drawdowns.

Fractional Kelly

Stake = (Fraction * f*) * Bankroll

If full Kelly suggests 8% and you use ½ Kelly, you stake 4%.

Practical Fraction Choices

Fraction	When Used	Rationale
1 (Full)	Rare; highly validated quant models	Max theoretical growth when edge precise
0.5	Experienced, stable calibration	Smoother growth vs full
0.25	Developing models / moderate error	Protective compromise
0.1–0.2	High uncertainty / exploratory	Capital preservation first

Multiple Simultaneous Bets

Kelly is derived for isolated wagers. With overlapping markets or correlated outcomes, summing raw Kelly stakes can overexpose you.

True “portfolio Kelly” needs return covariance. Shortcut: apply fractional Kelly per bet and impose a maximum simultaneous exposure cap (e.g., never have >15% total bankroll risked across unsettled positions).

Practical Guidelines

• Start with ¼ Kelly until you have stable probability calibration.
• Cap max stake % to avoid tail risk.
• Recalculate bankroll periodically (weekly / monthly) rather than after every bet to reduce churn.
• Log: probability, odds, stake fraction, result.
• Track realized vs suggested Kelly to audit discipline.
• Use a hard monetary max per wager to mitigate data/input errors.

Drawdowns & Volatility

Full Kelly commonly experiences 30–50% peak-to-trough drawdowns even with a genuine edge. Fractional Kelly reduces variance roughly proportional to the fraction (variance scales with f, growth penalty is sublinear).

Edge Uncertainty Adjustment

When probability estimate uncertainty is high, shrink f*. Simple heuristic:

Adjusted f = f* * ConfidenceFactor

Where ConfidenceFactor ∈ (0,1). Err on conservatism; overbetting destroys the log-optimal property faster than underbetting.

Compounding Cadence

To avoid over-reactive micro changes, update bankroll reference weekly or after ±10% swings rather than after every result.

FAQs

f is negative – what now?* Skip the bet; edge is negative. Consider if opposing side is available at positive EV.

Is flat staking inferior? For a known steady edge, yes; but with noisy estimates, conservative flat or fractional Kelly may dominate in realized Sharpe (risk-adjusted return).

Can I exceed Kelly because I’m certain? Confidence bias; systematically overestimation leads to ruin. Do not exceed full Kelly.

Is Martingale related to Kelly? No. Martingale ignores EV and bankroll risk; Kelly is grounded in information theory.

Takeaway

Kelly provides a rational framework for sizing—but prudence (fractional Kelly + caps) protects against noisy inputs and black swans.