04 // Interactive Formula Compendium // FINANCEf* = (b · p − q) / b = p − (1 − p) / b
Kelly Criterion Formula — Optimal Capital Allocation & Geometric Growth
The Kelly Criterion f* = (bp − q) / b determines the optimal capital fraction f* to wager when win probability is p, loss probability is q = 1 − p, and net payout odds are b-to-1, maximizing expected logarithmic wealth E[ln W].
Formula Reference
Main Formula
f* = (b · p − q) / b = p − (1 − p) / b
Rearranged Forms
f*_continuous = (μ − r) / σ² (Merton-Kelly Continuous Portfolio Fraction)
g(f) = p · ln(1 + b·f) + q · ln(1 − f) (Expected Logarithmic Growth Rate)
f_half = 0.5 · f* (Half-Kelly: 75% of Growth Rate with 50% Volatility)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| f* | Optimal Kelly Bankroll Fraction (%) | % |
| p | Win Probability (p, 0 to 1) | 1 |
| b | Net Payout Odds Ratio (b = Win / Risk) | 1 |
Inputs & Parameters
[1]
[1]
Calculated Result (Optimal Kelly Bankroll Fraction (%))
f* =10%
Formula Notes & Background
- After N independent trials with W wins and L losses, wealth is W_N = W₀ (1 + bf)^W (1 − f)^L.
- Maximize the expected logarithmic growth rate per trial g(f) = p ln(1 + bf) + q ln(1 − f) by setting dg/df = 0.
- Solving pb / (1 + bf*) − q / (1 − f*) = 0 gives pb(1 − f*) = q(1 + bf*) ⇒ f* = (bp − q) / b.
Step-by-Step Examples
#1: Find the optimal Kelly fraction for an even-money trade (b = 1) with a 55% win rate (p = 0.55).
p = 0.55q = 0.45b = 1.0
f* = (1.0 × 0.55 − 0.45) / 1.0 = 0.10
→ f* = 10.0% of bankroll (Half-Kelly = 5.0%)
#2: Find the optimal Kelly fraction for an asymmetric 2-to-1 reward/risk trade (b = 2.0) with a 45% win rate.
p = 0.45q = 0.55b = 2.0
f* = (2.0 × 0.45 − 0.55) / 2.0 = 0.35 / 2.0
→ f* = 17.5% of bankroll
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Frequently Asked Questions
What happens if you bet more than twice the Kelly fraction (f > 2f*)?+
Even on a positive-expected-value game, betting above 2f* makes the logarithmic growth rate g(f) negative, guaranteeing eventual ruin as N → ∞ due to volatility drag.
Why do quantitative traders use Half-Kelly (0.5 f*) in practice?+
Because true win probabilities p are estimated with error; Half-Kelly captures 75% of the theoretical maximum growth rate while cutting portfolio variance in half.
What does a negative Kelly fraction (f* ≤ 0) mean?+
It means expected value bp − q ≤ 0; you have no positive edge and should wager 0% (or take the opposite side if frictionless).