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Kelly Criterion

Properties2
descriptionA formula for the position size / leverage that maximizes long-term compounded growth — including the multi-strategy covariance form, its link to the Sharpe ratio, and half-Kelly betting.
tagstrading, finance, quant

2 min read

Kelly Criterion

Definition

The Kelly criterion (Kelly formula) gives the bet size or leverage that maximizes the long-term compounded growth rate of wealth. Because ruin makes long-term wealth zero, the objective implicitly rules out bet sizes that risk going to zero.

For a single bet with win probability W and win/loss ratio R:

K% = W − (1 − W)/R

Core Ideas

Continuous (return-based) form

For a strategy with mean excess return μ and return variance σ², the optimal leverage is:

f* = μ / σ²

Note this uses variance (σ²), not standard deviation. The Kelly f is independent of time scale, so annualizing return and standard deviation does not change it.

Multi-strategy form (Thorp)

For multiple strategies, the optimal leverage vector is:

F = C⁻¹ · M

where C is the covariance matrix of strategy returns and M is the vector of mean returns. Assuming the strategies are statistically independent, C becomes diagonal and this reduces to fᵢ = mᵢ / sᵢ² per strategy. The formula thus doubles as a capital-allocation rule across strategies.

The maximum compounded growth rate of a single Gaussian strategy is:

g = r + S²/2

with r the risk-free rate and S the Sharpe Ratio. Multi-strategy: g = r + F*ᵀCF*/2 and S = √(F*ᵀCF*). Since μ = Sharpe × σ, substituting into f* = μ/σ² gives:

Kelly leverage = Sharpe / Volatility

So higher Sharpe → more leverage; higher volatility → less. Kelly penalizes volatility more aggressively than the Sharpe ratio does.

Half-Kelly and practical use

Because return distributions are not truly Gaussian and parameter estimates are uncertain, practitioners cut the recommended leverage to half-Kelly (or quarter-Kelly) for safety — full Kelly is aggressive and sensitive to estimation error. Retail accounts capped at 2x/4x scale each fᵢ down by a common factor. Kelly requires continuously re-sizing as equity changes (at least daily): if equity drops 10%, you cut position size to keep leverage at f*. Use a lookback of ~6 months for short-holding strategies, and it is always safer to stay below the Kelly leverage.

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