Methodology5 min read

Kelly Criterion and Fractional Kelly

The Kelly criterion is a position-sizing rule that maximizes the expected logarithmic growth rate of capital given a known edge and payoff distribution. It tells you the fraction of bankroll to risk on each independent bet to compound wealth fastest in the long run — and equally important, it defines the boundary beyond which leverage destroys equity even when the underlying edge is real.

The formula

For a binary bet with win probability p, loss probability q = 1 - p, and payoff ratio b (units won per unit risked), the Kelly fraction is:

f* = (b·p - q) / b

For a continuous return stream with mean excess return μ and variance σ², the Gaussian-approximation form used in portfolio sizing is:

f* = μ / σ²

Fractional Kelly scales this by a constant k between 0 and 1:

f_frac = k · f*

Typical values for k are 0.25 to 0.5. Half-Kelly (k = 0.5) gives roughly three-quarters of the expected growth rate of full Kelly with about half the variance of log-equity — a trade-off most practitioners accept without hesitation.

Interpretation and ranges

A positive f* confirms a positive expected log-growth strategy at that sizing. A negative f* means the bet has negative expectancy and should not be taken at any size — Kelly does not rescue a losing edge by shrinking it. An f* greater than 1 implies leverage, and an f* greater than 2 in the continuous case typically signals an inflated edge estimate rather than a genuine opportunity.

For systematic equity strategies with realistic Sharpe ratios of 0.5 to 1.5 on daily data, raw Kelly fractions land in the 2x to 10x leverage range. This is almost never tradeable in practice because the inputs μ and σ² are estimated, not known. Estimation error in μ alone is enough to push realized growth deep into negative territory at full Kelly.

Doubling the Kelly fraction past f* produces zero expected log-growth — the same as not trading at all. Any sizing beyond 2·f* has negative expected log-growth despite a positive-edge strategy.

What Kelly does not capture

Kelly assumes the edge parameters are known exactly. In backtested strategies they are estimated from finite samples with substantial standard errors, and the optimal sizing under parameter uncertainty is strictly below the point estimate of f*. This alone justifies fractional Kelly on Bayesian grounds, independent of any risk-aversion argument.

Kelly is indifferent to drawdown path. A full-Kelly strategy on a true edge will experience a drawdown of 50% or more with probability approaching 1 over a long enough horizon. Investors with finite tolerance for drawdown, redemption risk, or psychological capacity for loss require a smaller fraction regardless of growth-rate considerations.

The formula assumes serially independent returns and stationary distributions. Real strategies exhibit autocorrelated returns, regime shifts, and fat tails — all of which inflate the realized variance above the sample estimate and break the Gaussian approximation. Kelly sizing on a non-stationary edge converts a model error into a sizing error and compounds it.

Full Kelly on a backtested Sharpe is almost always wrong. The combination of overfit μ, underestimated σ in the live regime, and unmodeled tail events means realized growth at full-Kelly sizing is typically far below — and often negative relative to — fractional Kelly with k around 0.25.

Finally, Kelly is a single-asset, single-strategy rule. Multi-strategy portfolios require the matrix form f* = Σ⁻¹·μ, which introduces estimation problems an order of magnitude harder than the scalar case. Covariance estimation error dominates the result, and shrinkage estimators are mandatory rather than optional.

In Kestrel Signal

Kestrel Signal reports the empirical Kelly fraction derived from the in-sample return distribution alongside its out-of-sample counterpart, and flags any divergence greater than 30% between the two. Position-sizing modules expose k as an explicit parameter rather than defaulting to full Kelly, and the equity curve simulator displays drawdown distributions at k = 0.25, 0.5, and 1.0 so the path-risk implications are visible before capital is committed.