Kelly criterion

Definition

The Kelly criterion (Kelly 1956) derives the allocation fraction that maximizes the long-run geometric growth rate of capital when the return distribution is KNOWN. In the binary model with win probability p and win/loss payout ratio b, the growth- optimal fraction is f* = p − (1 − p)/b. Its platform role is a CEILING and a diagnostic — full Kelly is where sizing theory says growth peaks under perfect knowledge; practical sizing lives far below it because knowledge is never perfect.

How it works / structure

• Continuous approximation: for small edges, f* ≈ μ/σ² (expected excess return over variance) — the form used for market strategies.
• Properties (derived, not empirical): f* maximizes expected log growth; sizing at 2× f* drives growth to zero; beyond that, expected growth is negative even with a positive- edge strategy — oversizing converts edge into decay. Full Kelly’s drawdown profile is violent: the probability of halving equity before doubling it is substantial even with the true f*.
• Fractional Kelly: sizing at k × f* (k = 0.25-0.5 typical) sacrifices little growth for large drawdown relief — the standard practical compromise (MacLean-Thorp-Ziemba formalize the trade-off).
• Parameters (engine-executable): the estimate source for μ/σ² or (p, b) — REPLAY distributions, never hopes; the k multiplier; and the reconciliation check against risk-fixed-fractional’s declared f (Kelly arithmetic run on replay stats tells the engine whether a strategy’s f is above its estimated ceiling).

When it applies

As the sizing sanity check on every ratified strategy: estimate f* from replay, compare declared f, flag any f above ~0.5 × estimated f*. As a design tool for understanding WHY overtrading size destroys positive-edge strategies — the platform’s canonical answer to “the strategy works, size up.”

Risk profile & failure modes

• Estimation error dominates: f* computed from estimated edges inherits their noise, and the loss function is asymmetric — oversizing from an optimistic estimate is ruinous, undersizing from a pessimistic one is cheap; this asymmetry is THE argument for fractional Kelly.
• Non-stationarity: edges decay (lens-quantitative); yesterday’s f* overstates today’s.
• Correlated repetition: Kelly logic assumes sequential independent opportunities; simultaneous correlated positions need the portfolio version, not per-position f* summed (risk-correlation-exposure).
• Psychological cover: “Kelly says size up” is overconfidence with a formula (bias-overconfidence); the criterion’s own arithmetic says the penalty for optimism is worse than for caution.

Evidence & limits

Kelly (1956) is the derivation; MacLean-Thorp-Ziemba (2011) collects the theory, Thorp’s practical application history, and the fractional-Kelly trade-off results. All of it is mathematics conditional on known distributions — the platform’s use (ceiling + diagnostic, never target) is the standard institutional adaptation to the fact that distributions are estimated.

Falsifiable-thesis examples

Illustrations only, not signals:

• “Strategy S’s replay-estimated μ/σ² implies f* above its declared f = 1% (the declared size is inside the Kelly ceiling)” — falsified by the computed estimate.
• “Running S at 0.25 × estimated f* keeps replay max drawdown under half of the full-f* replay’s drawdown while retaining at least 60% of its growth” — falsified by the paired replay.

Cross-references

• The rule it bounds: risk-fixed-fractional
• The budget it must respect: risk-max-drawdown-budget
• Why estimates decay: lens-quantitative
• The bias it gets used to justify: bias-overconfidence