The Kelly criterion prescribes the bet (or position) size that maximizes the long-run geometric growth rate of capital. "Full Kelly" means betting exactly that optimal fraction, f\; "fractional Kelly" means deliberately betting a constant multiple of it — most commonly half (½ Kelly) or a quarter (¼ Kelly). The central tension is this: full Kelly is theoretically growth-optimal only if your edge and odds are known exactly*, but it produces violent, often unbearable drawdowns and is acutely sensitive to estimation error. Fractional Kelly trades away a small slice of theoretical growth for a large reduction in volatility and ruin risk — which is why almost no professional bets full Kelly in practice.

How it's calculated

For a continuous return process, the expected log-growth rate as a function of the fraction f of capital allocated is approximately:

g(f) = f·m − ½·f²·s²

where m is the expected (excess) return and s² the variance. Maximizing over f gives the full-Kelly fraction f\ = m / s², which is the same quantity as the optimal mean-variance leverage. (Atlas Peak Research; Wikipedia, Kelly criterion*.)

Fractional Kelly bets a multiple c of f\, i.e. f = c·f\. Substituting into g(f) yields a clean and important result:

g(c·f\) = c·(2 − c)·g(f\)

This is the growth curve as a parabola peaking at c = 1. Two consequences follow directly:

• Half Kelly (c = ½): retains ½·(2 − ½) = ¾ of the full-Kelly growth rate while cutting the bet size in half — and because variance scales with f², it roughly halves volatility. This "75% of the growth at half the risk" figure is the most-cited justification for fractional Kelly (Thorp; MacLean, Thorp & Ziemba, The Kelly Capital Growth Investment Criterion; Wikipedia).
• Quarter Kelly (c = ¼): retains ¼·(2 − ¼) ≈ 44% of full-Kelly growth (commonly rounded to "about half") at roughly a quarter of the bet size.

The curve is symmetric in growth: betting at c = 1.5 (overbetting by 50%) gives the same growth as c = 0.5. But overbetting is far more dangerous because it pushes you toward the right side where g can turn negative (at c = 2, growth is zero; beyond that, capital declines almost surely). This asymmetry in risk — not in growth — is the heart of the argument.

How it's used in practice

A practitioner estimates win rate and payoff (or m and s² from a backtest/return series), computes f\, then applies a haircut: typically bet c·f\ with c between ¼ and ½. The discount serves three distinct purposes:

1. Drawdown tolerance. Full Kelly is mathematically prone to large drawdowns. Thorp's continuous-approximation results (the probability the bankroll ever falls to a fraction x of its value is x^(2/c−1) at fraction c of Kelly): under full Kelly the probability of your capital ever halving at some point is about ½, dropping to about ⅛ under half Kelly. (A separate, often-confused statistic is the probability of halving before doubling, which Thorp gives as ⅓ for full Kelly and ⅑ for half Kelly.) (Thorp, "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market," 2006, §7.) Fractional Kelly is how you make the strategy psychologically survivable. 2. Estimation error. f\ depends on inputs you do not actually know. Because the growth curve falls off steeply on the overbet side, if you overestimate your edge (the common direction of error), full Kelly can silently become an overbet with negative growth. Fractional Kelly is a margin of safety against this. Thorp argued the main* reason to bet fractionally is the human "background tendency to overestimate the chances of winning" — i.e. overbetting is worse than underbetting. 3. Real-world frictions. Costs, slippage, leverage limits, non-continuous rebalancing, fat tails, and career/redemption risk all violate the clean assumptions, pushing the robust optimum below the nominal f\*.

For multiple simultaneous positions, Kelly generalizes to a vector solution resembling f\* = Σ⁻¹·μ (the inverse-covariance times expected-return vector) — structurally identical to a mean-variance optimal portfolio — and the same fractional haircut is applied to the whole vector.

Adoption, debate & evidence

Fractional Kelly is broadly endorsed by its own intellectual lineage. Edward Thorp (who ran Kelly-style sizing in blackjack and at Princeton-Newport) and the academic Kelly-growth literature (Ziemba, MacLean) all advocate betting below full Kelly in any real application. Industry commentary consistently reports that quantitative funds run roughly ¼-to-½ Kelly, and half Kelly is the common default starting point. There is genuine consensus that full Kelly is too aggressive for live money.

The deeper, contested questions are: (1) whether Kelly's log-utility objective is even the right goal — Paul Samuelson famously and repeatedly attacked the "Kelly fallacy," arguing maximizing expected log wealth is not optimal for anyone whose utility isn't logarithmic, and that "in the long run it dominates" is mathematically misleading for finite horizons; and (2) whether f\ can be estimated reliably enough to be actionable at all, given that small input errors map to large sizing errors. The honest "folklore vs measured" line: the parabolic growth math (¾ growth at half size) is exact and well established; the claim that any given trading strategy has a stable, estimable f\ worth betting is the weak link, because edges drift and are usually overestimated.

Strengths & limitations

Strengths. Provides a principled, non-arbitrary anchor for position size tied directly to edge and variance; the fractional haircut converts a fragile optimum into a robust one; the favorable trade-off (lose ~25% of growth, shed ~50% of volatility at half Kelly) is mathematically real.

Limitations / when it fails. Full Kelly assumes known probabilities, continuous rebalancing, no transaction costs, and infinite horizon — all false in markets. It is non-robust: overestimating edge silently turns it into a ruinous overbet. It says nothing about regime change or fat tails. The #1 misuse is plugging an overfit backtest's win rate and payoff into the formula and betting the raw f\ — this combines an inflated edge estimate with the most dangerous (overbet) side of the curve. The practical antidote is to estimate conservatively and* apply a fractional multiplier, treating the two as independent layers of safety.

Sources

• Edward O. Thorp, "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market" (2006), §7 — the drawdown formula P(ever reach fraction x) = x^(2/c−1): full Kelly P(ever halve) = ½, half Kelly = ⅛; and the separate halve-before-double figures (⅓ full, ⅑ half). Also in MacLean, Thorp & Ziemba (eds.), The Kelly Capital Growth Investment Criterion (2011).
• Wikipedia, Kelly criterion — half/quarter Kelly definitions, growth-vs-fraction parabola, overbetting increases risk of ruin.
• Atlas Peak Research, "The Kelly Criterion in Financial Markets" — growth formula g = f·m − ½f²s², conditions under which full Kelly is optimal, robust-Kelly framing.
• Matthew Downey, "Why fractional Kelly?" — simulations on estimation uncertainty / downside-risk justification for fractional sizing.
• Frontiers in Applied Mathematics & Statistics (2020), "Practical Implementation of the Kelly Criterion" — empirical equity-portfolio Kelly results.
• Disputed: Samuelson's critique of the log-utility objective (the "Kelly fallacy" debate) remains genuinely unresolved at the level of whether growth-optimality is the right goal; this doc treats the fractional-Kelly math as settled and the applicability to real edges as contested.