The Kelly Criterion
Trading Intelligence
8 min read
Apply the Kelly formula and its fractional variants to find the theoretically optimal bet size that maximizes geometric growth rate.
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8 min read
Apply the Kelly formula and its fractional variants to find the theoretically optimal bet size that maximizes geometric growth rate.
The Kelly Criterion is a position-sizing formula that picks the bet fraction maximizing long-run geometric wealth growth. For binary outcomes:
f* = (bp − q)/b. For continuous trading returns:f* = μ/σ². Most professionals run half- or quarter-Kelly because edge estimates are noisy and full Kelly produces brutal drawdowns.
Risk sizing is the silent killer of most trading accounts.
Kelly gives you a mathematically defensible position size — balancing growth and ruin risk. It is how professional gamblers, quant traders, and algorithmic funds scale once they have measured their edge. (See Kelly Jr., 1956, A New Interpretation of Information Rate; Thorp, 2006, The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market.)
This lesson assumes you already know your per-trade risk policy — if not, start with Risk Per Trade & Position Sizing.
The Kelly Criterion is the bet fraction f* that maximizes the expected logarithm of wealth — equivalently, the long-run geometric (compounded) growth rate of an account.
It takes two forms depending on how your payoffs are shaped:
f* = (bp − q)/bf* = μ/σ²Kelly's f* is the bet fraction that maximizes E[log(wealth)]. Why log? Because a 50% loss requires a 100% gain to recover — arithmetic average return is misleading once you compound. Maximizing E[log W] is the correct objective for any trader who actually compounds capital.
For a bet with one fixed win size and one fixed loss size:
f* = (b*p - q) / b
where: p = probability of a win q = 1 - p = probability of a loss b = payoff ratio = average win / average loss (the "R" in R-multiples) f* = optimal fraction of capital to risk on this bet
Note the division by b. Skipping it is the single most common Kelly error online — and it inflates the answer by exactly the payoff ratio.
You have:
p = 0.55b = 2 (you make 2R when you win, lose 1R when you lose)q = 0.45Plug it in:
f* = (2 * 0.55 - 0.45) / 2 = (1.10 - 0.45) / 2 = 0.65 / 2 = 0.325
Full Kelly says risk 32.5% of capital per trade. Still way too aggressive — a 30%+ position on a single trade compounds drawdowns into ruin under any realistic edge uncertainty. That is why pros never use full Kelly.
Binary Kelly only applies to fixed-odds bets. Real trading PnL is continuous and asymmetric — you do not have one win size and one loss size; you have a distribution. The form that matters for trading is:
f* = mu / sigma^2
where: mu = mean per-trade (or per-period) return, in account units sigma^2 = variance of that return
See also variance and standard deviation for the variance term.
A strategy with mean trade return μ = 0.4% and standard deviation σ = 2% (so σ² = 0.0004) gives:
mu = 0.004, sigma = 0.02 so sigma^2 = 0.0004
f* = 0.004 / 0.0004 f* = 10
Ten times leverage on the strategy. Clearly insane — which is exactly why we use fractional Kelly and clamp the result against edge-uncertainty bounds. The continuous formula is honest: it tells you how much your edge "wants" you to bet. The fractional discount tells you how much your uncertainty about the edge forces you to give back.
If your computed Kelly is above 25%, your edge estimate is almost certainly overfit. Recompute with a haircut: shrink your estimated mean by 30–50% before sizing. Ed Thorp called this Kelly with regularization.
Kelly was derived for fixed-odds games — blackjack, horse racing — where the edge is known exactly and the rules are stationary. Trading violates both assumptions:
p, b (or μ, σ) propagates super-linearly into f*.Under full Kelly, the long-run probability of seeing a peak-to-trough drawdown of x% is approximately x% itself: a 50% chance of a 50% drawdown, a 90% chance of a 10% drawdown, eventually. That's acceptable for a casino with infinite bankroll horizons. It is not acceptable for a retail account with career risk.
Half-Kelly captures roughly 75% of full-Kelly's geometric growth while cutting expected drawdown roughly in half. Under full Kelly, edge estimates that are biased upward by even a small amount turn the sized bet into negative-EV territory; half- and quarter-Kelly buy back robustness against that bias. This is why Thorp, Ziemba, and most blackjack/quant pros run quarter- to half-Kelly (see MacLean, Thorp, Ziemba, eds., The Kelly Capital Growth Investment Criterion, 2011).
Fractional Kelly trade-off
Relative geometric growth vs relative drawdown (full Kelly = 100)
So traders typically use:
| Method | Formula / Rule | Long-run growth | Expected drawdown | Edge-uncertainty robust | Best for |
|---|---|---|---|---|---|
| Full Kelly | f* = (bp−q)/b or μ/σ² | Maximum (theoretical) | Brutal — ~50% chance of 50% DD | No | Casinos with known edge, infinite horizon |
| Half-Kelly | 0.5 · f* | ~75% of full | ~½ of full Kelly | Some | Measured strategies, 200+ trade samples |
| Quarter-Kelly | 0.25 · f* | ~44% of full | ~¼ of full Kelly | Yes | High variance, fat-tailed markets, smaller samples |
| Fixed Fractional (1–2%) | Constant % of equity per trade | Sub-Kelly | Bounded, predictable | Yes | Beginners, unproven systems, prop-firm rules |
| Volatility Targeting | Size = target_vol / realized_vol | Comparable to ¼–½ Kelly | Stable across regimes | Yes | Strategies whose vol changes regime-to-regime |
Don't use Kelly sizing if:
In these cases, static sizing (fixed % risk per trade) is safer until your data matures.
pbμ and standard deviation σf* = (bp − q)/bf* = μ/σ²0.25 · f* to 0.5 · f* instead of full KellyNo — Kelly says bet your edge scaled by your variance. A 1% edge in a 1%-variance market is a different bet from a 1% edge in a 10%-variance market. Confusing edge with optimal bet fraction is the #1 retail Kelly error.
The full-Kelly number is too aggressive given uncertainty. The Kelly principle — maximize log-wealth, size proportional to edge over variance — is exactly right for any compounding trader. Half- and quarter-Kelly are still Kelly; they just acknowledge that you don't know μ and σ exactly.
Kelly the binary formula was for blackjack. Kelly the principle (f* = μ/σ² from log-wealth maximization) generalizes cleanly to continuous trading returns. Don't throw out the principle because the toy formula doesn't fit.
The Kelly Criterion has two forms. For a binary bet with win probability p, loss probability q = 1−p, and payoff ratio b (average win divided by average loss): f* = (bp − q)/b. For continuous trading returns with mean μ and variance σ²: f* = μ/σ². Both pick the bet fraction that maximizes the long-run geometric growth rate of capital.
Skip Kelly sizing when your sample is under ~100 trades, when you haven't measured the variance of your returns, when you're trading fat-tailed instruments where Kelly's normality assumption breaks (crypto around news, micro-caps), or when your execution is emotionally driven. In those regimes, fixed-fractional sizing (1–2% per trade) is more robust because it does not amplify estimation error.
Because full Kelly assumes you know your edge exactly. In reality, p, b, μ, and σ are sample estimates, and any upward bias in those estimates turns full Kelly into a negative-EV bet. Half-Kelly captures roughly 75% of full Kelly's growth at roughly half the drawdown, which is a better trade-off once you account for edge uncertainty. Most quant and blackjack pros (Thorp, Ziemba) run quarter- to half-Kelly for that reason.
Kelly is not magic. It is a mathematical framework to stop you from being emotional about risk.
It tells you:
Don't risk based on confidence. Risk based on math — and then halve it for the math you don't have.