Guide

Kelly criterion explained

Harbor Capital's systematic sleeve had a genuine edge on a mean-reversion signal — positive expectancy over hundreds of back-tested trades — yet two analysts sized the same strategy differently. One allocated 4% of equity per trade using a fixed fractional rule; the other pushed 18% because recent winners “proved” the model worked. The second analyst hit a six-trade losing streak within a month and breached the fund's maximum drawdown limit. The first survived the same streak with half the peak-to-trough loss. The difference was not the signal; it was bet sizing. The Kelly criterion, developed by John L. Kelly Jr. at Bell Labs in 1956, answers a precise question: given known (or estimated) win probability and payoff odds, what fraction of bankroll maximizes the long-run geometric growth rate? This guide explains the discrete Kelly formula, the continuous investing version, why practitioners use fractional Kelly, how to estimate edge and variance without fooling yourself, a Harbor Capital allocator worked example, a sizing-method decision table, common pitfalls, and a production checklist. It complements broader risk management and position sizing without replacing stop-loss discipline or portfolio diversification.

What the Kelly criterion optimizes

Most traders think in arithmetic averages: “this strategy makes 0.3% per trade on average.” Compounding investors should think geometrically: a 50% gain followed by a 50% loss leaves you at 75% of starting capital, not break-even. The Kelly criterion maximizes expected log wealth — the growth rate you would observe over many independent bets when profits reinvest.

Kelly is not a stock-picking model. It assumes you already have an edge (positive expected value) and asks only: how much of your bankroll to risk on each opportunity so that compounding works for you rather than against you. Bet below Kelly and you grow slower than optimal; bet above Kelly and you guarantee ruin in the long run even with a positive edge.

Discrete Kelly formula (binary bets)

For a bet that wins with probability p, loses with probability q = 1 − p, and pays b units net profit per unit risked on a win (even-money would be b = 1):

f* = (bp − q) / b

f* is the fraction of bankroll to wager. Example: 55% win rate on even-money coin flips (b = 1, p = 0.55): f* = (1 × 0.55 − 0.45) / 1 = 0.10 — bet 10% of bankroll each round. A 60% win rate at 1:1 odds yields f* = 0.20.

Continuous Kelly (investing approximation)

For continuously compounded returns with expected excess return μ (above the risk-free rate) and variance σ²:

f* ≈ μ / σ²

This is the leverage fraction that maximizes geometric growth under Gaussian return assumptions. A strategy with 8% expected annual excess return and 16% annualized volatility (σ = 0.16, σ² = 0.0256) implies f* ≈ 0.08 / 0.0256 ≈ 3.1 — over 300% notional exposure. That number is why raw continuous Kelly horrifies risk managers: it assumes perfect knowledge of μ and σ, independent identically distributed returns, and no transaction costs.

Full Kelly vs fractional Kelly

Full Kelly (betting f* exactly) maximizes long-run growth rate but produces brutal intermediate drawdowns. Simulations of a 55% edge even-money game at full Kelly routinely see 50%+ peak-to-trough equity drops. Most professionals use fractional Kelly — betting a fixed fraction of the Kelly amount:

Half-Kelly (0.5 f*) — sacrifices roughly 25% of optimal growth rate but cuts variance roughly in half. The most common institutional compromise.
Quarter-Kelly (0.25 f*) — conservative; favored when edge estimates are noisy or tail risk is fat (crypto, options, early-stage strategies).
Fixed fractional unrelated to Kelly — e.g. risk 1% of equity per trade via stop distance. Simpler, but not growth-optimal when edge varies by setup.

Ed Thorp, who popularized Kelly in blackjack and hedge funds, has long advocated half-Kelly or less when model error exists. The criterion is a ceiling, not a target.

Estimating edge and variance without self-deception

Kelly output is only as good as inputs. Garbage edge estimates produce garbage bet sizes — often overbetting, which is far more dangerous than underbetting.

Win rate and payoff ratio

For discretionary traders, record at least 30–50 trades (100+ preferred) with entry, exit, and R-multiple (profit divided by initial risk). Win rate alone is insufficient: a 70% win rate with 1:3 risk/reward loses money. Kelly needs both p and b.

Mean and variance for systematic strategies

Use out-of-sample returns, not in-sample backtest peaks. Walk-forward validation, purged cross-validation for overlapping labels, and realistic slippage/fees shrink μ dramatically. For volatility, use a horizon that matches your rebalance frequency; annualizing daily σ with √252 assumes i.i.d. daily returns that rarely hold in crisis weeks.

Correlation and multiple simultaneous bets

Independent Kelly bets add: if you run two uncorrelated 10% Kelly strategies, total allocation is not 20% unless capital is partitioned. Correlated bets require a multivariate Kelly solution or simpler heuristics: scale each leg down when portfolio correlation rises. See modern portfolio theory for the diversification angle.

Worked example: Harbor Capital mean-reversion sleeve

Harbor Capital's quant team back-tested a short-horizon mean-reversion signal on large-cap U.S. equities over five years of walk-forward windows (not a single in-sample fit). Out-of-sample results after costs:

Win rate: 58%
Average winner: +1.4R (1.4× initial risk)
Average loser: −1.0R
Effective payoff ratio b ≈ 1.4 on wins, but blended expectancy uses win/loss magnitudes

For Kelly's discrete form with variable payoffs, use the generalized formula: f* = (p × W − q × L) / (W × L) where W and L are average win and loss fractions of bankroll at risk — or estimate via Monte Carlo on trade R-multiples. Harbor's simulation yielded f* ≈ 12% of allocated sleeve capital per signal.

The risk committee approved half-Kelly at 6% per trade, capped further by a 2% daily portfolio heat limit across all open positions. During a March drawdown (nine consecutive losers, a 5.4% probability event under model assumptions), sleeve drawdown reached 8.2% vs the 22% simulated at full Kelly on the same sequence. The allocator stayed inside mandate; the analyst who had been running 18% discretionary sizing did not.

Harbor also reports Kelly-implied leverage alongside Sharpe ratio and value at risk in monthly risk packs — Kelly answers growth optimality, Sharpe answers risk-adjusted efficiency, VaR answers tail loss at a confidence level. No single metric is sufficient.

Sizing method decision table

Your situation	Prefer	Avoid
Repeated i.i.d. bets, stable edge, casino/market-making	Fractional Kelly from measured p and b	Full Kelly without drawdown tolerance
Discretionary trades with defined stop and target	Fixed % risk per trade; compare implied f to Kelly as sanity check	Scaling up after wins without recalculating edge
Long-only investing, multi-year horizon	Strategic asset allocation + rebalance bands	Continuous Kelly leverage on estimated μ/σ²
Crypto / fat tails / model uncertainty	Quarter-Kelly or lower; hard notional caps	Backtest-optimal f* applied live without slippage haircut
Multiple correlated strategies	Partitioned bankrolls or scaled-down per-leg Kelly	Summing independent Kelly fractions naively
Retail account, psychological drawdown limit 15%	1–2% risk per trade regardless of Kelly ceiling	Any sizing that implies 30%+ peak drawdown

Common pitfalls

Overestimating edge — in-sample backtests inflate win rate and μ. Kelly overbetting from optimistic inputs is the fastest path to ruin with a real edge.
Ignoring fat tails — Kelly assumes known distributions; black-swan gaps blow up Gaussian σ. Use fractional Kelly and hard loss limits.
Betting above Kelly — even with positive expectancy, over-Kelly bets have negative geometric growth. More leverage is not more edge.
Changing size on emotion — doubling after wins (martingale tilt) or revenge trading after losses violates the fixed-fraction assumption Kelly requires.
Confusing arithmetic and geometric returns — +10% then −10% is not flat. Size for geometric compounding, especially with leverage.
Single-trade Kelly on overlapping positions — five concurrent 10% Kelly bets are not independent; aggregate exposure matters.

Production checklist

Estimate edge and payoff from out-of-sample data with realistic costs.
Compute full Kelly f*; default to half-Kelly or less for live deployment.
Cross-check Kelly fraction against per-trade stop-risk and daily heat limits.
Stress-test sizing on worst historical streaks and synthetic fat-tail scenarios.
Document assumptions (μ, σ, win rate, payoff) and review quarterly.
Cap notional leverage regardless of Kelly output when using margin or perps.
Report geometric CAGR alongside arithmetic average return in performance reviews.
Reduce fraction immediately when realized drawdown exceeds model expectations.

Key takeaways

Kelly maximizes long-run geometric growth — not short-run comfort or arithmetic average return.
Full Kelly is almost always too aggressive live — fractional Kelly trades growth for survivability.
Edge estimation error dominates — conservative inputs beat optimistic Kelly math.
Kelly complements, not replaces, risk rules — stops, heat limits, and diversification still apply.
Correlated bets need scaled sizing — never treat overlapping positions as independent Kelly legs.