Kelly Criterion¶

OVERVIEW¶

The Kelly Criterion is a position-sizing formula that determines the optimal fraction of capital to risk on each trade in order to maximize the long-term geometric growth rate of the portfolio. Developed by Bell Labs scientist John L. Kelly Jr. in 1956 and popularized in trading by Edward Thorp, it answers the question that risk-of-ruin and fixed-percent sizing cannot: given a measurable edge, exactly how much should you bet?

HOW THE KELLY CRITERION IS CALCULATED¶

Binary bet (win or lose with fixed payouts):

f* = (bp − q) / b

Variable	Definition
`f*`	Fraction of bankroll to wager
`b`	Net odds received on the wager (win amount per unit risked)
`p`	Probability of winning
`q`	Probability of losing (1 − p)

Continuous outcomes (practical trading form):

f* = W − [(1 − W) / R]

Variable	Definition
`W`	Historical win rate (decimal)
`R`	Average winner ÷ average loser (in dollars)

A negative f* means no edge — do not trade. A positive f* is the percentage of equity to allocate per trade. f* > 1 implies leverage is mathematically optimal, but in practice this is rarely deployed because the underlying assumptions (stable parameters, independent trades) are too brittle to bet a portfolio on.

WORKED EXAMPLE: SIZING A BREAKOUT SYSTEM¶

System: SPY breakout setups (backtest)

Input	Value
Win rate (W)	0.55
Average winner	$420
Average loser	$210
Reward/Risk ratio (R)	2.0

Calculation:

f* = 0.55 − (0.45 / 2.0)
f* = 0.55 − 0.225
f* = 0.325

Full Kelly says size each trade at 32.5% of equity. On a $50,000 account that is $16,250 of risk per trade — a level most traders find intolerable on volatility grounds. Standard practice is half-Kelly (16.25%) or quarter-Kelly (8.1%), which sacrifices some compounding rate in exchange for materially lower drawdowns and robustness to parameter error.

Kelly Fraction	Allocation %	Dollar Risk per Trade
Full Kelly	32.5%	$16,250
Half Kelly	16.25%	$8,125
Quarter Kelly	8.125%	$4,063

WHEN TRADERS USE KELLY¶

Scenario	Why Kelly Applies
Systematic strategies	Stable, measurable edges (stat arb, trend-following, options-selling)
Bankroll-style frameworks	Each trade is approximately independent
Cross-strategy comparison	Higher f* signals higher-edge system before live deployment
Drawdown forecasting	Monte Carlo simulations use Kelly fractions to estimate drawdown distributions

Note

Renaissance Technologies, Susquehanna, and most quantitative shops use Kelly-derived sizing internally, almost always at fractional Kelly. Edward Thorp ran his blackjack card-counting system and his Princeton-Newport hedge fund on the formula explicitly.

LIMITATIONS AND COMMON MISCONCEPTIONS¶

Limitation	Detail
Input sensitivity	Overestimating W or R by 5% produces dramatically oversized positions
Drawdown exposure	Full Kelly drawdowns can exceed 50% even with correctly specified parameters
Independence assumption	Real trades are correlated; regimes shift; win rate drifts over time
Asymptotic guarantees	Kelly's mathematical guarantees apply across hundreds of trades, not a single position
Not a stop-loss	Kelly sizes the position only — it does not determine exits

Most professionals use one-quarter to one-half Kelly because:

Half-Kelly captures roughly 75% of the geometric growth at one-quarter the variance
Drawdowns scale roughly linearly with the Kelly fraction
Fractional Kelly compensates for parameter uncertainty without rebuilding the formula

Note

Common misconceptions: Kelly is not a stop-loss — it sizes the position, not the exit. It does not work on a single trade; its guarantees are asymptotic across hundreds of trades. It cannot rescue a negative-expectancy strategy — if W and R produce f* ≤ 0, the only correct size is zero.