The Kelly Criterion

The Kelly criterion answers a question position sizing usually hand-waves: given a genuine edge, how much should you bet? Its answer maximizes the long-run growth rate of capital, and understanding both why it works and why almost no one bets full Kelly is a rite of passage for traders.

Why not maximize expected wealth?

Consider a favorable bet you can repeat. Maximizing expected wealth tells you to bet everything every time, but a single loss then ruins you, and with repeated play ruin is almost sure. Expected wealth is dominated by vanishingly rare paths where you win every time. The resolution: wealth compounds multiplicatively, so the object that governs long-run outcomes is not the arithmetic mean of returns but the geometric mean. Kelly maximizes the geometric growth rate, equivalently the expected logarithm of wealth.

The discrete derivation

Take a bet that wins with probability $p$, paying $b$-to-$1$ odds, and loses (your stake) with probability $q = 1 - p$. Bet a fraction $f$ of current wealth $W$ each round. After one round wealth is multiplied by $(1 + bf)$ on a win or $(1 - f)$ on a loss. After $n$ rounds with $w$ wins,

$W_n = W_0\,(1 + bf)^{w}\,(1 - f)^{n - w}.$

The per-round growth rate is $\frac{1}{n}\log(W_n/W_0)$. By the The Law of Large Numbers, $w/n \to p$, so the almost-sure long-run growth rate is

$g(f) = p\log(1 + bf) + q\log(1 - f).$

Maximize over $f$:

$g'(f) = \frac{pb}{1 + bf} - \frac{q}{1 - f} = 0.$

Solving gives the Kelly fraction:

$\boxed{f^\star = \frac{pb - q}{b} = \frac{p(b+1) - 1}{b}.}$

The numerator $pb - q$ is exactly the bet's expected profit per unit staked, its edge. So Kelly says: bet edge over odds. With even money ($b = 1$), $f^\star = p - q = 2p - 1$: with a 55% coin you bet 10% of your bankroll. If the edge is zero or negative, $f^\star \le 0$, don't bet.

The continuous / Gaussian case

For an asset with log-returns of drift $\mu$ and variance $\sigma^2$ (over some horizon), levering a fraction $f$ gives portfolio growth rate, to second order,

$g(f) = f\mu - \tfrac{1}{2} f^2 \sigma^2.$

This is the workhorse form. Maximizing gives

$f^\star = \frac{\mu}{\sigma^2},$

and the growth rate achieved at the optimum is

$g(f^\star) = \frac{\mu^2}{2\sigma^2} = \frac{1}{2}\,\text{SR}^2,$

where $\text{SR} = \mu/\sigma$ is the Sharpe Ratio. Two profound facts fall out. First, optimal leverage is the ratio of drift to variance, half the reciprocal of variance, a quantity extremely sensitive to your volatility estimate. Second, the maximum achievable growth rate is half the Sharpe ratio squared: Sharpe is not just a risk-adjusted return number, it is the currency of compound growth. Doubling your Sharpe quadruples your growth rate.

Multi-asset Kelly

With a vector of expected excess returns $\mu$ and covariance $\Sigma$, the growth rate is $g(f) = f^\top \mu - \tfrac12 f^\top \Sigma f$, maximized at

$f^\star = \Sigma^{-1}\mu.$

This is the (unnormalized) tangency portfolio of mean-variance theory, Kelly and Markowitz meet here. Which exposes Kelly's Achilles heel directly: it inherits the notorious instability of $\Sigma^{-1}\mu$, magnifying estimation error in both the mean and the covariance.

Why practitioners bet fractional Kelly

Full Kelly is growth-optimal but brutal to live through, for three compounding reasons:

• Drawdowns. Under full Kelly, the probability of your wealth ever dropping to a fraction $x$ of its peak is approximately $x$, a 50% drawdown at some point is roughly a coin flip. The growth rate near the optimum is flat (it's a maximum, so $g'(f^\star) = 0$), so betting half Kelly gives about three-quarters of the growth for far less than half the variance. That asymmetry is why "half Kelly" is a folk standard.
• Estimation error. $f^\star = \mu/\sigma^2$ assumes you know $\mu$ and $\sigma$. You don't. Overestimating your edge makes you over-bet, and over-betting past the growth-optimal point is severely punished, beyond $f = 2f^\star$ in the Gaussian case, growth turns negative. Because edges are estimated with large error and drift is nearly impossible to measure, betting a fraction builds in a margin of safety. Fractional Kelly can be shown to be the Bayesian-optimal response to parameter uncertainty.
• Non-log utility. Kelly is optimal only for a log-utility investor (relative risk aversion of exactly 1). Most investors are more risk-averse; a power-utility investor with risk aversion $\gamma$ optimally bets $f^\star/\gamma$, i.e., fractional Kelly is the optimal full-Kelly-scaled-down rule for higher risk aversion.

Worked example

You can buy an asset with expected annual excess return $\mu = 8\%$ and volatility $\sigma = 20\%$. Full Kelly leverage is $f^\star = 0.08 / 0.20^2 = 0.08/0.04 = 2.0$, twice your capital. The achieved growth rate is $\mu^2/2\sigma^2 = 0.0064/0.08 = 8\%$ per year, matching $\tfrac12 \text{SR}^2 = \tfrac12 (0.4)^2 = 0.08$. Most managers would find 2× leverage on a 0.4-Sharpe strategy terrifying, and rightly, given the estimate risk, so they'd run half Kelly ($1.0×$), keeping ~75% of the growth. This is exactly the logic behind Vol Targeting: size to a volatility budget rather than to a fragile point estimate of the mean.

Failure modes

• Garbage in. Kelly is only as good as $\mu$ and $\sigma$; a modestly overstated edge produces dangerous over-betting.
• Non-stationarity. Edges decay and regimes shift (Alpha Decay, Regime Detection); a fraction sized to yesterday's edge over-bets today's.
• Fat tails and gaps. The Gaussian approximation understates ruin risk when returns jump; the continuous formula assumes you can continuously rebalance, which you can't through a gap.
• Simultaneous bets. Correlations make independent per-bet Kelly sizing over-lever the book; the multi-asset $\Sigma^{-1}\mu$ form is required.

In interviews

The classic prompt is the biased-coin betting game: "a coin lands heads 60% of the time, you can bet any fraction of your bankroll at even odds, what fraction maximizes long-run growth?" The answer is $f^\star = 2(0.6) - 1 = 0.2$, and the expected-value trap is that maximizing expected wealth says bet everything. Be ready to derive $f^\star$ from maximizing $\mathbb{E}[\log W]$, to state the continuous result $f^\star = \mu/\sigma^2$ and $g = \tfrac12\text{SR}^2$, and to explain crisply why real desks bet a fraction, flat growth near the optimum plus estimation error. See Position Sizing for how this connects to real risk budgets.