Position Sizing & Leverage: Kelly Criterion Strategy › The Kelly Criterion

The Kelly Criterion, developed by John L. Kelly Jr. at Bell Labs in 1956, answers a simple question: Given an edge in a repeated game, what fraction of capital should be wagered to maximize long-term wealth growth? The same growth-rate logic, originally applied to information transmission over noisy channels, applies to repeated trading and investment decisions. The criterion emerges from a fundamental tension in betting: bet too small and you fail to exploit your edge, but bet too large and you risk catastrophic losses that compound negatively over time. Kelly's mathematical framework resolves this tension by identifying the unique betting fraction that maximizes the expected geometric growth rate of capital.

Consider a simple gambling scenario where you repeatedly face a bet with the following characteristics:

• Win probability: $p$
• Loss probability: $q = 1 - p$
• Win payoff: $b$ (for every dollar wagered, you receive $b$ dollars profit)
• Loss payoff: $-1$ (you lose your entire wager)

To understand why position sizing matters so critically, consider what happens when you repeatedly face this bet. If you bet fraction $f$ of your current capital on each round, the multiplicative nature of returns creates a fundamentally different dynamic than additive returns. After a win, your capital multiplies by $(1 + bf)$, and after a loss it multiplies by $(1 - f)$. This multiplicative structure means that the sequence of wins and losses matters far less than you might expect: what matters is the geometric growth rate.

After $n$ trials with $w$ wins and $l = n - w$ losses, your final wealth $W_n$ starting from initial wealth $W_0$ is:

$$W_n = W_0 (1 + bf)^w (1 - f)^l$$

where:

• $W_n$: final wealth after $n$ trials
• $W_0$: initial wealth
• $n$: number of trials
• $b$: win payoff (profit per dollar wagered)
• $f$: fraction of capital wagered per trial
• $w$: number of wins
• $l$: number of losses

This formula captures the essence of compounding. Notice that wealth is a product of factors, not a sum. This multiplicative structure means that a single devastating loss can overwhelm many small gains. If you bet your entire capital ($f = 1$) and lose even once, you're wiped out regardless of how many wins preceded or follow that loss.

Taking logarithms turns the multiplicative relationship into an additive one, which proves essential for analysis. The logarithm of wealth growth becomes:

$$\log\left(\frac{W_n}{W_0}\right) = w \log(1 + bf) + l \log(1 - f)$$

where:

• $W_n$: final wealth
• $W_0$: initial wealth
• $w$: number of wins
• $l$: number of losses
• $b$: win payoff
• $f$: fraction of capital wagered

This transformation reveals why logarithmic returns are natural for analyzing betting and investment: they turn compound growth into a sum of independent contributions from each trial.

The expected log growth per trial, which we call the geometric growth rate, captures the long-run behavior of the strategy. By the law of large numbers, over many trials the actual growth rate converges to this expected value. The geometric growth rate is:

$$G(f) = \mathbb{E}\left[\frac{1}{n}\log\left(\frac{W_n}{W_0}\right)\right] = p \log(1 + bf) + q \log(1 - f)$$

where:

• $G(f)$: expected geometric growth rate per trial
• $p$: probability of winning
• $q$: probability of losing ($1-p$)
• $b$: win payoff
• $f$: fraction of capital wagered

This function $G(f)$ is the central object of Kelly's analysis. It is concave in $f$, meaning it rises to a unique maximum and then falls. Betting nothing ($f = 0$) yields zero growth, but betting everything ($f = 1$) also yields negative expected growth for any realistic bet because the logarithm of zero (which you face after a loss with $f = 1$) is negative infinity.

To find the optimal fraction $f^*$ that maximizes $G(f)$, we apply standard calculus, differentiating with respect to $f$ and setting the derivative to zero:

$$\begin{aligned} \frac{dG}{df} &= \frac{d}{df}\left(p \log(1 + bf) + q \log(1 - f)\right) \\ &= \frac{pb}{1 + bf} - \frac{q}{1 - f} \end{aligned}$$

The first term represents the marginal benefit of increased betting: with probability $p$, you win, and increasing your bet increases your logarithmic gain. The second term represents the marginal cost: with probability $q$, you lose, and increased betting amplifies your logarithmic loss. At the optimal point, these marginal effects balance perfectly.

Setting the derivative to zero gives us the first-order condition:

$$\frac{pb}{1 + bf} - \frac{q}{1 - f} = 0$$

where:

• $G$: expected growth rate
• $f$: fraction of capital wagered
• $p$: win probability
• $q$: loss probability
• $b$: win payoff

Solving for $f$ requires straightforward algebraic manipulation. We rearrange to isolate $f^*$:

$$\begin{aligned} \frac{pb}{1 + bf^*} &= \frac{q}{1 - f^*} \\ \implies pb(1 - f^*) &= q(1 + bf^*) && \text{(cross-multiply)} \\ \implies pb - pbf^* &= q + qbf^* && \text{(expand terms)} \\ \implies pb - q &= pbf^* + qbf^* && \text{(group constants and variables)} \\ \implies pb - q &= bf^*(p + q) && \text{(factor out } bf^* \text{)} \\ \implies pb - q &= bf^* && \text{(since } p + q = 1 \text{)} \end{aligned}$$

The final step uses the fundamental probability constraint that win and loss probabilities must sum to one. This yields the celebrated Kelly formula:

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

where:

• $f^*$: optimal fraction of capital to wager
• $p$: win probability
• $q$: loss probability
• $b$: win payoff (odds)

This result admits a clear interpretation. The numerator $pb - q$ represents the expected profit per dollar wagered, often called the "edge." The denominator $b$ represents the odds. The optimal bet fraction is simply the edge divided by the odds. When the odds are generous (large $b$), you can afford to bet more conservatively. When the odds are stingy (small $b$), you must bet more aggressively to exploit a given edge.

This is the Kelly Criterion for a simple bet. For the special case of even odds ($b = 1$), the formula simplifies even further:

$$f^* = 2p - 1 = p - q$$

where:

• $f^*$: optimal fraction for even odds
• $p$: win probability
• $q$: loss probability

The result is intuitive: bet a fraction equal to the edge (win probability minus loss probability). If you have a 60% chance of winning an even-money bet, you should wager 20% of your capital. This provides a concrete, actionable rule that balances the benefit of exploiting your edge against the risk of overbetting.

Real trading doesn't involve discrete binary outcomes. Instead, returns are continuous and approximately normally distributed over short horizons. We can derive a continuous version of the Kelly formula that applies directly to realistic trading scenarios where positions may gain or lose varying amounts.

Suppose a strategy has expected return $\mu$ and volatility $\sigma$ per period. If you apply leverage $\ell$ (meaning you bet $\ell$ times your capital), your leveraged return is $r_\ell = \ell \cdot r$ where $r \sim N(\mu, \sigma^2)$. Leverage scales both the expected return and the volatility. The expected leveraged return is $\mathbb{E}[r_\ell] = \ell\mu$ and the variance is $\text{Var}(r_\ell) = \ell^2\sigma^2$. Notice that variance scales with the square of leverage, which foreshadows the extreme danger of high leverage.

For small returns, which is a reasonable approximation for daily or weekly trading, the expected log return, which determines geometric growth, is approximately:

$$\begin{aligned} \mathbb{E}[\log(1 + r_\ell)] &\approx \mathbb{E}[r_\ell] - \frac{1}{2}\text{Var}(r_\ell) \\ &= \ell\mu - \frac{1}{2}\ell^2\sigma^2 \end{aligned}$$

where:

• $r_\ell$: leveraged return
• $\ell$: leverage ratio
• $\mu$: expected return of the strategy
• $\sigma^2$: variance of the strategy

This approximation uses the Taylor expansion $\log(1 + x) \approx x - \frac{x^2}{2}$ for small $x$. The formula separates expected return from the cost of variance: the first term $\ell\mu$ is the expected arithmetic return, which increases linearly with leverage. The second term $\frac{1}{2}\ell^2\sigma^2$ is the variance penalty, sometimes called "volatility drag," which increases quadratically with leverage. This quadratic penalty is why unlimited leverage is disastrous: at high enough leverage, the variance penalty overwhelms any expected return.

To maximize this geometric growth rate, we differentiate with respect to $\ell$:

$$\begin{aligned} \frac{d}{d\ell}\left(\ell\mu - \frac{1}{2}\ell^2\sigma^2\right) &= \frac{d}{d\ell}(\ell\mu) - \frac{1}{2}\sigma^2 \frac{d}{d\ell}(\ell^2) \\ &= \mu - \frac{1}{2}\sigma^2(2\ell) \\ &= \mu - \ell\sigma^2 \end{aligned}$$

The derivative has two terms: $\mu$ represents the marginal benefit of increased leverage (more expected return), while $\ell\sigma^2$ represents the marginal cost (more variance penalty). Setting the derivative to zero identifies where these opposing forces balance:

$$\mu - \ell\sigma^2 = 0$$

where:

• $\ell$: leverage ratio
• $\mu$: expected return
• $\sigma^2$: return variance

Solving this simple equation yields the continuous Kelly leverage formula:

$$\ell^* = \frac{\mu}{\sigma^2}$$

where:

• $\ell^*$: optimal leverage ratio
• $\mu$: expected return
• $\sigma^2$: return variance

This formula is intuitive. Optimal leverage increases with expected return, as you should bet more aggressively when the edge is larger. Optimal leverage decreases with variance, as you should bet more conservatively when uncertainty is higher. The formula can be rewritten using the Sharpe ratio $S = \mu/\sigma$:

$$\ell^* = \frac{\mu}{\sigma^2} = \frac{S}{\sigma}$$

where:

• $\ell^*$: optimal leverage ratio
• $\mu$: expected return
• $S$: Sharpe ratio ($\mu/\sigma$)
• $\sigma$: return volatility

This alternative form reveals that optimal leverage is the Sharpe ratio divided by volatility. A strategy with higher Sharpe ratio warrants more aggressive position sizing because the edge relative to risk is larger. A strategy with lower volatility also warrants higher leverage because the same proportional bet involves less absolute risk. This formula provides the foundational insight for sizing trading positions optimally.

Kelly betting has several important mathematical properties that make it theoretically attractive:

• Maximizes geometric growth rate: No other fixed-fraction strategy achieves higher long-term wealth growth. This optimality is exact under the model assumptions.
• Never goes bankrupt: Since you only bet a fraction of capital, you always have something left, though it can become arbitrarily small. This contrasts with fixed-dollar betting, which can lead to complete ruin.
• Variance increases with leverage: At Kelly leverage, the variance of log returns equals the squared Sharpe ratio: $\text{Var}(\log W) = S^2$ per unit time. This provides a direct link between strategy quality and wealth volatility.

However, Kelly betting also has significant drawbacks that limit its practical applicability:

• Extreme drawdowns: Full Kelly can produce drawdowns of 50% or more with high probability, even for profitable strategies. These drawdowns are mathematically expected, not rare events, and they create severe psychological challenges for you.
• Parameter sensitivity: The optimal fraction depends critically on $\mu$ and $\sigma$, which must be estimated. Overestimating $\mu$ or underestimating $\sigma$ leads to overbetting, which can be catastrophic because overbetting beyond Kelly has worse expected growth than underbetting by the same proportional amount.
• Assumes ergodicity: Kelly assumes you face the same bet repeatedly forever with unchanging parameters. Finite horizons or changing conditions violate this assumption, and the infinite-horizon optimal strategy may perform poorly over realistic investment horizons.

Given the practical dangers of full Kelly betting, most practitioners use fractional Kelly, betting a fraction (typically 0.25 to 0.5) of the Kelly-optimal amount. This sacrifices some expected growth for substantially reduced variance and drawdown risk. The rationale is that we never know the true parameters $\mu$ and $\sigma$, so using full Kelly based on estimates is almost certainly betting too aggressively.

If we denote the Kelly fraction multiplier as $\kappa \in (0, 1]$, the leveraged position becomes $\ell = \kappa \cdot \ell^* = \kappa \frac{\mu}{\sigma^2}$. The multiplier $\kappa$ represents our conservatism: $\kappa = 1$ is full Kelly, $\kappa = 0.5$ is half-Kelly, and $\kappa = 0.25$ is quarter-Kelly.

The expected geometric growth rate at fractional Kelly follows from substituting the scaled leverage into our growth rate formula:

$$\begin{aligned} G(\kappa) &= \kappa \ell^* \mu - \frac{1}{2}(\kappa \ell^*)^2 \sigma^2 \\ &= \kappa \frac{\mu^2}{\sigma^2} - \frac{1}{2}\kappa^2 \frac{\mu^2}{\sigma^2} && \text{(substitute } \ell^* = \mu/\sigma^2 \text{)} \\ &= \frac{\mu^2}{\sigma^2}\left(\kappa - \frac{\kappa^2}{2}\right) && \text{(factor out } \mu^2/\sigma^2 \text{)} \end{aligned}$$

where:

• $G(\kappa)$: expected geometric growth rate with fractional Kelly
• $\kappa$: Kelly fraction multiplier ($\kappa \in (0, 1]$)
• $\ell^*$: optimal full Kelly leverage
• $\mu$: expected return
• $\sigma$: volatility

This formula reveals the fundamental tradeoff in fractional Kelly. The term $\kappa$ in the parentheses represents growth that increases linearly with aggressiveness, while the term $\kappa^2/2$ represents the variance penalty that increases quadratically. As $\kappa$ increases from zero, growth initially increases faster than the penalty, but eventually the penalty dominates.

At full Kelly ($\kappa = 1$), the growth rate is $G(1) = \frac{\mu^2}{2\sigma^2}$. At half-Kelly ($\kappa = 0.5$), the growth rate is:

$$\begin{aligned} G(0.5) &= \frac{\mu^2}{\sigma^2}\left(0.5 - \frac{0.5^2}{2}\right) \\ &= \frac{\mu^2}{\sigma^2}(0.5 - 0.125) \\ &= 0.375 \frac{\mu^2}{\sigma^2} \\ &= 0.75 \left( \frac{\mu^2}{2\sigma^2} \right) \\ &= 0.75 G(1) \end{aligned}$$

where:

• $G(0.5)$: expected geometric growth rate at half-Kelly
• $G(1)$: expected geometric growth rate at full Kelly
• $\mu$: expected return
• $\sigma$: volatility

This calculation shows the efficiency of half-Kelly: it achieves 75% of the growth rate but with only 25% of the variance, since variance scales as $\kappa^2$. This tradeoff is highly attractive for risk-averse investors who value smoother wealth paths over maximum expected growth. The insight that you can sacrifice only 25% of growth while eliminating 75% of variance explains why fractional Kelly dominates practical applications.