Position Sizing Under Uncertainty

In 1956, John Kelly, a physicist at Bell Labs, published a paper titled "A New Interpretation of Information Rate" (Kelly, 1956). The paper was about information theory, not investing. But it contained a formula that would eventually change how quantitative investors think about bet size. The Kelly Criterion specifies the fraction of capital an investor should allocate to a favorable bet in order to maximize the long-run geometric growth rate of wealth.

The formula itself is compact. For a binary bet with probability p of winning and probability (1-p) of losing, the optimal fraction is f* = (bp - q) / b, where b is the payoff ratio and q = 1 - p. For a bet with a 60% chance of winning an equal amount, the Kelly fraction is 20% of capital. Not 50%. Not 5%. Twenty percent. The number surprises people when they first encounter it, in part because the intuition for sizing does not come naturally, even to trained professionals.

That last point is not conjecture. In 2016, Victor Haghani and Richard Dewey ran an experiment that made it concrete (Haghani and Dewey, 2016). They gave 61 finance professionals, quantitative analysts, portfolio managers, and traders, a simple computer game. A biased coin with a 60% chance of heads. Starting bankroll of $25. Maximum payout of $250. Thirty minutes to play.

The expected value of each flip was unambiguously positive. The optimal strategy was to bet a fixed fraction of current wealth, in the neighborhood of 10-20%, depending on the exact assumptions. What happened instead was striking. Twenty-eight percent of participants went bust. Only 21% reached the $250 cap. Many bet their entire bankroll on a single flip. Others increased their stakes after losses, a pattern that more closely resembles the gambler's fallacy than a sizing framework. The median payout was $91, well below the roughly $240 that a disciplined fractional approach would have produced.

The Kelly Criterion is mathematically optimal, but only under a condition that financial markets do not satisfy. The formula assumes the bettor knows the true probability of winning. In a casino, a card counter can estimate that probability with reasonable precision, which is how Ed Thorp, the mathematician who brought Kelly's work into investing, generated returns in the 1960s (Thorp, 2006). In financial markets, probabilities are estimated from data that is noisy, non-stationary, and subject to regime shifts. When the inputs carry estimation error, full Kelly sizing amplifies that error. A modest overestimate of edge at full Kelly leverage can produce a drawdown from which the portfolio may not recover. This is why Thorp himself, and most practitioners familiar with the framework, recommend betting a fraction of the optimal amount, typically one-quarter to one-half Kelly.

There is an asymmetry in how the investment industry allocates attention that is worth acknowledging. Security selection, the question of what to buy, absorbs the bulk of research budgets. Position sizing, the question of how much to buy, tends to receive less scrutiny. Ralph Vince (1990) demonstrated through simulation that the same set of trades, applied with different sizing rules, can produce terminal outcomes ranging from ruin to substantial wealth. The trades are held constant. Only the fraction of capital allocated changes. The implication is that sizing may explain more of the variance in portfolio outcomes than is commonly appreciated, though the relative contribution is difficult to disentangle from selection in live portfolios.

The Kelly framework is not a trading rule. It is a way of thinking about the relationship between edge, variance, and the growth rate of capital. It forces the investor to quantify what they believe their advantage is, to account for the uncertainty around that estimate, and to recognize that the relationship between bet size and long-run compounding is not linear. Betting too much, in the framework, is as costly as betting too little, and often more so, because the damage from overleveraging compounds faster than the opportunity cost of underleveraging. Whether or not one uses the formula directly, the intuition behind it is difficult to dismiss.