Third-Party Research & Methodology Only

This section shares summaries of third-party academic research and descriptions of quantitative models. The content represents the findings of the original researchers, not the opinions or recommendations of Foxholm Financial. Foxholm Financial does not publish hypothetical or backtested performance metrics on its quantitative research pages. All content is restricted to methodology, signal construction, factor logic, and risk architecture. SEC rules require that investment advisers not present misleading performance data, and our methodology-only approach reflects that standard and the firm's fiduciary obligations.

Kelly Criterion

Risk Management Position Sizing Method Academic Finance

The Kelly criterion is a formula for deciding how much to stake on a bet or an investment position in order to maximize the long-run growth rate of wealth. Given the odds and the probability of winning, it prescribes a fraction of capital to commit. The central insight is that the size that grows wealth fastest over many repeated, independent opportunities is neither the largest possible stake nor a tiny one, but a specific fraction tied to the strength of the edge.

Introduced by John L. Kelly Jr. in 1956, the criterion frames sizing as a growth problem rather than a single-period gain problem. Staking too little leaves growth on the table; staking too much invites ruinous swings that compound poorly over time. The formula identifies the fraction that balances these two forces, which is why it has become a reference point in position sizing discussions across both gambling and investing.

Definition

The Kelly criterion is a rule that sets the fraction of capital to allocate to an opportunity so as to maximize the expected logarithm of wealth, which is mathematically equivalent to maximizing the long-run compound growth rate. Maximizing the logarithm matters because wealth compounds multiplicatively: a gain and a loss of the same percentage do not cancel, so the geometric (compounded) outcome, not the simple average, is what accumulates over many periods.

Because it targets growth over a long sequence of bets, the criterion is most directly applicable when opportunities repeat and are roughly independent. It is a sizing rule, not a selection rule: it answers how much to commit once an edge has been identified, and it says nothing about whether the edge is real. That separation makes it a natural companion to position sizing frameworks.

Key Principle

The prescribed fraction is proportional to the edge divided by the odds. In words, the larger the advantage relative to what is risked, the larger the suggested stake; the longer the odds, the smaller the stake for a given edge. The rule deliberately scales exposure to the quality of the opportunity rather than committing a fixed amount regardless of how favorable the situation is.

How It Works

Conceptually, the Kelly fraction is the edge divided by the odds. The edge is the expected gain per unit staked, reflecting both the probability of winning and the payoff structure. The odds describe how much is won relative to how much is risked on a winning outcome. When the edge is large and the odds are short, the formula points to a larger fraction; when the edge is thin or the odds are long, it points to a smaller one. When the expected edge is zero or negative, the formula prescribes no position at all.

Input	What It Represents	Effect on the Fraction
Probability of winning	How often the favorable outcome is expected to occur	Higher probability raises the suggested stake
Payoff odds	How much is gained relative to what is risked on a win	Longer odds lower the suggested stake for a given edge
Edge	Expected gain per unit staked, combining probability and payoff	A larger edge raises the suggested stake; zero edge means no position

In an investing context, the discrete win/loss inputs are replaced by estimates of an asset's expected excess return and its volatility (the dispersion of its returns). The continuous form of the rule then ties the recommended exposure to the ratio of expected excess return to variance, so a higher expected reward per unit of risk supports a larger position. The inputs are estimates, which is the source of much of the rule's practical fragility, discussed below.

Fractional Kelly

Because the full Kelly stake can produce large swings in capital, many practitioners use a fraction of it, an approach commonly called fractional Kelly. A half-Kelly or quarter-Kelly position commits a set portion of what the full formula suggests. The motivation is that the growth-versus-risk tradeoff is asymmetric near the peak: moving moderately below the full Kelly stake gives up relatively little long-run growth while substantially reducing the size of interim drawdowns (peak-to-trough declines).

Fractional Kelly also acts as a buffer against estimation error. The full formula assumes the inputs are known precisely, but in practice the probability of winning, the payoff, and the volatility are all estimated with uncertainty. Overstating the edge leads the full rule to oversize positions, so scaling down provides a margin against inputs that turn out to be too optimistic. The choice of fraction reflects a tradeoff between growth and tolerance for swings rather than a single correct answer.

Known Limitations

Limitations to Keep in Mind

Inputs are rarely known precisely. The formula assumes accurate estimates of probabilities and payoffs. In markets these are uncertain and unstable, and overstating the edge causes the rule to recommend positions that are too large.
Full Kelly can be highly volatile. Even with correct inputs, the full stake produces large fluctuations in capital and deep interim drawdowns. Many investors find these swings difficult to hold through, which is part of why fractional approaches are common.
It assumes many repeated, independent bets. The growth-maximizing logic relies on a long sequence of similar opportunities. For one-off decisions or short horizons, the long-run rationale does not apply cleanly, and the prescribed size may be ill-suited.
It ignores personal risk tolerance and goals. The criterion maximizes long-run growth, not utility for a specific investor. Someone with a near-term need for capital or a low tolerance for volatility may rationally hold far less than the formula suggests.
It assumes outcomes can be modeled. The rule needs a defined distribution of results. Rare, severe events that are poorly captured by the assumed model can produce losses larger than the framework anticipates.

Academic Origin

The criterion originated in a 1956 paper by John L. Kelly Jr., a researcher at Bell Labs, who derived it while studying information theory and the rate at which information can be transmitted over a noisy channel. Kelly showed that a gambler with private, partially reliable information should size bets to maximize the expected growth rate of capital, drawing a formal link between information and the mathematically ideal stake size.

The rule later moved from communications theory into gambling and finance, with Edward O. Thorp among its most prominent developers and popularizers across blackjack, sports betting, and securities markets. In the investment literature it is studied as a benchmark for growth-maximizing position sizing, usually accompanied by the caveats around estimation error and volatility that motivate fractional Kelly in practice.