The Kelly criterion: optimal position sizing when you have a quantifiable edge
The Kelly criterion is a formula for determining the optimal fraction of a portfolio to allocate to a position when the probability of winning and the payoff ratio are known. Correctly applied, it maximises the long-run expected growth rate of the portfolio. Over-betting—allocating more than the Kelly fraction—produces lower long-run growth than the optimum and can lead to ruin; under-betting produces sub-optimal growth but is safe. The criterion was developed by John L. Kelly Jr. at Bell Labs in 1956 and has since become foundational in both gambling theory and systematic investing.
What the Kelly criterion calculates
The Kelly criterion identifies the position size that maximises the expected logarithm of wealth—which is equivalent to maximising long-run geometric growth. The key insight is that the optimal fraction is not the one that maximises expected arithmetic return, but the one that maximises the geometric mean of returns across many bets. Because portfolio compounding is multiplicative rather than additive, maximising the geometric mean of individual outcomes produces better long-run outcomes than maximising the arithmetic mean.
At its core, the Kelly calculation asks: given the probability of a positive outcome and the size of that outcome relative to the loss if wrong, what fraction of the portfolio should be committed to this position? A bet with a 60% probability of doubling the stake and a 40% probability of losing the entire stake has a very different optimal size than a bet with the same win probability but a smaller payoff.
The formula
Formula (simple binary case)
f* = (bp − q) / b
Where:
f* = the optimal fraction of the portfolio to allocate
p = probability of a win
q = probability of a loss (q = 1 − p)
b = the ratio of win size to loss size (net odds)
If a trade has a 55% probability of winning and profits and losses are equal in size (b = 1), the Kelly fraction is (1 × 0.55 − 0.45) / 1 = 0.10, or 10% of the portfolio.
Full Kelly versus fractional Kelly
Full Kelly is theoretically optimal but practically aggressive. It assumes perfectly accurate estimates of p and b—which are never perfectly known in real markets. Estimation error in these inputs causes the calculated Kelly fraction to be too large, leading to over-betting and larger drawdowns than expected. For this reason, practitioners almost universally use a fraction of the full Kelly bet—typically half Kelly (50%) or quarter Kelly (25%)—which sacrifices some expected return in exchange for substantially smaller drawdowns and better resilience to model error.
Half Kelly retains approximately 75% of the expected growth rate of full Kelly while reducing the variance of outcomes significantly. This trade-off is considered favourable by most practitioners: the cost in expected growth is modest; the benefit in reduced drawdown and reduced sensitivity to estimation error is material.
Kelly in a multi-asset portfolio
The binary Kelly formula extends to continuous return distributions: the Kelly fraction is the expected excess return divided by the variance of returns, which is equivalent to the Sharpe ratio divided by the portfolio's volatility. This formulation highlights the link between Kelly sizing and standard risk metrics: a position with a higher Sharpe ratio relative to its volatility deserves a larger Kelly allocation.
In a multi-asset portfolio, the full Kelly solution requires estimating the mean return vector and the full covariance matrix of all assets—the same inputs as mean-variance optimisation. The two frameworks are closely related: Kelly maximises geometric growth; mean-variance maximises expected return for a given variance. For a risk-averse investor, the constrained mean-variance solution is typically more appropriate than full Kelly.
Limitations
The Kelly criterion assumes that the probability and payoff estimates are correct. In financial markets, both are uncertain—expected returns are notoriously difficult to estimate, and the distribution of outcomes is not binary or even well-defined. Using the Kelly criterion with overconfident input estimates produces aggressive position sizes that generate severe drawdowns when the estimates are wrong. See risk tolerance in investing for context on how to calibrate position sizes to actual investor psychology rather than theoretical optimality.
The criterion also assumes that bets are independent. Sequential positions in a portfolio are often correlated—a drawdown in one position frequently coincides with drawdowns in related positions. This correlation means that the aggregate Kelly fraction for a portfolio of positions may be lower than the sum of the individual Kelly fractions calculated in isolation.
Kelly criterion in pfolio
pfolio's systematic portfolio construction uses risk-controlled position sizing rather than full Kelly. Portfolio weights, volatility, and Sharpe ratio are visible in pfolio Insights, providing the inputs needed to evaluate position sizing relative to the theoretical Kelly framework. The metrics help article covers the relevant implementation detail.
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