Optimising Position Sizing Using Kelly Criterion Variants

1. The Kelly Problem

The Kelly criterion, first derived by John Kelly in 1956 for information transmission and later applied to gambling and investing, prescribes the fraction of capital to wager that maximises the long-run geometric growth rate. For a simple binary bet with win probability p and odds b, the Kelly fraction is:

where q = 1 − p. The elegance of this formula belies the practical difficulties of its application to trading. The Kelly criterion assumes known and stationary parameters — the probability of winning and the payoff ratio — neither of which is observable in financial markets. Estimation error in these parameters leads to over-betting, which can produce catastrophic drawdowns.

This paper examines several modifications to the Kelly criterion that address parameter uncertainty, and evaluates their performance in simulation under conditions designed to approximate real trading environments.

2. Kelly Variants

2.1 Fractional Kelly

The most common modification is to bet a fixed fraction of the Kelly amount. Half-Kelly (f = 0.5 · f*) is widely recommended in the practitioner literature. The geometric growth rate under fractional Kelly is:

At half-Kelly, the growth rate is approximately 75% of full Kelly, while the variance of outcomes is reduced by 50%. This is a favourable trade-off for most practitioners, but the choice of fraction is arbitrary and does not adapt to the degree of parameter uncertainty.

2.2 Bayesian Kelly

A more principled approach is to integrate the Kelly fraction over the posterior distribution of the parameters. Rather than estimating a point value for p and b and computing f*, The illustrative design maintains a posterior distribution over these parameters and compute the position size that maximises expected geometric growth under parameter uncertainty.

For a conjugate Beta-Binomial model of the win probability:

import numpy as np
from scipy.stats import beta
from scipy.optimize import minimize_scalar

def bayesian_kelly(wins, losses, avg_win, avg_loss, n_samples=10000):
    """
    Compute Bayesian Kelly fraction by integrating
    over the posterior distribution of win probability.
    """
    alpha = 1 + wins    # Beta posterior (uniform prior)
    beta_param = 1 + losses

    # Sample from posterior
    p_samples = beta.rvs(alpha, beta_param, size=n_samples)
    b = avg_win / avg_loss  # payoff ratio (point estimate)

    def neg_expected_growth(f):
        if f <= 0 or f >= 1:
            return 0
        growth = p_samples * np.log(1 + f * b) + \
                 (1 - p_samples) * np.log(1 - f)
        return -np.mean(growth)

    result = minimize_scalar(neg_expected_growth,
                            bounds=(0.001, 0.999),
                            method='bounded')
    return result.x

The Bayesian Kelly fraction is always smaller than the point-estimate Kelly fraction, because the concavity of the logarithmic utility function penalises over-betting more than it rewards under-betting (Jensen's inequality). The degree of shrinkage depends on the posterior variance, which in turn depends on the sample size.

2.3 Regularised Kelly

A computationally simpler alternative adds a quadratic penalty to the Kelly objective:

The regularisation parameter λ controls the degree of conservatism. This can be calibrated to a target maximum position size or a target drawdown level.

3. Simulation Design

We simulate 10,000 equity curves for each Kelly variant under the following conditions. The true win probability is drawn from Uniform(0.45, 0.65) — spanning strategies from marginally profitable to strong. The true payoff ratio is drawn from Uniform(1.0, 2.5). Each simulation runs for 500 trades. Transaction costs of 2 basis points per round trip are applied. The parameter estimates used by the Kelly variants are computed from a rolling window of the most recent 100 trades, introducing estimation error.

We measure terminal wealth, maximum drawdown, probability of ruin (defined as a drawdown exceeding 50%), and the realised geometric growth rate.

4. Results

The results confirm the well-known dangers of full Kelly: despite the highest median terminal wealth, the 28.4% ruin probability and 61.3% median maximum drawdown make it impractical for most traders. The psychological cost of a 60%+ drawdown — even if the long-run expectation is favourable — is a binding constraint that the Kelly framework ignores.

Bayesian Kelly achieves a higher growth rate than half-Kelly while maintaining a comparable drawdown profile. The advantage is most pronounced when the true parameters are far from the initial estimates — precisely the scenario where fixed fractional Kelly is most vulnerable to over-betting.

5. Sensitivity to Estimation Window

We repeat the simulation with estimation windows of 50, 100, 200, and 500 trades. Bayesian Kelly shows the least sensitivity to window length, because the posterior distribution automatically widens when less data is available, producing more conservative position sizes. Full Kelly's ruin probability increases monotonically as the window shrinks, reaching 41.2% at a 50-trade window.

6. Discussion

The practical lesson is straightforward: any Kelly-based position sizing scheme must account for parameter uncertainty. The specific method matters less than the principle. Bayesian Kelly is theoretically appealing but computationally heavier. Half-Kelly is simple and robust. The regularised variant offers a middle ground with a single tuning parameter.

For traders implementing systematic strategies, We recommend starting with half-Kelly as a baseline and moving to Bayesian Kelly only if the computational overhead is acceptable and the strategy generates sufficient trade data for meaningful posterior estimation. Quarter-Kelly is appropriate for strategies with short track records or high parameter uncertainty.

The key insight is that the optimal position size is a function of what you don't know, not just what you do know. Strategies with higher parameter uncertainty should use smaller fractions, regardless of the point estimate of edge.