Kelly Sizing for Non-Gaussian Return Distributions

Summary

The document examines how to choose a Kelly bet fraction when returns are not Gaussian. It notes that the familiar mean-over-variance estimate is exact only under Gaussian assumptions, and frames the general problem as maximizing expected logarithmic wealth growth, computed by integrating log wealth across the return distribution.

Because the logarithm is undefined when wealth becomes nonpositive, the discussion truncates the integral at the bankruptcy boundary. The author reports that this treatment produces a growth curve with no maximum and suggests the result may reflect modeling repeated betting without accounting for eventual ruin. Risk-Constrained Kelly is proposed as a possible way to restore a finite optimum, but the document gives no derivation, numerical example, or resolution. The issue therefore highlights the need to specify bankruptcy handling and tail behavior before interpreting an optimal fraction for non-Gaussian returns.

Key ideas

• The mean-over-variance Kelly approximation depends on Gaussian return assumptions.
• For general returns, Kelly sizing can be framed as maximizing expected logarithmic growth.
• The logarithmic growth calculation has a boundary where a loss makes wealth nonpositive.
• The proposed truncation yields no maximum in the author’s calculation.
• Risk-Constrained Kelly is raised as a possible way to impose a meaningful sizing limit.

Tags

# Calculating Kelly Bet Sizes for Non Gaussian Return Distributions


# Calculating Kelly Bet Sizes for Non Gaussian Return Distributions












Kelly position sizes are generally estimated using $\mu / \sigma^2$. This estimation is only accurate for a gaussian return distribution.

The exact formula is of course the argument at maximum of the growth rate as a function of the position.

This implies that the following integral must be evaluated to calculate average logarithmic growth rate per period for return distributions that are non gaussian:

$$ \int_{-\infty}^{+\infty} \log(1 + B r) \rho(r)\ dr $$

where $\rho(r)$ is the return distribution.

To calculate this integral numerically a truncation is required. This can be done in the following way:

$$\int_{-\frac{1}{B}}^{+\infty} \log(1 + B r) \rho(r)\ dr $$

However, with this truncation when $B$ is plotted on the X axis and geometric growth rate per period is plotted on the Y axis there exists no maximum with this truncation.

From my knowledge, this is because of the possible random walks which lead to bankruptcy but in the calculations it is assumed you can continue betting.

Has anyone dealt with such a problem?, i.e. finding kelly optimal sizing for non gaussian return distributions.

I am also trying to implement a ' Risk Constrained Kelly' approach as defined in the paper “Risk-Constrained Kelly Gambling” by Busetti, Ryu, and Boyd (2016) as I believe this will reestablish a maximum in this curve.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.