Numerical Kelly Sizing for Correlated Return Distributions

Summary

The post formulates Kelly sizing for multiple simultaneous bets whose returns may be correlated. It represents the outcomes as a joint return vector with a probability density and the bet sizes as a vector. Expected logarithmic wealth is then expressed as an integral over the joint distribution; the optimum satisfies a set of first-order conditions involving each asset’s return and the combined portfolio return. A Gaussian distribution is given as an example of a possible joint model.

The answer does not provide a closed-form solution or a worked allocation. Instead, it suggests maximizing the log-utility function numerically. It notes that constraints depend on the setup: classical long-only fractional betting imposes a total-stake bound, while leveraged long-short positions remove that particular bound. In practice, the result depends on the return distribution, estimates, and feasible constraints; the post does not address estimation error or show empirical performance.

Key ideas

• Model correlated bets with a joint distribution of asset returns.
• Choose a vector of bet sizes to maximize expected logarithmic wealth.
• The optimality conditions may require numerical optimization rather than a closed-form solution.
• Constraints differ between classical fractional bets and leveraged long-short portfolios.

Tags

# Kelly Criterion for Multiple Simultaneous Correlated Bets


# Kelly Criterion for Multiple Simultaneous Correlated Bets












I am looking for an equation for the optimal fractional bet sizing for N number of simultaneous correlated bets.

I am looking specifically for an equation for binary bets, but an equation for bets with distributional profit and loss outcomes would be welcome as well.

## Answer by Michael Isichenko (score 3)

https://quant.stackexchange.com/a/68308

A natural question which was likely studied in academic literature (even though Kelly is not particularly popular among portfolio managers). I guess you could generalize Eq. (6.87) of this book. If $f(R)$ is the joint probability density of the returns of your $N$ assets (so $R$ is $N$-dimensional vector), for example Gaussian, $$ f(R)\propto\exp(-(1/2)\sum_{ij}C^{-1}_{ij}(R_i-\mu_i)(R_j-\mu_j)), $$ and $x$ is an $N$-dimensional vector of your bets, the Kelly log utility is $$ U(x)=\int\log(1 + x\cdot R)f(R)dR. $$ The utility is maximized when $\partial U/\partial x_i=0$, or $$ 0 = \int\frac{R_i}{1+x\cdot R}f(R)dR. $$ I don't know if this can be solved in a closed form, but maximizing $U(x)$ numerically seems doable. Note that in the classical Kelly case you need to consider only the slab are $0\le\sum_ix_i\le1$, but if you are long/short and levered, the constraint is lifted.

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