How correlation shrinks your Kelly sizes

Question

You have two bets with identical edge \mu and volatility \sigma, but they are correlated with ho = 0.5. How much do you size each one, compared to sizing a single standalone bet? What does the answer become as ho 	o 1?

Accepted Answer

For correlated Gaussian bets the Kelly sizing is the vector generalization \mathbf{f}^ = \Sigma^{-1}\boldsymbol{\mu}, where \Sigma is the covariance matrix. For two symmetric bets, \Sigma = \sigma^2\begin{pmatrix}1 & ho\ ho & 1\end{pmatrix} and \boldsymbol{\mu} = (\mu, \mu). Inverting, \Sigma^{-1} = \frac{1}{\sigma^2(1-ho^2)}\begin{pmatrix}1 & -ho\ -ho & 1\end{pmatrix}, so each optimal fraction is fi^ = \frac{\mu(1-ho)}{\sigma^2(1-ho^2)} = \frac{\mu}{\sigma^2(1+ho)}. Compare to a standalone bet, which would get \mu/\sigma^2. The correlation scales each size by \frac{1}{1+h…

How correlation shrinks your Kelly sizes

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