# Kelly Criterion for Multiple Simultaneous Correlated Bets [closed]

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.

• Can you tell us about what you came across in your own research and why they did not work for you? Oct 9 at 11:11
• Kelly Criterion for a single or even multiple independent binary bets is easy to find, however; I can't really find anything regarding specifically what I asked for. Oct 9 at 19:29

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.