Kelly fraction for discrete distributions

Question

The Kelly fraction is $f^\star$ maximizing $\mathbb E[\log(1+f X)]$. For instance, if $$ X\sim\begin{cases} 1 & w.p. p\\ -1 & w.p. 1-p \end{cases}, $$ we get that $f^\star=2p-1$. I'm curious about closed-forms of $f^\star$ for discrete distributions with $X\in [-1,1]$. I wonder if such a closed-form is known in the economic literature.

I can derive a simple closed-form if $supp(x) = \{-1,0,1\}$ by rescaling $p$, but I'm interested in a larger (finite) support, say $\{-1,-\frac{1}{2},0,\frac{1}{2},1\}$.

Any ideas?

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Edit Given the comment, I'll clarify. The function $\mathbb E[\log(1+f X)]$ is strictly concave in $f$ for $f\in[0,1]$; hence, there exists only one maximum. Since it is also differentiable, that maximum is the root of its derivative.

Taylor series implies that $$ \log(1+y)=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {y^{n}}{n}}=y-{\frac {y^{2}}{2}}+{\frac {y^{3}}{3}}-\cdots $$ Hence, $$ \frac{d}{df}\mathbb E[\log(1+f X)]=\frac{d}{df}\mathbb E\left[\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {(fX)^{n}}{n}}\right]. $$ Invoking linearity of expectation, we have $$ \frac{d}{df}\mathbb E[\log(1+f X)]=\sum _{n=1}^{\infty }(-1)^{n+1}{f^{n-1} \mathbb E[ X^n]}. $$ The Kelly criterion is $f^\star$ is the root of the above equation. I'm wondering whether it has a nice closed-form for some discrete, non-Bernoulli distributions.

Answer 1

I’ve already made a comment about how to find the Kelly bet size, f, but I guess that misunderstood the question. I come by it honestly though, because it is so vague. There are an infinite number of possible discrete distributions with payoffs between -1 and 1 for certain probabilities. You know best what question you want answering, and if you know the tools, which “of course” you do, you can solve it.

Numerical methods can find an exact answer to the question too. I don’t understand why they are not applicable. You need numerical methods to find basic stuff like IRR. That function is super close to the derivative of your Kelly bet. Yet there’s no motivation for a closed form IRR formula because numerical methods are so accurate.

I also don’t understand why a Taylor Series expansion won’t work. Take it out to the fourth term. Then you have $O(x^5)$ error. Is that not precise enough, given any forward probability estimate in finance probably very likely contains WAY more error? One should not lose the forest through the trees.