Kelly criterion in general part I

General form for the return R:

$$R = \Pi_i (1+xa_i)^{p_i}$$

Optimal conditions:

$$\partial_x R = \sum_i \frac{p_i a_i}{1+xa_i}\Pi_j ((1+xa_j)^{p_j} = R \sum_i \frac{p_i a_i}{1+xa_i} = 0$$

Specializing to previous case, where i runs from 1 to 2:

$$\frac{p_1 a_1}{1+xa_1} + \frac{p_2 a_2}{1+xa_2} = 0$$

$$(p_1 a_1)(1+xa_2) + (p_2 a_2)(1+xa_1) = 0$$

$$p_1 a_1+ x(p_1+p_2)a_1a_2 + p_2 a_2 = 0$$

$$x = -\frac{p_1 a_1+p_2 a_2}{a_1a_2} = -\frac{\left<a\right>}{a_1a_2}$$

Notice that when $p_1\rightarrow 1$, $x \rightarrow -\frac{1}{a_2}$, so as disaster becomes ever more unlikely, you would bet a proportion of your wealth up to the loss ratio. This is a reflection of the Kelly criterion's tendency to never allow anything to go to zero, under any circumstance.

We'll consider why this is undesirable in some future post. For now, Let's make most use of the formulation, and try to find good ways of summarizing win/loss ratios and frequencies to fit into the form above.

The terms for each outcome ($\frac{p_i a_i}{1+xa_i} =\frac{p_i}{a_i^{-1}+x}$)  sum to zero for the optimal $x$. Since $x > 0$, These terms are either monotonically increasing or decreasing with $x$ depending on the sign of $a_i$.

The question then is how one would represent the two groups of monotonically increasing and decreasing terms so has to figure out which $x$ they net out at. This will be covered in the next post.