Kelly Ratio
We present here the derivation of the Kelly ratio in several cases.
Discrete Bet with +1 and -1 outcome
For a bet with payoff +1 with probability $p$ of success and payoff -1 with probability $q=1-p$ , the optimal fraction of wealth to invest is:
$$f = p-q$$
Single Asset Continuous Bet Given $\mu$ and $\sigma$
This formula first introduced for discrete bets specifies the optimal leverage given an investment simple return $\mu$ and standard deviation $\sigma$:
$$f = \frac{r}{\sigma^2}$$
It is related to the sharpe ratio $S$ by
$$f = S/\sigma$$
A derivation can be found here or here.
Multi Asset Continuous Bet $\hat{\mu}=\mu-r$ and $\Sigma$
Given the vector $\hat{\mu}$ of returns above the risk-free rate and the asset return covariance matrix $\Sigma$, the optimal portfolio weights are:
$$ \hat{w} = \Sigma^{-1} \hat{\mu} $$
The unleverage portfolio weights are $$ w = \frac{1}{1^t \hat{w}} \hat{w} $$
The Kelly ratio is the optimal leverage ratio relating the leveraged portfolio weights $\hat{w}$ and the unleveraged weights $w$: $$ f = 1^t \hat{w} = 1^t \Sigma^{-1} \hat{\mu} $$