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## 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 $w$: $$f = 1^t \hat{w} = 1^t \Sigma^{-1} \hat{\mu}$$