Kelly criterion and Sharpe ratio

Whats the relationship between the Kelly criterion and the Sharpe ratio?

$$ f=\frac{p(b+1)-1}{b} $$

where $f$ is a percentage of how much capital to place on a bet, $p$ is the probability of success, and $b$ is the payout odds (eg. 3 dollars for ever 1 dollar bet).

Is $b$ (the payout ratio) also the Sharpe ratio? I am having a hard time understanding what Ernie is refering to when he is connecting the two concepts.

Answer

The Sharpe ratio $S_i$ of a strategy indexed by $i$ is given by the ratio of the mean excess return $m_i$ to the standard deviation of returns $\sigma_i$,

The formula you have quoted is the discrete Kelly criterion. That's not so useful in trading, where the outcomes are continuous. The continuous Kelly criterion states that for every $i$th strategy with Sharpe ratio $S_i$ and standard deviation of returns $\sigma_i$, you should be leveraged $f_i = m_i/\sigma_i^2 = S_i/\sigma_i$.

Note of difference between the discrete and continuous criteria: The Kelly criterion is designed to protect your equity from "ruin", so it will never tell you to bet more than what you have in the discrete case - because when you "lose", you lose the complete bet you've placed. The leverage $f_i$ will always be $<1$ in the discrete case. On the other hand, in the continuous case, your leverage can be $>1$.

Let us assume we have a portfolio with an overall Sharpe ratio $S$. What Ernie is talking about is that the maximum compounded growth rate $g$ is given by $g = r + S^2/2$. We usually drop the risk-free rate (unless we post treasuries for margin), so we have $g = S^2/2$.