Relationship between Sharpe ratio, Kelly optimal fraction and volatility

NeroTulip · Post by **NeroTulip** » Mon Jan 28, 2013 8:20 am

Let's assume I have a strategy with a Sharpe ratio of 1. Kelly tells me that I can increase returns by increasing leverage up to the Kelly optimal fraction, but increasing leverage any further would actually damage long term returns.

In this very simplified model, what is the volatility that corresponds to the Kelly fraction? It would be useful to have a neat formula linking Sharpe ratio and optimal volatility.

PS: I would consider it to be an upper bound for volatility, but that's another discussion...

Praetorian · Post by **Praetorian** » Mon Jan 28, 2013 11:10 am

I'll give it a shot. Ignoring half kelly and riskless returns: Let Kelly be r/(vol^2) and Sharpe be r/vol. That means Kelly = sharpe/vol. Solving for vol gives: vol = sharpe/kelly. Is that the formula you would like to have?