So this is actually trivial and does not require calculus although one could do it with calculus, no doubt.
Kelly attempts to maximize growth which is given by:
g = risk free rate + (Sharpe ratio ^2)/2
When you succeed in finding optimal Kelly then you maximize g. Because of this simple formula it is obvious you are maximizing the Sharpe ratio when you maximize g (both being positive numbers and monotonically increasing).
This seems conclusive but troubling. All those years where Markowitz and Thorp debated the value (or lack of value) of the Kelly- criterion this connection was missed?
Oh well, ChatGPT 4 finally got it. I don’t see how it could be wrong within the relatively simple assumptions of the derivation of g in this formula (the usual Gaussian distribution etc)
This is not just an academic exercise for me. Portfolio Visualizer has some limitations in building a portfolio. I think there are some better ways that are not that hard mathematically, it seems. Several ways it seems now. That would be for another post, I think.
Jim