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If $Q$ is your covariance matrix, and $r$ is a vector of your expected returns, then the maximum Sharpe ratio is given by the following math program. $${\rm maximize} \frac{r^t x}{\sqrt{0.5 x^t Q x}}$$ subject to $$ 1^t x = m$$ $$ x \in \{0,1\}^n$$ Where $x$ is a vector of indicators of which of the $n$ assets are part of the $m$ selected assets. While the ...


For such a problem ("selecting n out of m") you can use optimisation heuristics. These algorithms work well even for large n and m, and they are flexible: you may as well select a portfolio that minimises some other function, for instance, the portfolio's drawdown. The downside is that you may have to do some programming yourself. An example very similar ...

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