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I am considering as my objective the Sortino ratio:

$\frac{\mu^{\top}x-R}{\sqrt{\mathbb{E}[(min\{0,(r-\mu)^{\top}x\})^2]}}$

In my textbook they state that this ratio just like the Sharpe ratio is neuther convex nor concave. I am stuggeling however to see why this is the case. Any ideas?

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    $\begingroup$ Convexity would mean: whenever you combine two portfolios with known SR into a bigger portfolio, the SR of the combined portfolio is always less than or equal the weighted average SR of the two. Unfortunately this "theorem" is not true as an example would show. Concavity would be "is always greater than or equal", unfortunately this is false also. It depends on the portfolios that are being combined, in other words on the risk reduction that takes place. $\endgroup$
    – nbbo2
    Mar 24 at 13:06

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