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Given the portfolio return $R$ and the benchmark return $B$, I want to define a risk indicator, measuring the ability to beat the benchmark ($R>B$), given the downside risk taken; the latter not intended as an absolute loss, but as the risk of falling below the benchmark.

This measure could be set as:

$$ \frac{\mathbb{E}R-\mathbb{E}B} {\int_{r\leq b}\, (r-b)^2 f_{R,B}(r,b)\mathrm{d}r\mathrm{d}b} $$

where $f_{R,B}$ is the joint density for $R$ and $B$.

The result is Sortino-like, but the target here is not a scalar, but the return of the benchmark.

Do you find any conceptual or formal error in this approach? or can you suggest a better way to implement the idea?

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