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Would it make sense to calculate the Sharpe Ratio with the semi-standard dev. So as to standardize/compare asset returns to their downside risk?

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    $\begingroup$ Something similar to this exists and is called the Sortino Ratio . You can find some discussion of it on this site, for ex. quant.stackexchange.com/questions/54383/… $\endgroup$
    – nbbo2
    Commented Apr 16, 2022 at 13:25
  • $\begingroup$ To compare to only downside (negative) risk returns, use Sortino ratio. The semi-standard deviation is the standard deviation of returns below the mean. $\endgroup$ Commented Apr 19, 2022 at 15:19

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Beside standard deviation there are many other risk measures as well. And of course Sharpe ratio can be generalized to use any risk measure:

$$ \text{Sharpe} = \frac{\Delta y}{\Delta x} = \frac{\mu_R - \mu_F}{\text{Risk}_R} $$

where $\mu_R$ is portfolio return and $\mu_F$ is risk-free interest rate. And if you plot available portfolios on a 2D risk-return plane, Sharpe ratio of a given portfolio is just a slope of the line connecting risk-free asset with that portfolio, i.e. $\frac{\Delta y}{\Delta x}$.

For instance in R package PerformanceAnalytics there is a function which calculates Sharpe ratio taking as risk measure Value-at-Risk, Expected Shortfall etc.

So there is no obstacle to use downside risk in generalized Sharpe ratio formula. In fact, such ratio is sometimes called Sortino Ratio.

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