Generalizing the Sharpe Ratio with Downside Risk

Summary

The discussion asks whether portfolio returns can be compared using semi-standard deviation, which focuses on downside variation, instead of ordinary standard deviation. The answer frames the Sharpe ratio more generally as excess return divided by a chosen measure of risk. On a risk-return plot, this ratio represents the slope from the risk-free asset to the portfolio, with the selected risk measure defining the horizontal axis.

Under this broader definition, downside risk can replace standard deviation, and the resulting measure is associated with the Sortino ratio. The same idea can extend to other risk measures, including value at risk and expected shortfall. This changes what the ratio penalizes, so values based on different risk definitions are not automatically interchangeable. The response gives a conceptual explanation rather than a calculation procedure, empirical comparison, or guidance on choosing a downside threshold; users must define the risk measure consistently for their intended comparison.

Key ideas

• A Sharpe-style measure can divide excess return by a chosen risk measure.
• Using downside variation emphasizes losses rather than all return variability.
• A downside-focused version is associated with the Sortino ratio.
• Value at risk and expected shortfall are other possible risk denominators.
• Comparisons depend on applying a clearly defined risk measure consistently.

Tags

# Calculating Sharpe Ratio with semi-standard deviation


# Calculating Sharpe Ratio with semi-standard deviation












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?

## Answer by SlavicDoomer (score 1, accepted)

https://quant.stackexchange.com/a/70561

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.