Target Downside Deviation and the Sortino Ratio Denominator
Summary
The document compares ways to calculate semivariance, focusing on the downside deviation used in the Sortino ratio. It reports a target downside deviation formula that measures each return’s shortfall from a target, replaces returns at or above the target with zero shortfall, squares the results, averages across the full sample, and takes the square root. Under this method, observations meeting or exceeding the target remain in the denominator through the full sample count.
The explanation argues that excluding those zero shortfalls would erase information about how often returns avoid falling below the target. It illustrates this with two return streams that have the same worst shortfall but different frequencies of below-target outcomes. The document also notes that published formulas differ, including some that omit zero observations, so it does not resolve every convention or establish that all sources use the same standard. It recommends matching the calculation to the intended Sortino definition and target return.
Key ideas
- Target downside deviation measures shortfalls relative to a target return.
- The presented Sortino method assigns zero shortfall to returns at or above the target and averages across all observations.
- Keeping zero shortfalls in the sample count captures how frequently returns meet the target.
- Excluding zero shortfalls can change the denominator and produce a different ratio.
- The document notes that sources use differing semivariance conventions.
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Full text
# Semivariance calculation (downside deviation)
# Semivariance calculation (downside deviation)
what is the accurate formula for semivariance? I see two versions up to now:
- this version which considers as N (denominator) all the numbers over/under the mean-or any other number. This is the same of a version of CFA (book: Quantitative Methods for Investment Analysis - 2004 page 136). This is the formula:
- Another version (stated in another CFA´s book) shows a different formula. This is the formula (taken for another source):
.
The first difference is in the numerator (which is Min between "a","b") and the second is in the denominator (where N is over the entire sample).
Which one is the correct one and why? I want to use this in the Sortino ratio
PD: In addition, I found this other comment which summarizes what I meant (link):
> ShaktiRathore It was my understanding that the downside deviation (i.e., denominator in Sortino) does not include the zeros; i.e., when PMAR, these positive excess values are EXCLUDED, not treated as zero. Although I had understood this to be the GIPS-compliant method (at the time I sat for the CIPM but this was several years ago ....), it seems to be controversial...
Thank you very much!
## Answer by Pithit (score 4, accepted)
https://quant.stackexchange.com/a/39007
I am interested in Semivariance because I want to use it to compute the Sortino Ratio. I found an article on Sortino which answers to my question. Here is the link "Sortino ratio: A better measure of risk, by Tom Rollinger and Scott Hoffman", Futures Magazine 2013.
In this article the Sortino Ratio is defined as $$SR=\frac{R-T}{TDD}$$ where R is the average period return, T is the target or required rate of return, and TDD is the Target Downside Deviation, which is found as $$TDD=\sqrt{\frac{1}{N}\sum_{i-1}^N[\min(0,X_i-T)]^2}$$
From a practical point of view, the calculation must take into account all the data (substituting a zero for those values above or equal to your target), not just the observations below the target. This is so the Sortino ratio will return a higher value that Sharpe ratio when there are many observations above the target. If you consider a reduced sample (excluding zeros in the denominator) the Sortino will be lower than Sharpe, which is not the idea of this ratio.
> "The Sortino ratio takes into account both the frequency of below-target returns as well as the magnitude of them. Throwing away the zero underperformance data points removes the ratio’s sensitivity to frequency of underperformance."
Example from the link:
> "Consider the following underperformance return streams: `[0, 0, 0, –10]` and `[–10, –10, –10, –10]`. Throwing away the zero underperformance data points results in the same target downside deviation for both return streams, but clearly the first return stream has much less downside risk than the second."
So, I should not discard any zero because I will be reducing the data, which results in a lower Sortino than Sharpe ratio.
Finally, I would like to point it out that I found papers and books (one was from the CFA) that compute semivariance using the first method (throwing away the zeros). So I think confusion around it is still there.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.