Constructing Semi-Covariance for Downside Risk
Summary
The discussion asks how downside semivariance can be extended to measure co-movement between assets and form a covariance-like matrix. It presents two candidate constructions: multiply each return after truncating it below a chosen reference level, or truncate the return product itself. These definitions capture different aspects of downside co-movement, and the responses say there is no settled consensus about which definition is correct or whether the resulting matrix will be positive semidefinite.
One proposed workaround is to retain a conventional correlation matrix and scale it on both sides using a diagonal matrix of semistandard deviations, provided those deviations are nonzero. The discussion also notes an optimization difficulty: when the reference is the mean, changing portfolio choices can change which observations count as downside. A fixed target avoids that moving threshold but assumes investors maintain the same target return. The replies point to heuristic optimization and later research on decomposing semivariances, but provide no comparative empirical evidence or universally accepted method.
Key ideas
- Downside co-movement can be defined by truncating returns individually or by truncating their product.
- The candidate semi-covariance definitions differ and do not have a universally accepted choice.
- A downside matrix may fail to be positive semidefinite, which complicates its use in optimization.
- Scaling a standard correlation matrix by semistandard deviations is one suggested construction.
- Using the mean as the downside threshold makes the observations counted as downside change with the portfolio.
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Full text
# Semi-variance/Downside Risk, what about the rest of the covariance matrix?
# Semi-variance/Downside Risk, what about the rest of the covariance matrix?
I just bumped into a rather interesting article from wikipedia :
http://en.wikipedia.org/wiki/Downside_risk
where they define the semi-variance also called Downside risk, which bascially only considers the "negative" variation with respect to some set level e.g. mean.
My question is : Is is possible to extend this also for the covariance, in order to obtain something like the covariance matrix ?
Thanks in advance
## Answer by Kumar (score 4)
https://quant.stackexchange.com/a/17192
There are 2 issues that come to mind
- What is the correct definition of semi-covariance $$ \frac{1}{n}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\min \left( {{r_i},0} \right)} } \min \left( {{r_j},0} \right) $$
$$ \frac{1}{n}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\min \left( {{r_i}{r_j},0} \right)} } $$ 2. Can you get a positive semi-definite covariance matrix with this definition?
These questions are tricky and there is no consensus.
## Answer by Steve Satchell (score 3)
https://quant.stackexchange.com/a/37807
one solution that works is set up the usual correlation matrix and pre- and post multiply by a diagonal matrix with semi standard deviations down the diagonal taking care that they are not zero
## Answer by Downer (score 2)
https://quant.stackexchange.com/a/19441
This is the challenge for below-mean semivariance in optimization. Since the mean becomes a moving target, the observations that impact the min function change. Estrada proposed a heuristic method for optimization and Beach(2011) discusses the decomposition and semi covariances. Below target semivariance assumes investors do not change their target return, if you believe that one.
## Answer by J. Steed Huang (score 0)
https://quant.stackexchange.com/a/66569
Yes, we extended Downside risk to Downside "co-risk" as shown below:
https://www.mdpi.com/1911-8074/14/4/172
Multi-Factorized Semi-Covariance of Stock Markets and Gold PriceShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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