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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

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There are 2 issues that come to mind

  1. 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.

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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

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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.

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  • $\begingroup$ Thanks, it is indeed quite heuristic. However, I lack the intuition of how this matrix looks like in order to be able to prove that it full fills the condition in order to justify its use in optimization, i.e. positive definiteness. I applied some tests in R to assess this property, e.g. Cholesky decomposition and it seems to work. But is there any formal way to prove this ? $\endgroup$
    – Jonkie
    Aug 24 '15 at 8:53
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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 Price

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