Semi-variance/Downside Risk, what about the rest of the covariance matrix?

Question

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 1

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.

Answer 2

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 3

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.