Downside deviation (semivariance) in m.v. portfolio optimization

Question

Currently I am considering the downside deviation or semivariance in a m.v. optimization framework.

For this specific measure of risk I have found in papers different formulae. The majority of them are regarding the single asset rather the entire portfolio.

So my question is "Which of the following is the most adequate approach to compute and use this measure for portfolio optimization?"

1. Given a $NxT$ returns series one sets to $0.$ all the returns that are above some specific target and then one computes the covariance matrix with the modified series. Then portfolio variance (semivariance in this case) is computed as $w^T\Sigma w$.
2. Given a $NxT$ returns series one computes the covariance matrix using the whole series then sets its diagonal with the assets semvariance.
3. One should compute the DD directly for the whole portfolio considering weights (i.e. in the objective function). Please consider the following formula from "Robust Portfolio Optimization and Management by Fabozzi":

$$\sigma^2_{P,\ min} = E\bigg(min\bigg(\sum^n_{i=1}w_iR_i-\sum^n_{i=1}w_i\mu_i, \ 0\bigg)\bigg)^2$$

I suspect the first approach is not ideal due to the fact that some useful information is missing when computing the covariance matrix.

Answer 1

The third approach is the correct one. In general, one cannot aggregate partial moments of single assets into partial moments of the portfolio, as discussed for instance in this paper:

@ARTICLE{Grootveld1999,
  author       = {Henk Grootveld and Winfried Hallerbach},
  title        = {Variance vs downside risk: Is there really that
                  much difference?},
  journal      = {European Journal of Operational Research},
  year         = 1999,
  volume       = 114,
  pages        = {304--319},
  number       = 2,
}

However, approximations may be possible and may even work well when the distributions are reasonably symmetric. (In which case there might be little use for partial moments, anyway.) Computationally, there should be no need to use approximations because the third, direct approach can easily be handled; see for instance the last example in CVAR alternatives for optimization .