Computing Portfolio Semivariance for Mean-Variance Optimization
Summary
The document compares ways to incorporate downside deviation, or semivariance, into portfolio optimization. It considers modifying each asset’s returns by replacing observations above a target with zero, using a covariance matrix whose diagonal contains asset semivariances, and calculating downside risk directly from the weighted portfolio returns.
The answer favors calculating the portfolio’s downside deviations after applying portfolio weights. Its reasoning is that individual assets’ partial moments generally cannot be combined to recover the portfolio’s partial moment, because portfolio downside outcomes depend on how asset returns move together. A cited academic paper discusses the broader comparison between variance and downside risk. The answer notes that approximations may work when return distributions are reasonably symmetric, but argues that direct calculation is computationally manageable. It does not provide implementation details or empirical comparisons of optimization results.
Key ideas
- Portfolio semivariance should generally be calculated from weighted portfolio returns.
- Asset-level downside measures do not generally aggregate into the portfolio’s downside risk.
- A covariance matrix built from individually clipped returns may omit information needed to describe portfolio downside outcomes.
- Approximations may be reasonable for sufficiently symmetric return distributions, though they may offer little benefit in that case.
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# Downside deviation (semivariance) in m.v. portfolio optimization
# Downside deviation (semivariance) in m.v. portfolio optimization
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?"
- 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 $w^T\Sigma w$.
- Given a $NxT$ returns series one computes the covariance matrix using the whole series then sets its diagonal with the assets semvariance.
- 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 by Enrico Schumann (score 2)
https://quant.stackexchange.com/a/60331
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 .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.