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Constructing a Downside-Risk Efficient Frontier with Sortino Optimization

Article Quant Q&A · Author: OvermanZarathustra
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Summary

The document asks how to choose portfolio weights for a Post Modern Portfolio Theory efficient frontier, where risk is measured by downside loss rather than variance. It contrasts calculating downside risk for an existing portfolio with the harder task of selecting asset weights to optimize it.

The response points to a formulation based on the Sortino ratio and suggests that quadratic optimization tools may help implement it. It does not provide the optimization equations, constraints, or a worked portfolio example, and the cited supporting papers are not reproduced in the text. The proposed approach is therefore a starting point rather than a complete recipe; the appropriate downside-risk measure and return target would need to be specified for a practical implementation.

Key ideas

  • PMPT evaluates risk using downside outcomes rather than variance.
  • Choosing asset weights to optimize downside risk is more involved than measuring risk for a fixed portfolio.
  • The response identifies a Sortino-based formulation as a possible basis for optimization.
  • Quadratic optimization software may be useful, but the document does not provide a complete algorithm.

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Portfolio constructionRisk managementStatistics
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# Efficient frontier using Post Modern Portfolio theory


# Efficient frontier using Post Modern Portfolio theory












I have been trying to find a way to create the efficient frontier using Post Modern Portfolio Theory (PMPT), but have failed to come across a source that mentions how to do so. I know PMPT uses downside risk as opposed variance (MPT), so somehow I need to find a method to minimize downside risk I suppose.

According to this research paper: http://www.ecocyb.ase.ro/nr_2013_pdf/Geambasu%20Cristina,%20Robert%20Sova.pdf,

"It is simple to compute for a share or for a portfolio already formed, for historical or predictive data, but things became more complicated if we intend to use the PMPT model in determining the portfolio assets structure"

So is there not a way to find an optimal set of asset weights to minimize downside risk?

In this paper by Rom and Ferguson, http://www.actuaries.org/AFIR/Colloquia/Orlando/Ferguson_Rom.pdf, they mention

"The PMPT efficient frontier is calculated using an algorithm for downside risk developed by The Pension Research Institute applied to the expected return, standard deviation and skewness values" and they also provide an efficient frontier calculated by using PMPT on p.12, but the algorithms I'm assuming have not been made public.

So my question is, does anyone know what the algorithm is/might be or how can I go about creating it?

## Answer by mark leeds (score 1)

https://quant.stackexchange.com/a/49458

Hi: Footnote 15 of the paper at this link explains what the formulation is (in brief: it is based on the Sortino Ratio). It sounds like something that can be programmed as a quadratic optimization. R has a lot of facilities for doing that sort of thing.

Addendum: I didn't read it but this paper provides a lot more detail than the one above.

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.