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I just read some articles about $MAD$ as a measure of risk in finance.

Is the following formulation a correct way to implement a $MAD$ portfolio optimization model which minimizes risk without considering expected return?

Assuming returns to be Gaussian distributed one can use $MAD=E(|X|)=\sigma \sqrt{\frac{2}{\pi}}$. The problem then can be written:

$$ w^* = {{\underset{w}{\mathrm{arg\ min}}} = \sqrt{w^T\Sigma w}\cdot \sqrt{\frac{2}{\pi}}}\\ s.t.,\ 1^Tw=1 $$

Once the assumption of Gaussian distributed returns is removed how the model can be formulated using matrix notation?

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    $\begingroup$ There is quite a bit of literature on MAD portfolio optimization, which I don't know very well. There is a somewhat famous paper in this area: Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science, 37(5), 519-531. And here is a comparison beween Markowitsz and MAD scholar.rose-hulman.edu/cgi/… $\endgroup$
    – noob2
    Dec 20 '20 at 21:34
  • $\begingroup$ Thank-you for the suggestion. Unfortunately I have not found a free access to the paper. In any case all the articles I considered provide formulations of the MAD model that require returns estimation (which is not the case for the above one) $\endgroup$
    – Nipper
    Dec 21 '20 at 11:40
  • $\begingroup$ This is not the full paper. $\endgroup$
    – Nipper
    Dec 21 '20 at 18:28
  • $\begingroup$ Sorry, my fault. $\endgroup$
    – noob2
    Dec 21 '20 at 18:40
  • $\begingroup$ @noob2 no problem ;) $\endgroup$
    – Nipper
    Dec 21 '20 at 23:42
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You can handle this problem with scenario optimization: assume a matrix $R$ of returns, in which the rows are the scenarios and the columns are assets. For given portfolio weights $w$, you can compute the portfolio returns as $Rw$. You can now evaluate an objective function such as the MAD, so your objective becomes $\min\ \mathrm{mean}(|Rw|)$. Now feed this model to an appropriate solver.

The paper mentioned by @noob2,

@ARTICLE{Konno1991,
  author       = {Konno, Hiroshi and Yamazaki, Hiroaki},
  title        = {Mean-Absolute Deviation Portfolio Optimization Model
                  and Its Applications to {T}okyo Stock Market},
  journal      = {Management Science},
  year         = 1991,
  volume       = 37,
  pages        = {519--531},
  number       = 5,
}

describes how to solve this model via linear programming.

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