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?

  • 2
    $\begingroup$ 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/… $\endgroup$
    – nbbo2
    Dec 20, 2020 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, 2020 at 11:40
  • $\begingroup$ This is not the full paper. $\endgroup$
    – Nipper
    Dec 21, 2020 at 18:28
  • $\begingroup$ Sorry, my fault. $\endgroup$
    – nbbo2
    Dec 21, 2020 at 18:40
  • $\begingroup$ @noob2 no problem ;) $\endgroup$
    – Nipper
    Dec 21, 2020 at 23:42

1 Answer 1


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,

  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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.