# Mean Absolute Deviation in m.v. portfolio optimization

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?

• 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/… Dec 20 '20 at 21:34
• 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) Dec 21 '20 at 11:40
• This is not the full paper. Dec 21 '20 at 18:28
• Sorry, my fault. Dec 21 '20 at 18:40
• @noob2 no problem ;) Dec 21 '20 at 23:42

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.