Mean Absolute Deviation in m.v. portfolio optimization

Question

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?

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,

@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.