Scenario-Based Mean Absolute Deviation Portfolio Optimization

Summary

The document explains how to formulate a portfolio that minimizes mean absolute deviation (MAD) without an expected-return objective. Under normally distributed returns, it relates MAD to portfolio volatility, so minimizing MAD is equivalent to minimizing the standard deviation implied by the covariance matrix, subject to weights summing to one.

For returns without a Gaussian assumption, the answer proposes scenario optimization. Arrange observed or assumed returns in a matrix with scenarios in rows and assets in columns, calculate each scenario’s portfolio return from the weight vector, and minimize the average absolute portfolio return. The cited work by Konno and Yamazaki describes a linear programming approach to this model. The document does not discuss additional portfolio constraints, expected-return targets, or how scenarios should be selected. Its scenario formulation also depends on whether absolute portfolio returns, rather than deviations from their mean, match the intended definition of risk.

Key ideas

• With Gaussian returns, MAD is proportional to portfolio standard deviation.
• A minimum-risk formulation can minimize portfolio volatility subject to weights summing to one.
• Scenario returns can be arranged in a matrix and multiplied by portfolio weights.
• The proposed scenario objective minimizes the mean absolute portfolio return.
• The cited research describes solving the MAD model with linear programming.

Tags

# Mean Absolute Deviation in m.v. portfolio optimization


# 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?

## Answer by Enrico Schumann (score 1)

https://quant.stackexchange.com/a/60338

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.

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.