The Markowitz problem is an optimization problem of a series of
Gaussian distributions (symmetric) with a variance-covariance matrix
This is a common misunderstanding. Markowitz (mean-variance = MV) model do not require the Normal distribution of returns, even if such condition is optimal in some sense. The only necessary distributional condition is the finiteness of the second moments.
I wonder, however, if I can use distributional and asymmetric
information of the returns to solve the problem
You can ignore them and continue to use properly the MV allocation model. If you think that distributional features that go beyond mean and variance must be considered, then, you have to move beyond MV model. In this case I suggest you the Mean Conditional VaR model (M-CVaR).
For instance, I have a process that follows Frechet distribution, and
it does have a finite variance
In this case M-CVaR model can be used but MV model can be used as well; the solution(s) will be different. About those differences asymmetries matters a lot.
How can we do this for a series of Frechet distributions (asymmetric),
would it basically be the same? Or would there be certain precautions?
If our distributions would not be stable to have variance, how can we
decide to choose other measures of risk, how can we check for the
The M-CVaR model is a good alternative in most case. In any case I suggest to consider the following steps as separated:
- Choose the allocation model (for example: MV or MCVaR)
- Choose/find the algorithm for solve the problem at point 1
- Choose an estimation/simulation thecnique for parameters/distributions draw
All those points can be linked and all choice can depend from the same, or similar, considerations. However in my experience to face all problems together can bring in confusion.
I was wondering if something like that can be achieved by using
certain libraries, or if you can provide any pointer for me in this
In the case of alpha stable distributions (infinite variance) some solution that look like MV was proposed; there the scale parameter play the role of variance one. See: Financial Risk and Heavy Tails - Bradley and Taqqu (2001).
Finally, the literature about Optimal Portfolios with Skewed and Heavy-Tailed Distributions is vast. Related books are:
Quantitative Risk Management Concepts, Techniques and Tools – McNeil, Frey, Embrechts
Risk and Asset Allocation – Meucci - Springer
several related software exist, for example in Matlab: