# Optimal Portfolios with Skewed and Heavy-Tailed Distributions

I am learning about portfolio theory and been using Markowitz. I wondered, however, if I can use distributional and asymmetric information of the returns to solve the problem. For instance, I have a process that follows Frechet distribution, and it does have a finite variance, for which such optimization would be very helpful.

I searched online a lot to find a solution, but things I found so far are quite math extensive, and for what I have figured, have not solved the optimization problem. I found several interesting sources including the following most interesting one:

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 quest.

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 subadditivity?

Thanks

• Certain formulations of Meucci's Copula Opinion Pooling admit optimization (not necessarily in closed form). I say this as a comment rather than an answer because COP is definitely "math-heavy". Finding an answer that is not math heavy might be a quixotic pursuit. Feb 4, 2021 at 14:25
• Can you think of a good reference on it, perhaps an example? Thanks a lot! Feb 4, 2021 at 14:27
• My other idea is that there must be a numerical solution for this, considering a Dirichlet draw for weights. Does that make sense to you? Feb 4, 2021 at 14:33

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 subadditivity?

The M-CVaR model is a good alternative in most case. In any case I suggest to consider the following steps as separated:

1. Choose the allocation model (for example: MV or MCVaR)
2. Choose/find the algorithm for solve the problem at point 1
3. 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 quest.

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

https://www.amazon.com/Quantitative-Risk-Management-Techniques-Princeton/dp/0691166277

Risk and Asset Allocation – Meucci - Springer

https://www.springer.com/gp/book/9783540222132

several related software exist, for example in Matlab:

https://it.mathworks.com/videos/cvar-portfolio-optimization-1538481411693.html

• Thanks a lot for a very descriptive answer! Feb 6, 2021 at 21:00