Portfolio Optimization with Skewed and Heavy-Tailed Returns
Summary
The document explains that mean-variance portfolio optimization does not require normally distributed returns; it requires finite second moments. For returns with finite variance, including the stated Fréchet example, mean-variance allocation remains available, though it may produce different allocations from a model that accounts for asymmetry and tail behavior.
For investors who want to incorporate those distributional features, the response recommends mean-CVaR optimization. It separates the task into choosing an allocation objective, selecting a solver, and choosing an estimation or simulation method for the return distributions. It also notes that alternatives resembling mean-variance methods have been proposed for alpha-stable returns with infinite variance, using scale in place of variance. The discussion gives conceptual guidance and references but no derivation, empirical comparison, or specific procedure for checking subadditivity, so risk-measure choice and model implementation remain open questions.
Key ideas
- Mean-variance optimization can be used with non-normal returns when their second moments are finite.
- Mean-CVaR optimization is suggested when portfolio decisions should reflect asymmetry or tail risk beyond mean and variance.
- Objective choice, optimization algorithm, and distribution estimation or simulation are distinct decisions that should be considered separately.
- For infinite-variance alpha-stable returns, the response mentions approaches that use the scale parameter in a role similar to variance.
- The document does not give a concrete method for evaluating subadditivity or comparing the resulting allocations.
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# Optimal Portfolios with Skewed and Heavy-Tailed Distributions # 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: - https://www.actapress.com/Abstract.aspx?paperId=28635 - https://link.springer.com/article/10.1007/s001860050092 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 ## Answer by markowitz (score 2) https://quant.stackexchange.com/a/60974 > 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: - 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 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
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