Assumptions and Limits of Higher-Moment Portfolio Optimization
Summary
The document discusses how portfolio optimization changes when skewness and kurtosis are added to mean and variance. It emphasizes that mean-variance optimization does not itself require normally distributed returns: the chosen portfolio can differ across distributions that share similar parameters, as higher distributional features affect portfolio outcomes. The claim that mean-variance optimization assumes normality is therefore treated as an imprecise shorthand for its limited treatment of distribution shape.
For higher moments, the response warns that estimating co-kurtosis matrices may be impractical when covariance estimation is already difficult. It suggests scenario-based returns as an alternative: calculate portfolio outcomes in simulation and measure their skewness and kurtosis directly. This avoids relying on analytical higher-moment formulas, but requires investors to specify preferences for trading off those moments. The response presents CVaR or Expected Shortfall as a more tractable way to address tail risk, while offering no formal comparison or universal assumptions for all higher-moment methods.
Key ideas
- Mean-variance optimization does not inherently require normally distributed returns.
- Matching mean and covariance does not ensure that different return distributions produce the same optimized portfolio.
- Estimating co-kurtosis can be difficult when covariance estimation is already unreliable.
- Scenario-based optimization can measure portfolio skewness and kurtosis from simulated outcomes.
- Higher-moment objectives require preferences about how those moments trade off, while CVaR or ES offers a tail-risk alternative.
Tags
Full text
# What are the assumptions of portfolio optimisation with higher moments? # What are the assumptions of portfolio optimisation with higher moments? I was wondering whether there are a set of assumptions for portfolio optimisation with higher moments (including kurtosis and skewness) as there are for regular mean-variance optimisation? ## Answer by John (score 4) https://quant.stackexchange.com/a/11171 There are many papers on this subject (try googling portfolio optimization skewness kurtosis) that can describe the assumptions of including skewness and kurtosis in a utility function (if that's what you're interested in). I would highlight two main points. - Mean-variance optimization does not make an assumption of normality. Assume returns are distributed by a multivariate normal distribution with particular mean and covariance parameters and then construct a portfolio. Then perform the same optimization assuming a multivariate t distribution with the same parameters and a degree of freedom parameter. The result will be different portfolios that will become more alike as you increase the degrees of freedom. This is because the parameter for covariance in a multivariate normal distribution matches the covariance as a moment, but that is not true for a multivariate t (or more general multivariate distributions). So people saying mean-variance assumes normality is disingenuous. It's probably more accurate to say that it doesn't incorporate the full shape of a distribution of portfolio returns as well as other techniques can. - When you say portfolio optimization with higher moments, there are many different options. I tend to askew the techniques that rely on analytical techniques. If estimating a covariance matrix is hard, a co-kurtosis matrix is a waste of time. This means that you are best off relying on a scenario-based representation of returns. For instance, in the example in point 1, you could randomly generate returns from both the multivariate normal and t. In the optimization, you could use an analytical formula to get the mean and covariance (calculated from the scenarios), but then you could calculate the actual portfolio returns under the simulation and calculate the skewness and kurtosis of those, instead of relying on an analytical formula. The downside of this approach is that it requires you to specify preferences on portfolio skewness and kurtosis. I find it hard enough to be confident in my preferences for a trade-off between mean and variance, so I'm not inclined to try to have even more opinions on how these moments are traded off. The most common alternative is to rely on techniques like CVaR/ES that try to capture tail risks. CVaR has many useful properties that make it easier to deal with than adding in skewness and kurtosis terms to utility functions.
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.