# 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?

## 1 Answer

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.

1. 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.
2. 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.