# How can higher co-moments be applied to portfolio optimization in an asset allocation context?

Traditional portfolio optimization involves mean variance optimization, where only the mean and covariance matrix of returns are estimated. What asset allocation and portfolio optimization techniques make use of the higher order co-moments of the returns distribution? Also, how would one compute the higher order matrices?

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This is my contribution to the weekly topic. Let me know what you guys think. If you do not like it I can remove it. If you need hints or additional help in answering or citation to formulate your answer I am happy to provide this. – Suminda Sirinath Salpitikorala Jan 4 '12 at 17:21
This is a good question, although it needed some editing for clarity. This is generally not a problem, so long as you make an effort to express your question as clearly as possible, others can come in later to improve the English. – Tal Fishman Jan 4 '12 at 17:24
Can I additionally ask a question about the risk implication of using higher moments. I think this should be a separate question. – Suminda Sirinath Salpitikorala Jan 4 '12 at 17:29
Yes, you certainly can. I do recommend you wait a couple days until people have had a chance to digest this question, though. Or submit your own thoughts on this question first, as an answer. – Tal Fishman Jan 4 '12 at 17:31
Guys. How do you calibrate the matrices? Give me at lease two ways to do this. This is also part of the question. – Suminda Sirinath Salpitikorala Jan 5 '12 at 5:59

The blog post http://www.portfolioprobe.com/2011/10/03/predictability-of-kurtosis-and-skewness-in-sp-constituents/ suggests that there is some predictability in kurtosis, but it isn't clear (to me at least) that there is enough predictabiilty to be useful.

If there is a place for higher moments, my guess is that it is in asset allocation problems where there are only a few assets rather than in equity portfolios.

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Yes, this is what the idea behind Omega as a portfolio optimization objective is all about.

Keating and Shadwick (2002a, 2002b) first introduced this notion. An introduction by Keating is here.

In fact, the Performance Analytics package in R includes a function to calculate Omega.

For your second question, one can compute the moments of higher orders using linear algebra, although this is not computationally practical if you are optimizing a large portfolio on the order of say, 500 assets, since the terms required are approximately (excluding symmetries) N raised to the order of the moment where N is the number of assets (i.e. mean = N, standard deviation = N^2, skew = N^3 ...). I would suggest using the Omega function in Performance Analytics instead.

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Morningstar recently came out with a piece entitled The Real World is Not Normal: Introducing the new frontier: an alternative to the mean-variance optimizer. It essentially summarizes their views on Mean-CVaR optimization, based on Xiong and Idzorek (2011).

This research piece also contains their estimates for the first four moments (but does not list correlations, co-skewness, or co-kurtosis) for all the major asset classes typically considered in asset allocation.

A Mean-CVaR Optimization will put much lower weight on investment grade bonds and hedge funds and much higher weight on international bonds, real estate, and small cap stocks.

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Adding a bit to the references mentioned by Quant Guy - apart from the aforementioned paper by Keating and Shadwick, Kazemi et al. introduce an alternative formulation of the Omega ratio (Sharpe-Omega) similar to the Sharpe ratio.

As noted by Patrick Burns, higher moments could have some use when instruments other than equity are involved (hedge fund portfolios seem to be the most popular research topic: Togher & Barsbay, Favre-Bulle and Pache). However, you can also encounter some skeptical views concerning the whole Omega ratio concept.

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For an small demonstration of the calculation of higher order co-skewness and co-kurtosis tensor matrices in Excel and VBA see;

Portfolio-Analytics-Coskew-and-CoKurt-VBA3

available from enter link description here

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