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9.5 Summary

  • Markowitz’s portfolio is formulated in terms of the mean and variance of the returns (first and second moments), but financial data is not Gaussian distributed and higher orders may be necessary for a proper characterization.

  • High-order portfolios attempt to capture the non-Gaussianity by incorporating the skewness and kurtosis (third and fourth moments) in the formulation to better model the asymmetry and heavy tails of the distribution.

  • High-order portfolios go back to the 1960s. However, the estimation and manipulation of such high-order moments was an impossibility in those early days. For a universe of \(N\) assets, the number of parameters increases at a rate of \(N^4\), which rapidly becomes unmanageable in terms of computational complexity and memory storage. In addition, the portfolio formulations are nonconvex, adding to the difficulty of designing optimal portfolios.

  • There is a wide variety of portfolio formulations incorporating high orders, such as the MVSK portfolios, portfolio tilting, polynomial-goal formulations, and even using the alternative linear moments (L-moments).

  • Efficient algorithms are now readily available based on mature iterative algorithmic frameworks.

  • Consequently, after over half a century of research by the scientific community, all the challenges have been overcome, and designing high-order portfolios can now be easily achieved with hundreds or even thousands of assets. The decision to incorporate high-order moments now rests in the hands of the trader.