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Higher-Order Moments of Multivariate Lognormal Returns

Article Quant Q&A · Author: Marc F
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Summary

The document raises the problem of calculating coskewness and cokurtosis for a multivariate lognormal distribution from the parameters of an underlying multivariate normal distribution of log returns. The motivation is portfolio optimization: these higher-order moments are needed to estimate conditional value at risk, which the author uses as an objective. The text notes that formulas for univariate lognormal moments and covariance are available in a cited risk-management reference, while corresponding multivariate higher moments were not located.

No formulas, derivation, numerical example, or resolution are provided. Consequently, the document identifies a gap in the author’s moment calculations rather than teaching a completed method. It also does not specify the exact moment conventions, portfolio setup, or CVaR estimator being used. Readers seeking to apply the idea would need to derive or obtain the cross-moments and verify that their definitions match the chosen risk optimization framework.

Key ideas

  • The author seeks multivariate lognormal coskewness and cokurtosis from normally distributed log returns.
  • The higher-order moments are intended to support CVaR estimation in portfolio optimization.
  • The document mentions existing references for univariate moments and covariance.
  • It supplies no multivariate formulas, derivation, or worked example.

Tags

StatisticsRisk managementPortfolio construction
Full text
# Formula for coskewness and cokurtosis of LogN to project linear returns


# Formula for coskewness and cokurtosis of LogN to project linear returns












I want to find the coskewness and cokurtosis of the multivariate LogN(mu, sigma) distribution from the moments of a normally distributed multivariate distribution (ie: log returns). These higher order moments are required to estimate the CVaR that I use as one of the objectives in my optimisation function. I have the formula to estimate the univariate moments from Meucci (2005) as well as that for covariance (see explanation of page 268 in Meucci's Exercises in Advanced Risk and Portfolio Management) but I could not find anything on the coskewness and cokurtosis of the multivariate LogN.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.