# 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.

• Calculating the cokurtosis is a huge waste of time. Its size grows at like N^4. Instead, simulate from the distribution and calculate the CVaR from the scenarios.
– John
Commented Aug 30, 2020 at 18:46
• I am passing m3 and m4 as custom moments in the R package PortfolioAnalytics. I could generate those from rlnorm, but I would rather have the analytical form. I only have a handful of assets so computation efficiency in not an issue. Commented Aug 30, 2020 at 21:30
• I would recommend calculating the sample coskewness or cokurtosis before trying to derive an analytical formula for coskewness or cokurtosis for a multivariate log normal.
– John
Commented Aug 31, 2020 at 1:01
• My estimated distribution of invariants is based on log returns and I need to convert those into linear before optimisation. Alternatively I could also project the log distribution into prices and then calculate the linear moments of those. Commented Aug 31, 2020 at 9:37
• You can simulate from the multivariate normal invariants, convert to multivariate log normal and then calculate sample coskewness and cokurtosis from that (and note that since you have the simulated multivariate log normal returns, you can easily calculate CVaR directly). I don't think converting to prices would help you there though.
– John
Commented Aug 31, 2020 at 12:59