Annualizing Co-Skewness and Co-Kurtosis in Portfolio Models

Summary

The document raises an unresolved scaling question for dynamic portfolio optimization using the Modified Sharpe ratio. Because that ratio depends on portfolio skewness and kurtosis, the author needs co-skewness and co-kurtosis arrays to construct the higher-order portfolio moments. The prompt notes commonly cited annualization relationships for univariate skewness and kurtosis, then asks whether corresponding adjustments apply to multivariate co-moment arrays.

No answer or derivation is included, so the document does not establish an annualization rule for co-skewness or co-kurtosis. Applying the stated univariate relationships directly to array elements is not justified by the discussion. The practical lesson is that moment scaling must be derived for the particular joint-moment definition, return aggregation process, and time-series assumptions used in the portfolio model. The question is useful as a research issue, but readers must consult a suitable derivation before implementing annualized co-moments.

Key ideas

• Modified Sharpe ratio portfolio optimization may require co-skewness and co-kurtosis arrays.
• The prompt cites scaling rules for univariate skewness and kurtosis but provides no co-moment formula.
• Univariate moment annualization rules cannot be assumed to apply directly to joint moments.
• Derive scaling based on the co-moment definition and assumptions about return aggregation.

Tags

# Annualization of higher-order co-moments (coskewness and cokurtosis arrays)


# Annualization of higher-order co-moments (coskewness and cokurtosis arrays)












I'm developing a dynamic portfolio optimization procedure based on the implementation of the Modified sharpe ratio. The mentioned ratio depends, among other factors, on the skewness and kurtosis of the portfolio.

To build the skewness and kurtosis I need to compute the co-skewness and co-kurtosis arrays. In financial literature, there is a great deal of studies which describe how to annualize higher order moments in general, such as

annualized skewness$=\frac{skewness}{\sqrt{n}}$

and

annualized kurtosis$=\frac{kurtosis}{n},$

but not how to do it for co-moments such as the co-skewness and co-kurtosis arrays. To sum up: would anyone know how to annualize higher order co-moments?

Thanks in advance

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