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Why Return Standardization Leaves Skewness and Kurtosis Unchanged

Article Quant Q&A · Author: develarist
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Summary

The document asks whether de-meaning returns and dividing by their standard deviation changes their skewness or kurtosis. It distinguishes the first two moments, which standardization sets to zero and one, from the higher standardized moments used to describe asymmetry and tail shape.

The accepted response says skewness and kurtosis are defined using standardized moments, so an affine change of location and scale does not alter them. This is a concise conceptual explanation rather than an empirical study or a detailed derivation. It does not discuss estimation from finite samples, alternative definitions of kurtosis, or transformations beyond centering and scaling, so those issues require separate treatment.

Key ideas

  • Centering returns changes their mean but does not change skewness.
  • Dividing returns by their standard deviation sets the scale without changing kurtosis.
  • Skewness and kurtosis are themselves based on standardized moments.
  • The explanation addresses location and scale changes, not other transformations of returns.

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# Does standardizing/normalizing asset returns change their skewness and kurtosis?


# Does standardizing/normalizing asset returns change their skewness and kurtosis?












Asset returns are obtained by log-differencing prices. Standardizing or normalizing/scaling asset returns can be carried out by de-meaning the returns and dividing them by their standard deviation, causing their mean to change to 0 and standard deviation to be 1. Can we also expect the asset returns' skewness and kurtosis to also change? If not, how can it be that the first two moments (mean and standard deviation) change, whereas the third and fourth moments (skew and kurtosis) do not?

## Answer by Kermittfrog (score 4, accepted)

https://quant.stackexchange.com/a/55172

From the wikipedia on skewness and kurtosis, both are defined as expectations of standardised moments of the respective distributions. Hence, no.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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