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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?

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

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