Annualizing Skewness and Kurtosis from Daily Returns

Summary

The discussion compares two ways to estimate annual return skewness and kurtosis from a long series of daily continuously compounded returns. One approach is to aggregate daily returns into annual returns and calculate the statistics from those annual observations. This gives ex-post estimates tied directly to the observed years, but the sample is very small, so estimation error may be substantial.

The alternative is to calculate moments at a more frequent interval, such as monthly, and apply scaling rules under assumptions about the return process. The answer recommends monthly observations for the stated data history and emphasizes that scaling depends on returns being stationary and independent. It also notes that using a lower frequency can help avoid spurious autocorrelation associated with trading hours and related effects. The exchange does not provide the scaling formulas themselves or validate either estimate empirically; the appropriate method depends on the assumptions and the purpose of the estimate.

Key ideas

• Annual aggregation estimates moments from the observed yearly returns but may leave too few observations for precise estimates.
• Scaling higher-frequency moments relies on assumptions about the return series, including stationarity and independence.
• Using weekly or monthly returns may reduce spurious autocorrelation associated with trading hours.
• The choice between direct annual estimates and scaled estimates depends on the data and the intended interpretation.

Tags

# How to annualize skewness and kurtosis based on daily returns


# How to annualize skewness and kurtosis based on daily returns












I'm trying to annualize the four moments based on a string of daily returns (continuously compounded) for 11 years.

The formulas for the annualization of the mean and the standard deviation I did find, but unfortunately the formulas for the skewness and kurtosis and the way to apply them not.

Can anybody help me? I'd would be appreciated a lot!

## Answer by Richi Wa (score 6, accepted)

https://quant.stackexchange.com/a/3957

In my opinion you have two choices:

- You calculate annual returns from the daily returns that you have - I guess it is clear how. Subsequently you calculate your statistics on these $11$ data points. When I look at your comment above, this could be what you want to achieve. Then you have the ex-post statistics on your data. The drawback is that $11$ data points are not that many and the error of your estimator is rather large.

- The other approach that people mostly do is to use data at a higher frequency (e.g. daily, weekly or monthly) and then apply the scaling rules that are mentioned in Bob Jansens comment. In order to avoid any spurious auto-correlations that could result from trading hours and similar effects often a lower frequency than daily is used - e.g. weekly if there have not been gathered that many data points yet, or monthly. In your example I would use monthly data and then apply the scaling rules with $n=12$.

The assumption of this approach is that your weekly/monthly returns are stationary and independent (in fact a bit less is assumed but this would go too far).

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