# 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!

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Have a look at this question: possible duplicate of Skewness and Kurtosis under aggregation –  Bob Jansen Aug 16 '12 at 14:46
Thanks for your comment, Bob. Unfortunately, the answer is not really sufficient for my problem. I may have to clarify: I just need to calculate the annual skew and the only data I have are daily returns. To achieve that, can I just calculate the yearly returns (based on the daily returns) for each year individually and use the outcome as input to calculate the skewness? thanks in advance for the answer! –  Stephan Aug 16 '12 at 15:05

In my opinion you have two choices:

1. 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.

2. 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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Thanx for your answer. Especially for the second part. After reviewing Bob Jansen's post, I do see the link and the solution to my problem. Although, regarding the practical implementation, I still have some ambiguity: If I would apply the formula given for skewness, what would correspond to Z1? Or will this be replaced by Z12 as it initially is Zn? Additionally, as I will have more than n = 12 observations (i.e. 131 monthly returns in this case), can I still just apply this formula by summarizing the all 131 monthly returns and using n = 12? Thanx for the answer! –  Stephan Aug 17 '12 at 8:10
@Stephan If you have $131$ monthly observations then you can calculate monthly statistics on these $131$ observations which will lead to small confidence intervals. Thus skewness and kurtosis of monthly returns. The sum of $12$ monthly returns is an annual return (this is true with log returns and approximately true with geometric returns). Thus you can apply the findings for scaling with $n=12$. The result of th link above is that you divide monthly skewness by $\sqrt{12}$ to get annual skewnes and you devide ex. kurtosis by $12$. –  Richard Aug 17 '12 at 9:34
@Stephan Keep in mind that the assumption of this procedure is that returns are independent. The sum of independent random variables converges to something Gaussian which has skewness and ex.kurtosis zero. Thus for $n$ to infinity skewness and kurtosis vanish. –  Richard Aug 17 '12 at 9:36
Thanks again for your answers. and your indication about the assumption of i.i.d. Just to check, if I got that right: if I would base my computations on daily returns, the scaling would be n = 252 (or whatever no. of days I use)!? I am so sorry to ask so simple questions... –  Stephan Aug 17 '12 at 12:10
@Stephan $n$ is the number of summands. A year has $12$ months and it is often assumed that there are $252$ business days in a year. So this is correct. –  Richard Aug 17 '12 at 12:39