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How Return Frequency and Aggregation Affect Skewness and Kurtosis

Article Quant Q&A · Author: Masher
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Summary

The document considers why the same asset-allocation strategy can show different skewness and kurtosis when evaluated with weekly rather than monthly index returns. The responses say that the reported weekly values are plausible, since higher-frequency returns are often less normal. Aggregating returns over longer horizons can move the distribution toward normality under the central limit theorem, although convergence may be slow when tails are heavy.

The discussion also notes that portfolio formation can diversify some asset-level skewness and kurtosis, while co-skewness and co-kurtosis describe cross-asset dependence in higher moments. It points to academic work on the pricing and portfolio implications of these statistics. The replies offer broad context rather than a diagnosis of the user's calculations or a test of the specific strategy. Return aggregation does not guarantee normality, and portfolio construction may still produce pronounced higher moments even at low frequencies.

Key ideas

  • Return distributions are often less normal at higher observation frequencies.
  • Aggregating returns can move skewness and kurtosis toward normal-distribution values, but heavy tails can slow this process.
  • Diversification may reduce asset-level skewness and kurtosis in a portfolio.
  • Co-skewness and co-kurtosis capture higher-moment dependence among assets.
  • Asset allocation can retain substantial skewness and kurtosis at lower frequencies.

Tags

StatisticsMulti-assetPortfolio construction
Full text
# High values of skewness and kurtosis of realized protfolio returns


# High values of skewness and kurtosis of realized protfolio returns












I am investigating some asset allocation strategies and I am wondering about the results I obtain. I am working on monthly and weekly data of the same stock indices (SP500, FTSE 100 etc). And when I compute the summary statistics of the realized returns I observe that for the very same strategy those statistics vary greatly between weekly and monthly data. For the monthly f.e. I obtain skewness = 0.4 and kurtosis = 5, while for the weekly frequency skewness = 1.5 and kurtosis = 25.

Are the results for the weekly data plausible? And is it possible that the difference in the frequency of data results in such differences in the summary statistics? All computations are identical, the only difference is the data frequency.

I hope the question is not too general and it is possible to give some insight based on my description.

## Answer by Freddorick (score 3)

https://quant.stackexchange.com/a/22102

The skewness and kurtosis values you obtain appear to be of realistic magnitude.

In general higher frequencies are more non-normal, i.e. have higher skewness and kurtosis. If non-normal returns are aggregated the central limit theorem starts working and the return distribution coverges to a normal. Convergence can be quite slow under fat tails. You can try yearly returns, skewness will be closer to 0 and kurtosis closer to 3. If you form portfolios of stocks with skewed returns, skewness and kurtosis will diversify away to some extent. Co-skewness and co-kurtosis, similar to covariance, are relevant in this case. There is quite a lot of academic literature on the asset pricing implications of skewness and co-skewness, most notably Harvey and Siddique (2000). Portfolio selection under skewness is explored, for example, by De Miguel et.al (2012).

## Answer by phdstudent (score 2)

https://quant.stackexchange.com/a/22119

The only thing weird is skewness not being lower for the weekly vs daily. In any case, take a look at table 1.1 from Campbell, Lo and Mackinlay, and check that your values are not far off the ballpark. Actually, with annual data, you should have nearly zero skewness and zero excess kurtosis (on the market). However, asset allocation might lead to severe skewness and kurtosis even at low frequencies.

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.