# 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.

• Hard to say what is plausible without knowing your strategies but skewness and kurtosis tend to their normal values under temporal aggregation. – Bob Jansen Dec 2 '15 at 21:52
• hum....that's sounds a bit dodgyfor vanilla indexes although some differences are expected but not that huge (focusing on the kurtosis side). Please make me a favor by re-checking your weekly return series to see if there are no outliers, errors there. Alternatively specify your time horizon... As a matter of fact got 6.559... (weekly) for spx from jan-99 til nov-15 vs 10.63..(weekly) for ukx from jan-98 til nov-15. hope you understand more about my reluctance here. Please quickly use Excel kurt() on your data and keep me posted as I am curious.   – owner Dec 2 '15 at 22:03
• @owner I am solving the asset allocation problem on excess returns and the values of kurtosis for spx is 6.66 and for ftse 7.72. The values for non-excess returns are very similar. And I think there are no mistakes in the data, I have not omitted any outliers, as research on which I base mine didn't omit any either. The most intriguing thing is the difference between monthly and weekly data, as the computation process is the same in both cases, only the data changes. – Masher Dec 2 '15 at 22:30
• @Masher: fair enough... I had seen such a big gap in the past on an academic paper focusing on intra-day calculations vs other frequencies on fx markets, and am fine to discover something unexpected a priori from your asset allocation problem. – owner Dec 2 '15 at 22:46
• @owner do you remember the paper which you have mentioned? It would be great to show that such a case did not only happen in my research but in other works too. I am still wondering how is it possible, since there are no indications in the data or the calculations that the results would be so extreme in case of the weekly data. – Masher Dec 2 '15 at 22:54

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).

• Yes, the results are obtained over an identical period. And is it plausible that the skewness and kurtosis of the weekly stock returns are so much different that the skewness and kurtosis of the realized portfolio returns created out of those weekly data? In general of course, f.e. for a mean variance portfolio. – Masher Dec 4 '15 at 18:47
• "If you form portfolios of stocks with skewed returns, skewness and kurtosis will diversify away to some extent". Have there been any studies proving this? based on the mean-variance model's mathematics, how is it actually possible to diversify away the higher moments? – develarist Dec 28 '20 at 16:12

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.