I am calculating the skewness of returns of 10,000 randomly selected portfolios with different portfolio sizes(where assets range from 2 to 100 by an increment of 5 . i.e 2,5,10,15,..100). When I increase the portfolio size the kurtosis decreases which is favorable for investors. However, i am puzzled by the fact that when I take the kurtsosis of a 10,000 randomly selected portfolios that consist of N-1 assets (which means if I have 30,000 assets I build randomly 10,000 portfolios where each consists of 29,999 assets) the Kurtosis is much higher than the kurtsosis of portfolios with sizes ranging from 2 to 100. I don't understand how the kurtosis decrease when i increase the diversification but when I diversify to the maximum kurtosis increases and the monotic decrease disappears.
I hope someone could explain this point to me.


TS Contributor
Its not quite clear what the random variable is here, but I think it is what you call 'returns'. It also appears that this 'returns' is dependent on the size of the portfolio. So, if this is the case, I think I can explain this phenomenon. We know that kurtosis is a measure of how peaked the frequency distribution is. That is, if the histogram of the data is tall and narrow, then the kurtosis will be high. When the histogram appears more spread out, the kurtosis will be lower. This appears to be the case here. When the portfolio sizes are varied, I imagine that the 'returns' are also varied, and we can expect the kurtosis to be low. Now, when you have a large sample of portfolios of all the same size, the 'returns' will also be quite similar, and the histogram will appear very spiked and narrow, hence a larger kurtosis.

Hope this helped.