How Skewness and Kurtosis Scale When Returns Are Aggregated

# Skewness and Kurtosis under aggregation


# Skewness and Kurtosis under aggregation












Returns possess non-zero skewness and excess kurtosis. If these assets are temporally aggregated both will disappear due to the law of large numbers. To be exact, if we assume IID returns skewness scales with $\frac{1}{\sqrt{n}}$ and kurtosis with $\frac{1}{n}$.

I'm interested in a concise, clear and openly accessible proof for the above statement preferably for all higher moments.

This question is inspired by this question by Richard which deals with, among other things, the behaviour of the higher moments of returns under temporal aggregation. I know about two papers that answer this question. Hawawini (1980) is wrong and Hon-Shiang and Wingender (1989) is behind a paywall and a bit inscrutable.

## Answer by JL344 (score 24, accepted)

https://quant.stackexchange.com/a/3678

Just to be painfully clear, it only seems to make sense to consider the logarithm of returns, i.e. $X=\log (1+\frac r{100})$ for a simple return of $r\%$ in an arbitrary period because this is what sums when returns are temporally aggregated. A basic property of cumulants is that cumulants of all orders are additive under convolution, for which a proof can be found here here.

So if $X_1$, $X_2$, ... $X_n$ are i.i.d., then all the cumulants of $$Y_n = \sum_{i=1}^nX_i$$ scale linearly with $n$, i.e. $$\kappa_k(Y_n)=n\kappa_k(Y_1).$$ However, I suspect that you are normalizing this sum so that the variance (or volatility) remains constant with increasing $n$. So instead let us consider $$Z_n=\frac{Y_n}{\sqrt n}= \frac 1 {\sqrt n} \sum_{i=1}^nX_i.$$ Another basic property of cumulants is that the $k$th cumulant is homogeneous of order $k$ as to scale. Using both properties together we have $$\kappa_k(Z_n)=\left(\frac 1 {\sqrt n}\right)^k\kappa_k(Y_n)=\left(\frac 1 {\sqrt n}\right)^kn\kappa_k(Y_1)=\frac {\kappa_k(Z_1)}{n^{(k-2)/2}}.$$

(Don't forget that $Z_1=Y_1=X_1$.) Now we can show that the statistics scale as you have described: $$\textrm{variance}=\kappa_2(Z_n)=\kappa_2(Z_1)\propto 1;$$ $$\textrm{skewness} =\frac{\kappa_3(Z_n)}{\kappa_2(Z_n)^{3/2}}=\frac{\frac{1}{n^{1/2}}\kappa_3(Z_1)}{\kappa_2(Z_1)^{3/2}}\propto \frac 1{\sqrt n};$$ $$\textrm{ex. kurtosis}=\frac{\kappa_4(Z_n)}{\kappa_2(Z_n)^2}=\frac{\frac{1}{n}\kappa_4(Z_1)}{\kappa_2(Z_1)^{2}}\propto \frac 1 n.$$

There is no reason this cannot be extended to higher orders, although it works out more directly in terms of cumulants than of moments.