How Skewness and Kurtosis Scale Across Return Frequencies

Summary

The answer explains how the skewness and kurtosis of cumulative returns change with the number of periods when returns are independent and identically distributed. For a sum of such returns, variance grows in proportion to the number of observations, while the third and fourth centered moments also grow proportionally under the stated assumptions. Standardizing those moments yields skewness that declines with the square root of the period count and kurtosis that declines in proportion to the inverse period count.

This scaling helps explain why aggregated returns may appear more normal over longer horizons. The result depends on independent, identically distributed return contributions; dependence or changing distributions can invalidate the simple conversion. The discussion does not provide an empirical test or address alternative definitions such as excess kurtosis, so the formulas should be applied with care to the convention used in a particular analysis.

Key ideas

• For independent, identically distributed returns, cumulative variance increases in proportion to the number of periods.
• Cumulative skewness falls in proportion to the inverse square root of the number of periods.
• Cumulative kurtosis falls in proportion to the inverse of the number of periods under the stated convention.
• The scaling assumes independent and identically distributed return contributions.
• Longer aggregation horizons can make returns appear more nearly normal under these assumptions.

Tags

# Is there such a thing as monthly/yearly skewness and kurtosis?


# Is there such a thing as monthly/yearly skewness and kurtosis?












As the title suggests, when performing regression analysis or portfolio optimization, one must adjust the frequency of his variables to match the frequency of other variables in the problem. For example, a regression of a monthly return against the Black-Scholes IV, the annualized Black-Scholes IV must be divided by $\sqrt{12}$ to arrive at a monthly Black-Scholes IV.

Therefore, is there such a thing as a monthly/yearly skewness or kurtosis? How does one convert the daily skewness/kurtosis calculated from a set of daily returns to that of a monthly frequency?

## Answer by Kermittfrog (score 5, accepted)

https://quant.stackexchange.com/a/81127

Assuming $n$ independent and identically distributed returns $X_i$, all moments of the distribution of the cumulative return $Y=\sum_{i=1}^n x_i$ scale with $n$:

$$ \mathrm{E}(Y^k)\propto n\mathrm{E}(X^k) $$

For the skewness, this implies $$ S(Y)\equiv \frac{\mathrm{E}\left(\left(Y-\mathrm{E}(Y)\right)^3\right)}{\left.\mathrm{E}\left(\left(Y-\mathrm{E}(Y)\right)^2\right)\right.^{1.5}}\propto\frac{n}{n^{1.5}}S(X)=\frac{1}{\sqrt{n}}S(X) $$ For kurtosis, $K(Y)\propto \frac{1}{n}K(X)$.

Hence the notion that - assuming independent return contributions - monthly/quarterly/annual returns will look more and more normal.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.