Return Moments and Cornish-Fisher Risk Calculations
Summary
The document explains which observations enter the moments used in a Cornish-Fisher calculation. Each r_i is an individual asset return, often a daily observation. The count of returns is M0, the arithmetic mean is M1, and M2, M3, and M4 are averages of the second, third, and fourth powers of deviations from that mean. The response then gives formulas for skewness and kurtosis based on those moments.
It flags a degrees-of-freedom caveat: dividing each centered sum by the observation count is not the same as using the stated adjusted denominators, particularly for a small sample. The answer also notes that Cornish-Fisher and Edgeworth methods rely on advanced statistics and points readers toward further study. It does not work through the referenced PRIIPs example or explain all regulatory conventions, so the formulas should not be treated as a complete implementation guide.
Key ideas
- Each r_i denotes one observed asset return in the sample.
- M1 is the sample mean, while M2, M3, and M4 summarize centered powers of returns.
- Skewness and kurtosis are formed by scaling the third and fourth moments by powers of M2.
- Using sample-size adjustments can matter, especially when the return history is short.
- The response does not fully specify regulatory conventions for a PRIIPs calculation.
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# PRIIPs category 2 Cornish-Fisher : how to calculate
# PRIIPs category 2 Cornish-Fisher : how to calculate
i am not very good at finance ,but i have been trying to calculate the example from this link. https://www.dropbox.com/s/egfx0ktolojfsek/3.PRIIPs%20Workshop%20-%20Risk%20Reward%20Methodology.pdf?dl=0
According to the regulation we have to calculate $M1, M2, M3$ and $M4$. For example $M2=(r_i-M1)^2/M0$, $M3=(r_i-M1)^3/M0$
What i dont understand is, which "$r_i$" we have to take for $M2, M3$ and $M4$
Thank you in advance
## Answer by kurtosis (score 1)
https://quant.stackexchange.com/a/57307
You have made a few typos. The basic idea is for asset returns $r_i$ where $i$ denotes the observation (often, $i$ indexes days), you have: $$ \begin{align} M0 &= \sum_i 1_{r_i} = \text{# of returns} \\ M1 &= \sum_i r_i/M0 = \text{mean return} \\ M2 &= \sum_i (r_i-M1)^2/M0 = \text{variance of returns}^* \\ M3 &= \sum_i (r_i-M1)^3/M0 = \text{third moment}^* \\ M4 &= \sum_i (r_i-M1)^4/M0 = \text{fourth moment}^* \end{align} $$ with the * meaning a big caveat that these are actually wrong in terms of not being the correct degrees of freedom: the M2 sum should be divided by $M0-1$; the M3 and M4 sums should be divided by $M0-2$. (Is dividing by M0 alone gravely wrong? If you have a small dataset, yes.)
The skewness is then $M3/M2^{3/2}$ and kurtosis is $M4/M2^2$.
Finally... if you are "not very good at finance," you should realize that you have jumped into some of the more complicated material in finance and what is definitely graduate-level statistics. Playing with Cornish-Fisher and Edgeworth expansions is not something I would advise when you are just trying to understand the variables being referred to (e.g. $r_i$).
I highly recommend you consult a text that can explain at least a little of this to you. Chapter 8 of A Quantitative Primer on Investments with $R$ covers these expansions and other methods and Kolassa's Series Approximation Methods in Statistics is a more in-depth reference (with McCullagh's Tensor Methods in Statistics going into even greater depth).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.