PRIIPs category 2 Cornish-Fisher : how to calculate

Question

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 1

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).