# Distributional assumptions in PRIIPs

And yet another question to discuss the assumptions in PRIIPs. It is remarkable that in these legal documents a Cornish-Fisher expansion including skewness and kurtosis is used.

Looking at the very recent version of the document we find on page 27 the following formula for the moderate scenario (Which is, if I read it correctly, supposed to be the 50% quantile):

$$\exp(M_1 \cdot N - \sigma \mu_1/6 - 0.5 \sigma^2 N ),$$ where $N$ is the number of days (more details are not necessary here), $M_1$ is the first moment of the log returns observed, $\sigma$ is the standard deviation and $\mu_1$ is the skewness measured.

I have one question: I see that $- \sigma \mu_1/6$ enters if we put in $0$ for the "z-value". Thus there is something that remains from skewness.

But is it ok to have the average return $M_1$ if we model in a risk-neutral world?

If $M_1$ is the average of log-returns then we have $M_1 = \tilde{\mu} + \sigma^2/2$ where $\tilde{\mu}$ is the "true" mean and $\sigma^2/2$ is the convexity that we have in the log-normal case. This is corrected in the last part of the formula by the term $- 0.5 \sigma^2 N$. This formula is different from the others where there is usually just an expected return of $-\sigma^2/2 N$ which makes the expected growth zero (see e.g. page 28 point 11).

In short: is it really consistent to have the $M_1$ term above? Any comments are really appreciated!