# 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!

I think the reason is that the performance scenarios should not be based on the risk-neutral world.

You can clearly see that for Category 3, where in Annex IV, p. 12a the regulation requires that "the expected return for each asset or assets shall be the return observed over" the underlying history, hence one does not correct for impact of the mean as required for the MRM. Similarly, the Category 2 formulas, also include the mean for the regular performance scenarios.

• The major problem with not adjusting for risk neutrality is that it produces overly flattering estimates of investor risk-reward due to the pro-cyclical nature of investment products. By this I mean that bullish products tend to be written on assets that have gone up recently but also have relatively low forwards. So without adjusting the diffusion to match closer to the market forward, the Performance Scenarios look unreasonably positive. It's a major issue that may require a deviation from the literal read of the RTS to avoid understating risk to investors. – James Spencer-Lavan Aug 14 '17 at 19:26