Why PRIIPs Performance Scenarios Use Historical Returns
Summary
The document questions a PRIIPs moderate performance scenario formula that includes the observed mean of log returns, volatility, and skewness. It focuses on whether using the historical average is consistent with risk-neutral valuation, noting that in a lognormal setting the log-return mean and the underlying asset’s expected growth differ by a volatility adjustment. The formula’s variance term appears to offset that adjustment, while skewness still contributes to the quantile calculation.
The response explains that performance scenarios are intended to reflect observed or expected performance rather than risk-neutral pricing. It points to regulatory provisions for certain product categories that use returns observed over the underlying history, and notes that related category formulas also retain the mean for performance scenarios. The exchange provides a regulatory rationale, not a complete derivation of the cited formula or a full treatment of every PRIIPs category. Its conclusion should therefore be read in the context of scenario disclosure and the particular regulatory formulas discussed, rather than as a general rule for derivative valuation.
Key ideas
- The question concerns the use of historical mean log returns in a PRIIPs moderate scenario formula.
- A risk-neutral drift used for pricing serves a different purpose from a performance scenario estimate.
- The response says some PRIIPs categories base scenarios on returns observed in the underlying history.
- The discussion does not derive the formula or cover all product categories.
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# Distributional assumptions in PRIIPs
# 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!
## Answer by Boris (score 1)
https://quant.stackexchange.com/a/35598
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.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.