Risk-Neutral Adjustment of Bootstrapped Returns for PRIIPs VaR
Article Quant Q&A · Author: Branislav Cuchran
Summary
The document examines how to adjust bootstrapped historical log returns when calculating value at risk for category III packaged products under the PRIIPs framework. The example samples historical daily returns to form simulated multi-day paths, selects a high return quantile, and then considers how to convert that result into a risk-neutral return using a risk-free rate, the historical mean, volatility, and a variance adjustment.
The questioner’s calculation comes close to a later return value in the cited regulatory presentation after accounting for a likely typo and interpreting the displayed result through a logarithm. A response says the proposed mean replacement method matches the presentation under those assumptions, while noting ambiguity in the source example and in the number of trading days per year. The document therefore offers a worked interpretation rather than a definitive regulatory derivation; the exact conventions and input assumptions need careful checking.
Key ideas
- The described procedure bootstraps historical log returns to simulate multi-day paths and estimate a tail quantile.
- The proposed risk-neutral adjustment replaces the historical drift with a risk-free return and includes volatility and variance terms.
- A later response attributes the discrepancy with the stated example to a likely presentation typo and logarithmic interpretation.
- The assumed annual trading-day count is uncertain, so the numerical adjustment depends on conventions.
Tags
Full text
# Risk neutrality correction for Monte Carlo Bootstrapping according to PRIIP regulation for products of category III
# Risk neutrality correction for Monte Carlo Bootstrapping according to PRIIP regulation for products of category III
The PRIIP (packaged products) regulation prescribes Monte Carlo bootstrapping simulation for calculation of VaR for products of category III (non-linearly leveraged products). The idea is based on Geometric Brownian Motion. The bank shall for each product realize Monte Carlo simulation of future returns and disclose the total-life 97.5% VaR.
There is one example published (https://ec.europa.eu/info/system/files/risk-section-kid-11072016_en.pdf). This example follows this logic:
1) History of 9 daily log-returns is available, with the sample mean of $\mu$=-0,00227252 and standard deviation $\sigma$=0,011551773.
2) Bootstrapping of the returns is performed via random sampling out of these 9 historical log-returns. N=10 000 paths, each T=10 days long are boostrapped. Total return for each path is calculated as sum of sampled log-returns.
3) The path responding to 97.5% quantile is selected, with total log-return of $r_{bootstrapped}^{97,5\%}$=0,029342.
4) The risk-risk free rate of $r_{free}$=1.2% per annum is assumed.
Now the intermediary result is clear, the VaR log-return $r_{bootstrapped}^{97,5\%}$=0,029342 is however not risk-neutral. The regulation prescribes to derive a risk neutral return, based on the risk-free rate $r_{free}$ and historical mean $\mu$ and sample deviation $\sigma$.
How could the risk-neutral return be calculated ?
The example states deemed result of $r_{neutral}^{97,5\%}$=0,055447241. I am however not able to derive to this results. The regulation states formula
$Return = E[Return_{risk-neutral}] - E[Return_{measured}] - 0.5\sigma^2T$
which is very unclear. The calculation I have tried (assuming 261 business days, which is however information not given directly in the example) is based on the logic, that we have to replace the historical mean with risk-free rate. I tried to do it in following way, not sure if correct (?):
$r_{neutral}^{97,5\%}=r_{bootstrapped}^{97,5\%}+(r_{free}*T/261-0,5*\sigma^2*T-\mu*T)=0,029342+(0,012*10/261-0,5*0,011551773^2*10-(-0,00227252*10)) )= 0,05185976$
Is this approach correct or how should one derive risk-neutral return from the bootstrapping Monte Carlo simulation ?
## Answer by user29205 (score 2)
https://quant.stackexchange.com/a/35629
Isnt't the simulated return of 1,057013249 just a typo? Next to it on the presentation you can spot no. "3" - I suppose it is a return ID which should further correspond to what is shown 2 slides forward. However, the return presented there under ID 3 is 1,053225 (ln(1,053225)=0,05185689 which is very close to the number you stated, maybe it's a matter of rounding). Does it make sense?
## Answer by Fokko (score 2)
https://quant.stackexchange.com/a/44553
Better late than never. Although the question is rather old I think the topic is still important for some people. I had the same question how to deal with that formula and arrived here. So I did some calculations on the proposed method of OP - and the method matches the example perfectly if all comments are considered.
First of all, I think no one could find a reason why they get 0,055447241 for risk neutral return - because it is wrong. @E.B. is right, seems to be a typo. If one take the method of OP and the remark of @Tim Wilding, than one will get exactly (rounded) the result ranking on 3 on slide 26. So yes, the proposed method is correct and could be used.
The risk-free return of 1.2% might be an example. However, in this technical advice for funds the risk-free return is referred to the interest rate swap (Box 8, p. 12). Once upon a time, there existed a EURIBOR swap with 1.2% p.a. ;) But again - the remark regarding the logarithm is important.
Regarding the 261 days - I don't know how they come up with this (because the OP's assumption seems to be correct). I would go for 252 as a typical number of trading days. Anyway this should not be a showkiller as the regulation is as unprecise as it could be in so many ways.
Finally, thanks for the question and the considerations. That helped me a lot!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.