# 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 ?

• Saying the formula is very unclear is a flowery understatement. If you look at it nearer, it will show you that the second term is (somehow) the return of the PRiiP, meaning that if the PRiiP performs better and better, the 'corrected' return will be lower and lower, and reversely. The same with VEV vs. VaR. – Philonous Feb 1 '18 at 12:03

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?

• Could be a typo indeed. Otherwise, I think the formula and the approach above is correct (in the sense that it follows the PRIIP regulation). The problem can also be in the calculation of the risk-free return from the risk-free rate of 1.2%. I am not sure how the risk-free return has to be calculated, probably my formula (risk_free_rate*T/261) is not correct. Does anyone have idea how the risk-free return is to be calculated correctly ? – Branislav Cuchran Oct 11 '17 at 8:59
• Given that the regulations are dealing with logarithmic returns, I would suggest that you use the following formula to calculate the required risk-free return $=log(1+r_f) \times T / 261$. Given the size of the risk-free rate, this will only change the third significant figure of your calculation. Risk-free return over 10 days = 0.04571% using logarithms, and 0.04598% using your calculation. – Tim Wilding Apr 3 '18 at 11:38

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!