# Can a VaR equivalent Volatility (VEV) be negative?

As from title, can a VaR equivalent Volatility (VEV) as defined by KID/PRIIPS law be negative?

The formula for VaR equivalent volatility (from here) is :

$$\frac{\sqrt{3.842 - 2 \ln{\mathrm{VaR}}} - 1.96}{\sqrt{T}}$$

which looks like this (for T=1):

where the x axis is VaR.

Since VaR is bounded between 0 and 1, no you cannot have a negative VaR equivalent volatility.

Indeed, from the paper I have stated above, it is written on page 29:

"The VaR is the percentage of the amount paid that is returned to the retail investor"

Therefore, you can say something like this:

\begin{align} \frac{\sqrt{3.842 - 2 \ln{\mathrm{VaR}}} - 1.96}{\sqrt{T}} &\geqslant 0 \\ 3.842 - 2 \ln{\mathrm{VaR}} &\geqslant 1.96^2 \\ - 2 \ln{\mathrm{VaR}} &\geqslant 1.96^2 - 3.842\\ - 2 \ln{\mathrm{VaR}} &\geqslant -0.0004\\ \ln{\mathrm{VaR}} &\leqslant 0.0002\\ \mathrm{VaR} &\leqslant e^{0.0002}\\ \mathrm{VaR} &\leqslant 1.00040008001\\ \end{align} Implying that while $\mathrm{VaR} \leqslant 1.00040008001$, the VaR implied vol. is positive.

• Thanks but VaR cannnot be outside the 0-1 interval if it is computed with Cornish Fisher expansion, am I right? – Thegamer23 May 11 '17 at 9:20
• @Simons123 VaR is the probability of a loss, so if you're computing it to be outside of this integral, then you've made a mistake... – will May 11 '17 at 9:27
• Ok I got your point but I do not agree. Actually that is because you are giving a theoretical answer. But if you take the VaR as the percentile of an historical distribution over a given horizon you might have (e.g. in a series of all positive returns) a positive VaR. – Thegamer23 May 11 '17 at 10:14
• So is your question "If i use a flawed method to calculate VaR such that i end up with it outside of the range of actually attainable values of VaR, can this formula result in a negative VEV?" ? – will May 11 '17 at 10:18
• Actually it is the law that requires that methodology cfr. "as defined by..." btw I agree with you – Thegamer23 May 11 '17 at 10:27

VeV under PRIIPS will be negative if the "loss" at the 2.5% cutoff is actually a gain

Imagine a product where you receive a small positive coupon C with probability 0.99, full capital loss with probability 0.01

This product will have a negative VeV and thus score MRM1

VEV can also be negative in the following case (please correct me if I'm wrong):

for a category 3 product characterized by an uconditional protection of capital, you have to calc the pV. With negative risk free rates, dfs are >1 and thus VaR as a percentage of the invested amount is >1. i.e:

Amount= 10,000 ,Residual= 145 , Risk free= -0.295% , PV= 10,011.75 , VaR [Price Space]= 1.001174886, VEV= -0.079%

Does it look OK to you?