When PRIIPs VaR Equivalent Volatility Can Be Negative
Summary
The thread examines whether VaR equivalent volatility (VEV), a PRIIPs key information document measure, can be negative. It gives the formula used to convert a VaR measure into VEV and argues that if VaR is interpreted as the fraction of the amount invested that is returned, and therefore stays at or below one, the formula produces a nonnegative result. The derivation notes a small rounding discrepancy in the constants, yielding a threshold slightly above one.
Other replies describe circumstances that can produce negative VEV under particular interpretations or product conditions. One says a negative value can arise when the relevant tail outcome is a gain, using a product with a small coupon in most scenarios and a rare capital loss. Another suggests that, for a capital-protected product, negative risk-free rates can make discounted residual capital exceed the invested amount, so the stated VaR ratio is above one. These are forum answers rather than a settled regulatory interpretation; the result depends on how VaR is defined and on the applicable PRIIPs calculation conventions.
Key ideas
- VEV is calculated by transforming a VaR measure under the PRIIPs framework.
- The claim that VEV cannot be negative depends on VaR being bounded by one.
- A tail outcome that represents a gain may lead to negative VEV under a different interpretation.
- Negative discount rates can raise the present value of protected capital above the invested amount.
- Interpretation depends on the VaR definition and the calculation rules used for the product.
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Full text
# Can a VaR equivalent Volatility (VEV) be negative?
# 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?
## Answer by will (score 2)
https://quant.stackexchange.com/a/34150
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.
## Answer by James Spencer-Lavan (score 1)
https://quant.stackexchange.com/a/34156
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
## Answer by tgeorge (score 1)
https://quant.stackexchange.com/a/36403
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%
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.