# PRIIPs KID: if VaR (Return Space) < -1, how to compute VEV (VaR-equivalent volatility)?

The PRIIPs regulation does not specify how to compute the VaR-equivalent volatility if $$VaR_{Return Space} < -1$$. What would you do in the following case?

I have the following moments from the historical daily log-returns of a stock:

$$M1 = 0.0019$$ (Mean)

$$\sigma = 0.0378$$ (Standard Deviation)

$$\mu_1 = 0.9201$$ (Skewness)

$$\mu_2 = 12.068$$ (Excess Kurtosis)

Assume

$$T = 1$$ (asset’s holding period in years)

$$N = 256$$ (number of trading periods in days)

Then, the Cornish-Fisher VaR  is: $$\begin{eqnarray} VaR_{Return Space} &=& \sigma \sqrt{N} * (− 1.96 + 0.474 * \mu_1/\sqrt{N} - 0.0687 * \mu_2/N + 0.146* \mu_{1}^2/N) − 0.5 \sigma^2 N \\ &=& -1.3534 \end{eqnarray}$$

Given that $$VaR_{Return Space}$$ is below - 1, which of these two VEVs would be the correct one:

(1) Simply apply the formula and obtain: $$\begin{eqnarray} VEV &=& (\sqrt{3.842-2*VaR_{Return Space}}-1.96)/\sqrt{T} \\ &=& \sqrt{3.842-2*(-1.3534)}-1.96 \\ &=& 0.5990 \end{eqnarray}$$

(2) Since an investor cannot lose more than the initial investment, put a floor to $$VaR_{Return Space} = -1$$ and get $$\begin{eqnarray} VEV &=& \sqrt{3.842-2*(-1)}-1.96 \\ &=& 0.4570 \end{eqnarray}$$