VaR-equivalent volatility (VEV)

The VEV turns a VaR into a volatility. Annex II of Delegated Regulation (EU) 2017/653 starts from a quantile, the 97.5% VaR over a recommended holding period of T years, and looks for the volatility σ such that a lognormal return with no drift gives the same loss. The relation -1.96 σ√T - 0.5 σ²T = VaR solves to σ = (√(3.842 - 2 × VaR) - 1.96) / √T, where 3.842 rounds 1.96². The text writes it in two forms. In category 2, VaR is computed in return space through a Cornish-Fisher expansion on observed returns: VaR = σ√N × (-1.96 + 0.474 μ1/√N - 0.0687 μ2/N + 0.146 μ1²/N) - 0.5 σ²N, with N the number of pricing periods in the holding period, μ1 the skew and μ2 the excess kurtosis. In category 3, VaR is read in price space: at least 10,000 final values are simulated by bootstrap, the 2.5th percentile is kept, discounted and expressed relative to the amount invested, and the VEV equals (√(3.842 - 2 ln VaR) - 1.96) / √T. The coefficients 0.474, 0.0687 and 0.146 are those of the Cornish-Fisher expansion at the -1.96 quantile. The VEV has the merit of putting an equity fund and an autocall on a single scale. The flaw follows from that: a 97.5% VaR sees nothing beyond the quantile, and two products with the same VEV can have very different extreme losses, which a TVaR would show. Annex IV does not change this, since performance scenarios belong to a separate calculation.