Classical VaR assumes returns are normal, but real markets have fat tails and skewness. A 99% VaR from a normal fit can understate true tail risk by 30-50% during crises. Practitioners use Historical Simulation (re-sort actual returns), Cornish-Fisher expansion (adjusts for skew/kurtosis), or Extreme Value Theory (fit a Generalized Pareto to the tail).

Key mechanics:

Historical VaR: take the 1st percentile of the empirical return distribution — no distributional assumption.
Cornish-Fisher: z_cf = z + (z²−1)·S/6 + (z³−3z)·K/24 − (2z³−5z)·S²/36, where S=skew, K=excess kurtosis.
EVT: model only losses beyond a threshold u with GPD; captures tail shape parameter ξ.

Method	99% VaR (daily)	Assumption
Normal	2.33σ	Gaussian tails
Cornish-Fisher	2.33σ + adj	Skew/kurtosis
Historical	empirical 1%	Past repeats
EVT	u + (β/ξ)[(n/Nu·α)^−ξ − 1]	Pareto tail