Methods for computing value at risk

Explain the main ways of calculating Value at Risk for a portfolio.

VaR is a loss quantile: the loss not exceeded with probability $\alpha$ over a horizon. It can be computed parametrically (variance-covariance: assume normal returns, so VaR is about $z_\alpha\sigma\sqrt{h}$ times portfolio value, with $\sigma$ from a covariance matrix and position deltas), by historical simulation (revalue today's book under past daily moves and read off the empirical quantile), or by Monte Carlo (simulate risk factors from a fitted model and revalue). Mention its weaknesses: fat tails, non-subadditivity and silence about losses beyond the quantile.