What value at risk measures and where it falls short

Explain what value at risk (VaR) is as a measure of portfolio risk, and discuss its main weaknesses.

VaR is a loss threshold: the amount a portfolio will not lose more than with confidence $\alpha$ over a set horizon (a quantile of the loss distribution). Its drawbacks: it says nothing about how bad losses are beyond the threshold, it is not subadditive so diversification can appear to raise risk, it depends heavily on distributional and correlation assumptions (often normal and stable), and it can be gamed by positions with rare large losses. Expected shortfall addresses several of these.