Value-at-Risk

Value-at-Risk (VaR) is the loss level that won't be exceeded with a given probability over a given horizon. A 1-day $95\%$ VaR of 1M USD means: $95\%$ of days, losses are at most 1M USD.

Three calculation methods:

Historical: rank actual past P&L, take the appropriate percentile. Captures real-world distributional features but assumes the past represents the future.Parametric: assume a distribution (often Gaussian), compute the percentile analytically. Fast but blind to fat tails.Monte Carlo: simulate many paths, compute P&L on each, take the percentile. Flexible but expensive.

VaR's most important shortcoming: it says nothing about how bad losses get past the threshold. The $5\%$ "worst" days could be $-1.1$M USD each or $-10$M USD each — VaR can't tell. That's why Expected Shortfall is increasingly preferred.

Regulators (Basel) use VaR for capital requirements and have started transitioning to Expected Shortfall.