Definition
\[\text{VaR}_\alpha(L) = \inf\{l : P(L > l) \leq 1-\alpha\} = F_L^{-1}(\alpha)\]The $\alpha = 99\%$ VaR is the 99th percentile of the loss distribution — losses exceed this level on 1% of days.
Calculation methods
Parametric (variance-covariance): assume losses are normally distributed:
\[\text{VaR}_\alpha = \mu_L + z_\alpha\,\sigma_L\]Fast but misses fat tails. For a portfolio: $\sigma_L^2 = \mathbf{w}^\top\Sigma\mathbf{w}$ (in return space).
Historical simulation: sort the last $T$ daily P&L observations; the $\alpha$-VaR is the $(1-\alpha)T$-th worst loss. Non-parametric, captures fat tails, but limited by history.
Monte Carlo: simulate thousands of scenarios; compute the empirical quantile. Flexible but computationally expensive.
Limitations and Expected Shortfall
VaR fails the subadditivity axiom: $\text{VaR}(A+B) > \text{VaR}(A) + \text{VaR}(B)$ is possible — diversification can appear to increase risk.
Expected Shortfall (CVaR) at level $\alpha$:
\[\text{ES}_\alpha = \mathbb{E}[L \mid L > \text{VaR}_\alpha]\]ES is subadditive and coherent. The Basel III framework replaced VaR with ES at $\alpha = 97.5\%$ for trading book capital requirements.