Quantitative Risk Management

Value at Risk (VaR) at level α

VaR_α(L) = inf{x : P(L > x) ≤ 1 − α} = inf{x : F_L(x) ≥ α}. The smallest loss threshold exceeded with probability at mos

Historical-simulation VaR

Use the empirical distribution of past P&L. The α-VaR is the (1 − α)-quantile of historical losses — no distributional a

Parametric (variance–covariance) VaR

Assume losses are normal: VaR_α = μ + σ · Φ⁻¹(α). Fast and analytic, but understates tail risk when returns are heavy-ta

Monte Carlo VaR

Simulate N P&L scenarios from a posited (possibly multivariate, non-normal) model of risk factors, then take the empiric

Expected Shortfall (ES / CVaR)

ES_α(L) = E[L | L ≥ VaR_α(L)] = (1/(1−α)) ∫_α¹ VaR_u(L) du. Average loss in the worst (1 − α) tail. Always ≥ VaR_α.