Quantitative Risk Management

VaR_α(L) = inf{x : P(L > x) ≤ 1 − α} = inf{x : F_L(x) ≥ α}. The smallest loss threshold exceeded with probability at most 1 − α (e.g., α = 0.99 ⇒ exceeded only 1% of the time).

Use the empirical distribution of past P&L. The α-VaR is the (1 − α)-quantile of historical losses — no distributional assumption, but limited by sample length and assumes the past represents the future.

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

Simulate N P&L scenarios from a posited (possibly multivariate, non-normal) model of risk factors, then take the empirical (1 − α)-quantile. Flexible for path-dependent / non-linear portfolios; cost is computational.

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.

A map ρ on losses is coherent iff it satisfies: (1) monotonicity — L₁ ≤ L₂ ⇒ ρ(L₁) ≤ ρ(L₂); (2) translation invariance — ρ(L + c) = ρ(L) + c; (3) positive homogeneity — ρ(λL) = λ ρ(L), λ ≥ 0; (4) subadditivity — ρ(L₁ + L₂) ≤ ρ(L₁) + ρ(L₂).