Component VaR

Component VaR

Component VaR decomposes total portfolio VaR into the contribution from each individual position:

CVaR_i = w_i × MVaR_i

A key property: component VaRs sum to total portfolio VaR, making this a perfect decomposition tool.

Two-Asset Formulas

Given weights w1, w2, vols σ1, σ2, correlation ρ, and multiplier z:

σ_p = sqrt(w1²σ1² + w2²σ2² + 2·w1·w2·σ1·σ2·ρ)

MVaR₁ = z · (w1·σ1² + w2·σ1·σ2·ρ) / σ_p
MVaR₂ = z · (w2·σ2² + w1·σ1·σ2·ρ) / σ_p

CVaR₁ = w1 × MVaR₁
CVaR₂ = w2 × MVaR₂

CVaR₁ + CVaR₂ = z · σ_p  (total portfolio VaR)

Example

w1=0.5, w2=0.5, σ1=0.02, σ2=0.03, ρ=0.4, z=1.645
Total VaR ≈ 0.0347
CVaR₁ ≈ 0.0125, CVaR₂ ≈ 0.0222
Sum = 0.0347 ✓