Marginal VaR

Marginal VaR

Marginal VaR measures how portfolio VaR changes when you increase a position by a small amount. It is the partial derivative of portfolio VaR with respect to a position weight.

Two-Asset Portfolio

For a portfolio with weights w1, w2 and asset volatilities σ1, σ2 and correlation ρ:

Portfolio volatility:

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

Marginal VaR w.r.t. w1:

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

where z is the confidence-level multiplier (default 1.645 for 95%).

Interpretation

A marginal VaR of 0.025 means that increasing the allocation to asset 1 by 1 percentage point adds approximately 0.025 × 0.01 = 0.025 basis points of VaR.

Example

w1=0.5, w2=0.5, σ1=0.02, σ2=0.03, ρ=0.4, z=1.645
σ_p = sqrt(0.25×0.0004 + 0.25×0.0009 + 2×0.25×0.0006×0.4) = sqrt(0.000445)
MVaR₁ = 1.645 × (0.5×0.0004 + 0.5×0.02×0.03×0.4) / σ_p ≈ 0.025