What is marginal VaR and how does it relate to optimal portfolio construction?

Marginal VaR (MVaR) is the rate of change of portfolio VaR with respect to a small change in a position's weight. It's the derivative (in the calculus sense) of portfolio VaR with respect to position size.

Formula:

MVaRᵢ = ∂(Portfolio VaR) / ∂(wᵢ)

For a parametric VaR under normality:

MVaRᵢ = z_α × (Cov(Rᵢ, Rp) / σp)

Or equivalently:

MVaRᵢ = βᵢ × VaR_p / Portfolio Value

Relationship to Component VaR:

CVaRᵢ = wᵢ × MVaRᵢ × Portfolio Value

Marginal VaR is per-unit risk; component VaR is the total contribution.

Why marginal VaR matters for optimization:

At the optimal portfolio (minimum VaR for a given return), the ratio of expected excess return to marginal VaR should be equal across all positions:

E(Rᵢ) - Rf / MVaRᵢ = constant for all i

This is the equal marginal risk-return condition — the analog of the tangency portfolio in mean-variance optimization.

Intuition: If one position offers higher return per unit of marginal risk than another, you should increase that position and decrease the other until the ratios equalize.

Example:

Analysis: Hedge Funds have the highest ratio (3.08) — they offer the best return per unit of marginal risk. Gold has the lowest (1.88). To optimize, increase hedge fund allocation and decrease gold until ratios converge.

Practical applications:

1. Risk budgeting: Allocate risk efficiently by equalizing return-to-MVaR ratios
2. Rebalancing signals: If ratios diverge significantly, rebalancing is warranted
3. Marginal contribution analysis: Understand which positions are most/least risk-efficient

Common confusion: Marginal vs. Incremental:

Exam tip: FRM Part II tests the optimization condition (equal return-to-MVaR ratios), the relationship between MVaR and CVaR, and when to use MVaR vs. IVaR.

Practice VaR decomposition on AcadiFi's FRM materials.