Marginal, Incremental & Component VaR: Euler Portfolio Risk Attribution Math

Standalone desk Value-at-Risk numbers never add up to firm-wide diversified VaR: if Desk A carries $4.00M VaR and Desk B carries $3.00M VaR with pairwise correlation $\rho = 0.00$, firm-wide diversified VaR equals $\sqrt{4^2 + 3^2} = \$5.00\text{M}$, leaving a $-$2.00\text{M}$ diversification benefit that risk committees must allocate fairly across desks. Quantitative risk architects solve this capital-allocation problem using Euler's Homogeneous Function Theorem, splitting portfolio risk into Marginal VaR ($\Delta VaR_i$), Component VaR ($CVaR_i^{comp}$), and Incremental VaR ($IVaR_i$). Benchmark single-asset quantiles first in our Value at Risk (VaR) calculator and see how covariance structures integrate into our Parametric VaR & Expected Shortfall calibration guide.

Exact Matrix Algebra of Marginal, Component, and Incremental VaR#

Let dollar position vector $x = (x1, \dots, xN)^\top = W \cdot w$ have return covariance matrix $\Sigma$, so portfolio dollar variance is $\sigmaP^2(\$) = x^\top \Sigma x$ and parametric VaR (or Gaussian/Student-$t$ Basel III FRTB 97.5% Expected Shortfall) equals $VaRP = k\alpha \sqrt{x^\top \Sigma x}$. Because scaling every dollar position by $\lambda > 0$ scales $VaRP(\lambda x) = \lambda VaRP(x)$, $VaRP$ is homogeneous of degree 1. Differentiating with respect to $xi$ yields Marginal VaR, and multiplying by $xi$ recovers Euler's exact additive decomposition:

$$\Delta VaRi = \frac{\partial VaRP}{\partial xi} = k\alpha \frac{(\Sigma x)i}{\sqrt{x^\top \Sigma x}} = \frac{VaRP}{W}\,\betai, \qquad \text{where } \betai = \frac{\text{Cov}(ri, rP)}{\sigma_P^2}$$

$$\text{Component } VaRi = xi \cdot \Delta VaRi = wi \betai VaRP \implies \sum{i=1}^N \text{Component } VaRi = VaR_P$$

Worked 3-Asset Euler Component VaR & Marginal Hedge Audit ($100M)#

Study Step 3 and Step 4 carefully! Desk 3 (the $20M Treasury position) carries a positive standalone VaR of $$325,689$, yet its Component VaR is negative ($-$54,097$) because its $-$42.0\text{B}$ negative covariance with US equities outweighs its own variance! Furthermore, Component VaR ($-$54,097$) does not equal exact Incremental VaR ($-$80,241$) because Component VaR is a local first-order tangent approximation ($x3 \cdot \partial VaRP / \partial x_3$), whereas deleting the entire $20M sleeve moves nonlinearly along the convex risk surface!

Best Hedge Sizing and Minimum-VaR Portfolio Optimality Conditions#

Marginal and Incremental VaR govern two daily trading-desk decisions:

1. Risk-Minimizing Position Cut: Which desk should reduce exposure by $$1\text{M}$ to shrink firm-wide VaR fastest? Always trim the position with the highest Marginal VaR ($\Delta VaR_i$)—here Desk 1 ($\Delta VaR_1 = \$1.632\text{M}/\$50\text{M} = 3.265\%$ per dollar), not necessarily the asset with the highest standalone volatility!
2. *Optimal Hedge Trade Size ($\Delta x_i^$): Adding $\Delta x_3$ of a hedging instrument reduces portfolio VaR until its new Marginal VaR hits zero** ($\beta3^{new} = 0$). Adding Treasuries beyond the point where $\text{Cov}(r3, rP^{new}) = 0$ over-hedges the book and pushes $VaRP$ back up along a quadratic parabola!

Comparing Marginal, Component, and Incremental VaR Risk Metrics#

Extending Euler Attribution to Monte Carlo Expected Shortfall#

In a Cholesky Multi-Asset Correlated Monte Carlo Path Simulator, computing numerical finite-difference derivatives across 500 positions would require 501 full simulations. Tasche (2000) proved a remarkable shortcut for Expected Shortfall: the Euler Component ES of asset $i$ is simply the average loss of asset $i$ across the exact tail scenarios where the total portfolio breaches $VaR_\alpha$! For VaR itself, kernel smoothing around the quantile window prevents Monte Carlo step noise. Pair this decomposition with Ledoit-Wolf covariance shrinkage, stress-test correlation jumps in stressed-beta VaR amplification, and browse the Parametric VaR & Expected Shortfall Topic Hub.