Marginal VaR
Marginal VaR (MVaR) is the change in a portfolio's total Value-at-Risk resulting from a small increase in the exposure to a specific position or asset, measuring each position's marginal contribution to total portfolio risk and enabling optimal risk allocation and capital efficiency analysis.
Key takeaways
- Marginal VaR identifies which positions contribute most to portfolio-level VaR, distinguishing between large positions that are well-diversified (low Marginal VaR) and smaller positions that are highly correlated to existing risk (high Marginal VaR).
- The sum of all positions' marginal VaR contributions (component VaR) equals the portfolio VaR, enabling exact attribution of total portfolio risk to individual positions or risk factors.
- A position with negative Marginal VaR reduces total portfolio VaR when size increases—it acts as a hedge against the rest of the portfolio.
- Marginal VaR is the theoretical basis for risk-budgeting: by equalizing Marginal VaR across positions (after adjusting for expected returns), a portfolio manager can construct a risk-efficient portfolio where each unit of risk is equally compensated.
- Marginal VaR is sensitive to the correlation assumptions used in the VaR model; in stress periods when correlations spike toward 1.0, positions with historically low Marginal VaR can suddenly contribute much more to portfolio VaR than the model predicts.
Explanation
Marginal VaR is one of three risk decomposition measures derived from portfolio VaR analysis, alongside Component VaR (the contribution of each position to total portfolio VaR) and Incremental VaR (the change in portfolio VaR from adding or removing an entire position). The distinction matters: Marginal VaR assumes an infinitesimally small change in position size, Component VaR is the proportional attribution that sums to total VaR, and Incremental VaR measures the discrete impact of full position addition or removal. For risk management purposes, all three are useful in different contexts, but Marginal VaR and Component VaR are the most analytically tractable.
The mathematical derivation of Marginal VaR begins with the observation that portfolio VaR, under the assumption of normally distributed returns, can be expressed as: VaR_P = z_α × σ_P, where z_α is the standard normal critical value for confidence level α and σ_P is portfolio standard deviation. The Marginal VaR of asset i is the partial derivative of portfolio VaR with respect to the weight of asset i: MVaR_i = ∂VaR_P / ∂w_i = z_α × (∂σ_P / ∂w_i) = z_α × Cov(R_i, R_P) / σ_P = z_α × ρ_{i,P} × σ_i. Here, Cov(R_i, R_P) is the covariance of asset i's return with the portfolio return, ρ_{i,P} is their correlation, and σ_i is the standard deviation of asset i. This formula reveals that Marginal VaR depends not on the asset's standalone volatility but on its covariance with the existing portfolio—a position with high standalone volatility but low correlation to the portfolio may have a smaller Marginal VaR than a lower-volatility position that is highly correlated with existing holdings.
Component VaR—the product of each position's Marginal VaR and its portfolio weight—is the most useful decomposition for risk attribution and risk budgeting. By the Euler decomposition theorem for homogeneous functions, the portfolio VaR equals the sum of all component VaR values: VaR_P = Σ w_i × MVaR_i = Σ CVaR_i. This additive property is critical for practical risk management: it allows a risk manager to attribute 100% of portfolio VaR to individual positions, asset classes, sector exposures, or risk factors. A risk report showing each position's component VaR contribution and percentage share of total portfolio VaR provides a clear roadmap for where portfolio risk is concentrated and where diversification is effectively reducing aggregate risk.
Risk budgeting frameworks use Marginal VaR as the objective function for portfolio optimization. The risk parity approach, for example, targets equal Component VaR contributions from each asset class in the portfolio: if fixed income, equities, commodities, and alternatives each contribute 25% of portfolio VaR, the portfolio is said to have equal risk budgets. In a mean-variance optimal framework, the marginal contribution to portfolio VaR from any position should be proportional to its expected return divided by its Marginal VaR (the ratio known as the VaR-adjusted Sharpe ratio); if the ratios are unequal, expected return per unit of marginal risk can be improved by reallocating from low-ratio to high-ratio positions.
The primary limitations of Marginal VaR as a risk measure relate to those of VaR more broadly. VaR is not subadditive (it can underestimate tail risk when positions are non-linearly related), and Marginal VaR based on variance-covariance matrices assumes returns are elliptically distributed, which fails to capture fat tails and skewness. For options-heavy portfolios, where the distribution of returns is inherently asymmetric, historical simulation or Monte Carlo-based VaR with full repricing is more appropriate for Marginal VaR calculation, though computationally more demanding. Additionally, correlations used in Marginal VaR computation are not stable—they typically increase in stress scenarios, meaning that positions with low Marginal VaR in normal markets may have high Marginal VaR during market crises.
Formula
MVaR_i = z_α × Cov(R_i, R_P) / σ_P = z_α × ρ_{i,P} × σ_i; Component VaR_i = w_i × MVaR_i; Portfolio VaR = Σ Component VaR_i
Example
A risk manager analyzes a $500 million multi-asset portfolio with a 95% daily portfolio VaR of $8.2 million. The portfolio contains five major positions: US Large Cap Equities ($200M), Investment Grade Credit ($150M), Emerging Market Equities ($75M), US Treasuries ($50M), and Gold ($25M). Marginal VaR calculation (using variance-covariance method) yields: US Equities MVaR = $12.5 per $1M increase; IG Credit MVaR = $8.2; EM Equities MVaR = $18.1; US Treasuries MVaR = −$4.3 (negative—acts as a hedge); Gold MVaR = $6.1. Component VaR: US Equities = 200 × $12.5 = $2,500K (30.5%); IG Credit = 150 × $8.2 = $1,230K (15.0%); EM Equities = 75 × $18.1 = $1,358K (16.6%); Treasuries = 50 × (−$4.3) = −$215K (−2.6%); Gold = 25 × $6.1 = $153K (1.9%). Residual/rounding = $1.174M. Total component VaR = $6.986M + rounding ≈ $8.2M. The risk manager observes that EM Equities has the highest Marginal VaR despite being only 15% of the portfolio, suggesting it is either highly volatile, highly correlated with the rest of the portfolio, or both. A decision to reduce EM Equities from $75M to $50M would reduce portfolio VaR by approximately 25 × $18.1 = $452,500, lowering VaR from $8.2M to approximately $7.75M—a 5.5% reduction in risk for a 5% reduction in position size, reflecting above-average marginal risk contribution.
Related terms
Cap Component Var Correlation Covariance Diversification Expected Shortfall Fat Tails Gold Greeks Hedging Incremental Var Investment Grade Model Risk