What is the difference between marginal VaR, component VaR, and incremental VaR?

Marginal VaR (MVaR) measures the change in portfolio VaR per unit change in an asset's weight. Component VaR (CVaR) is the portion of total portfolio VaR attributable to each asset, computed as weight times marginal VaR times portfolio value. Incremental VaR (IVaR) measures the exact change in portfolio VaR when an asset is completely removed and remaining weights are renormalized. Component VaRs sum exactly to total portfolio VaR, making them ideal for risk budgeting.

Why do component VaR values sum to total portfolio VaR?

Component VaR is defined as CVaR_i = w_i x MVaR_i x V_P, where MVaR_i = z x (covariance matrix times weight vector)_i / portfolio volatility. By Euler's theorem for homogeneous functions, the sum of w_i x (partial derivative of VaR with respect to w_i) equals VaR itself. This additive decomposition property makes component VaR the standard tool for risk attribution and risk budgeting in portfolio management.

What does a negative component VaR mean?

A negative component VaR means that asset actually reduces the portfolio's total VaR. This occurs when an asset is negatively correlated with the rest of the portfolio, effectively hedging portfolio risk. Adding more of this asset would decrease total portfolio VaR. Assets with negative component VaR are valuable diversifiers.

How do portfolio managers use VaR decomposition?

Portfolio managers use VaR decomposition for risk budgeting (allocating risk limits to each position), identifying risk concentrations (which positions contribute most to portfolio risk), evaluating diversification effectiveness, and making position sizing decisions. Component VaR helps managers understand whether a position's risk contribution is proportional to its expected return contribution.

What are the limitations of parametric VaR decomposition?

Parametric VaR decomposition assumes returns are normally distributed, which underestimates tail risk (fat tails). It assumes stable correlations and volatilities, which can break down during market stress. It treats positions as linear (no optionality). It uses historical parameters to estimate future risk. For portfolios with options or other nonlinear instruments, Monte Carlo VaR decomposition may be more appropriate.

VaR Decomposition Calculator | Marginal & Component VaR

Portfolio Inputs

Number of Assets

Total Portfolio Value

Total portfolio market value

Confidence Level

Time Horizon

trading days

1 day standard; 10 days for Basel regulatory VaR

Correlation Matrix

Model Assumptions

Returns are normally distributed (parametric VaR)
Correlations and volatilities are stable over the time horizon
Positions can be perfectly liquidated at current prices (no liquidity adjustment)
Linear portfolio (no optionality or convexity in positions)
Daily volatilities derived from annual: σ_daily = σ_annual / √252

For educational purposes. Not financial advice. Market conventions simplified.

Calculator by Ryan O'Connell, CFA

VaR Decomposition Results

Portfolio VaR $15,481 154.8 bps

Portfolio Vol (Daily) 0.9411%

Undiversified VaR $18,653

Diversification Benefit $3,172

Diversification Ratio 83.0%

Per-Asset VaR Decomposition

Asset	Weight	Individual VaR	Marginal VaR	Component VaR	CVaR %	Incremental VaR

Component VaR Breakdown

Formula Breakdown

VaR_P = z_α × σ_P × V_P

Component VaR_i = w_i × MVaR_i × V_P

When to Use This Calculator

The existing VaR Calculator computes a portfolio's total Value at Risk as a single number. This calculator extends that analysis by decomposing VaR into per-asset contributions — answering "which position contributes most to portfolio risk?"

VaR Calculator: Use for basic total portfolio VaR (parametric, historical, or Monte Carlo methods)
VaR Decomposition Calculator: Use when you need to understand risk concentration, identify hedging assets, and allocate risk budgets across positions

For related portfolio risk analysis, see the Portfolio Variance Calculator, Correlation Calculator, Covariance Calculator, and Portfolio Beta Calculator. For dynamic risk management strategies, explore the CPPI Calculator and Pension Funded Status Calculator.

Understanding VaR Decomposition

What is VaR Decomposition?

VaR decomposition breaks down a portfolio's total Value at Risk into the risk contribution of each individual position. This allows portfolio managers to identify which assets contribute the most (or least) to overall portfolio risk, enabling better risk budgeting and position sizing decisions.

Key Formulas

Portfolio VaR: VaR_P = z_α × σ_P × V_P
Marginal VaR: MVaR_i = z_α × (Σ × w)_i / σ_P
Component VaR: CVaR_i = w_i × MVaR_i × V_P
ΣCVaR_i = VaR_P (additive decomposition)

Three Types of VaR Measures

Marginal VaR

Sensitivity of portfolio VaR to a small change in an asset's weight. Used for optimal portfolio construction.

Component VaR

Each asset's contribution to total VaR. Components sum exactly to total VaR. Standard for risk budgeting.

Incremental VaR

Exact change in VaR from completely removing an asset (with weight renormalization). Useful for position elimination decisions.

Diversification Benefit

Difference between undiversified VaR (sum of individual VaRs) and actual portfolio VaR. Measures the risk reduction from diversification.

Risk Budgeting: In an optimally diversified portfolio, each asset's ratio of expected excess return to marginal VaR should be equal across all positions. Learn more about how portfolio diversification reduces risk.

Frequently Asked Questions

Marginal VaR (MVaR) measures the change in portfolio VaR per unit change in an asset's weight. It is a sensitivity measure used for optimal portfolio construction. Component VaR (CVaR) is the portion of total portfolio VaR attributable to each asset, computed as weight times marginal VaR times portfolio value. The key property is that component VaRs sum exactly to total portfolio VaR, making them ideal for risk budgeting. Incremental VaR (IVaR) measures the exact change in portfolio VaR when an asset is completely removed and remaining weights are renormalized to 100%. Unlike marginal VaR (which is a derivative), incremental VaR is a discrete, finite change.

This property follows from Euler's theorem for homogeneous functions. Portfolio VaR is a positively homogeneous function of degree 1 in the portfolio weights: if you scale all weights by a factor k, VaR scales by k. Euler's theorem states that for such functions, the sum of each variable times its partial derivative equals the function value: Σ w_i × (∂VaR/∂w_i) = VaR. Since component VaR is defined as CVaR_i = w_i × MVaR_i × V_P, and MVaR_i captures the partial derivative, the sum of all component VaRs equals total portfolio VaR.

Diversification reduces VaR because portfolio volatility is typically less than the weighted sum of individual asset volatilities when correlations are below 1.0. The diversification benefit equals the undiversified VaR (sum of individual asset VaRs) minus the actual portfolio VaR. This benefit increases with: (1) lower correlations between assets, (2) more assets in the portfolio, and (3) more equal weighting across assets. When all correlations equal 1.0, there is zero diversification benefit and the portfolio VaR equals the undiversified VaR.

A negative component VaR means that asset actually reduces the portfolio's total VaR. This occurs when an asset has a sufficiently negative correlation with the rest of the portfolio that its beta to the portfolio is negative, effectively hedging portfolio risk. For example, a short position in a highly correlated asset, or a long position in an asset negatively correlated with the portfolio, can have negative component VaR. Adding more of this asset would decrease total portfolio VaR. Such assets are valuable diversifiers.

Portfolio managers use VaR decomposition for several purposes: (1) Risk budgeting — allocating risk limits to each position and ensuring no single position dominates portfolio risk; (2) Risk concentration analysis — identifying positions that contribute disproportionately to total risk; (3) Diversification assessment — evaluating whether the portfolio is effectively diversified; (4) Position sizing — adjusting weights so that each position's risk contribution is proportional to its expected return contribution; (5) What-if analysis — using incremental VaR to evaluate the impact of adding or removing positions.

Parametric VaR decomposition has several limitations: (1) It assumes normally distributed returns, which underestimates tail risk (real returns exhibit fat tails and skewness); (2) It assumes stable correlations and volatilities, which can break down during market stress when correlations tend to spike toward 1.0; (3) It treats positions as linear (no optionality), so portfolios with options or other nonlinear instruments may not be accurately decomposed; (4) It uses historical parameters to estimate future risk; (5) The square-root-of-time scaling for multi-day VaR assumes returns are i.i.d., which may not hold in practice.

Disclaimer

This calculator is for educational purposes only. It assumes normally distributed returns and stable parameters. Actual portfolio risk involves fat tails, time-varying correlations, liquidity risk, and non-linear instruments. For precise risk measurement, use institutional risk management systems. This tool should not be used for trading or investment decisions.

VaR Decomposition Calculator