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Key Takeaways
- Marginal VaR is the partial derivative of portfolio Value at Risk with respect to one position size, so it is a rate: dollars of portfolio VaR added per dollar added to that position.
- For a differentiable VaR function that is homogeneous of degree one in position sizes, component VaR equals position size times its marginal VaR, and the components sum to portfolio VaR.
- Incremental VaR recomputes the portfolio before and after a finite trade. State whether it measures an addition or the reduction from deleting a position.
- In a zero-mean delta-normal model, marginal VaR is proportional to covariance with the whole portfolio. Other models calculate the corresponding quantile sensitivity.
- For the example’s positive positions and zero-mean delta-normal model, B has negative marginal VaR when its covariance with the whole portfolio is negative.
What is marginal VaR?
Marginal VaR is the partial derivative of portfolio Value at Risk with respect to one position’s size: the rate at which portfolio VaR changes as that position grows by a small amount while the other positions stay fixed. For dollar exposures, its units are VaR dollars per added dollar. A negative marginal VaR means a small addition lowers portfolio VaR.
Component VaR multiplies that rate by the current position size. Incremental VaR recalculates the change for a finite trade. Standalone VaR measures the position by itself.
How is marginal VaR calculated in a delta-normal model?
In a zero-mean delta-normal model, marginal VaR is z × Cov(Ri, Rp) / sigma_p. It also equals portfolio VaR per dollar times the position’s beta to the portfolio.
With a nonzero expected unit return mu_i, subtract mu_i from that covariance term. The zero-mean formula applies to the worked example.
Formula
Zero-mean delta-normal MVaR(i) = z × Cov(Ri, Rp) / σ_p = [VaR / V] × beta_i. Component VaR(i) = x_i × MVaR(i). With nonzero mean return mu_i, MVaR(i) = z × Cov(Ri, Rp) / σ_p − mu_i.- z is the confidence z-score, about 1.645 at 95% and about 2.326 at 99%.
- Cov(Ri, Rp) is the covariance between the return of position i and the return of the whole portfolio over the chosen horizon.
- σ_p is the portfolio return standard deviation over that horizon, and portfolio VaR equals z × σ_p × portfolio value when the expected return is set to zero.
- beta_i is Cov(Ri, Rp) divided by portfolio variance, the sensitivity of position i to the portfolio rather than to a market index.
- Marginal VaR is a rate with units of VaR dollars per dollar of exposure; component VaR is a dollar amount.
- Euler components sum to portfolio VaR when VaR is differentiable and positively homogeneous of degree one in the selected position sizes.
Marginal VaR example: a two-position book
A $1,000,000 book holds $600,000 of position A with 2.00% daily volatility and $400,000 of position B with 1.00% daily volatility. The two returns have a correlation of 0.30. The desk measures one-day 95% VaR, so z is 1.645, and wants to know what B is really costing in risk terms.
- Portfolio volatility: √(0.012² + 0.004² + 2 × 0.6 × 0.4 × 0.30 × 0.02 × 0.01) = √(0.0001888) = 1.37% per day
- Portfolio 95% VaR: 1.645 × √(0.0001888) × $1,000,000 = $22,603
- Beta of B to the portfolio: Cov(B, portfolio) / portfolio variance = 0.000076 / 0.0001888 = 0.403
- Marginal VaR of B: $0.00909868 per added dollar. A $10,000 addition gives a $90.99 first-order estimate; a full recalculation gives $91.40.
- Component VaR: B = $3,639.47, or 16.1%; A = $18,963.57, or 83.9%. The components total $22,603.04.
- Standalone 95% VaR of B alone: 1.645 × 1.00% × $400,000 = $6,580
- Incremental VaR of deleting B: $22,603 − (1.645 × 2.00% × $600,000 = $19,740) = $2,863
B has standalone VaR $6,580, component VaR $3,639.47 and a $2,863.04 reduction in portfolio VaR when removed. Its marginal VaR is $0.00909868 per added dollar. The 0.30 correlation is between A and B; B’s correlation with the whole portfolio is 0.55. A $10,000 addition raises portfolio VaR by $91.40.
B’s beta to the portfolio is 0.403. Its marginal VaR is about $91 for each $10,000 added. Its component VaR is $3,639.47, or 16.1% of the total. A contributes $18,963.57.
B’s standalone VaR is $6,580. Removing B reduces portfolio VaR by $2,863.04. These dollar figures answer different questions from the per-dollar marginal rate.
The 0.30 correlation belongs to A and B. B’s correlation with the whole portfolio is 0.55.
Marginal VaR vs incremental VaR
Multiplying B’s marginal rate by $10,000 predicts a $90.99 increase. Recomputing the portfolio with $410,000 of B gives a $91.40 increase.
For the convex delta-normal risk function, the increase from an addition is at least its current tangent estimate. The reduction from a removal is at most the current rate times the removed amount. Addition and removal therefore need distinct descriptions.
Component VaR and Euler allocation
For a differentiable function homogeneous of degree one, Euler’s theorem gives portfolio VaR as the sum of position size times its derivative. That is the component allocation.
A fixed scenario loss that is linear in position quantities preserves positive homogeneity even when the option payoff is nonlinear in underlying price. Differentiability is a separate requirement. Use the selected model’s derivatives or an explicit allocation rule at a nondifferentiable point.
The example’s standalone figures sum to $26,320. Its Euler components sum to $22,603.04. A universal ordering of standalone and combined VaR does not follow across all loss distributions.
How correlation changes marginal VaR
For B, covariance with the book includes B’s own weighted variance and its covariance with A. At A/B correlation 0, B’s marginal VaR is $0.0052 per dollar. At −0.30 it is $0.0006. It reaches zero at −1/3 and is negative below that level in this example.
Recalculate the derivative after changing position size. For a simulated portfolio, keep the scenario set fixed while comparing the original and bumped books.
Using marginal VaR to size a position
Rank the next-dollar sensitivities, inspect each position’s current component contribution, then recompute the full proposed trade. A rate of $0.03 per dollar and $50,000 of available VaR budget gives a first-order size estimate of $1,666,667; the final size comes from the recalculated portfolio.
Limitations of marginal VaR
- Recompute the full portfolio for a real trade size. Marginal VaR is a local, first-order rate: in the convex delta-normal case the tangent estimate is a floor for the VaR a finite purchase adds and a ceiling for the relief a sale provides, and removing B frees $2,863.04, less than its $3,639.47 component.
- Check the model’s assumptions against the book. The delta-normal version relies on normal returns, linear exposures and a stable covariance matrix, so an options-heavy or fat-tailed book needs a simulation instead.
- The rate shifts when positions change, so recalculate it after every resize.
- Confirm that VaR is differentiable and homogeneous of degree one in position sizes before summing components. Only then do they add up to portfolio VaR.
- It prices the loss quantile. For the size of losses beyond it, run expected shortfall or a stress test.
Bottom line
Marginal VaR shows how much portfolio risk the next dollar in a position adds or removes, which makes it useful for ranking where to add or trim. Use component VaR for the current allocation, and recompute the full portfolio for any real trade size before acting on the estimate.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute: Measuring and Managing Market Risk
- Tasche (2008), Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle
- Hallerbach (1999), Decomposing Portfolio Value-at-Risk: A General Analysis
How entries are sourced, calculated and updated: editorial methodology.