FRM Exam Part II · Portfolio Risk: Analytical Methods

Marginal, Incremental and Component VaR Explained for FRM Part 2

Updated 11 October 2026 · Fact-checked

Marginal VaR is the change in portfolio VaR for a small extra unit of a position. Component VaR is position size times marginal VaR, and the components add up to total VaR. Incremental VaR is the exact change in VaR when you add or remove a whole position. Find each one from the position's covariance with the portfolio.

Understand Marginal, Incremental and Component VaR

Portfolio VaR is one number for the whole book. Risk managers also need to know which positions drive it and what happens if a position changes. Three related measures answer this.

Marginal VaR asks: if I add one more unit (one dollar) to position i, how much does portfolio VaR rise? It is a slope, the partial derivative of VaR with respect to the position. It is high for positions that move closely with the portfolio, and low or even negative for hedges.

Component VaR answers: how much of today's VaR is due to position i? It is the position size times its marginal VaR. Portfolio VaR is a homogeneous function of degree one in the positions (double all positions and VaR doubles). Because of that, the component VaRs add up exactly to portfolio VaR. This is why component VaR is the standard tool for VaR decomposition and risk budgeting.

Incremental VaR answers: how much does VaR change if I add a new position or remove an existing one? It is the difference between VaR after and VaR before. For a small change, marginal VaR times the change is a good approximation (often called delta VaR). For a large change it is not, because the portfolio volatility is not linear in the position size.

The link between them is beta. A position's beta to the portfolio is Cov(Ri, Rp) ÷ σp². Component VaR is portfolio VaR times the position weight times that beta. A position with beta above 1 contributes more than its weight. A position with negative beta has negative component VaR, so it reduces portfolio risk.

These results are exact for the normal (parametric) VaR. For other VaR methods the same ideas hold, but the numbers come from simulation, and the components still sum to total VaR.

Key formulas to remember

Portfolio VaR (normal): VaR = z × σp × W, with σp² = Σi Σj wi wj σi σj ρij; z is the normal quantile (1.645 at 95%, 2.326 at 99%). W is portfolio value. Use weights that sum to 1.
Beta of position i to the portfolio: βi = Cov(Ri, Rp) ÷ σp², where Cov(Ri, Rp) = Σj wj σi σj ρij; Σi wi βi = 1. The weighted average beta of the portfolio is always 1.
Marginal VaR (per unit of currency): Marginal VaR_i = ∂VaR/∂(position i) = z × Cov(Ri, Rp) ÷ σp = (VaR ÷ W) × βi; Gives the VaR change for one extra unit of position i. Some texts quote it per unit of weight; check the unit.
Component VaR: Component VaR_i = position_i × Marginal VaR_i = VaR × wi × βi; Same currency as VaR. Can be negative for hedges.
Additivity (Euler) property: Σi Component VaR_i = Portfolio VaR; Use this to check your work. Percent contribution = wi × βi.
Incremental VaR: Incremental VaR = VaR(portfolio with change) − VaR(portfolio before); Exact for any size of change. Needs two full VaR calculations.
Delta VaR approximation: Incremental VaR ≈ Marginal VaR × change in position; Good for small changes only. It equals the component VaR scaled by the fractional change in the position.

How to solve Marginal, Incremental and Component VaR questions

Use this method for any question on marginal, incremental or component VaR. It works with weights or with currency amounts.

1Read what is asked: marginal (per unit), component (share of total) or incremental (effect of a change). Note the confidence level and the horizon.
2Write down weights or position amounts, volatilities and correlations. Convert volatilities to the VaR horizon if needed.
3Compute the covariance of each position with the portfolio: Cov(Ri, Rp) = Σj wj σi σj ρij.
4Compute portfolio variance as Σi wi Cov(Ri, Rp), then σp = √variance, then portfolio VaR = z × σp × W.
5Compute beta (Cov ÷ σp²) or marginal VaR (z × Cov ÷ σp) for each position.
6For component VaR, multiply: position × marginal VaR, or VaR × weight × beta.
7Check that the components add up to portfolio VaR. If not, find the arithmetic slip.
8For incremental VaR of a small change, multiply marginal VaR by the change. For a large change or a new position, recompute full VaR and subtract. State the sign and interpret it.

Quickest way: Beta shortcut for component VaR

When to use it: Use when the question gives portfolio VaR and betas or covariances, or when you only need percent contributions.

Compute wi × Cov(Ri, Rp) for each position. These add up to the portfolio variance.
Percent contribution of i = wi × Cov(Ri, Rp) ÷ portfolio variance. This equals wi × βi.
Component VaR = percent contribution × portfolio VaR.
For small changes, incremental VaR ≈ component VaR × (change ÷ current position).
Eliminate answer options where components do not sum to VaR, or where a clearly hedging position has a positive contribution.

Common mistakes in Marginal, Incremental and Component VaR

Treating marginal VaR and component VaR as the same thing.
Both come from the same derivative and the names sound alike.
Fix: Marginal VaR is per unit and does not add up. Component VaR is marginal VaR times position size, and it adds up to total VaR.
Assuming incremental VaR equals component VaR.
Both describe the effect of a position on VaR.
Fix: Component VaR is a decomposition of current VaR. Incremental VaR is the actual change after adding or removing the position. They match only approximately, for small changes.
Using the position's own volatility instead of its covariance with the portfolio.
Students think a riskier asset always contributes more.
Fix: Contribution depends on σi × ρ with the portfolio. A volatile asset that hedges the book can contribute little or negative risk.
Concluding that a position with negative component VaR has negative stand-alone VaR.
Mixing up stand-alone risk and risk contribution.
Fix: Stand-alone VaR is never negative. Negative component VaR means the position reduces portfolio VaR at the margin.
Using the delta approximation for a large trade.
It is quick, and students forget it is a linear estimate.
Fix: If the question says close out a large part of the position or add a new asset, compute VaR before and after and take the difference.
Mixing weights and currency amounts, or forgetting the z-value.
Marginal VaR can be quoted per unit of weight or per unit of currency.
Fix: Check units. Multiply by z and by W once only. Confirm the sum of components equals total VaR.

Worked examples

Example 1

A USD 100 million portfolio holds USD 60 million in asset A and USD 40 million in asset B. Annual volatilities are 20% for A and 10% for B, and the correlation is 0.5. Using one-year 95% normal VaR (z = 1.645), find the portfolio VaR and the component VaR of each asset.

Show the solution

Weights: wA = 0.6, wB = 0.4.
Portfolio variance = 0.6² × 0.04 + 0.4² × 0.01 + 2 × 0.6 × 0.4 × 0.5 × 0.2 × 0.1 = 0.0144 + 0.0016 + 0.0048 = 0.0208.
σp = √0.0208 = 0.14422. Portfolio VaR = 1.645 × 0.14422 × 100 = USD 23.72 million.
Cov(RA, Rp) = 0.6 × 0.04 + 0.4 × 0.5 × 0.2 × 0.1 = 0.024 + 0.004 = 0.028. Cov(RB, Rp) = 0.6 × 0.01 + 0.4 × 0.01 = 0.010.
Check: 0.6 × 0.028 + 0.4 × 0.010 = 0.0168 + 0.0040 = 0.0208, which matches the variance.
Percent contributions: A = 0.0168 ÷ 0.0208 = 80.77%. B = 0.0040 ÷ 0.0208 = 19.23%.
Component VaR A = 23.72 × 0.8077 = USD 19.16 million. Component VaR B = 23.72 × 0.1923 = USD 4.56 million. Sum = USD 23.72 million.

Answer: Portfolio VaR is about USD 23.72 million. Component VaR is about USD 19.16 million for A (80.8%) and USD 4.56 million for B (19.2%). A is 60% of the capital but about 81% of the risk.

Example 2

For the same portfolio, the manager sells USD 5 million of asset A. Estimate the change in VaR using marginal VaR, then compute the exact incremental VaR and compare.

Show the solution

Marginal VaR of A per USD = z × Cov(RA, Rp) ÷ σp = 1.645 × 0.028 ÷ 0.14422 = 0.3194.
Delta VaR ≈ 0.3194 × (−5) = −USD 1.60 million (VaR falls by about 1.60 million).
Exact: new positions are USD 55 million in A and USD 40 million in B. Variance in USD million squared = 55² × 0.04 + 40² × 0.01 + 2 × 55 × 40 × 0.5 × 0.2 × 0.1 = 121 + 16 + 44 = 181.
σ = √181 = 13.454. New VaR = 1.645 × 13.454 = USD 22.13 million.
Incremental VaR = 22.13 − 23.72 = −USD 1.59 million.
The approximation (−1.60) is close to the exact answer (−1.59). It overstates the fall slightly because VaR is not linear in the position size.

Answer: The marginal VaR estimate is a fall of about USD 1.60 million. The exact incremental VaR is a fall of about USD 1.59 million, from USD 23.72 million to USD 22.13 million.

Exam tips

Questions often give betas or covariances with the portfolio. Go straight to VaR × weight × beta and skip the full variance calculation.
Use the sum rule as a check. Component VaRs must add to portfolio VaR, and weighted betas must add to 1.
Read the wording. Add or remove a position points to incremental VaR. Share of risk or decomposition points to component VaR. Per extra dollar points to marginal VaR.
For interpretation items, say what the number means: a hedge has negative marginal and component VaR, and a position with beta above 1 contributes more than its weight.
Watch whether the question wants a result in currency or in percent, and whether volatility needs scaling to the VaR horizon.

Practice questions from Portfolio Risk: Analytical Methods

Marginal, Incremental and Component VaR in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IIVaR Extensions: CVaR, Incremental and Marginal VaR

Marginal, Incremental and Component VaR: frequently asked questions

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

Marginal VaR is the VaR change per extra unit of a position. Component VaR is position times marginal VaR, and the components sum to total VaR. Incremental VaR is the actual change in VaR after adding or removing a position, found by recomputing VaR.

What is the component VaR formula for FRM Part II?

Component VaR for position i equals position size times marginal VaR. Equivalently, it is portfolio VaR × wi × βi, where βi = Cov(Ri, Rp) ÷ σp². The sum of all component VaRs equals portfolio VaR.

Can component VaR be negative?

Yes. If a position has negative covariance with the rest of the portfolio, its beta is negative. Adding more of it lowers portfolio VaR at the margin, so its component VaR is negative.

When is delta VaR a good estimate of incremental VaR?

It works for small changes in a position, because marginal VaR is a local slope. For large trades or new positions, VaR is not linear in size, so compute VaR before and after and take the difference.