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FRM Exam Part II · VaR and Risk Budgeting in Investment Management

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 tiny change in one position. Incremental VaR is the change in VaR when you add or remove a whole position. Component VaR is marginal VaR times position size, and component VaRs add up to total portfolio VaR. Use them to find which positions drive risk.

Understand Marginal, Incremental and Component VaR

Portfolio VaR is not the sum of stand-alone VaRs, because diversification lowers risk. So when a risk manager asks which position is responsible for the VaR, a simple split is not enough. You need a way to share the diversified VaR across positions.

Marginal VaR answers: if I raise this position by a very small amount, how much does portfolio VaR rise per unit of money added? It is a derivative. Under normal (parametric) VaR, it is the position's covariance with the portfolio divided by the portfolio volatility, times the z-score.

Component VaR turns that rate into a rupee or dollar contribution. Multiply marginal VaR by the amount held in the position. Because VaR under the normal model is homogeneous of degree one, the component VaRs sum exactly to portfolio VaR. This is why component VaR is the standard tool for risk budgeting.

Incremental VaR is different. It is the actual change in portfolio VaR when a position is added, removed or resized by a large amount: VaR with the position minus VaR without it. It needs two full VaR calculations. Incremental VaR is used for trade decisions. Component VaR is an approximation of incremental VaR for small moves, and the two are not equal for a large position.

A position can have a negative component VaR. This means it hedges the rest of the portfolio: it has negative covariance with the portfolio, so holding it reduces total VaR.

Key formulas to remember

Portfolio VaR (normal)
VaR_p = z × σ_p × V
z is the confidence z-score (1.645 at 95%, 2.326 at 99%). V is portfolio value. Mean assumed zero unless told otherwise.
Beta of position to portfolio
β_i = Cov(R_i, R_p) ÷ σ_p²
Measures how much position i moves with the portfolio.
Marginal VaR
MVaR_i = z × Cov(R_i, R_p) ÷ σ_p = β_i × VaR_p ÷ V
VaR change per unit of money added to position i. Often quoted as β_i × (VaR_p ÷ V).
Component VaR
CVaR_i = MVaR_i × (w_i × V) = w_i × β_i × VaR_p
w_i is the portfolio weight. Can be negative for a hedging position.
Additivity
Σ CVaR_i = VaR_p
Holds because Σ w_i × β_i = 1. Use it as a check on your answer.
Percentage contribution
CVaR_i ÷ VaR_p = w_i × β_i
Compare with the weight w_i to see whether a position takes more or less than its share of risk.
Incremental VaR
IVaR_i = VaR_with position − VaR_without position
An exact change from a large trade. Approximately CVaR_i for small positions only.

How to solve Marginal, Incremental and Component VaR questions

Use this order for any question on decomposing VaR. It works whether the data are covariances, betas or volatilities and correlations.

  1. 1Identify what is asked: marginal (per unit, small change), component (contribution that adds to total) or incremental (full add or remove).
  2. 2Write down the confidence level and z-score, the horizon and the portfolio value.
  3. 3Compute portfolio volatility σ_p from weights, volatilities and correlations, then VaR_p = z × σ_p × V.
  4. 4For marginal or component VaR, find each position's covariance with the portfolio (or its beta). Marginal VaR = z × Cov(R_i, R_p) ÷ σ_p.
  5. 5Multiply marginal VaR by the position's amount to get component VaR. Check that all components sum to VaR_p.
  6. 6For incremental VaR, recompute portfolio VaR with and without the position and subtract.
  7. 7Interpret: compare each component share with its weight, flag negative components as hedges, and state the decision.

Quickest way: Beta shortcut for component VaR

When to use it: Use when the question gives or lets you easily find the portfolio VaR and each position's beta to the portfolio, or the covariance with the portfolio.

  1. Get VaR_p first. Everything else scales from it.
  2. Compute β_i = Cov(R_i, R_p) ÷ σ_p².
  3. Component VaR = w_i × β_i × VaR_p. Marginal VaR = β_i × VaR_p ÷ V.
  4. Check that Σ w_i × β_i = 1. If not, you made an arithmetic error.
  5. For incremental VaR, do not use the shortcut for a large trade. Recompute VaR with and without.

Common mistakes in Marginal, Incremental and Component VaR

  • Treating marginal VaR and incremental VaR as the same thing.

    Both measure a change in VaR and the names sound alike.

    Fix: Marginal is a rate for a tiny change. Incremental is the actual change from a full add or removal. Marginal is an approximation tool.

  • Expecting component VaRs to sum to the sum of stand-alone VaRs.

    Students forget diversification.

    Fix: Component VaRs sum to the diversified portfolio VaR. Stand-alone VaRs sum to a larger number unless correlations are all 1.

  • Forgetting to multiply marginal VaR by the position size.

    Marginal VaR looks like a final answer.

    Fix: Component VaR = marginal VaR × amount invested. Marginal VaR is per unit of money.

  • Assuming a larger position always has a larger component VaR.

    Weight is confused with risk contribution.

    Fix: Contribution is w_i × β_i. A small position with high beta can contribute more than a large one with low beta.

  • Treating a negative component VaR as an error.

    Risk is thought to be always positive.

    Fix: Negative means the position has negative covariance with the portfolio. It is a hedge and lowers total VaR.

  • Using σ_p instead of σ_p² when computing beta.

    Mixing the marginal VaR formula (÷ σ_p) with the beta formula (÷ σ_p²).

    Fix: Beta divides covariance by variance. Marginal VaR divides covariance by volatility and multiplies by z.

Worked examples

Example 1

A ₹/USD-neutral portfolio is worth USD 100 million with two positions: A = USD 60 million and B = USD 40 million. Daily volatilities are 2% for A and 3% for B, correlation 0.5. Using 99% one-day normal VaR (z = 2.33), find portfolio VaR and the component VaR of each position.

Show the solution
  1. Weights: w_A = 0.6, w_B = 0.4.
  2. Variance = (0.6×0.02)² + (0.4×0.03)² + 2×0.5×(0.6×0.02)×(0.4×0.03).
  3. = 0.012² + 0.012² + 0.012×0.012 = 0.000144 + 0.000144 + 0.000144 = 0.000432.
  4. σ_p = √0.000432 = 0.020785 (2.0785%).
  5. VaR_p = 2.33 × 0.020785 × 100 = USD 4.843 million.
  6. Cov(A, p) = w_A σ_A² + w_B ρ σ_A σ_B = 0.6×0.0004 + 0.4×0.5×0.0006 = 0.00024 + 0.00012 = 0.00036.
  7. Cov(B, p) = w_A ρ σ_A σ_B + w_B σ_B² = 0.6×0.0003 + 0.4×0.0009 → 0.6×0.5×0.0006 = 0.00018; 0.4×0.0009 = 0.00036; total 0.00054. Check: 0.6×0.00036 + 0.4×0.00054 = 0.000216 + 0.000216 = 0.000432, matches variance.
  8. β_A = 0.00036 ÷ 0.000432 = 0.8333; β_B = 0.00054 ÷ 0.000432 = 1.25.
  9. CVaR_A = 0.6 × 0.8333 × 4.843 = 0.5 × 4.843 = USD 2.42 million.
  10. CVaR_B = 0.4 × 1.25 × 4.843 = 0.5 × 4.843 = USD 2.42 million.

Answer: Portfolio VaR is about USD 4.84 million. Each position contributes about USD 2.42 million (50% each), even though B is only 40% of the capital. The components sum to the total.

Example 2

A portfolio has 99% one-day VaR of USD 10.0 million. After removing a position, the VaR of the remaining portfolio is USD 7.5 million. The removed position had a component VaR of USD 3.2 million. What is the incremental VaR of the position, and why does it differ from component VaR?

Show the solution
  1. Incremental VaR = VaR with position − VaR without position.
  2. = 10.0 − 7.5 = USD 2.5 million.
  3. Component VaR is USD 3.2 million, a linear estimate based on the marginal rate at the current size.
  4. Removing a whole position is a large change. VaR is not linear in position size across the whole range, so the exact change differs from the estimate.
  5. The component VaR of 3.2 overstates the VaR saving from removal by 0.7.

Answer: Incremental VaR is USD 2.5 million. It is lower than the USD 3.2 million component VaR because component VaR is only a first-order approximation, accurate for small changes, not for removing an entire position.

Exam tips

  • Read the verb. 'Per unit' or 'small change' means marginal. 'Contribution' or 'sums to total' means component. 'Add the trade' or 'remove the desk' means incremental.
  • Always check that your components sum to portfolio VaR. It catches most arithmetic slips in seconds.
  • Expect interpretation questions: which position takes a bigger share of risk than its weight, and which one is a hedge.
  • Incremental VaR needs two VaR numbers. If the question gives only betas, it is probably asking for component or marginal VaR.
  • Do not carry rounding too early. Keep σ_p to four or more decimals before multiplying by z.

Practice questions from VaR and Risk Budgeting in Investment Management

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.

Marginal, Incremental and Component VaR: frequently asked questions

What is the difference between marginal VaR and incremental VaR?

Marginal VaR is the rate of change of portfolio VaR for a very small increase in a position. Incremental VaR is the actual change in VaR when a position is added or removed in full. Marginal is an approximation tool, incremental is the exact before-and-after difference.

How do you calculate component VaR of a portfolio?

Compute portfolio VaR, then each position's beta to the portfolio (covariance with the portfolio divided by portfolio variance). Component VaR is weight × beta × portfolio VaR. The components add up to the portfolio VaR.

Why do component VaRs sum to total VaR?

Under the normal model, VaR scales in proportion to position size. By Euler's theorem for such functions, the sum of each position times its marginal VaR equals the total. Equivalently, the weighted betas sum to one.

Can component VaR be negative?

Yes. A position with negative covariance with the rest of the portfolio reduces total VaR, so its contribution is negative. It acts as a hedge.