CFA Level II Exam · Measuring and Managing Market Risk

CVaR, Incremental VaR and Marginal VaR Explained

Updated 7 October 2026 · Fact-checked

Conditional VaR (expected shortfall) is the average loss in the worst tail beyond the VaR cutoff, so it is always at least as large as VaR. Incremental VaR is the actual change in portfolio VaR from adding or removing a position. Marginal VaR is the change per small unit change in a position. Relative VaR measures loss versus a benchmark.

Understand VaR Extensions: CVaR, Incremental and Marginal VaR

VaR gives a loss threshold: with a stated probability over a stated period, losses will not exceed this amount. Its weakness is that it says nothing about how bad losses are once they pass the threshold. The extensions below fix that and also help you see how each position drives portfolio risk.

Conditional VaR (CVaR), also called expected shortfall, is the expected loss given that the loss is at least as large as the VaR. Think of it as the average of the tail. Because it averages losses beyond the cutoff, CVaR is at least as large as VaR at the same confidence level. It is more informative for fat-tailed returns. It can be estimated from historical, parametric or Monte Carlo methods, but it still depends on the quality of the tail data and the model.

Incremental VaR (IVaR) is the change in portfolio VaR when a position is added, removed or resized by a meaningful amount. You compute it by running VaR twice, with and without the change, and taking the difference. It captures diversification: a position that is highly correlated with the rest of the portfolio adds more risk than one that is not. It can be negative if the position acts as a hedge.

Marginal VaR (MVaR) is the change in portfolio VaR for a very small change in a position, like a derivative or a per-unit sensitivity. It is a first-order approximation. Incremental VaR is for larger, real changes; marginal VaR is for tiny ones. A related idea is component VaR, which splits portfolio VaR across positions so the parts add up exactly to total VaR. Component VaR is the position's value times its marginal VaR, where marginal VaR is measured per unit of that position's value. Using weights instead of values gives the same result when marginal VaR is defined per unit of portfolio value, so keep the definitions consistent.

Relative VaR (also called tracking VaR) is the VaR of the difference between portfolio and benchmark returns, so it measures the risk of falling behind the benchmark. Absolute VaR measures the loss in currency or percentage terms with no benchmark. Benchmarked managers are usually judged on relative VaR.

Key formulas to remember

Conditional VaR (expected shortfall): CVaR = E[Loss | Loss ≥ VaR]; Average loss in the tail beyond the VaR cutoff. CVaR ≥ VaR at the same confidence level.
Incremental VaR: IVaR = VaR(portfolio with change) − VaR(portfolio without change); Use for sizeable changes such as adding or removing a position. Can be negative for a hedge.
Marginal VaR (concept): MVaR ≈ ΔVaR ÷ Δposition, for a very small change in position; A rate of change per unit of position value. A first-order approximation, not a full revaluation.
Component VaR: Component VaR(i) = position value(i) × MVaR(i); Σ component VaRs = portfolio VaR; MVaR is per unit of position value. Allocates total VaR across positions. The components add up exactly to the total. If MVaR is instead defined per unit of portfolio value, multiply by weight(i) rather than position value.
Relative VaR: Relative VaR = VaR of (portfolio return − benchmark return); Measures risk of underperforming the benchmark. Absolute VaR ignores the benchmark.

How to solve VaR Extensions: CVaR, Incremental and Marginal VaR questions

Identify which measure the vignette asks for, then pull only the numbers that measure needs.

1Read the question stem first and name the measure: CVaR, incremental, marginal, component or relative VaR.
2Find the confidence level, time horizon and currency in the vignette or exhibit. Check that they match across all numbers.
3For CVaR, locate the tail losses beyond the VaR cutoff and average them, using any probabilities given.
4For incremental VaR, find portfolio VaR before and after the change and subtract: after minus before.
5For marginal or component VaR, use the stated per-unit change or position value times marginal VaR, and check that the components sum to portfolio VaR.
6For relative VaR, make sure the inputs are about the active return (portfolio minus benchmark), not the portfolio alone.
7Check the sign and reasonableness: CVaR is not below VaR, and a hedge should give a negative incremental VaR.
8Match your result to the option, checking units (currency or percent).

Quickest way: Match the measure to the question words

When to use it: Use when time is short and the options differ by concept rather than by calculation.

Words like 'average loss beyond VaR' or 'tail' point to CVaR, which is larger than VaR.
'Adding a position' or 'removing a position' points to incremental VaR: after minus before.
'Very small change' or 'per unit' points to marginal VaR.
'Allocation that sums to total' points to component VaR.
'Against a benchmark' or 'tracking' points to relative VaR.
Eliminate any option where CVaR is less than VaR or where components do not add up.

Common mistakes in VaR Extensions: CVaR, Incremental and Marginal VaR

Treating CVaR as the loss at a higher confidence level.
Both look at the tail, so they seem alike.
Fix: CVaR is an average of all losses beyond the VaR cutoff, not a single percentile.
Using marginal VaR for a large position change.
The names sound interchangeable.
Fix: Marginal VaR is for tiny changes. For sizeable changes, recompute VaR and use incremental VaR.
Subtracting in the wrong order for incremental VaR.
Candidates rush and take before minus after.
Fix: Always compute after minus before. A positive answer means the change adds risk.
Assuming incremental VaR is always positive.
Adding a position feels like adding risk.
Fix: A position that offsets existing exposure can lower portfolio VaR, giving a negative incremental VaR.
Using portfolio return instead of active return for relative VaR.
Confusing relative VaR with absolute VaR.
Fix: Relative VaR is built from the difference between portfolio and benchmark returns.
Adding stand-alone position VaRs to get portfolio VaR.
Ignoring correlation.
Fix: With imperfect correlation, stand-alone VaRs typically sum to more than portfolio VaR under normal assumptions. They equal it only with perfect positive correlation in the parametric case. Component VaRs are the ones that add up exactly to portfolio VaR.

Worked examples

Example 1

A risk report shows a portfolio's one-day 5% VaR is ₹4,00,000. The simulated outcomes beyond the VaR cutoff are five equally likely losses, each larger than the VaR: ₹4,20,000, ₹4,40,000, ₹4,80,000, ₹5,20,000 and ₹6,40,000. (1) What is the CVaR? (2) How does it compare with VaR? Choices for (1): A. ₹4,00,000 B. ₹5,00,000 C. ₹5,20,000 D. ₹6,40,000

Show the solution

CVaR is the average of the tail losses beyond the cutoff, here the five stated losses.
Sum: 4,20,000 + 4,40,000 + 4,80,000 + 5,20,000 + 6,40,000 = 25,00,000.
Divide by 5: 25,00,000 ÷ 5 = 5,00,000.
Match to the options: ₹5,00,000 is choice B. A is just the VaR, C is the fourth loss, and D is the largest loss, not the average.
Part (2): CVaR of ₹5,00,000 is ₹1,00,000 above the VaR of ₹4,00,000, as it must be, since CVaR is at least as large as VaR.

Answer: B. CVaR = ₹5,00,000, which is above the VaR of ₹4,00,000.

Example 2

A fund's one-day 95% VaR is ₹20,00,000. A manager considers adding a position. The pro forma VaR with the position is ₹22,50,000. Separately, as an alternative to adding, removing a different position from the original ₹20,00,000 portfolio would lower VaR to ₹18,00,000. The two changes are not made together. Which statement is correct? A. Adding the position lowers risk. B. Removing the other position adds ₹2,00,000 of risk. C. Adding the position adds ₹2,50,000 of risk. D. Removing the other position reduces VaR by ₹4,00,000.

Show the solution

Incremental VaR = VaR after − VaR before.
Adding: 22,50,000 − 20,00,000 = +2,50,000, so it adds risk of ₹2,50,000.
Removing (from the original ₹20,00,000 portfolio, taken on its own): 18,00,000 − 20,00,000 = −2,00,000, so it reduces VaR by ₹2,00,000.
Check the options: A is false because adding raises VaR. B reverses the sign of the removal effect. C matches the +₹2,50,000 result. D overstates the reduction, which is ₹2,00,000, not ₹4,00,000.

Answer: C. Adding the position adds ₹2,50,000 of risk (incremental VaR = +₹2,50,000). Removing the other position from the original portfolio gives −₹2,00,000.

Exam tips

Write 'after minus before' beside any incremental VaR question before you calculate.
Check the confidence level and horizon in the exhibit. Mismatched horizons are a common trap.
Remember the ordering: CVaR ≥ VaR at the same confidence level, and use it to eliminate options quickly.
Know the use case of each measure: marginal for small changes, incremental for large ones, component for allocation, relative for benchmarked mandates.
State limits when asked: CVaR still depends on the model and tail data, and VaR measures do not predict the worst case.

VaR Extensions: CVaR, Incremental and Marginal VaR in other exams

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

FRM Part IIMarginal, Incremental and Component VaR

VaR Extensions: CVaR, Incremental and Marginal VaR: frequently asked questions

What is the difference between marginal VaR and incremental VaR?

Marginal VaR is the change in portfolio VaR for a very small change in a position, expressed per unit. Incremental VaR is the actual change in VaR when you add, remove or resize a position by a meaningful amount, found by recomputing VaR.

How do you calculate CVaR?

Identify the losses beyond the VaR cutoff and average them, weighting by probability if given. For a parametric model you use the distribution formula instead. The result is at least as large as VaR.

What is relative VaR versus absolute VaR?

Absolute VaR is the potential loss of the portfolio itself. Relative VaR is the potential shortfall against a benchmark, based on the portfolio return minus the benchmark return.

Why is CVaR preferred over VaR for fat tails?

VaR ignores the size of losses beyond its cutoff. CVaR averages those tail losses, so it shows more of the severity when returns have fat tails or extreme outcomes.