Skip to content

FRM Exam Part II · Risk Capital Attribution and Risk-Adjusted Performance Measurement

Risk Capital Attribution Methods: Stand-Alone, Incremental and Euler

Updated 11 October 2026 · Fact-checked

Risk capital attribution splits a firm's total risk capital across business units. Stand-alone capital ignores diversification. Incremental capital is the firm's capital minus its capital without the unit. Marginal (Euler) allocation uses each unit's contribution to portfolio risk, so the parts add exactly to the total. Compute each, then check the sum against firm capital.

Understand Risk Capital Attribution Methods

A bank holds one pool of risk capital for the whole firm. Management wants to know how much of it each business unit uses, so it can set limits and judge returns. That split is risk capital attribution.

The difficulty is diversification. Units rarely lose at the same time, so firm capital is lower than the sum of the units' stand-alone capital. The gap is the diversification benefit. Someone has to be credited with it, and the method you choose decides who.

There are three main methods.

  • Stand-alone: measure each unit's capital as if it were a separate firm. It is simple, but the amounts add up to more than firm capital. You must scale them down, usually pro rata, to reconcile.
  • Incremental: capital of the whole firm minus capital of the firm without that unit. It answers a what-if question, such as whether to close or add a unit. The amounts usually do not add to firm capital.
  • Marginal (Euler): the unit's size times the rate at which firm risk changes when that unit grows a little. It is also called component or contribution allocation. For a risk measure that is homogeneous of degree one (scale the portfolio by k and the risk scales by k), the pieces add exactly to total capital. Standard deviation and expected shortfall have this property. VaR does too, under the usual smoothness conditions.

Under Euler with a volatility-based measure, a unit's share relative to its stand-alone capital equals its correlation with the whole portfolio, ρ_iP. A unit with a low ρ_iP gets a smaller share relative to its stand-alone capital, because it adds less to total risk. A unit with ρ_iP close to 1 gets a share close to its stand-alone capital. Euler is therefore the method that rewards diversification in a way that is consistent and fully allocated.

On the exam, name the method, compute the number, and say what it is used for. Stand-alone suits limits and ignores diversification. Incremental suits decisions about adding or removing a unit. Euler suits full allocation and performance measurement.

Key formulas to remember

Diversification benefit
DB = Σ K_i(stand-alone) − K(firm)
Positive when units are not perfectly correlated. For a volatility-based measure, DB is zero only when every pair of units has a correlation of 1 (perfect positive correlation).
Pro rata scaling of stand-alone capital
K_i(allocated) = K_i(stand-alone) ÷ Σ K_j(stand-alone) × K(firm)
Spreads the diversification benefit in proportion to stand-alone size. It ignores each unit's correlation with the rest of the firm.
Incremental capital
K_i(incremental) = K(firm) − K(firm without unit i)
Sum across units is generally not equal to K(firm). Use it for add or remove decisions.
Euler allocation
K_i = w_i × ∂K/∂w_i, and Σ K_i = K
Holds when the risk measure is homogeneous of degree one and differentiable. The full-allocation property is the reason it is preferred.
Euler contribution with volatility
K_i = k × Cov(R_i, R_P) ÷ σ_P = k × ρ_iP × σ_i
Here capital K = k × σ_P (k is a multiplier) and R_i is the unit's loss or return contribution. For two units, Cov(R_A, R_P) = σ_A² + Cov(A, B). The Euler share is below stand-alone capital (k × σ_i) when ρ_iP < 1. Here ρ_iP is the unit's correlation with the whole portfolio, not with another unit.
Portfolio volatility, two units
σ_P = √(σ_A² + σ_B² + 2ρσ_Aσ_B)
Needed to find the firm capital and the Euler shares.

How to solve Risk Capital Attribution Methods questions

Use this order for any question on attributing risk capital.

  1. 1Identify the risk measure and the capital rule: standard deviation times a multiplier, VaR, or expected shortfall.
  2. 2Compute each unit's stand-alone capital and add them up.
  3. 3Compute firm capital with diversification. Use the portfolio volatility formula if you are given volatilities and correlations.
  4. 4Find the diversification benefit: sum of stand-alone capital minus firm capital.
  5. 5Apply the requested method: pro rata scaling for stand-alone, firm minus firm-without-unit for incremental, or covariance divided by portfolio volatility for Euler.
  6. 6Check the sum. Euler and scaled stand-alone must equal firm capital. Unscaled stand-alone should be higher. Incremental usually falls short.
  7. 7Interpret: say who receives the diversification benefit and which decision the method suits.

Quickest way: Sum check and covariance shortcut

When to use it: Use it when a multiple-choice question gives volatilities and a correlation and asks for one unit's allocated capital.

  1. Get the portfolio σ_P first. Many exam numbers are chosen so it is a round value.
  2. For unit i, compute σ_i² plus the covariance with the other units, then divide by σ_P. Multiply by the capital multiplier.
  3. Check that your unit amounts add to the firm figure. If they do not, you used the wrong method or made an arithmetic slip.
  4. Eliminate options that equal stand-alone capital. For a volatility-based measure, the Euler share is lower than stand-alone when the unit's correlation with the portfolio (ρ_iP) is below 1, so a figure above stand-alone is wrong.
  5. For incremental questions, just subtract. Compute capital without the unit and take the difference.

Common mistakes in Risk Capital Attribution Methods

  • Treating stand-alone capital as the allocated capital.

    It is the easiest number to compute, and it looks like each unit's own risk.

    Fix: Stand-alone amounts sum to more than firm capital. Scale them or use Euler if the question wants a full allocation.

  • Assuming incremental capital amounts add up to firm capital.

    Students assume any sensible allocation must be additive.

    Fix: Incremental amounts generally do not sum to the total. Say so, and use Euler when full allocation is required.

  • Dividing covariance by variance instead of by portfolio standard deviation in the Euler formula.

    Beta uses variance, and the formulas look alike.

    Fix: The Euler share of σ_P is Cov(R_i, R_P) ÷ σ_P. Check that your shares add to σ_P.

  • Forgetting the diversification benefit must be allocated, or giving every unit the same percentage regardless of correlation.

    Pro rata scaling is simple and gets confused with Euler.

    Fix: Pro rata ignores correlation. Under Euler, a unit with a lower correlation to the portfolio gets a smaller share relative to its stand-alone capital.

  • Applying the Euler full-allocation result to any risk measure.

    The condition is easy to skip.

    Fix: State the condition: homogeneous of degree one and differentiable. Standard deviation and expected shortfall qualify, and VaR qualifies under the usual smoothness conditions.

  • Reading marginal as incremental, the discrete change from removing a whole unit.

    Both terms sound like a change in capital.

    Fix: Marginal is the derivative for a very small change in position size. Incremental is the discrete effect of the whole unit.

Worked examples

Example 1

A bank has two units. Stand-alone economic capital is USD 60 million for Unit A and USD 40 million for Unit B. Firm-level economic capital, with diversification, is USD 80 million. Find the diversification benefit and each unit's capital if stand-alone capital is scaled pro rata.

Show the solution
  1. Sum of stand-alone capital = 60 + 40 = USD 100 million.
  2. Diversification benefit = 100 − 80 = USD 20 million.
  3. Scaling factor = 80 ÷ 100 = 0.8.
  4. Unit A = 60 × 0.8 = USD 48 million.
  5. Unit B = 40 × 0.8 = USD 32 million.
  6. Check: 48 + 32 = 80, which equals firm capital.

Answer: Diversification benefit is USD 20 million. Allocated capital is USD 48 million to Unit A and USD 32 million to Unit B.

Example 2

Two units have loss volatilities σ_A = USD 30 million and σ_B = USD 40 million, with zero correlation. Capital is 3 × the portfolio volatility. Compute the firm capital, the Euler allocation to each unit, and the incremental capital of Unit A. Comment on the sums.

Show the solution
  1. Portfolio σ_P = √(30² + 40²) = √(900 + 1,600) = √2,500 = USD 50 million.
  2. Firm capital = 3 × 50 = USD 150 million. Stand-alone capital: A = 90, B = 120, total 210, so diversification benefit = 210 − 150 = USD 60 million.
  3. Euler for A: Cov(A, P) = σ_A² + Cov(A, B) = 900 + 0 = 900. Divide by σ_P: 900 ÷ 50 = 18. Capital = 3 × 18 = USD 54 million.
  4. Euler for B: 1,600 ÷ 50 = 32. Capital = 3 × 32 = USD 96 million.
  5. Check: 54 + 96 = 150, equal to firm capital.
  6. Incremental for A: firm without A is Unit B alone, capital = 3 × 40 = 120. Incremental = 150 − 120 = USD 30 million.
  7. Incremental for B: firm without B has capital 3 × 30 = 90, so B = 150 − 90 = 60. The incremental amounts sum to 30 + 60 = 90, which is less than 150.

Answer: Firm capital is USD 150 million. Euler allocates USD 54 million to A and USD 96 million to B, which sums to the total. Incremental capital for A is USD 30 million, and the incremental amounts (30 and 60) do not sum to firm capital.

Exam tips

  • Always check whether the question asks for a full allocation. If it does, the answer is Euler or a scaled method, not raw incremental or raw stand-alone.
  • The diversification benefit, stand-alone sum minus firm capital, is a quantity a question may ask for directly. Compute it as a check even when it is not requested.
  • Many questions test interpretation: which method suits closing a unit (incremental), setting limits (stand-alone) or performance measurement (Euler).
  • Know the condition for Euler: homogeneous of degree one. Be ready for a question asking why the shares add to the total.
  • Do the arithmetic with round numbers. Exam volatilities are usually chosen so that √ gives a whole number.

Practice questions from Risk Capital Attribution and Risk-Adjusted Performance Measurement

Risk Capital Attribution Methods: frequently asked questions

What is the difference between stand-alone, incremental and marginal capital allocation?

Stand-alone measures each unit as a separate firm and ignores diversification. Incremental is firm capital minus firm capital without the unit. Marginal (Euler) uses the unit's contribution to firm risk, so the pieces add to total capital.

How do you allocate the diversification benefit to business units?

Either spread it pro rata to stand-alone capital, or use Euler allocation, which allocates it according to each unit's covariance with the portfolio. With a volatility-based measure, Euler gives a smaller share relative to stand-alone capital to units whose correlation with the portfolio is lower.

Why does Euler allocation add up exactly to total capital?

For a risk measure that is homogeneous of degree one and differentiable, Euler's theorem says the sum of each position size times its partial derivative equals the total. That is why the contributions are fully additive.

Which method is best for deciding whether to close a business unit?

Incremental capital, because it measures what the firm's capital would fall by if the unit were removed. It does not give a set of amounts that sum to the total, so it is not used for full allocation.