Common Errors in Capital Attribution

Why Capital Attribution Errors Matter

A capital allocation is useful only if its components have a clear interpretation and reconcile to the aggregate capital being allocated. Many apparent calculation errors arise because different quantities are treated as though they were interchangeable: standalone capital, marginal capital, component allocations, and diversification benefit. Even an arithmetically correct number can be misleading if it answers the wrong question or is combined with an adjustment that has already been included.

A disciplined check starts by stating the aggregate capital measure, the allocation method, and the portfolio being measured. It then asks whether each component is defined on the same basis, whether diversification is counted once, and whether the component amounts add to the intended total. The checks below focus on diagnosing these errors rather than selecting a capital measure.

Definition: Let \(C\) denote aggregate capital for a portfolio of \(n\) components. A component allocation is a set of amounts \(A_1,\ldots,A_n\) assigned to those components. The allocation is fully allocated when \(\sum_{i=1}^n A_i=C\). Standalone capital is the capital for a component considered by itself; marginal capital is the change in aggregate capital per unit change in a component's exposure; allocated capital is the amount assigned to that component under a specified allocation method. These quantities need not be equal.

A Three-Part Attribution Check

Before interpreting an allocation, verify its basis. First, confirm that the aggregate and component amounts use the same capital definition, time horizon, confidence level, and exposure basis. Second, distinguish the allocation output from any reported diversification benefit. Third, reconcile the component amounts to the aggregate total, tracing any difference to a specific cause rather than forcing the numbers to agree.

Name the quantity.
Label each input as standalone capital, aggregate capital, marginal capital, allocated capital, or a diversification adjustment. Do not compare quantities merely because they are expressed in the same units.

Check what is already included.
Determine whether the allocation method already reflects portfolio diversification. If it does, do not deduct a separate diversification credit from the resulting allocations unless the method explicitly requires that adjustment.

Reconcile on a consistent basis.
Compute the sum of the component allocations and compare it with the intended aggregate capital. Investigate differences from scope, method, duplicated adjustments, or rounding before applying any residual rule.

When Euler Contributions Reconcile

A common source of confusion is the relationship between marginal capital and an allocated amount. Under specific conditions, Euler allocation converts marginal capital into component capital contributions. The conditions matter: the aggregate capital function must be differentiable at the portfolio being evaluated and homogeneous of degree one in the exposures.

Theorem: Let \(C(x_1,\ldots,x_n)\) be differentiable and homogeneous of degree one, so that \(C(tx_1,\ldots,tx_n)=tC(x_1,\ldots,x_n)\) for \(t>0\). The Euler component allocation is \(A_i=x_i\frac{\partial C}{\partial x_i}\). Under these conditions, \(\sum_{i=1}^n A_i=C(x_1,\ldots,x_n)\). This conclusion does not automatically apply to a non-differentiable capital function or to an allocation method that is not Euler allocation.

The justification follows directly from scaling the whole portfolio. Define \(g(t)=C(tx_1,\ldots,tx_n)\). Homogeneity gives \(g(t)=tC(x_1,\ldots,x_n)\), so \(g'(1)=C(x_1,\ldots,x_n)\). Applying the chain rule instead gives \(g'(1)=\sum_{i=1}^n x_i\frac{\partial C}{\partial x_i}\). Equating the two expressions proves full allocation. The factors \(x_i\) are essential: a marginal derivative alone is a rate of change per unit exposure, not the component's allocated capital.

Worked Example: Subtracting Diversification Twice

A hypothetical two-line portfolio uses a covariance-based capital measure. The standalone capital amounts are 100 and 80, and their correlation is 0.25. Calculate aggregate capital, the Euler component allocations, and the diversification benefit. Then diagnose an analyst's proposal to deduct half the diversification benefit from each Euler allocation.

Method and assumptions. Treat the two standalone capital amounts as standard deviations in a quadratic aggregation model. For exposures of one unit in each line, aggregate capital is the square root of the sum of the two variances and twice their covariance. The covariance is the correlation times the product of the standalone amounts.

C=\sqrt{100^2+80^2+2(0.25)(100)(80)} =\sqrt{10{,}000+6{,}400+4{,}000} =\sqrt{20{,}400} \approx 142.83.

The sum of standalone capital is \(100+80=180\). Thus, the diversification benefit is \(180-142.83=37.17\), with the calculation retained unrounded until the final display.

For this quadratic model, the Euler contribution for the first line is its variance plus its covariance with the second line, divided by aggregate capital. The corresponding numerator for the second line is its variance plus the same covariance:

A_1=\frac{100^2+(0.25)(100)(80)}{C} =\frac{12{,}000}{\sqrt{20{,}400}} \approx 84.02, \qquad A_2=\frac{80^2+(0.25)(100)(80)}{C} =\frac{8{,}400}{\sqrt{20{,}400}} \approx 58.81.

The unrounded contributions sum to \(20{,}400/\sqrt{20{,}400}=\sqrt{20{,}400}=C\). The displayed rounded values also sum to \(84.02+58.81=142.83\). This confirms full allocation.

Common mistake. The analyst subtracts half the diversification benefit from each Euler contribution. That produces \(84.02-37.17/2=65.44\) and \(58.81-37.17/2=40.23\), totaling \(105.67\), not \(142.83\). The Euler contributions already reflect the covariance and hence portfolio diversification. The separate benefit can be reported alongside them, but subtracting it again is double counting. A method that starts with standalone amounts and then allocates the diversification benefit is a different construction; it must not be layered on top of these Euler amounts.

Do Not Confuse Marginal and Allocated Capital

A marginal amount answers a sensitivity question: how much would aggregate capital change for a small increase in a specified exposure? An allocation answers an attribution question: what amount of the current aggregate is assigned to that exposure? Under Euler allocation, the component amount is exposure multiplied by the marginal derivative. If exposure units differ, simply summing marginal derivatives has no reason to reproduce aggregate capital.

Worked Example: Multiplying Marginals by Exposure

A hypothetical capital model for two exposures \(x_1\) and \(x_2\) is \(C(x_1,x_2)=\sqrt{(100x_1)^2+(80x_2)^2+2(0.25)(100x_1)(80x_2)}\). At \(x_1=2\) and \(x_2=1\), calculate aggregate capital, the two marginal derivatives, and the Euler allocations. Explain why the derivatives alone do not add to capital.

Method and assumptions. Use the stated differentiable, degree-one homogeneous model and calculate the derivatives at the current exposures. The exposures are measured in their given units; do not treat a derivative as an allocated amount without multiplying by its exposure.

First evaluate aggregate capital:

C(2,1)=\sqrt{(200)^2+(80)^2+2(0.25)(200)(80)} =\sqrt{40{,}000+6{,}400+8{,}000} =\sqrt{54{,}400} \approx 233.24.

For \(x_1\), differentiating the square root gives the numerator \(10{,}000x_1+2{,}000x_2\), divided by \(C\). At \((2,1)\), this is \((20{,}000+2{,}000)/C=22{,}000/C\), or approximately 94.32. For \(x_2\), the numerator is \(6{,}400x_2+2{,}000x_1=(6{,}400+4{,}000)=10{,}400\), so the derivative is \(10{,}400/C\), or approximately 44.59.

The marginal derivatives sum to about \(94.32+44.59=138.91\), which is not aggregate capital. The Euler component amounts include the current exposures:

A_1=2\left(\frac{22{,}000}{\sqrt{54{,}400}}\right)\approx 188.65, \qquad A_2=1\left(\frac{10{,}400}{\sqrt{54{,}400}}\right)\approx 44.59.

Using unrounded values, their sum is \((44{,}000+10{,}400)/\sqrt{54{,}400}=\sqrt{54{,}400}\), so it equals aggregate capital. Rounded to cents, \(188.65+44.59=233.24\). The common error is to report the two marginal derivatives as component allocations; the correction is to multiply each derivative by its exposure before checking full allocation.

Reconciliation Is a Diagnostic, Not a Cosmetic Adjustment

A mismatch between a top-down total and a bottom-up sum is evidence to investigate. Possible causes include different measurement bases, a component omitted from one view, duplicated diversification adjustments, or rounding. A balancing residual can be appropriate when the method explicitly defines one, but it should be identified and governed. Silently scaling every component to force a match may conceal the error and change the allocation's intended meaning.

For a hierarchical allocation, check both levels: the business-unit amounts should sum to aggregate capital, and the risk-type amounts within each business unit should sum to that unit's allocation. A match at the top level does not prove that the suballocations are correct. Likewise, matching every business row does not ensure that the risk totals are based on a consistent method.

Worked Example: Finding a Duplicated Adjustment in a Hierarchy

A hypothetical insurer's aggregate capital is 200. A top-down method assigns 120 to Business A and 80 to Business B. A bottom-up table reports A's risk-type amounts as 70 for market, 35 for credit, and 15 for operational risk. It reports B's amounts as 35, 30, and 20, respectively. The B operational figure includes a 5 adjustment that was already credited at the business-unit level. Diagnose the mismatch and give the corrected reconciliation.

Method. First sum each business unit's risk-type amounts, then compare those sums with the top-down allocations. Use the stated information about the duplicate adjustment to correct its source; do not proportionally rescale unrelated risk-type amounts.

Business A's bottom-up amount is \(70+35+15=120\), which agrees with its top-down allocation. Business B's reported amount is \(35+30+20=85\), which exceeds its top-down allocation of 80 by 5. The full bottom-up table totals \(120+85=205\), exceeding aggregate capital of 200 by the same amount.

The problem specifies the cause: 5 in B's operational figure is a duplicate of an adjustment already credited at the business-unit level. Remove that duplicate from the B operational amount: \(20-5=15\). The corrected B allocation is \(35+30+15=80\).

The reconciled business totals are now \(120+80=200\), and the corrected risk-type amounts across both businesses also sum to \(70+35+15+35+30+15=200\). Both levels reconcile. The common mistake would be to scale all six cells by \(200/205\); that would force the total to 200 but distort five correct cells instead of fixing the known duplicated adjustment.

Common Errors to Check Before Reporting

Deducting diversification twice. Check whether the selected allocation already incorporates diversification. Do not subtract a separate benefit from allocations that already reflect it.
Calling a marginal derivative an allocation. A derivative is a change per unit of exposure. Under Euler allocation, multiply by the current exposure to obtain the component contribution.
Assuming every method fully allocates. Full allocation follows for Euler contributions under differentiability and degree-one homogeneity. Other methods require their own full-allocation property or an explicit reconciliation rule.
Accepting a total match as sufficient evidence. Verify the allocation at each hierarchy level and check that components cover the intended risks and use consistent bases.
Using a residual to hide an unexplained gap. Identify whether a discrepancy is a model difference, an omitted item, a duplicated adjustment, or rounding before deciding how to treat it.

Key takeaway: For every attribution, name the quantity being allocated, establish whether diversification is already included, and reconcile component amounts to the correct aggregate. A numerical match is necessary for full allocation, but the method and basis must also be defensible.

Check Your Understanding

For each problem, diagnose the allocation issue or calculate the requested amounts. Use unrounded values in intermediate steps.

A two-component covariance model has standalone capitals 60 and 50 with correlation 0.20. Calculate aggregate capital and diversification benefit. (Answer: Aggregate capital is \(\sqrt{7{,}300}\approx 85.44\); diversification benefit is approximately 24.56.)
A differentiable homogeneous capital function is \(C(x,y)=\sqrt{81x^2+144y^2+72xy}\). At \((1,2)\), calculate the Euler allocations using \(A_x=x\frac{\partial C}{\partial x}\) and \(A_y=y\frac{\partial C}{\partial y}\), and verify the total. (Answer: \(C=\sqrt{801}\approx 28.30\), \(A_x=153/\sqrt{801}\approx 5.41\), and \(A_y=648/\sqrt{801}\approx 22.90\); the unrounded allocations sum to \(C\).)
Aggregate capital is 150, with top-down business allocations of 90 and 60. The first business's bottom-up risk amounts are 48 and 44; the second's are 25 and 35. A source check confirms that the first business's first risk amount contains a duplicated adjustment of 2. Give the corrected amounts and verify the total. (Answer: Correct the first amount to 46; the business totals are \(46+44=90\) and \(25+35=60\), totaling 150.)
At a current portfolio, two marginal derivatives are 12 and 9, and the corresponding exposures are 3 and 2. Under Euler allocation, calculate the component contributions and aggregate capital. (Answer: Contributions are 36 and 18; aggregate capital is 54.)
Standalone capitals are 70 and 40, while aggregate capital is 90. A standalone-based allocation assigns the diversification benefit in the ratio 3:1. Calculate the two allocations and check full allocation. (Answer: The benefit is 20, assigned as 15 and 5; allocations are 55 and 35, totaling 90.)

What You'll Learn