Worked Covariance-Based Capital Allocation

Why Allocate Capital Using Covariance?

A portfolio’s aggregate capital reflects both the size of each risk and how the risks move together. Two portfolios can have the same standalone capital amounts but different aggregate capital if their correlations differ. Allocation should reflect that difference: a risk that is more strongly associated with the rest of the portfolio can receive a different allocation from one with the same standalone capital but weaker or negative association.

This tutorial works through the covariance aggregation model and its Euler allocation. The calculations are appropriate when the standalone capital amounts are on a common volatility scale: specifically, when each is the same multiple of its risk’s standard deviation. If standalone figures are based on different measures or different scaling factors, the covariance formula below does not automatically apply.

Definition: Let $C_i$ denote the standalone capital for risk $i$, and let $\rho_{ij}$ denote the correlation between risks $i$ and $j$. Under covariance aggregation, portfolio capital is $$ C=\sqrt{\sum_i C_i^2+2\sum_{i<j}\rho_{ij}C_iC_j}. $$ The Euler allocation to risk $i$ is $$ A_i=\frac{C_i\left(C_i+\sum_{j\ne i}\rho_{ij}C_j\right)}{C}. $$ The formula assumes $C>0$, a common volatility scaling for the standalone amounts, and a valid correlation matrix.

The $C_i$ values are standalone capital amounts, not the portfolio allocations. The $A_i$ values are the allocations from the portfolio capital $C$. Both are expressed in the same units, such as dollars or millions of dollars. Correlations must be consistent as a set: every pairwise value must lie between $-1$ and $1$, and the full correlation matrix must be positive semidefinite. Checking pairs individually is not enough to establish that a multi-risk matrix is valid.

Deriving the Euler Allocation

To see why the allocation takes this form, let $w_i$ be a scale applied to risk $i$. Under covariance aggregation, portfolio capital as a function of the scales is

C(\mathbf{w})= \sqrt{\sum_i w_i^2 C_i^2+ 2\sum_{i<j}\rho_{ij}w_iw_jC_iC_j}.

The marginal capital contribution of risk $i$ is the partial derivative of $C(\mathbf{w})$ with respect to $w_i$. Differentiating the expression under the square root gives

\frac{\partial C(\mathbf{w})}{\partial w_i} = \frac{w_i C_i^2+\sum_{j\ne i}\rho_{ij}w_jC_iC_j} {C(\mathbf{w})}.

At the actual portfolio, every $w_i=1$. Multiplying the derivative by the current scale $w_i=1$ gives the Euler allocation $A_i$ stated above. Each pairwise covariance term appears in two allocations—once for each risk in the pair—so the allocations add to the full portfolio variance numerator before division by $C$:

\sum_i A_i = \frac{\sum_i C_i^2+2\sum_{i<j}\rho_{ij}C_iC_j}{C} = \frac{C^2}{C} = C.

This identity is the reconciliation check: before rounding, the Euler allocations sum exactly to aggregate capital. It also explains why the allocation uses each risk’s covariance relationships with the other risks rather than allocating in proportion to standalone capital alone.

Worked Calculations

Worked Example: Two Risks With Positive Correlation

A hypothetical insurer models two risk categories with standalone capital of $60 million and $80 million. Their correlation is $0.25$. Calculate aggregate capital and the Euler allocation to each risk. Report amounts in millions of dollars to two decimal places, rounding only at the final step.

Method. Apply the two-risk covariance formula, then use each risk’s covariance-weighted numerator for the Euler allocation.

Step 1: Calculate aggregate capital. The squared standalone amounts are $60^2=3{,}600$ and $80^2=6{,}400$. The covariance contribution inside the portfolio variance is $2(0.25)(60)(80)=2{,}400$. Therefore,

C=\sqrt{3{,}600+6{,}400+2{,}400} =\sqrt{12{,}400} =111.355287\text{ million}.

Step 2: Allocate to the first risk. Its covariance-weighted numerator is $60(60+0.25(80))=60(80)=4{,}800$. Thus,

A_1=\frac{4{,}800}{111.355287} =43.105\ldots \approx \$43.11\text{ million}.

Step 3: Allocate to the second risk. Its numerator is $80(80+0.25(60))=80(95)=7{,}600$. Thus,

A_2=\frac{7{,}600}{111.355287} =68.250\ldots \approx \$68.25\text{ million}.

Check. The unrounded allocations sum to $(4{,}800+7{,}600)/111.355287=111.355287$ million, equal to aggregate capital. The rounded amounts also sum to $111.36 million. A common mistake avoided here is allocating in proportion to the standalone amounts; that would ignore the different covariance contributions.

Worked Example: Three Risks With an Offset From Negative Correlation

A hypothetical portfolio has three risks, each with standalone capital of $30 million. The correlation between risks 1 and 2 is $0.5$; between risks 1 and 3 it is $-0.5$; and between risks 2 and 3 it is $0$. Calculate aggregate capital and all three Euler allocations. Report amounts in millions of dollars to two decimal places, rounding only at the final step.

Method. Include all three pairwise covariance terms in aggregate capital, then calculate each risk’s covariance-weighted numerator. First check that the assumed correlations are collectively possible. The determinant of the correlation matrix is $1-0.5^2-(-0.5)^2=0.5$, which is nonnegative; its principal $2$-by-$2$ minors are also nonnegative. The matrix is positive semidefinite.

Step 1: Calculate aggregate capital. The diagonal terms sum to $3(30^2)=2{,}700$. The pairwise covariance contributions are $2(0.5)(30)(30)=900$, $2(-0.5)(30)(30)=-900$, and $2(0)(30)(30)=0$. Hence,

C=\sqrt{2{,}700+900-900+0} =\sqrt{2{,}700} =51.961524\text{ million}.

Step 2: Allocate to risk 1. Its covariance-weighted numerator is $30(30+0.5(30)-0.5(30))=900$, so $A_1=900/51.961524=17.320508\ldots$, or $17.32 million.

Step 3: Allocate to risk 2. Its numerator is $30(30+0.5(30)+0(30))=1{,}350$, so $A_2=1{,}350/51.961524=25.980762\ldots$, or $25.98 million.

Step 4: Allocate to risk 3. Its numerator is $30(30-0.5(30)+0(30))=450$, so $A_3=450/51.961524=8.660254\ldots$, or $8.66 million.

Check. The unrounded numerators total $900+1{,}350+450=2{,}700$, so the unrounded allocations total $\sqrt{2{,}700}=C$. The negative correlation reduces aggregate variance through its negative pairwise term. It does not mean that every allocation must be negative.

Worked Example: A Three-Risk Allocation With Unequal Standalone Capital

A hypothetical insurer has standalone capital amounts of $30 million, $40 million, and $50 million. The pairwise correlations are $\rho_{12}=0$, $\rho_{13}=0.2$, and $\rho_{23}=0.4$. Calculate aggregate capital and the allocation to each risk, in millions of dollars to two decimal places. Round only at the final step.

Method. Sum the three squared standalone amounts and twice each distinct pairwise covariance contribution. Then use the allocation formula with all relevant correlations for each risk.

Step 1: Calculate aggregate capital. The squared amounts sum to $30^2+40^2+50^2=900+1{,}600+2{,}500=5{,}000$. The distinct pairwise covariance products are $0(30)(40)=0$, $0.2(30)(50)=300$, and $0.4(40)(50)=800$. The formula doubles their sum:

C=\sqrt{5{,}000+2(0+300+800)} =\sqrt{7{,}200} =84.852814\text{ million}.

Step 2: Calculate each allocation. For risk 1, the covariance-weighted capital inside the parentheses is $30+0(40)+0.2(50)=40$, so its numerator is $30(40)=1{,}200$. For risk 2, that amount is $40+0(30)+0.4(50)=60$, giving a numerator of $40(60)=2{,}400$. For risk 3, it is $50+0.2(30)+0.4(40)=72$, giving a numerator of $50(72)=3{,}600$. Dividing each numerator by aggregate capital gives

A_1=\frac{1{,}200}{84.852814}=14.142136\ldots\approx \$14.14\text{ million},

A_2=\frac{2{,}400}{84.852814}=28.284271\ldots\approx \$28.28\text{ million},

A_3=\frac{3{,}600}{84.852814}=42.426407\ldots\approx \$42.43\text{ million}.

Check. The allocation numerators sum to $1{,}200+2{,}400+3{,}600=7{,}200=C^2$. Dividing by $C$ confirms that the unrounded allocations sum to $C$. The rounded amounts sum to $84.85 million, matching aggregate capital rounded to the same precision.

Interpreting and Checking the Allocation

The Euler allocation assigns each risk its standalone squared-capital term plus its share of covariance with the other risks, all divided by aggregate capital. For example, in a two-risk portfolio,

A_1=\frac{C_1^2+\rho_{12}C_1C_2}{C}, \qquad A_2=\frac{C_2^2+\rho_{12}C_1C_2}{C}.

Each risk receives half of the portfolio’s total covariance contribution in this two-risk case when the allocations are added together: each allocation includes one copy of $\rho_{12}C_1C_2$, and together they include the doubled term in $C^2$. With more risks, the same accounting holds pair by pair.

A frequent exam trap is to use proportional allocation, $C_iC/\sum_j C_j$, when the question asks for covariance-based Euler allocation. Proportional allocation reconciles by construction, but generally ignores the correlation structure. For the first worked example, it would allocate $111.355287(60/140)=47.72$ million to risk 1 and $111.355287(80/140)=63.63$ million to risk 2. These add to aggregate capital, but they are not the Euler allocations of $43.11 million and $68.25 million. Reconciliation alone does not prove that the intended allocation method was used.

Before finalizing a calculation, check the model conditions and the arithmetic:

Confirm that the standalone capital amounts use the same volatility scale and that all amounts use consistent units.
Confirm that the correlation matrix is valid, especially for portfolios with three or more risks.
Include every pairwise covariance term in aggregate variance, with a factor of two when summing only distinct pairs.
Use the risk-specific covariance-weighted numerator for each Euler allocation.
Verify the unrounded allocations sum to aggregate capital. Rounding can create a small reported discrepancy; if it does, retain precision during calculations and apply the organization’s stated rounding reconciliation policy.

Covariance-based Euler allocation is useful when the capital model uses a common volatility basis and the goal is to attribute aggregate capital consistently with modeled dependence. It is not a general substitute for an allocation method based on a different risk measure or a different set of modeling assumptions.

Check Your Understanding

For each problem, calculate using covariance aggregation and Euler allocation where requested. Treat all capital amounts as being on a common volatility scale. Report money in millions of dollars to the nearest cent.

A two-risk portfolio has standalone capital of $20 million and $25 million, with correlation $0.2$. Find aggregate capital and both Euler allocations. (Answer: Aggregate capital $35.00 million; allocations $14.29 million and $20.71 million.)
Two risks have standalone capital of $30 million and $40 million and correlation $0.5$. Find aggregate capital and the Euler allocation to the $30 million risk. (Answer: Aggregate capital $60.83 million; allocation to the $30 million risk $24.66 million.)
Three risks have standalone capital of $25 million, $35 million, and $40 million. Their correlations are $\rho_{12}=0.2$, $\rho_{13}=0$, and $\rho_{23}=0.3$. Calculate aggregate capital. (Answer: $68.12 million.)
Two risks have standalone capital of $30 million and $50 million and correlation $-0.2$. Calculate aggregate capital and both Euler allocations. (Answer: Aggregate capital $52.92 million; allocations approximately $11.34 million and $41.58 million.)
Three risks have standalone capital of $20 million, $30 million, and $40 million, with $\rho_{12}=0.5$, $\rho_{13}=0$, and $\rho_{23}=0.25$. Calculate aggregate capital and the three Euler allocations. (Answer: Aggregate capital $64.03 million; allocations approximately $10.93 million, $23.43 million, and $29.67 million.)

What You'll Learn