Euler Allocation Under Covariance Aggregation

Why Allocate Covariance-Based Capital?

A portfolio’s standard-deviation capital reflects both the scale of each component and the way components move together. An allocation method should therefore recognize more than each component’s standalone risk: it should also reflect each component’s covariance with the whole portfolio. Euler allocation does this by multiplying a component’s exposure by its marginal effect on portfolio capital.

This tutorial considers capital defined by a covariance aggregation formula. It derives the component allocations, explains the assumptions behind them, and shows how to check that they sum to total capital. The treatment is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let \(X_1,\ldots,X_n\) be component losses and let \(a=(a_1,\ldots,a_n)^\mathsf{T}\) be their exposures. The portfolio loss is \(X(a)=\sum_{i=1}^n a_iX_i\). Write \(\Sigma\) for the covariance matrix with entry \(\Sigma_{ij}=\operatorname{Cov}(X_i,X_j)\). Covariance-aggregated standard-deviation capital is \(K(a)=\sqrt{\operatorname{Var}(X(a))}=\sqrt{a^\mathsf{T}\Sigma a}\).

The covariance matrix is symmetric and positive semidefinite. For the allocation formulas below, assume \(\Sigma\) is fixed as exposures vary and \(K(a)>0\). A positive semidefinite matrix ensures that \(a^\mathsf{T}\Sigma a\) is nonnegative; the strict inequality ensures that the square-root capital function is differentiable at the portfolio being evaluated. The exposures are typically nonnegative business volumes, although the formulas themselves do not require that restriction.

This is a standard-deviation capital measure, not a claim that standard deviation alone describes every feature of a loss distribution. Covariance aggregation requires finite second moments. The formulas below do not require normally distributed losses; when an example specifies a multivariate normal model, that assumption gives a complete distributional description but is not needed to calculate standard-deviation capital.

From Covariances to Euler Allocations

For a fixed covariance matrix, expanding the portfolio variance gives the covariance aggregation formula. The diagonal terms are the component variances; the off-diagonal terms account for pairwise covariance.

K(a)^2=a^\mathsf{T}\Sigma a =\sum_{i=1}^n\sum_{j=1}^n a_i a_j\operatorname{Cov}(X_i,X_j).

The marginal capital for component \(i\) is the partial derivative of \(K\) with respect to its exposure \(a_i\). Euler allocation assigns that component its exposure multiplied by this marginal capital. For a differentiable, positively homogeneous capital function, the earlier material on Euler allocation establishes that these contributions sum to total capital. Here, both the derivative and the reconciliation can be shown directly.

Euler allocation under covariance aggregation: If \(\Sigma\) is symmetric and fixed and \(K(a)=\sqrt{a^\mathsf{T}\Sigma a}>0\), then the marginal capital of component \(i\) is \(\partial K/\partial a_i=(\Sigma a)_i/K(a)\). Its Euler allocation is \(A_i=a_i(\Sigma a)_i/K(a)\), and \(\sum_{i=1}^n A_i=K(a)\).

To see why the derivative has this form, differentiate the quadratic form. Symmetry makes the derivative of its two appearances of \(a_i\) combine into twice the \(i\)th entry of \(\Sigma a\). Applying the chain rule to the square root cancels the factor of two. The allocation identity then follows by summing the component contributions.

\frac{\partial}{\partial a_i}(a^\mathsf{T}\Sigma a) =2(\Sigma a)_i, \qquad \frac{\partial K}{\partial a_i} =\frac{1}{2\sqrt{a^\mathsf{T}\Sigma a}}\;2(\Sigma a)_i =\frac{(\Sigma a)_i}{K(a)}.

\sum_{i=1}^n A_i =\sum_{i=1}^n a_i\frac{(\Sigma a)_i}{K(a)} =\frac{a^\mathsf{T}\Sigma a}{K(a)} =\frac{K(a)^2}{K(a)} =K(a).

The quantity \((\Sigma a)_i\) is the covariance of component loss \(X_i\) with portfolio loss \(X(a)\): \(\operatorname{Cov}(X_i,X(a))=\sum_j a_j\operatorname{Cov}(X_i,X_j)=(\Sigma a)_i\). Thus the marginal capital is that covariance divided by portfolio standard deviation. This interpretation makes clear why the allocation depends on both component exposure and portfolio interaction.

Worked Calculations

Worked Example: Two Components and a Matrix-Product Check

A hypothetical portfolio has two zero-mean component losses with a multivariate normal distribution and covariance matrix, in squared thousands of dollars, \(\Sigma=\begin{pmatrix}4&1\\1&6\end{pmatrix}\). Its exposures are \(a=(1,3)^\mathsf{T}\). Calculate covariance-aggregated standard-deviation capital and the Euler allocations. There is no interest discounting or mortality assumption in this loss model.

Method. First calculate \(\Sigma a\), then the portfolio variance \(a^\mathsf{T}\Sigma a\), and finally \(a_i(\Sigma a)_i/K(a)\). The covariance matrix and exposure vector are the model parameters.

Step 1: Multiply the matrix by the exposure vector. \(\Sigma a=(4(1)+1(3),\,1(1)+6(3))^\mathsf{T}=(7,19)^\mathsf{T}\).

Step 2: Calculate variance and capital. \(a^\mathsf{T}\Sigma a=1(7)+3(19)=7+57=64\), so \(K(a)=\sqrt{64}=8\) thousand dollars.

Step 3: Calculate allocations. \(A_1=1(7)/8=0.875\) thousand dollars and \(A_2=3(19)/8=57/8=7.125\) thousand dollars.

Check and common mistake avoided. The allocations sum to \(0.875+7.125=8\) thousand dollars, the portfolio capital. In the first entry of \(\Sigma a\), the calculation is \(4(1)+1(3)=7\), not 12; substituting an incorrect matrix product would also corrupt the variance and allocation.

Worked Example: Building the Covariance Matrix from Correlations

A hypothetical portfolio has three zero-mean component losses with a multivariate normal distribution. Their standalone standard deviations are \(2\), \(1\), and \(3\) thousand dollars. The pairwise correlations are \(\rho_{12}=0.5\), \(\rho_{13}=0\), and \(\rho_{23}=1/3\). Exposures are \(a=(1,2,1)^\mathsf{T}\). Find the standard-deviation capital and allocations. No interest rate or mortality assumption applies.

Method. Convert correlations to covariances using \(\Sigma_{ij}=\rho_{ij}\sigma_i\sigma_j\), where \(\sigma_i\) is component \(i\)'s standalone standard deviation. The diagonal entries are variances.

Step 1: Construct the covariance matrix. The variances are \(2^2=4\), \(1^2=1\), and \(3^2=9\). The off-diagonal covariances are \(0.5(2)(1)=1\), \(0(2)(3)=0\), and \((1/3)(1)(3)=1\). Therefore, \(\Sigma=\begin{pmatrix}4&1&0\\1&1&1\\0&1&9\end{pmatrix}\), in squared thousands of dollars.

Step 2: Multiply by exposures. \(\Sigma a=(4(1)+1(2)+0(1),\,1(1)+1(2)+1(1),\,0(1)+1(2)+9(1))^\mathsf{T}=(6,4,11)^\mathsf{T}\).

Step 3: Calculate capital. \(a^\mathsf{T}\Sigma a=1(6)+2(4)+1(11)=6+8+11=25\), so \(K(a)=5\) thousand dollars.

Step 4: Allocate and check. \(A_1=1(6)/5=1.2\), \(A_2=2(4)/5=1.6\), and \(A_3=1(11)/5=2.2\), all in thousands of dollars. Their sum is \(1.2+1.6+2.2=5.0\) thousand dollars, matching portfolio capital.

Common mistake avoided. Correlation is not covariance. Using \(0.5\) itself as the covariance between components 1 and 2 would be wrong; the covariance is \(0.5(2)(1)=1\).

Worked Example: Applying a Capital Multiplier

A hypothetical two-component zero-mean loss vector is multivariate normal, with component standard deviations \(3\) and \(4\) thousand dollars and correlation \(0.5\). The exposures are \(a=(2,1)^\mathsf{T}\). Management defines capital as twice the portfolio standard deviation. Calculate total capital and Euler allocations. There is no interest discounting or mortality assumption.

Method. First apply the standard-deviation allocation formula. Because the capital measure is \(C(a)=2K(a)\), its marginal capital and Euler allocations are twice those for \(K\).

Step 1: Form the covariance matrix. The variances are \(3^2=9\) and \(4^2=16\), and the covariance is \(0.5(3)(4)=6\). Thus \(\Sigma=\begin{pmatrix}9&6\\6&16\end{pmatrix}\), in squared thousands of dollars.

Step 2: Calculate the portfolio covariance vector and variance. \(\Sigma a=(9(2)+6(1),\,6(2)+16(1))^\mathsf{T}=(24,28)^\mathsf{T}\). Hence \(a^\mathsf{T}\Sigma a=2(24)+1(28)=76\), and \(K(a)=\sqrt{76}\approx8.7178\) thousand dollars.

Step 3: Apply the multiplier to capital and allocations. Total capital is \(C(a)=2\sqrt{76}\approx17.4356\) thousand dollars. The allocations are \(A_1=2\{2(24)/\sqrt{76}\}=96/\sqrt{76}\approx11.0120\) thousand dollars and \(A_2=2\{1(28)/\sqrt{76}\}=56/\sqrt{76}\approx6.4236\) thousand dollars. Values are rounded only at the final step, to four decimal places in thousands of dollars.

Check and common mistake avoided. The rounded allocations sum to \(11.0120+6.4236=17.4356\) thousand dollars, matching rounded total capital. Applying the multiplier to total capital but not to the component allocations would fail this reconciliation.

Interpretation, Conditions, and Exam Traps

Each allocation is the component’s exposure times its marginal contribution to portfolio standard deviation. In particular, the numerator is \(a_i(\Sigma a)_i\), not simply the component variance \(a_i^2\Sigma_{ii}\). A component can have a substantial standalone variance but a smaller allocated amount if its covariance with the rest of the portfolio is limited. Conversely, positive covariance with other components contributes to its portfolio allocation.

The full-allocation identity depends on using the same covariance matrix and exposure vector consistently in the capital and component calculations. A risk multiplier \(c>0\) scales total capital and every Euler allocation by \(c\). More generally, if capital is defined as \(C(a)=cK(a)\) for fixed \(c\), then \(A_i^C=cA_i\). This simple scaling is valid because \(c\) is constant with respect to exposures.

The formula is not defined when \(K(a)=0\), since its derivative has a zero denominator. If the covariance matrix changes with exposures, the derivative must also account for that dependence; the fixed-\(\Sigma\) formula cannot be used unchanged. And although positive semidefiniteness guarantees nonnegative portfolio variance, it does not by itself guarantee that every Euler allocation is positive. The allocation’s sign is determined by \(a_i(\Sigma a)_i\).

Confusing a covariance with a correlation. Convert using \(\operatorname{Cov}(X_i,X_j)=\rho_{ij}\sigma_i\sigma_j\).
Using a standalone variance as the numerator. The correct numerator is \(a_i(\Sigma a)_i\), which includes portfolio interactions.
Omitting an exposure. The marginal capital is \((\Sigma a)_i/K(a)\); the Euler allocation additionally multiplies it by \(a_i\).
Scaling only the total. If the stated capital measure includes a fixed multiplier, apply it to each component allocation too.
Rounding too early. Keep unrounded matrix products and capital through the allocation calculations, then round the final reported values.

Key takeaway: Under fixed-covariance standard-deviation aggregation, calculate \(\Sigma a\), divide each entry by portfolio capital to obtain marginal capital, and multiply by the corresponding exposure to obtain the Euler allocation. The allocations reconcile because their sum is \(a^\mathsf{T}\Sigma a/K(a)=K(a)\).

Check Your Understanding

For each problem, use the covariance-based Euler formula and check the allocations against the stated capital measure.

A hypothetical zero-mean two-component loss vector has a multivariate normal distribution with covariance matrix \(\begin{pmatrix}4&1\\1&9\end{pmatrix}\) and exposures \((1,2)^\mathsf{T}\). Find standard-deviation capital and both allocations, in the same units as the component losses. (Answer: \(K=\sqrt{44}\); allocations \(6/\sqrt{44}\) and \(38/\sqrt{44}\).)
Two hypothetical independent components have standard deviations \(2\) and \(3\) thousand dollars and exposures \((3,1)^\mathsf{T}\). Find standard-deviation capital and allocations. (Answer: \(K=\sqrt{45}=3\sqrt{5}\) thousand dollars; allocations \(36/\sqrt{45}\) and \(9/\sqrt{45}\) thousand dollars.)
A hypothetical two-component loss vector has covariance matrix \(\begin{pmatrix}4&-2\\-2&25\end{pmatrix}\) and exposures \((1,1)^\mathsf{T}\). Find standard-deviation capital and allocations. (Answer: \(K=5\); allocations \(2/5\) and \(23/5\), in the loss units.)
A hypothetical three-component portfolio has covariance matrix \(\begin{pmatrix}1&0.5&0\\0.5&4&1\\0&1&9\end{pmatrix}\) and exposures \((2,1,1)^\mathsf{T}\). Calculate standard-deviation capital and the three Euler allocations. (Answer: \(K=\sqrt{21}\); allocations \(5/\sqrt{21}\), \(6/\sqrt{21}\), and \(10/\sqrt{21}\), in the loss units.)
A hypothetical portfolio has covariance matrix \(\begin{pmatrix}9&3\\3&4\end{pmatrix}\), exposures \((1,2)^\mathsf{T}\), and capital equal to \(1.5\) times standard deviation. Find total capital and both allocations. (Answer: Total capital \(1.5\sqrt{37}\); allocations \(22.5/\sqrt{37}\) and \(33/\sqrt{37}\), in the loss units.)

What You'll Learn