Sensitivity and Stability of Capital Allocations

Why Allocation Sensitivity Matters

A capital allocation is a model-based attribution, not an independently observed quantity. It can change when the portfolio’s dependence assumptions, loss distribution, or aggregate capital target changes. A reported allocation may therefore be mathematically correct under one set of assumptions yet too fragile to support a decision without qualification.

Sensitivity analysis asks how much an allocation changes when an input changes. Stability analysis asks whether those changes are reasonable relative to the size of the input change, the allocation itself, and the uncertainty in the model. In particular, a small change in correlation or a fitted tail parameter should not be mistaken for a definite change in a business unit’s underlying risk.

Definition: Let $c_i(\theta)$ be the capital allocated to component $i$ under model input $\theta$. If the allocation is differentiable, its local sensitivity is $\partial c_i/\partial\theta$. For a finite change from $\theta_0$ to $\theta_1$, the allocation change is $c_i(\theta_1)-c_i(\theta_0)$. An attribution is unstable with respect to an input when a small, plausible change in that input causes a large or decision-relevant change in one or more component allocations.

Sensitivity is not a property of an allocation in isolation. It depends on the input being varied and the scale used to judge the response. A derivative describes a local response; a finite-difference comparison shows the response over a specified range. Both should be interpreted alongside the corresponding change in aggregate capital and the model’s limitations.

Correlation Sensitivity Under Covariance Aggregation

Consider two loss components whose standard deviations are $\sigma_1$ and $\sigma_2$, with correlation $\rho$. Under covariance aggregation, the standard deviation of their aggregate loss is

K(\rho)=\sqrt{\sigma_1^2+\sigma_2^2+2\rho\sigma_1\sigma_2}.

For a differentiable, positively homogeneous capital measure, Euler allocation assigns component $i$ its amount $c_i=w_i\,\partial K/\partial w_i$, where $w_i$ is its exposure scale. For the two-component covariance model, the allocations are

c_1(\rho)=\frac{\sigma_1^2+\rho\sigma_1\sigma_2}{K(\rho)},\qquad c_2(\rho)=\frac{\sigma_2^2+\rho\sigma_1\sigma_2}{K(\rho)}.

These formulas assume the covariance aggregation model applies and $K(\rho)>0$. They allocate the aggregate standard deviation, not a tail measure under an arbitrary distribution. Their sum is $K(\rho)$, since the two numerators sum to $K(\rho)^2$.

Derivation: Write $V=K^2=\sigma_1^2+\sigma_2^2+2\rho\sigma_1\sigma_2$. Since $c_1=(\sigma_1^2+\rho\sigma_1\sigma_2)V^{-1/2}$, differentiating with respect to $\rho$ gives $$ \frac{\partial c_1}{\partial\rho} =\frac{\sigma_1\sigma_2}{K} -\frac{(\sigma_1^2+\rho\sigma_1\sigma_2)\sigma_1\sigma_2}{K^3} =\frac{\sigma_1\sigma_2(\sigma_2^2+\rho\sigma_1\sigma_2)}{K^3}. $$ The first term is the direct increase in the numerator; the second is the effect of the changing aggregate denominator. Thus, differentiating only the numerator gives an incorrect sensitivity.

Worked Example: Measuring the Effect of a Correlation Change

A hypothetical portfolio has component standard deviations $\sigma_1=3$ and $\sigma_2=4$, in the same monetary units. Using covariance aggregation, calculate both Euler allocations at $\rho=0.25$ and $\rho=0.50$, and assess the allocation response. Also calculate the local sensitivity of $c_1$ at $\rho=0.25$.

Method. Substitute the standard deviations and each correlation into the aggregate variance and Euler formulas. This compares a finite change with the derivative at the starting point.

At $\rho=0.25$, the aggregate variance is $9+16+2(0.25)(3)(4)=31$, so $K=\sqrt{31}\approx5.5678$. The allocations are

c_1=\frac{9+3}{\sqrt{31}}=\frac{12}{\sqrt{31}}\approx2.1553,\qquad c_2=\frac{16+3}{\sqrt{31}}=\frac{19}{\sqrt{31}}\approx3.4125.

Their sum is $31/\sqrt{31}=\sqrt{31}\approx5.5678$, as required. At $\rho=0.50$, the variance is $9+16+12=37$, so $K=\sqrt{37}\approx6.0828$. The allocations are

c_1=\frac{15}{\sqrt{37}}\approx2.4660,\qquad c_2=\frac{22}{\sqrt{37}}\approx3.6168.

Their sum is $37/\sqrt{37}=\sqrt{37}\approx6.0828$. Over the correlation increase of $0.25$, the component changes are approximately $2.4660-2.1553=0.3107$ and $3.6168-3.4125=0.2043$. Their sum, $0.5150$, matches the aggregate increase $6.0828-5.5678=0.5150$, to the displayed precision.

For the local sensitivity, the derivative formula gives $\partial c_1/\partial\rho=12(16+3)/(31\sqrt{31})=228/(31\sqrt{31})\approx1.3210$ at $\rho=0.25$. This is an allocation change of about 1.3210 monetary units per unit increase in correlation, locally; it is not the change for every finite correlation increase.

Common mistake and correction. Differentiating only the numerator would give $12/\sqrt{31}\approx2.1553$ as the sensitivity. That omits the change in the denominator $K$. The full derivative is $12/\sqrt{31}-144/(31\sqrt{31})=228/(31\sqrt{31})\approx1.3210$. The allocation sum check also confirms that the two component changes must add to the aggregate change.

Distributional Assumptions and Tail Allocations

The same covariance information need not imply the same tail capital. A useful comparison is available for centered elliptical loss models, which include a multivariate normal model and a multivariate $t$ model. If $S$ is the aggregate loss, its $p$-level VaR in such a model can be written as $q_p$ times its standard deviation, where $q_p$ is the model’s standardized $p$-quantile. For a differentiable model, the Euler VaR allocation to component $i$ is then

c_i=q_p\frac{\operatorname{Cov}(X_i,S)}{\sqrt{\operatorname{Var}(S)}}.

This result depends on the centered elliptical assumption and a consistent covariance and quantile specification. It does not say that every loss distribution with the same covariance has the same VaR or allocation. Outside such a model, tail shape, skewness, or dependence in the extremes may change component contributions in ways that covariance alone cannot capture.

Worked Example: Comparing Normal and Heavy-Tailed VaR Allocations

Suppose two hypothetical, centered loss components have standard deviations 3 and 4 and correlation 0.50. Compare their 99% Euler VaR allocations under (a) a bivariate normal model with standardized quantile $q_{0.99}=2.3263$, and (b) a covariance-matched bivariate $t$ model with 5 degrees of freedom and standardized quantile $q_{0.99}\approx2.6065$. Use the covariance allocations from the preceding calculation as the unit-quantile contributions.

Method. The covariance contributions per unit of standardized quantile are $15/\sqrt{37}\approx2.4660$ and $22/\sqrt{37}\approx3.6168$. Multiply each by the quantile multiplier for the stated model.

For the normal model, the component allocations are

c_1=2.3263\frac{15}{\sqrt{37}}\approx5.7366,\qquad c_2=2.3263\frac{22}{\sqrt{37}}\approx8.4137.

The aggregate VaR is $2.3263\sqrt{37}\approx14.1503$, and the allocations sum to approximately $5.7366+8.4137=14.1503$.

For the standardized $t$ model, use the supplied quantile multiplier:

c_1=2.6065\frac{15}{\sqrt{37}}\approx6.4276,\qquad c_2=2.6065\frac{22}{\sqrt{37}}\approx9.4271.

The aggregate VaR is $2.6065\sqrt{37}\approx15.8547$, matching the sum $6.4276+9.4271=15.8547$. Relative to the normal model, the allocations rise by about $0.6910$ and $1.0134$, respectively. Because both models are elliptical and have the same covariance structure, the component proportions are unchanged here; the heavier-tailed model raises the overall quantile scale. A different, non-elliptical model could also alter the relative attribution.

Common mistake and correction. Reusing the normal quantile while changing to the $t$ model would leave the allocations unchanged and conceal the stated distributional change. Use the quantile associated with the model and standardization being assumed. The $t$ multiplier here is supplied as a rounded input; calculations are rounded to four decimal places.

Sensitivity to an Aggregate Capital Target

Sometimes an allocation must be adjusted to an externally specified aggregate target, for example when a model output is calibrated to a management or regulatory target. The adjustment rule must be stated: there is no unique way to distribute the difference between a baseline total and a new target.

One transparent rule starts with baseline component amounts $a_i$, whose sum is $K_0$, and assigns a fraction $\lambda_i$ of the target difference to component $i$, where $\sum_i\lambda_i=1$. Then

c_i(K^*)=a_i+\lambda_i(K^*-K_0),\qquad \frac{\partial c_i}{\partial K^*}=\lambda_i.

This rule fully allocates the revised target because summing the component formula gives $K_0+(K^*-K_0)=K^*$. It also makes each component’s target sensitivity explicit. The rule is an allocation convention, not a universal property of capital models.

Worked Example: Reallocating a Revised Capital Target

A hypothetical model produces baseline allocations $a_1=6$ and $a_2=4$, for $K_0=10$. Under a stated target-adjustment policy, components 1 and 2 receive 70% and 30% of any target difference. Calculate the allocations at targets $K^*=10.5$ and $K^*=11.5$, and determine the sensitivity to the target.

Method. Apply the stated adjustment rule with $\lambda_1=0.70$ and $\lambda_2=0.30$. The target differences are $10.5-10=0.5$ and $11.5-10=1.5$.

At $K^*=10.5$, the allocations are $c_1=6+0.70(0.5)=6.35$ and $c_2=4+0.30(0.5)=4.15$. They sum to $6.35+4.15=10.50$.

At $K^*=11.5$, the allocations are $c_1=6+0.70(1.5)=7.05$ and $c_2=4+0.30(1.5)=4.45$. They sum to $7.05+4.45=11.50$. From the derivative formula, a one-unit increase in the target increases the allocations by 0.70 and 0.30 units under this policy.

Common mistake and correction. Adding the entire target increase to both baseline allocations would double-count the adjustment. The component adjustments must sum to the target change: $0.70\Delta K+0.30\Delta K=\Delta K$.

Diagnosing Unstable Attributions

A practical review should distinguish three sources of movement: a genuine change in aggregate risk, a change in the model’s component risk relationships, and a change introduced by the allocation rule. A finite-difference table across plausible assumptions often makes this distinction visible. It is useful to report the aggregate capital, each component allocation, and each component’s share or change together rather than presenting isolated component numbers.

Signed contributions require particular care. If a rule rescales raw component scores $g_i$ to a target using $c_i=K^*g_i/\sum_jg_j$, then the sensitivity of the allocation to its raw score is

\frac{\partial c_i}{\partial g_k} =K^*\frac{\mathbf{1}_{i=k}\sum_jg_j-g_i}{(\sum_jg_j)^2}.

Here $\mathbf{1}_{i=k}$ equals 1 when $i=k$ and 0 otherwise. If the signed scores nearly cancel, the denominator is small and modest score changes can produce very large allocation changes. The method can still be evaluated algebraically, but the resulting shares should not be described as stable evidence of relative risk.

For a finite-difference assessment, choose plausible input values, recompute the aggregate and all allocations under the same rule, and verify that the allocations reconcile to the aggregate at every setting. A relative change such as $\lvert\Delta c_i\rvert/\lvert c_i\rvert$ can flag large movements, but it is uninformative when $c_i$ is near zero; in that case report an absolute change and the relevant monetary scale. Derivatives are useful locally, but they can mislead over a large change, especially near a kink, a boundary, or a change in which scenarios determine the tail.

State the perturbation. Identify whether correlation, a distribution parameter, an exposure, or the target changed, and by how much.
Recompute consistently. Use the same allocation definition and conventions at each setting; do not compare allocations produced by different rules as though only the model input changed.
Check full allocation. Confirm that component allocations sum to the aggregate capital at every setting, allowing only for stated rounding.
Separate model uncertainty from precision. A finely rounded result does not imply that the underlying assumptions are known with comparable precision.

Key takeaway: A useful allocation sensitivity analysis reports both what moved and why. Check the aggregate, the component responses, the allocation rule, and the plausibility of the perturbed assumptions before interpreting a change as a change in business risk. This treatment is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each problem, use the stated model and allocation rule, and check that component allocations reconcile to aggregate capital where applicable.

A two-component covariance model has standard deviations 2 and 5 and correlation 0.20. Calculate aggregate capital and both Euler allocations. (Answer: $K=\sqrt{33}\approx5.7446$; $c_1=6/\sqrt{33}\approx1.0445$, $c_2=27/\sqrt{33}\approx4.7001$; they sum to $K$.)
For standard deviations 3 and 4 under covariance aggregation, show that at correlation 0.25 the local sensitivity of $c_1$ is $228/(31\sqrt{31})$. (Answer: $\partial c_1/\partial\rho=228/(31\sqrt{31})\approx1.3210$.)
A centered elliptical model has aggregate standard deviation 8 and standardized 99% quantile 2.5. The covariance shares are 40% and 60%. Find the two Euler VaR allocations and verify the total. (Answer: $8(2.5)=20$; allocations are 8 and 12, summing to 20.)
Baseline allocations are 9 and 6, with baseline total 15. A stated target-adjustment rule assigns 20% of the target difference to component 1 and 80% to component 2. Find the allocations at target 16.25 and the sensitivity of each allocation to the target. (Answer: $c_1=9.25$, $c_2=7.00$; sensitivities are 0.20 and 0.80, and the allocations sum to 16.25.)
A signed-score normalization uses raw scores $g_1=5$ and $g_2=-4$ with target $K^*=2$. Calculate allocations, then recalculate if $g_2$ changes to $-3.9$. What indicates instability? (Answer: Initially the score total is 1, giving allocations 10 and $-8$. After the change the total is 1.1, giving approximately 9.0909 and $-7.0909$. A 0.1 score change produces a substantial shift because the signed total used for normalization is small.)

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