Capital Allocation Across Products

Why Allocate Capital Across Products?

A company may calculate one capital requirement for its whole portfolio, yet need product-level figures for planning, performance analysis, or internal reporting. Assigning that company-wide amount to products is a separate step from calculating the total: the allocation rule determines how the total is divided, not how much total capital the company requires.

Different rules answer different management questions. A proportional rule is straightforward when product risk measures are available. An Euler rule assigns each product its marginal contribution to a specified aggregate capital function. A company may also impose allocation floors and distribute the remaining capital by a stated weight. These are conventions for assigning a total; the resulting product amounts should not automatically be interpreted as independently required or legally ring-fenced capital.

Definition: Let $C$ be a stated company-wide capital requirement and let $c_j$ be the amount assigned to product $j$. A complete allocation reconciles when $\sum_j c_j=C$. The allocation method specifies how the $c_j$ are determined. Product standalone capital measures, where used, are measures of individual product risk and need not sum to $C$.

All amounts in the examples are hypothetical. Unless stated otherwise, amounts are in millions of dollars. In every method, first identify the company-wide total, the product-level inputs, and the exact allocation rule. Then calculate the assigned amounts and verify their sum.

Proportional Allocation

A proportional rule assigns capital in proportion to specified nonnegative product risk measures. If $r_j$ is product $j$'s measure and $\sum_j r_j>0$, the allocation is:

c_j=C\frac{r_j}{\sum_k r_k}.

The measures $r_j$ might be standalone capital amounts or another explicitly specified risk basis. The denominator is the sum of the measures across the products being allocated. This rule always reconciles when the stated conditions hold, because the product shares sum to one. It does not, by itself, account for diversification between products.

Worked Example: Proportional Allocation From Standalone Measures

A hypothetical insurer has a company-wide capital requirement of $30 million. Its three product groups have specified standalone capital measures of $12 million, $8 million, and $5 million. Allocate the company-wide amount in proportion to these measures.

Method. Use proportional allocation. The key is to divide by the sum of the standalone measures, not by the company-wide requirement.

Step 1: Find the total measure.

12+8+5=25 \text{ million}.

Step 2: Calculate each allocation. For the first group, the share is $12/25=0.48$, so its allocation is $30(0.48)=14.4$ million. For the second group, the share is $8/25=0.32$, so its allocation is $30(0.32)=9.6$ million. For the third group, the share is $5/25=0.20$, so its allocation is $30(0.20)=6.0$ million.

(c_1,c_2,c_3)=(14.4,9.6,6.0) \text{ million}.

Step 3: Reconcile. The assignments sum to $14.4+9.6+6.0=30.0$ million, equal to the company-wide requirement. The standalone measures sum to $25 million, not $30 million; they determine the proportions, while the stated company-wide total is the amount being allocated.

Common mistake avoided: Assigning the standalone measures themselves would allocate only $25 million. The rule scales those measures to the specified $30 million total.

Euler Allocation and Marginal Contributions

When the company-wide capital is calculated from a differentiable aggregate function of product risk measures, an Euler allocation can assign each product its marginal contribution at the current portfolio. This allocation is especially useful when the aggregate function is positively homogeneous of degree one: scaling every product input by the same positive factor scales aggregate capital by that factor.

Definition: Suppose $C(x_1,\ldots,x_n)$ is differentiable and positively homogeneous of degree one, so $C(tx_1,\ldots,tx_n)=tC(x_1,\ldots,x_n)$ for $t>0$. The Euler allocation to product $j$ is $c_j=x_j\frac{\partial C}{\partial x_j}$, with the derivative evaluated at the portfolio inputs.

The reconciliation property follows from Euler's theorem for homogeneous functions. To see why, differentiate $C(tx_1,\ldots,tx_n)=tC(x_1,\ldots,x_n)$ with respect to $t$. By the chain rule, the derivative of the left side is $\sum_j x_j\frac{\partial C(tx_1,\ldots,tx_n)}{\partial (tx_j)}$, and the derivative of the right side is $C(x_1,\ldots,x_n)$. Set $t=1$ to obtain:

\sum_j x_j\frac{\partial C}{\partial x_j}=C.

Thus the Euler allocations sum to aggregate capital, provided the function is differentiable and positively homogeneous as specified. If the function is not differentiable at the portfolio point, or does not satisfy this homogeneity condition, this result cannot simply be assumed; the chosen allocation method and reconciliation must be checked separately.

Worked Example: Euler Allocation Under Independent Quadratic Aggregation

A hypothetical two-product portfolio has risk inputs $x_1=6$ million and $x_2=8$ million. Its stated aggregate capital function is $C(x_1,x_2)=\sqrt{x_1^2+x_2^2}$, with no covariance term. Calculate the aggregate requirement and allocate it by the Euler rule.

Method. Differentiate the stated aggregate function, then multiply each partial derivative by that product's input. The function is differentiable at these positive inputs and homogeneous of degree one.

Step 1: Calculate company-wide capital.

C=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10 \text{ million}.

Step 2: Find the partial derivatives.

\frac{\partial C}{\partial x_1}=\frac{x_1}{\sqrt{x_1^2+x_2^2}}, \qquad \frac{\partial C}{\partial x_2}=\frac{x_2}{\sqrt{x_1^2+x_2^2}}.

At the given inputs, these derivatives are $6/10=0.6$ and $8/10=0.8$. The Euler allocations are therefore $c_1=6(0.6)=3.6$ million and $c_2=8(0.8)=6.4$ million.

c_1=3.6,\qquad c_2=6.4,\qquad c_1+c_2=10.0 \text{ million}.

Check. The allocations reconcile exactly to the aggregate requirement of $10 million. The larger input receives the larger allocation, but the amounts are not simply the standalone inputs: the aggregate function combines them.

Common mistake avoided: Allocating $6 million and $8 million directly would assign $14 million, rather than the $10 million produced by the stated aggregate function.

Euler Allocation With Correlated Product Risks

A specified correlation changes both aggregate capital and marginal contributions. For two product risk inputs $x_1,x_2$ and a fixed correlation parameter $\rho$, consider the quadratic function:

C(x_1,x_2)=\sqrt{x_1^2+x_2^2+2\rho x_1x_2}.

This is a model assumption for the example, not a universal capital formula. Use it only when the problem specifies this aggregation rule and its inputs and correlation. For a positive-definite two-risk quadratic form, $-1<\rho<1$; the examples below use $\rho=0.5$. Differentiating gives $\frac{\partial C}{\partial x_1}=(x_1+\rho x_2)/C$ and $\frac{\partial C}{\partial x_2}=(x_2+\rho x_1)/C$. Multiplication by the corresponding inputs gives Euler allocations that reconcile to $C$.

Worked Example: Correlated Euler Contributions

A hypothetical portfolio has two product risk inputs, $x_1=3$ million and $x_2=6$ million. The stated aggregation rule is the quadratic function above, with $\rho=0.5$. Calculate the total and allocate it by Euler contributions. Round allocations to three decimal places in millions.

Method. Include the correlation term in the aggregate function, then use its partial derivatives. Omitting that term would calculate a different portfolio requirement.

Step 1: Calculate aggregate capital.

C=\sqrt{3^2+6^2+2(0.5)(3)(6)} =\sqrt{9+36+18} =\sqrt{63} \approx 7.9372539 \text{ million}.

Step 2: Calculate the first contribution. The first partial derivative is $(x_1+\rho x_2)/C=(3+0.5(6))/\sqrt{63}=6/\sqrt{63}$. Thus $c_1=3(6/\sqrt{63})=18/\sqrt{63}\approx2.268$ million.

Step 3: Calculate the second contribution. The second partial derivative is $(x_2+\rho x_1)/C=(6+0.5(3))/\sqrt{63}=7.5/\sqrt{63}$. Thus $c_2=6(7.5/\sqrt{63})=45/\sqrt{63}\approx5.669$ million.

c_1+c_2=\frac{18+45}{\sqrt{63}}=\sqrt{63} \approx 7.937 \text{ million}.

Check. The unrounded contributions sum exactly to $C$; the rounded values sum to $7.937$ million. The second product has a larger contribution, reflecting both its larger risk input and its interaction with the first product under the specified rule.

Common mistake avoided: Do not allocate using $x_1^2/C$ and $x_2^2/C$ here. Those expressions omit the correlation term and are not the Euler derivatives of this function.

Floors and Residual Allocation

An organization may specify minimum internal assignments for products before allocating the remaining company-wide capital. This creates a floor-plus-residual rule. It is important to distinguish a floor imposed by the allocation convention from a separately calculated product capital requirement: the former is part of the chosen distribution method.

Worked Example: Applying Floors Before Distributing Residual Capital

A hypothetical insurer assigns a total of $24 million across three products. Its internal allocation rule first assigns floors of $3 million, $2 million, and $1 million. Any capital left after these floors is distributed in proportion to risk scores of 10, 6, and 4. Calculate the product assignments.

Method. Subtract the sum of the floors from the company-wide total. Allocate only this residual in proportion to the scores, then add each product's floor.

Step 1: Sum the floors and find the residual. The floors total $3+2+1=6$ million. The amount still to allocate is $24-6=18$ million.

Step 2: Find the sum of risk scores. The scores total $10+6+4=20$. The residual shares are therefore $10/20$, $6/20$, and $4/20$.

Step 3: Allocate the residual and add floors. Product 1 receives $18(10/20)=9$ million of residual capital, giving $3+9=12$ million in total. Product 2 receives $18(6/20)=5.4$ million of residual capital, giving $2+5.4=7.4$ million. Product 3 receives $18(4/20)=3.6$ million of residual capital, giving $1+3.6=4.6$ million.

(c_1,c_2,c_3)=(12.0,7.4,4.6) \text{ million}, \qquad 12.0+7.4+4.6=24.0 \text{ million}.

Check. Each assignment is at least its floor, and the allocations reconcile to $24 million.

Common mistake avoided: Applying the risk-score proportions to all $24 million and then adding the floors would allocate more than the company-wide total. The scores divide only the residual.

Interpreting and Checking Assigned Capital

A product assignment is meaningful only in relation to its allocation rule. A proportional assignment describes a product's share of the chosen risk measure. An Euler assignment describes its marginal contribution under the specified aggregate function at the current portfolio mix. A floor-plus-residual assignment also reflects the imposed floors. These amounts can therefore differ even when the company-wide requirement is unchanged.

An allocation is not itself evidence that each product could be capitalized independently at that amount, nor does an internal allocation automatically change the company's total requirement. Interpret the figure alongside its rule, inputs, and basis. If the portfolio composition or the capital function changes, marginal contributions and proportional shares may change as well.

For exam calculations and practical review, apply the following checks:

Confirm which figure is the company-wide total and which figures are product inputs or risk weights.
Write the allocation rule before substituting values, especially when floors or correlations are involved.
Check that the rule's conditions hold. For example, proportional allocation needs a positive sum of weights, while the Euler reconciliation above requires differentiability and degree-one homogeneity.
Sum the allocated amounts before rounding where possible. Explain any small difference caused by final rounding.

The central distinction is between measuring total capital and assigning that total across products. A correct allocation uses the stated rule, preserves the company-wide total, and is interpreted as a product-level assignment under that rule—not as a new company-wide requirement.

Check Your Understanding

For each problem, use the allocation rule stated and verify that the assigned amounts reconcile. Amounts are in millions unless indicated otherwise.

A company-wide requirement of $18 million is allocated proportionally using product measures 4, 3, and 2. What is each product's allocation? (Answer: $8 million, $6 million, and $4 million.)
Two independent product inputs are 5 and 12, and aggregate capital is $C=\sqrt{x_1^2+x_2^2}$. Find the Euler allocations. (Answer: $25/13\approx1.9231$ million and $144/13\approx11.0769$ million.)
Two product inputs are both 4, with $\rho=0.25$ under the stated quadratic aggregation rule. Find aggregate capital and the Euler allocation to each product. (Answer: Total $C=\sqrt{40}\approx6.3249$ million; each product receives $\sqrt{40}/2\approx3.1623$ million.)
A $20 million total is allocated using floors of 2, 2, and 1, then residual capital in proportion to scores 6, 3, and 1. Find the product assignments. (Answer: $11 million, $6.5 million, and $2.5 million.)
A stated total of $15 million is allocated proportionally using weights 2, 5, and 3. What share of the total, and what amount, go to the product with weight 5? (Answer: Share $5/10=50\%$; allocation $7.5 million.)

What You'll Learn