Allocation Consistency and Full Allocation

Why Allocation Properties Matter

A capital allocation assigns a portfolio’s required capital to its components, such as business units or risk components. The assigned amounts can inform performance measurement and management decisions, so an allocation should be checked against the capital measure it is intended to explain. A list of plausible-looking contributions is not enough: the amounts should add up correctly and treat genuinely equivalent components consistently.

Let $N=\{1,\ldots,n\}$ be the set of components, and let $C(N)$ denote the capital assigned to the full portfolio. Write $a_i$ for the capital allocated to component $i$. In a coalition model, $v(S)$ is the capital of the portfolio containing the subset $S\subseteq N$, with $v(\varnothing)=0$. This convention makes the coalition value the capital attributable to the included components, rather than including a separate fixed baseline.

Definition: An allocation is fully allocated (or efficient) when the sum of the component allocations equals the capital being allocated: $$ \sum_{i\in N} a_i=C(N). $$ For a coalition-value model with $v(\varnothing)=0$, the corresponding condition is $\sum_{i\in N}a_i=v(N)$. If a model includes a nonzero baseline $v(\varnothing)$, the allocation of incremental capital should instead sum to $v(N)-v(\varnothing)$.

Full allocation is an accounting condition, not a proof that the contributions are fair or useful for every decision. A rule can sum exactly to total capital and still treat identical components differently. Conversely, an attractive fairness rule can fail to allocate the entire amount unless its construction or a subsequent reconciliation ensures full allocation.

Three Tests of Allocation Consistency

The following properties provide distinct checks. They are often considered together, but passing one does not imply that the other two hold.

Full allocation: all component amounts add to the portfolio capital.
Symmetry: components that have identical roles in the capital model receive equal allocations.
Null component: a component that never changes coalition capital receives zero allocation.

For a coalition model, components $i$ and $j$ are symmetric when, for every coalition $S$ containing neither, adding $i$ or adding $j$ produces the same value: $v(S\cup\{i\})=v(S\cup\{j\})$. This is a property of their roles in the model, not merely their labels or standalone capital. Component $i$ is null when $v(S\cup\{i\})=v(S)$ for every $S\subseteq N\setminus\{i\}$. Checking only some coalitions does not establish nullity.

Shapley Contributions and the Three Properties

The Shapley allocation assigns each component its average marginal contribution over all possible orders in which the components could enter the portfolio. For $n$ components, the Shapley contribution of $i$ is

\phi_i(v)= \sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(n-|S|-1)!}{n!} \left[v(S\cup\{i\})-v(S)\right].

The bracketed term is the marginal contribution of $i$ when the components in $S$ are already present. The weight is the fraction of all $n!$ possible orders in which exactly the components in $S$ appear before $i$. The allocation therefore averages the component’s contribution over the possible contexts in which it could be added.

Full allocation follows because, in each ordering, the marginal contributions telescope. If the components in one order are $i_1,\ldots,i_n$, their contributions sum to

\sum_{k=1}^{n} \left[ v(\{i_1,\ldots,i_k\})-v(\{i_1,\ldots,i_{k-1}\}) \right] =v(N)-v(\varnothing)=v(N).

Averaging this equality over all orders preserves the sum, so the Shapley contributions are fully allocated. If two components are symmetric, swapping their labels gives a one-to-one pairing of orders in which their marginal contributions exchange; their averages are therefore equal. If component $i$ is null, every marginal contribution in its formula is zero, so $\phi_i(v)=0$. These conclusions depend on using the stated coalition values consistently; an inaccurate or incomplete set of values can invalidate the tests.

Worked Example: Checking Full Allocation from Coalition Values

A hypothetical three-component portfolio has capital values, in thousands of dollars, given by $v(\varnothing)=0$, $v(A)=40$, $v(B)=30$, $v(C)=20$, $v(AB)=60$, $v(AC)=55$, $v(BC)=45$, and $v(ABC)=70$. Calculate its Shapley contributions and test full allocation. Use the three-component Shapley weights: $1/3$ for the empty and two-component predecessor sets, and $1/6$ for each one-component predecessor set.

Method. For each component, list its marginal contribution in each possible predecessor coalition, multiply by the corresponding weight, and add.

For $A$, the marginal contributions when the predecessor coalition is $\varnothing$, $B$, $C$, and $BC$ are $40$, $60-30=30$, $55-20=35$, and $70-45=25$. Thus

\phi_A=\frac13(40)+\frac16(30)+\frac16(35)+\frac13(25) =13.3333+5+5.8333+8.3333=32.5.

For $B$, the corresponding marginal contributions are $30$, $60-40=20$, $45-20=25$, and $70-55=15$. Therefore

\phi_B=\frac13(30)+\frac16(20)+\frac16(25)+\frac13(15) =10+3.3333+4.1667+5=22.5.

For $C$, they are $20$, $55-40=15$, $45-30=15$, and $70-60=10$. Therefore

\phi_C=\frac13(20)+\frac16(15)+\frac16(15)+\frac13(10) =6.6667+2.5+2.5+3.3333=15.

The total is $32.5+22.5+15=70$ thousand dollars, equal to $v(ABC)=70$. The allocation is fully allocated. The calculation is checked against the portfolio total; small displayed decimal differences can arise from intermediate rounding, so the exact weighted sums determine the reported values. A common mistake is to allocate each component’s standalone capital directly: the standalone amounts sum to $40+30+20=90$, not the portfolio capital of $70$.

Symmetry: Equal Roles, Equal Allocations

Symmetry is a consistency requirement, not a direction to divide capital equally among all components. It applies only to components that are interchangeable in the capital model. Distinct standalone risks or distinct patterns of interaction with the rest of the portfolio can justify different allocations.

Worked Example: Testing Symmetry and Full Allocation

A hypothetical coalition model has $v(A)=12$, $v(B)=12$, $v(C)=8$, $v(AB)=20$, $v(AC)=17$, $v(BC)=17$, and $v(ABC)=25$, with $v(\varnothing)=0$, all in thousands of dollars. Determine whether $A$ and $B$ are symmetric and find their Shapley allocations.

Method. Compare the marginal effects of $A$ and $B$ in coalitions containing neither. With no other component, both have value $12$. With $C$ already present, adding either raises capital from $8$ to $17$, a marginal increase of $9$. Thus $A$ and $B$ are symmetric.

For $A$, its marginal contributions for predecessor sets $\varnothing$, $B$, $C$, and $BC$ are $12$, $20-12=8$, $17-8=9$, and $25-17=8$. Hence

\phi_A=\frac13(12)+\frac16(8)+\frac16(9)+\frac13(8) =4+\frac43+\frac32+\frac83=9.5.

For $B$, the marginal contributions are $12$, $20-12=8$, $17-8=9$, and $25-17=8$, so $\phi_B=9.5$ as well. For $C$, they are $8$, $17-12=5$, $17-12=5$, and $25-20=5$. Thus

\phi_C=\frac13(8)+\frac16(5)+\frac16(5)+\frac13(5)=6.

The allocations satisfy symmetry, since $\phi_A=\phi_B=9.5$, and full allocation, since $9.5+9.5+6=25=v(ABC)$. The common mistake is to infer that all three components should receive equal amounts because the method is “symmetric.” Only $A$ and $B$ have been shown to be symmetric; $C$ has different coalition effects.

Null Components: Every Coalition Must Be Checked

A null component adds no capital in any context. It is not enough for its standalone value to be zero, or for its marginal contribution to vanish in several selected coalitions. In a portfolio with three components $A$, $B$, and $Z$, testing whether $Z$ is null requires checking its marginal contribution when added to each possible coalition excluding $Z$: $\varnothing$, $A$, $B$, and $AB$.

Worked Example: Establishing a Null Component

A hypothetical portfolio has components $A$, $B$, and $Z$, and coalition values in thousands of dollars as follows: $v(\varnothing)=0$, $v(A)=40$, $v(B)=30$, $v(AB)=60$, $v(Z)=0$, $v(AZ)=40$, $v(BZ)=30$, and $v(ABZ)=60$. Determine whether $Z$ is null, then calculate the Shapley allocation.

Method. Compare each coalition value without $Z$ to the value obtained by adding $Z$. The four marginal contributions of $Z$ are $0-0=0$, $40-40=0$, $30-30=0$, and $60-60=0$. All are zero, so $Z$ is null.

Its Shapley contribution is consequently zero. For $A$, the marginal contributions across predecessor sets $\varnothing$, $B$, $Z$, and $BZ$ are $40$, $60-30=30$, $40-0=40$, and $60-30=30$. Therefore

\phi_A=\frac13(40)+\frac16(30)+\frac16(40)+\frac13(30) =13.3333+5+6.6667+10=35.

For $B$, the corresponding marginal contributions are $30$, $60-40=20$, $30-0=30$, and $60-40=20$, giving

\phi_B=\frac13(30)+\frac16(20)+\frac16(30)+\frac13(20) =10+3.3333+5+6.6667=25.

The resulting allocation is $(35,25,0)$ thousand dollars, and its sum is $60=v(ABZ)$. A common mistake is to check only $Z$'s standalone value and conclude it is null. The zero marginal contribution must hold in every coalition, including $AB$.

Full Allocation from Euler Contributions

Full allocation also arises from Euler allocation under specific conditions. Let $C(x_1,\ldots,x_n)$ be a capital function of component exposures, differentiable at the portfolio exposure vector $x$, and positively homogeneous of degree one. Positive homogeneity means $C(tx)=tC(x)$ for $t>0$. Define the Euler contribution of component $i$ as $a_i=x_i\,\partial C(x)/\partial x_i$. This result does not automatically apply at points where the required derivatives do not exist, or to a capital function that is not degree-one homogeneous.

To see why the contributions sum to capital, differentiate $C(tx)=tC(x)$ with respect to $t$ at $t=1$. The chain rule gives $\sum_i x_i\,\partial C(x)/\partial x_i=C(x)$, which is exactly the full-allocation identity for Euler contributions.

Worked Example: Verifying Full Allocation Under a Homogeneous Capital Function

A hypothetical two-component portfolio has exposures $x_1=3$ and $x_2=4$, measured in millions of dollars. Its capital function is $C(x_1,x_2)=\sqrt{x_1^2+x_2^2}$. Calculate the Euler contributions and check full allocation.

Method. The function is differentiable at $(3,4)$ and is positively homogeneous of degree one, since $C(tx_1,tx_2)=tC(x_1,x_2)$ for $t>0$. Its total capital is $C(3,4)=\sqrt{9+16}=5$ million dollars. The partial derivatives are $\partial C/\partial x_1=x_1/C$ and $\partial C/\partial x_2=x_2/C$. Thus

a_1=3\left(\frac35\right)=1.8,\qquad a_2=4\left(\frac45\right)=3.2

million dollars. Their sum is $1.8+3.2=5$, equal to total capital. This example also checks the derivative calculation directly. The common mistake is to assume that any derivative-based allocation must be fully allocated: here the conclusion follows from positive homogeneity and differentiability, not merely from using derivatives.

When a Rule Passes One Test but Fails Another

Consider two symmetric components with total portfolio capital of $100$ thousand dollars. An allocation of $70$ to the first and $30$ to the second is fully allocated because $70+30=100$, but it violates symmetry because the components have identical roles. An allocation of $50$ to each satisfies both tests. In a different setting, unequal allocations may be appropriate if the components are not symmetric.

These properties are useful diagnostics, not universal guarantees of suitability. For example, an allocation can be fully allocated and symmetric while still being sensitive to modelling choices or inappropriate for a particular management decision. The appropriate capital model and allocation purpose must be stated before interpreting the numbers. This tutorial focuses on internal consistency conditions; it does not establish that every allocation satisfying them is uniquely correct.

Key takeaway: Check the allocation against the capital measure in three separate ways: do the contributions sum to portfolio capital, do equivalent components receive equal amounts, and does every truly null component receive zero? For a null-component test in an $A,B,Z$ portfolio, the marginal effect of $Z$ in coalition $AB$ must be checked as well as its effects in $\varnothing$, $A$, and $B$.

Check Your Understanding

For each problem, apply the stated property to the complete set of information given.

A two-component portfolio has total capital $84$ thousand dollars and proposed allocations of $50$ and $34$ thousand dollars. Does it satisfy full allocation? (Answer: Yes; $50+34=84$.)
Two components have the same marginal effect in every coalition containing neither, but receive allocations of $18$ and $22$ thousand dollars under a proposed rule. Which property does the rule violate? (Answer: Symmetry.)
A portfolio contains components $A$, $B$, and $Z$. The marginal contributions of $Z$ are zero when added to $\varnothing$, $A$, and $B$, but its marginal contribution when added to $AB$ is $6$ thousand dollars. Is $Z$ null? (Answer: No; its contribution in coalition $AB$ is nonzero.)
A three-component model gives $v(ABC)=90$ thousand dollars and Shapley contributions of $38$, $31$, and $21$ thousand dollars. Does the allocation satisfy full allocation? (Answer: Yes; $38+31+21=90$.)
In a portfolio with components $A$, $B$, and $Z$, the marginal contributions of $Z$ are zero when added to $\varnothing$, $A$, and $B$. What additional coalition must be checked before concluding that $Z$ is null? (Answer: Check the marginal contribution of $Z$ when added to coalition $AB$.)

What You'll Learn