Why Average Across Portfolio Orderings?
A business unit’s incremental capital depends on which other units are already in the portfolio. Adding a unit to an empty portfolio may require its full standalone capital; adding the same unit after several other risks may require less because some risk is shared or diversified. An allocation based on just one order of addition can therefore make a unit appear more or less capital-intensive simply because of where it was placed in that order.
The Shapley allocation addresses this order effect by calculating each unit’s incremental contribution in every possible ordering of the portfolio and taking the average. It gives each ordering equal weight. The resulting allocation is useful when a capital model supplies capital for each possible subset of business units, or when a model can evaluate those subsets consistently.
Here, \(C(S)\) must use a consistent capital methodology for every subset. For example, if \(C(S)\) is based on a selected risk measure applied to the aggregate loss of the units in \(S\), the same risk measure and modeling basis should be used throughout. The method redistributes the model’s portfolio capital; it does not select the risk measure or replace the underlying capital model.
Formula and Rationale
For a finite portfolio with \(n\) units, let \(\Pi(N)\) be the set of all \(n!\) orderings of those units. For an ordering \(\pi\), let \(P_i^\pi\) be the set of units placed before \(i\). The Shapley allocation can be written as:
The same calculation can be organized by predecessor subsets rather than by full orderings. A particular subset \(S\), not containing \(i\), appears before \(i\) in a fraction of orderings determined by its size. There are \(|S|!\) ways to order the members of \(S\), and \((n-|S|-1)!\) ways to order the units after \(i\). Therefore:
This form makes clear that the method averages marginal contributions across portfolio contexts. It does not give every possible predecessor subset the same weight: the weight reflects how often that subset occurs before \(i\) in a uniformly selected ordering.
There is also a direct reason the Shapley allocations add to the portfolio’s capital. In any one ordering, the incremental contributions telescope: the first increment starts at \(C(\varnothing)=0\), and the final increment ends at \(C(N)\). Averaging those ordering totals leaves the same endpoint difference.
Since averaging preserves sums, \(\sum_{i\in N}\phi_i=C(N)\). If the standalone capital for unit \(i\) is \(c_i=C(\{i\})\), the total diversification benefit is \(\sum_i c_i-C(N)\). The amount attributed to unit \(i\) under this allocation is \(c_i-\phi_i\); adding those amounts gives the total benefit. Thus, the allocation distributes the benefit through the same averaged marginal-contribution calculation, rather than by imposing a separate equal split.
Worked Example: Averaging Contributions for Two Units
Method: Calculate each unit’s marginal contribution in both possible orderings, then average. A hypothetical insurer’s model reports standalone capital of $120 thousand for unit A, $80 thousand for unit B, and $150 thousand for the combined portfolio. Find the Shapley allocations and the diversification benefit attributed to each unit.
There are two orderings. In the order A, then B, A contributes $120 thousand and B contributes \(150-120=30\) thousand. In the order B, then A, B contributes $80 thousand and A contributes \(150-80=70\) thousand.
A’s average contribution is \((120+70)/2=95\) thousand. B’s average contribution is \((30+80)/2=55\) thousand. Their allocations total \(95+55=150\) thousand, the modeled combined capital.
Standalone capital totals \(120+80=200\) thousand, so the portfolio diversification benefit is \(200-150=50\) thousand. A’s attributed benefit is \(120-95=25\) thousand, and B’s is \(80-55=25\) thousand. The two attributed amounts sum to the $50 thousand portfolio benefit.
Check and common mistake avoided: The allocations reconcile to $150 thousand. Using only the A-then-B order would assign A $120 thousand and B $30 thousand; those figures are valid increments for that ordering, but they are not the Shapley averages.
Worked Example Across All Three-Unit Orderings
With three units, each unit may join in different-sized portfolios and face different marginal capital. A full ordering table is a transparent way to avoid overlooking any of these contexts.
Worked Example: Allocating Capital Among Three Units
Method: Enumerate all six orderings, record the marginal contribution at each step, and average by unit. A hypothetical capital model reports the following amounts, in thousands of dollars:
| Portfolio subset | Capital \(C(S)\), $000 |
|---|---|
| A | 120 |
| B | 90 |
| C | 70 |
| A and B | 170 |
| A and C | 150 |
| B and C | 130 |
| A, B, and C | 212 |
For the ordering A, B, C, the increments are \(120\), \(170-120=50\), and \(212-170=42\), respectively. Applying the same subtraction to each ordering gives:
| Ordering | Marginal for A, $000 | Marginal for B, $000 | Marginal for C, $000 |
|---|---|---|---|
| A, B, C | 120 | 50 | 42 |
| A, C, B | 120 | 62 | 30 |
| B, A, C | 80 | 90 | 42 |
| B, C, A | 82 | 90 | 40 |
| C, A, B | 80 | 62 | 70 |
| C, B, A | 82 | 60 | 70 |
For example, in the ordering B, C, A, A joins the portfolio containing B and C. Its increment is \(C(\{A,B,C\})-C(\{B,C\})=212-130=82\) thousand. The other entries use the same calculation: each unit’s increment is the capital of the subset after it joins minus the capital of the subset before it joins.
The total marginal contributions by unit are \(120+120+80+82+80+82=564\) for A, \(50+62+90+90+62+60=414\) for B, and \(42+30+42+40+70+70=294\) for C. Dividing each total by six gives Shapley allocations of \(564/6=94\), \(414/6=69\), and \(294/6=49\) thousand.
The check is \(94+69+49=212\) thousand, equal to the combined capital. Standalone capital totals \(120+90+70=280\) thousand, so the diversification benefit is \(280-212=68\) thousand. The amounts attributed to A, B, and C are \(120-94=26\), \(90-69=21\), and \(70-49=21\) thousand. They sum to \(68\) thousand.
Common mistake avoided: Averaging each unit’s marginal contributions requires dividing by the six orderings, not by the number of units. The six ordering contributions for A, for instance, produce \(564/6=94\), not \(564/3\).
Interpreting the Diversification Benefit
The Shapley allocation attributes diversification through reductions in allocated capital relative to standalone capital. A unit can have a comparatively large standalone requirement and still receive a smaller Shapley allocation if its marginal contribution tends to be lower when added to portfolios that already contain other units. Conversely, a unit whose inclusion often adds substantial capital can receive a larger allocation.
The approach is neutral to the order in which units are listed: every ordering is included with equal weight. It is not, however, a claim that the units literally enter the portfolio in random sequence, nor is it an estimate of the frequency of business growth paths. Orderings are a calculation device for averaging the effects of adding units in different contexts.
The calculation also depends on the chosen capital model. If the model’s subset capital values change, the marginal contributions and allocations can change. The Shapley rule determines how to average those values; it does not determine whether the underlying risk assumptions, dependencies, or capital measure are appropriate.
Pairwise Diversification Credits
A useful special case expresses subset capital as the sum of standalone amounts less credits for pairs of units that are both present. Under that model, the Shapley rule splits each pairwise credit equally between its two units. This follows because in half of all orderings one member of a pair arrives first, and in the other half the other member arrives first. The later member receives the reduction in marginal capital associated with that pair; averaging across orderings assigns half the credit to each.
Worked Example: Sharing Pairwise Diversification Credits
Method: Begin with standalone capital, then split each pairwise credit equally between its two units. A hypothetical six-unit portfolio has standalone capitals, in thousands of dollars, of 100, 90, 60, 50, 65, and 50 for units A through F. Their total is \(100+90+60+50+65+50=415\) thousand.
Suppose the model’s only diversification credits are $12 thousand for the pair A-B, $6 thousand for A-C, $8 thousand for B-C, $5 thousand for D-E, and $4 thousand for E-F. The total credit is \(12+6+8+5+4=35\) thousand. Under the stated pairwise model, combined capital is \(415-35=380\) thousand.
A receives half of the A-B credit and half of the A-C credit, so its allocation is \(100-12/2-6/2=91\) thousand. B’s allocation is \(90-12/2-8/2=80\) thousand. C’s is \(60-6/2-8/2=53\) thousand.
D receives half of the D-E credit, giving \(50-5/2=47.5\) thousand. E receives half of each credit involving E, giving \(65-5/2-4/2=60.5\) thousand. F receives half of the E-F credit, giving \(50-4/2=48\) thousand.
The allocations total \(91+80+53+47.5+60.5+48=380\) thousand. The attributed benefits are 9, 10, 7, 2.5, 4.5, and 2 thousand, respectively, which sum to \(35\) thousand. Both checks match the model’s combined capital and total diversification credit.
Common mistake avoided: Subtracting each full pairwise credit from both participating units would count that credit twice. For example, the A-B credit is $12 thousand, so this model assigns $6 thousand of benefit to A and $6 thousand to B.
Practical Checks Before Reporting an Allocation
A careful calculation begins by stating what \(C(S)\) means and confirming that values for different subsets use a consistent model basis. Then calculate the marginal change using the same before-and-after subset convention throughout. For a small portfolio, explicitly listing orderings makes it easier to spot omitted cases or a subtraction taken in the wrong direction.
Define the units, the subset capital \(C(S)\), and \(C(\varnothing)=0\). State the units in which capital is reported.
For each ordering, subtract the capital before a unit joins from the capital after it joins.
Sum each unit’s increments over all orderings and divide by the number of orderings.
Confirm that allocations sum to \(C(N)\), and that standalone capital minus allocated capital sums to the total diversification benefit.
A common exam trap is to use one convenient ordering as if it were the Shapley result. Another is to distribute the total diversification benefit equally without checking the marginal contributions. Both shortcuts ignore how a unit’s effect changes with the portfolio it joins. The defining calculation is the average over all orderings, using the supplied capital values consistently.
Check Your Understanding
For each problem, use the stated capital model and calculate the requested Shapley result.
- A hypothetical two-unit portfolio has standalone capital of $75 thousand for A and $45 thousand for B, and combined capital of $100 thousand. Find both Shapley allocations and the total diversification benefit. (Answer: A = $65 thousand; B = $35 thousand; total benefit = $20 thousand.)
- A model gives \(C(A)=60\), \(C(B)=50\), \(C(C)=40\), \(C(AB)=90\), \(C(AC)=80\), \(C(BC)=70\), and \(C(ABC)=105\), all in thousands of dollars. Average over the six orderings to find each allocation and check the total. (Answer: A = $45 thousand; B = $35 thousand; C = $25 thousand; total = $105 thousand.)
- Three units have standalone capital of $80, $70, and $50 thousand. The only pairwise credits are $12 thousand for A-B, $6 thousand for A-C, and $8 thousand for B-C. Under the pairwise-credit model, find each Shapley allocation and the total benefit. (Answer: A = $71 thousand; B = $60 thousand; C = $43 thousand; total benefit = $26 thousand.)
- For a four-unit portfolio, the sums of unit A’s marginal contributions over predecessor subsets of sizes 0, 1, 2, and 3 are respectively 100, 240, 150, and 30 thousand dollars. Use the Shapley subset weights to find A’s allocation. (Answer: $65 thousand.)
- Six units have standalone capitals of 100, 90, 60, 50, 65, and 50 thousand dollars. Pairwise credits are $12 thousand for A-B, $6 thousand for A-C, $8 thousand for B-C, $5 thousand for D-E, and $4 thousand for E-F, with no other credits. Under the pairwise-credit model, find the combined capital and allocation to E. (Answer: Combined capital = $380 thousand; E’s allocation = $60.5 thousand.)