Why Hierarchical Allocation Requires Reconciliation
An insurer may want to attribute aggregate economic capital both to business units and to risk categories. A two-dimensional allocation assigns capital to each business–risk combination, such as property catastrophe in the commercial unit. Those cell amounts can then be added across risk categories to obtain business totals, or across businesses to obtain risk totals.
A key control is whether the subtotals implied by the cells agree with the amounts calculated for each dimension. A sequential method first allocates capital to one dimension and then divides each subtotal across the other. A direct method allocates to all cells in one step. Both can fully allocate aggregate capital, yet they need not produce the same cell amounts or the same subtotals.
All monetary amounts in the examples are hypothetical and stated in thousands of dollars. The methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
A direct allocation is fully allocated if the sum of all cell amounts equals aggregate capital. Its business subtotal for unit \(b\) is the sum across that unit’s risk categories; its risk subtotal for category \(r\) is the sum across businesses. A sequential allocation is fully allocated if its first-stage subtotals sum to \(C\) and each second-stage allocation sums to its first-stage subtotal.
The conditional shares in a sequential allocation must be defined relative to the subtotal being split. For a business-first allocation, \(w_{r\mid b}\) is the share of business \(b\)'s allocated capital assigned to risk category \(r\), so the shares for each business sum to one. The resulting cell amount is \(c_{br}^{(B)}=C_b w_{r\mid b}\). In a risk-first allocation, the conditional business shares instead sum to one within each risk category.
Why a Consistent Sequential Allocation Reconciles
The reconciliation of a sequential allocation follows directly from how its conditional shares are defined. For each business, its cell allocations add back to the subtotal assigned to that business. Adding those business subtotals then recovers aggregate capital.
This equality requires \(\sum_r w_{r\mid b}=1\) for each business. Summing over businesses gives \(\sum_b\sum_r c_{br}^{(B)}=\sum_b C_b=C\), provided the first-stage business allocations also sum to \(C\). Thus full allocation is guaranteed by the two sets of sum-to-one conditions.
That argument establishes reconciliation to the chosen first-stage subtotals. It does not guarantee that the risk totals implied by the cells agree with risk totals calculated independently. Nor does it show that a direct cell allocation or a risk-first procedure will produce the same cell amounts. Those comparisons require an explicit check.
Worked Example: Business-First Allocation and Direct Reconciliation
A hypothetical insurer has aggregate capital of $150 thousand. A selected allocation method assigns $90 thousand to Business A and $60 thousand to Business B. Within Business A, the conditional shares for catastrophe and market risk are 40% and 60%; within Business B, the shares are 25% and 75%. Calculate the cells and subtotals, then compare them with a direct allocation reported as follows.
| Business | Catastrophe, direct | Market, direct | Direct business subtotal |
|---|---|---|---|
| A | 36 | 54 | 90 |
| B | 15 | 45 | 60 |
| Risk subtotal | 51 | 99 | 150 |
Method. First multiply each business subtotal by its conditional risk shares. For Business A, the catastrophe cell is \(90(0.40)=36\), and the market cell is \(90(0.60)=54\). The check is \(36+54=90\), the Business A subtotal.
For Business B, the catastrophe cell is \(60(0.25)=15\), and the market cell is \(60(0.75)=45\). The check is \(15+45=60\), the Business B subtotal. The implied risk totals are \(36+15=51\) for catastrophe and \(54+45=99\) for market.
The business totals add to \(90+60=150\), and the risk totals add to \(51+99=150\). Each sequential cell agrees with the corresponding direct cell in the table, so both the business and risk subtotals reconcile. The common mistake to avoid is applying a conditional share to total capital: \(0.40(150)=60\) is not Business A’s catastrophe cell. The 40% share applies to Business A’s $90 thousand subtotal, giving $36 thousand.
Direct and Sequential Allocations Can Differ
A direct allocation can be fully allocated without matching the results of a separately specified sequential procedure. In that situation, the difference is not necessarily an arithmetic error. It may reflect a different allocation rule or a different set of conditional shares. Reconciliation tests whether the relevant subtotals agree; it does not make two distinct allocation methods equivalent.
Worked Example: Testing Sequential Results Against Direct Cells
A hypothetical insurer has $120 thousand of aggregate capital. Its direct cell allocation is $28 thousand to Business A’s catastrophe risk, $32 thousand to Business A’s market risk, $18 thousand to Business B’s catastrophe risk, and $42 thousand to Business B’s market risk. A separate business-first procedure assigns $60 thousand to each business and allocates Business A equally between the risks, while assigning 25% of Business B’s subtotal to catastrophe risk and 75% to market risk. Compare the two methods.
Direct subtotals. Business A receives \(28+32=60\), and Business B receives \(18+42=60\). Catastrophe receives \(28+18=46\), and market receives \(32+42=74\). The direct cells fully allocate capital because \(28+32+18+42=120\); the business totals and risk totals each also sum to 120.
Sequential cells. Business A’s cells are \(60(0.50)=30\) for catastrophe and \(60(0.50)=30\) for market. Business B’s cells are \(60(0.25)=15\) for catastrophe and \(60(0.75)=45\) for market. The sequential risk totals are therefore \(30+15=45\) for catastrophe and \(30+45=75\) for market.
The sequential method fully allocates capital: its four cells total \(30+30+15+45=120\), and its business subtotals are still $60 thousand each. However, its risk subtotals are $45 thousand and $75 thousand, not the direct allocation’s $46 thousand and $74 thousand. The discrepancy is $1 thousand in each risk category in opposite directions.
Check. The business subtotals reconcile across methods, but the risk subtotals do not. The direct cells and sequential cells differ because the conditional shares used in the sequential procedure do not reproduce the direct cell amounts. Full allocation alone does not establish agreement with every independently calculated subtotal.
Allocation Order and Cell-Level Results
A second source of difference is the order of allocation. Business-first and risk-first procedures can both preserve the same business and risk totals but distribute capital differently among the cells. A matching set of row and column totals does not uniquely determine every cell amount.
Worked Example: Business-First Versus Risk-First Allocation
A hypothetical portfolio has $120 thousand of capital, with business subtotals of $70 thousand for Business A and $50 thousand for Business B. Its risk subtotals are $60 thousand for catastrophe and $60 thousand for market. Under the business-first rule, catastrophe receives 60% of Business A and 36% of Business B. Under the risk-first rule, Business A receives five-sixths of catastrophe capital and one-third of market capital. Calculate both cell allocations.
Business-first. Business A’s catastrophe allocation is \(70(0.60)=42\), leaving \(70-42=28\) for market. Business B’s catastrophe allocation is \(50(0.36)=18\), leaving \(50-18=32\) for market. The cells are therefore A-catastrophe 42, A-market 28, B-catastrophe 18, and B-market 32.
The business checks are \(42+28=70\) and \(18+32=50\). The risk checks are \(42+18=60\) and \(28+32=60\). The total is \(70+50=120\).
Risk-first. Business A receives \(60(5/6)=50\) of catastrophe capital, leaving \(60-50=10\) for Business B. For market risk, Business A receives \(60(1/3)=20\), leaving \(60-20=40\) for Business B. The cells are A-catastrophe 50, A-market 20, B-catastrophe 10, and B-market 40.
The risk checks are \(50+10=60\) and \(20+40=60\). The business checks are \(50+20=70\) and \(10+40=50\), and the total remains $120 thousand. Both procedures agree on every business and risk subtotal, yet the A-catastrophe cell is $42 thousand under business-first allocation and $50 thousand under risk-first allocation. The cell difference is a consequence of the different conditional allocation rules, not a failure to sum to aggregate capital.
A Practical Reconciliation Process
A useful review separates three questions: whether the cells sum to aggregate capital, whether the business subtotals agree with their intended amounts, and whether the risk subtotals agree with their intended amounts. A “yes” to the first question does not imply a “yes” to the other two.
Record aggregate capital, the business units, the risk categories, and the amount assigned to each cell.
Confirm that every intended business–risk combination is included and that cell amounts use the same units.
Sum across risk categories for each business and across businesses for each risk category.
Report any difference explicitly. If two methods disagree, identify whether the discrepancy is at the business level, risk level, or cell level.
If all cell amounts are nonnegative, the cell totals and margins can also be displayed as a table whose row and column sums provide visible arithmetic checks. The same reconciliation logic applies if a selected allocation method permits negative contributions: use signed sums rather than silently replacing negative amounts with zero.
Common Mistakes and Interpretation
- Using the wrong base for a conditional share. A business-conditional share is applied to that business’s subtotal, not directly to aggregate capital.
- Checking only the grand total. Cells can sum to aggregate capital while failing to match one or more specified business or risk subtotals.
- Assuming that allocation order cannot matter. Different conditional rules can yield different cell amounts, even if they preserve identical business and risk margins.
- Treating a discrepancy as automatically an arithmetic error. First determine whether the methods use the same target subtotals and conditional allocation rules. If they do, recheck the calculations; if not, explain which rule caused the difference.
Check Your Understanding
For each hypothetical allocation, calculate the requested cells or subtotals and check whether the stated amounts reconcile.
- A business-first allocation assigns $120 thousand to Business A and $80 thousand to Business B. Business A allocates 25% to catastrophe and 75% to market; Business B splits equally between the two risks. Find all four cells and the risk subtotals. (Answer: A: 30 catastrophe, 90 market; B: 40 catastrophe, 40 market; risk totals 70 catastrophe and 130 market; total 200, all amounts in thousands.)
- A direct allocation has cells A-catastrophe 26, A-market 34, B-catastrophe 24, and B-market 36. A sequential method instead assigns $60 thousand to each business, with catastrophe shares of 40% for A and 30% for B. Find the sequential cells and compare the risk subtotals with the direct allocation. (Answer: Sequential cells are 24, 36, 18, and 42 in the order listed; sequential risk totals are 42 catastrophe and 78 market, compared with direct totals 50 and 70; both methods total 120 and have business subtotals 60 each.)
- Risk-first allocation assigns $30 thousand to catastrophe and $60 thousand to market. Business X receives two-thirds of catastrophe and one-third of market. Find the four cells, the business totals, and the aggregate total. (Answer: X receives 20 catastrophe and 20 market; Y receives 10 catastrophe and 40 market; business totals are 40 and 50; aggregate total is 90, all amounts in thousands.)
- A direct cell table has business rows A: 35 catastrophe and 45 market; B: 25 catastrophe and 45 market. Target business subtotals are 80 and 70, while target risk subtotals are 60 catastrophe and 90 market. Does it reconcile to aggregate capital and to both sets of target subtotals? (Answer: Yes. The rows are 80 and 70, the columns are 60 and 90, and all cells sum to 150 thousand.)
- Two allocation paths both produce business subtotals A = 70 and B = 50, and risk subtotals catastrophe = 60 and market = 60. One assigns 42 to A-catastrophe; the other assigns 50 to A-catastrophe. What must the other three cells be in each path, and what does the difference demonstrate? (Answer: With A-catastrophe 42, the other cells are A-market 28, B-catastrophe 18, B-market 32. With A-catastrophe 50, they are A-market 20, B-catastrophe 10, B-market 40. Both paths have the same margins and total 120 thousand, but different conditional rules produce different cell allocations.)