Why Allocate Capital Along Two Dimensions?
An insurer may want to understand capital by business unit and by risk type at the same time. A single allocation by business unit can show which units use capital, but not whether that capital arises from property, casualty, or market risk. A single allocation by risk type leaves the corresponding business-unit picture missing. A two-dimensional allocation puts both views in one table: each cell represents capital attributed to one business unit and one risk type.
The table is useful only if its parts reconcile. Adding across a business-unit row must give that unit’s allocated capital; adding down a risk-type column must give that risk type’s allocated capital. Adding all cells must give the aggregate capital being allocated. These are accounting identities, but obtaining meaningful cell values requires an allocation rule.
A model-based approach is to calculate a contribution for each business-unit/risk-type cell, then obtain row and column allocations by summing those contributions. The same cell values must be used for both views. Calculating one set of row contributions and a separate set of column contributions can produce inconsistent totals or fail to define how capital is split across the cells.
Cell Contributions Under Euler Allocation
Suppose aggregate capital is a differentiable, positively homogeneous function of degree one of the exposures \(x_{ij}\) in the cells. Positive homogeneity of degree one means that scaling every exposure by \(t>0\) scales capital by the same factor: \(K(tx)=tK(x)\). Under Euler allocation, the contribution assigned to a cell is its exposure multiplied by the marginal change in capital with respect to that exposure.
Euler’s theorem gives full allocation under these assumptions. It also explains why row and column totals reconcile with Euler allocations made directly at either dimension. If all exposures in business unit \(i\) are scaled together, the derivative with respect to that unit’s scale is the sum of its cell exposures times their marginal capital effects. That sum is precisely the row total. The same chain-rule argument applies when scaling all cells of risk type \(j\).
These equalities do not say that every possible allocation method produces the same cell values. They say that, for a given set of Euler cell contributions, the row totals, column totals, and aggregate total are consistent. The differentiability and homogeneity assumptions matter; where they do not hold, a different allocation convention may be required.
Worked Example: Independent Cell Risks
Worked Example: Independent Cell Risks
A hypothetical insurer has two business units, Alpha and Beta, and two risk types, Property and Casualty. The cell losses are independent, with standard deviations in thousands of dollars as follows: Alpha–Property \(3\), Alpha–Casualty \(4\), Beta–Property \(5\), and Beta–Casualty \(\sqrt{50}\). Use standard-deviation capital, \(K=\sqrt{\sum_{i,j}\sigma_{ij}^2}\), and Euler allocation. Construct the table and check its margins. The model assumes independent, mean-zero cell losses and positive total standard deviation.
Method. For independent cell losses, \(K\) is the square root of the sum of cell variances. Differentiating with respect to a cell standard deviation gives \(\partial K/\partial \sigma_{ij}=\sigma_{ij}/K\). Therefore its Euler contribution is \(\sigma_{ij}^2/K\).
First calculate aggregate capital:
Thus \(K=10\) thousand dollars. Each cell contribution is its variance divided by \(10\): Alpha–Property \(9/10=0.90\); Alpha–Casualty \(16/10=1.60\); Beta–Property \(25/10=2.50\); and Beta–Casualty \(50/10=5.00\), all in thousands of dollars.
| Business unit | Property ($000) | Casualty ($000) | Row total ($000) |
|---|---|---|---|
| Alpha | 0.90 | 1.60 | 2.50 |
| Beta | 2.50 | 5.00 | 7.50 |
| Column total | 3.40 | 6.60 | 10.00 |
Check. The row totals are \(0.90+1.60=2.50\) and \(2.50+5.00=7.50\). The column totals are \(0.90+2.50=3.40\) and \(1.60+5.00=6.60\). Both views sum to \(10.00\), matching aggregate capital. The common mistake this method avoids is assigning each cell its standalone standard deviation; those amounts would not, in general, sum to portfolio standard-deviation capital.
Correlated Cells and Reconciliation
For correlated cell losses, capital cannot generally be calculated by adding cell variances alone. Under covariance aggregation, let \(s\) be the vector of cell standard deviations and \(R\) their correlation matrix. Then \(K=\sqrt{s^\mathsf{T}Rs}\). For a positive \(K\), differentiating this expression gives the Euler contribution of cell \(k\) as \(s_k(Rs)_k/K\). Equivalently, if \(\Sigma\) is the covariance matrix of the cell losses and \(x\) is the vector of exposure scale factors, the contribution is \(x_k(\Sigma x)_k/K\).
The reason the contributions sum to capital is that \(s^\mathsf{T}Rs=K^2\). Consequently, summing \(s_k(Rs)_k/K\) over cells gives \(s^\mathsf{T}Rs/K=K\). This also shows why the covariance terms matter: a cell’s marginal capital effect depends on its relationships with the other cells, not only on its own standard deviation.
Worked Example: A Correlation Between Business Units
A hypothetical insurer has Property and Casualty cells in business units Alpha and Beta. Their standard deviations, in thousands of dollars, are Alpha–Property \(3\), Alpha–Casualty \(5\), Beta–Property \(3\), and Beta–Casualty \(0\). The two Property cells have correlation \(1/3\); every other pair is uncorrelated. Use covariance aggregation and Euler allocation to form the table. Assume the stated correlations define a valid covariance structure.
Method. The zero-standard-deviation cell contributes zero. The variance of aggregate loss is the two Property variances, the Casualty variance, and twice the covariance between the Property cells:
Thus \(K=7\) thousand dollars. For either Property cell, the covariance-weighted sum \((Rs)_k\) is \(3+(1/3)(3)=4\), since its only nonzero correlation with another cell is with the other Property cell. Its Euler contribution is \(3(4)/7=12/7=1.7143\) thousand dollars, rounded to four decimal places. The Alpha–Casualty cell has no covariance with another nonzero cell, so its contribution is \(5(5)/7=25/7=3.5714\) thousand dollars. Beta–Casualty contributes zero.
| Business unit | Property ($000) | Casualty ($000) | Row total ($000) |
|---|---|---|---|
| Alpha | 1.7143 | 3.5714 | 5.2857 |
| Beta | 1.7143 | 0.0000 | 1.7143 |
| Column total | 3.4286 | 3.5714 | 7.0000 |
Check. Before rounding, the four cell contributions sum to \((12+25+12+0)/7=49/7=7\). The Property column is \(24/7\), the Casualty column is \(25/7\), Alpha’s row is \(37/7\), and Beta’s row is \(12/7\); either set of margins sums to \(49/7\). The common mistake is to treat the Property cells as independent and report \(9/K\) for each. Their positive covariance increases each cell’s marginal contribution; the correct numerator is \(s_k(Rs)_k\), not just \(s_k^2\).
When Only Row and Column Totals Are Available
Sometimes an analyst is given business-unit allocations and risk-type allocations but no model-based cell contributions. If both sets of margins add to the same aggregate capital, a table can be constructed to match them. One simple convention is to allocate each row total across columns in proportion to the column totals. This is an explicit allocation convention, not a consequence of Euler allocation; it assumes each business unit has the same risk-type mix in its allocated capital.
Summing this formula across a row gives \(R_i(\sum_j Q_j)/K=R_i\), and summing down a column gives \(Q_j(\sum_i R_i)/K=Q_j\). Thus it reconciles by construction. But other tables can have the same margins. Unless the underlying model or another allocation rule supplies cell-level information, the margins alone do not uniquely determine the cells.
Worked Example: Constructing a Table From Given Margins
A hypothetical insurer has \(K=100\) thousand dollars of aggregate capital. Its approved business-unit allocations are North \(40\) and South \(60\), while its risk-type allocations are Property \(30\), Casualty \(50\), and Market \(20\), all in thousands. Construct a table using proportional allocation across the columns and verify all margins.
Method. The row and column totals both sum to \(100\), so the proportional formula applies. For example, North–Property is \(40(30)/100=12\). The other five cells are North–Casualty \(40(50)/100=20\), North–Market \(40(20)/100=8\), South–Property \(60(30)/100=18\), South–Casualty \(60(50)/100=30\), and South–Market \(60(20)/100=12\), in thousands.
| Business unit | Property ($000) | Casualty ($000) | Market ($000) | Row total ($000) |
|---|---|---|---|---|
| North | 12 | 20 | 8 | 40 |
| South | 18 | 30 | 12 | 60 |
| Column total | 30 | 50 | 20 | 100 |
Check. The row totals are \(12+20+8=40\) and \(18+30+12=60\). The column totals are \(12+18=30\), \(20+30=50\), and \(8+12=20\). The cells sum to \(100\), matching \(K\). The common mistake is to treat this proportional table as if it revealed the model’s actual cell contributions. It only reconciles the supplied margins under the stated convention.
Practical Checks and Common Traps
A two-dimensional allocation is both a model output and a reconciliation tool. Before interpreting it, establish what each cell measures, what capital measure generated it, and whether cell contributions were calculated directly or imputed from margins. A clean table does not by itself establish that the allocation rule is economically appropriate.
- Check the aggregate first. The business-unit allocations and the risk-type allocations must each sum to the same \(K\). If they do not, a single table cannot match both sets of margins without adjustment or an explicit reconciliation item.
- Keep units consistent. Do not combine cell amounts in dollars with margins in thousands of dollars. Make the unit clear in the table headings.
- Do not mix allocation rules silently. A model-based Euler row total should not be combined with unrelated proportional cells unless the table’s convention explains how those figures relate.
- Round after summing. Displayed rounded cells may not add exactly to displayed margins. Keep sufficient precision in calculations, then state whether any small display difference is due to rounding.
For an allocation based on a differentiable, degree-one homogeneous capital function, construct the cell contributions under one consistent rule and aggregate them across both dimensions. If only margins are known, state the additional convention used to fill the table. In either case, verify the row totals, column totals, and grand total separately. This approach is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
Check Your Understanding
For each problem, calculate the requested cell contributions or reconciliation and check the stated margins.
- A hypothetical portfolio has two independent cells with standard deviations \(6\) and \(8\) thousand dollars, assigned to Unit A–Property and Unit B–Casualty respectively. Under standard-deviation capital and Euler allocation, find \(K\) and both cell contributions. (Answer: \(K=10\) thousand dollars; contributions \(3.60\) and \(6.40\) thousand dollars.)
- For a hypothetical two-cell portfolio, standard deviations are \(4\) and \(3\) thousand dollars and correlation is \(0.5\). Calculate covariance-aggregation capital and the Euler contribution of each cell. (Answer: \(K=\sqrt{37}=6.0828\) thousand dollars; contributions \(22/ \sqrt{37}=3.6157\) and \(15/\sqrt{37}=2.4652\) thousand dollars, rounded to four decimals.)
- A hypothetical insurer has business-unit allocations \(A=25\) and \(B=75\) thousand dollars, and risk-type allocations \(P=40\) and \(C=60\) thousand dollars. Using the proportional-margin convention, calculate the four cells and verify the margins. (Answer: \(A\)-\(P=10\), \(A\)-\(C=15\), \(B\)-\(P=30\), \(B\)-\(C=45\) thousand dollars; row totals \(25,75\), column totals \(40,60\), aggregate \(100\).)
- Under covariance aggregation, two nonzero cells have standard deviations \(2\) and \(2\) thousand dollars and correlation \(0.5\); a third cell has standard deviation \(3\) and is independent of both. Find aggregate capital and all three Euler contributions. (Answer: \(K=\sqrt{4+4+9+4}= \sqrt{21}=4.5826\) thousand dollars; contributions \(6/\sqrt{21}=1.3093\), \(6/\sqrt{21}=1.3093\), and \(9/\sqrt{21}=1.9640\) thousand dollars, rounded to four decimals.)
- A proposed table has business-unit row totals \(32\) and \(48\) thousand dollars, but risk-type column totals \(35\) and \(50\) thousand dollars. Can one table have both sets of margins without a reconciliation adjustment? (Answer: No. The row totals sum to \(80\) thousand dollars and the column totals to \(85\) thousand dollars, so the margins must first be reconciled.)