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CAS Exam 9 · Capital and Performance Decisions · Tutorial 388 of 995

Business Performance Under Alternative Capital Allocations

Learn how a capital allocation changes reported line performance, how to check enterprise reconciliation, and why an allocation method can change business-line rankings.

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What You'll Learn

  • Calculate business-line RAROC and economic profit under alternative capital allocations
  • Distinguish raw marginal capital contributions from allocations that sum to total capital
  • Derive Euler allocations for a differentiable, positively homogeneous capital model
  • Reconcile business-line results to enterprise RAROC and economic profit
  • Identify when a change in reported line performance reflects allocation rather than changed economics

Why the Allocation Method Changes Reported Performance

An enterprise may have a single total capital requirement but several business lines to evaluate. To calculate each line’s reported return on capital, management must assign some portion of that total to each line. The assignment can change a line’s reported RAROC and economic profit even when its earnings, the enterprise’s total capital, and the enterprise’s total economic profit are unchanged.

The key distinction is between enterprise results and allocated line results. Enterprise results depend on total earnings and total capital. Line results depend additionally on the chosen allocation rule. A line’s reported performance is therefore not an allocation-free property of the line.

Definition: Let \(E_i\) be the earnings assigned to business line \(i\), let \(K_i\) be its allocated capital, let \(E=\sum_i E_i\) be total enterprise earnings, and let \(K\) be total enterprise capital. For positive allocated capital, line RAROC is \(E_i/K_i\). At a capital hurdle rate \(h\), line economic profit is \(E_i-hK_i\). Enterprise RAROC is \(E/K\), and enterprise economic profit is \(E-hK\).

These measures reconcile by addition only when the line capital allocations fully reconcile to enterprise capital: \(\sum_i K_i=K\). If earnings are also fully assigned, then

$$ \sum_i (E_i-hK_i)=E-hK. $$

Line RAROCs do not add. When \(K_i>0\) and allocations sum to \(K\), enterprise RAROC is their capital-weighted average:

$$ \frac{E}{K} =\frac{\sum_i E_i}{\sum_i K_i} =\sum_i \frac{K_i}{K}\frac{E_i}{K_i}. $$

Thus, changing allocations can change line RAROCs and their ranking without changing the enterprise ratio. For positive earnings, assigning a line more positive capital lowers its RAROC. For zero earnings, its RAROC remains zero; for negative earnings, assigning it more positive capital makes its RAROC less negative. For any fixed earnings and a positive hurdle, assigning more capital lowers economic profit because the capital charge increases.

Proportional Allocation: A Direct Reassignment

A proportional allocation distributes total capital using specified weights \(w_i\), where \(w_i\geq 0\) and \(\sum_i w_i=1\). The allocation is \(K_i=w_iK\). Weights may reflect a chosen exposure measure or another management basis; they are not automatically the same as each line’s contribution to portfolio risk. Once the weights are set, the results follow directly.

Worked Example: Changing Proportional Weights

A hypothetical insurer has two lines with annual earnings of \(12\) million dollars for Property and \(8\) million dollars for Specialty. Enterprise capital is \(100\) million dollars, and the capital hurdle is \(10\%\). Compare allocations of \(40/60\) and \(70/30\) between Property and Specialty. Use the allocated capital in millions of dollars.

Method. For each allocation, calculate line RAROC as earnings divided by allocated capital and line economic profit as earnings less \(0.10\) times allocated capital. Then check the enterprise totals.

Allocation 1: \(40/60\). Property receives \(0.40(100)=40\) million dollars; Specialty receives \(0.60(100)=60\) million dollars. Their RAROCs are \(12/40=0.3000\), or \(30.00\%\), and \(8/60=0.1333\), or \(13.33\%\). Their economic profits are \(12-0.10(40)=8\) million dollars and \(8-0.10(60)=2\) million dollars. The total is \(10\) million dollars.

Allocation 2: \(70/30\). Property receives \(70\) million dollars; Specialty receives \(30\) million dollars. Their RAROCs are \(12/70=0.1714\), or \(17.14\%\), and \(8/30=0.2667\), or \(26.67\%\). Their economic profits are \(12-7=5\) million dollars and \(8-3=5\) million dollars. The total remains \(10\) million dollars.

Check. In both cases, enterprise RAROC is \((12+8)/100=20.00\%\), and enterprise economic profit is \(20-0.10(100)=10\) million dollars. The line RAROCs and the line comparison change, but neither enterprise result changes.

Common mistake avoided. The change in Property’s RAROC does not mean its earnings or enterprise risk changed. Its earnings stayed at \(12\) million dollars; only its denominator changed. Do not interpret an allocation-driven change as new underlying business performance.

Marginal Contributions and the Reconciliation Issue

A common meaning of a line’s marginal capital contribution is the additional portfolio capital attributable to including that line, holding the rest of the portfolio fixed. If \(K(S)\) is the capital for a set of lines \(S\), the contribution of line \(i\) is \(K(S)-K(S\setminus\{i\})\). These leave-one-out contributions need not sum to \(K(S)\): diversification benefits and interactions can make the sum smaller or larger.

A different procedure is to add lines in a specified order, assigning the first line its standalone capital and each later line the increase in portfolio capital when it is added. Those sequential increments sum to total capital, but the allocation can depend on the order. Neither raw leave-one-out contributions nor sequential increments should be treated as interchangeable definitions.

Worked Example: Raw Marginal Contributions Versus a Reconciled Allocation

A hypothetical two-line portfolio has standalone capital of \(40\) million dollars for Line A and \(50\) million dollars for Line B. Combined portfolio capital is \(70\) million dollars. Annual earnings are \(10\) million dollars for A and \(12\) million dollars for B, and the hurdle rate is \(10\%\). Calculate the raw leave-one-out contributions, then compare them with sequential allocations in both possible orders.

Raw leave-one-out contributions. A’s contribution is \(70-50=20\) million dollars. B’s contribution is \(70-40=30\) million dollars. Together they total \(20+30=50\) million dollars, not the \(70\) million dollars of enterprise capital.

If these raw contributions are used as line denominators, A’s RAROC is \(10/20=50.00\%\), and B’s is \(12/30=40.00\%\). Their economic profits are \(10-0.10(20)=8\) million dollars and \(12-0.10(30)=9\) million dollars. The sum, \(17\) million dollars, is not enterprise economic profit: \(22-0.10(70)=15\) million dollars. The discrepancy occurs because only \(50\) million dollars of capital was allocated, so the line capital charges sum to \(5\) million dollars instead of the enterprise charge of \(7\) million dollars.

Sequential order A, then B. Assign A its standalone \(40\) million dollars. Adding B raises capital from \(40\) to \(70\), so assign B the increment \(30\) million dollars. The allocations sum to \(70\). The line RAROCs are \(10/40=25.00\%\) and \(12/30=40.00\%\); economic profits are \(10-4=6\) million dollars and \(12-3=9\) million dollars. Their sum is \(15\) million dollars.

Sequential order B, then A. Assign B its standalone \(50\) million dollars. Adding A raises capital from \(50\) to \(70\), so assign A \(20\) million dollars. The line RAROCs are \(10/20=50.00\%\) for A and \(12/50=24.00\%\) for B. Economic profits are \(10-2=8\) million dollars and \(12-5=7\) million dollars, again totaling \(15\) million dollars.

Check and lesson. Both sequential allocations sum to \(70\) million dollars and reconcile to enterprise economic profit. But the allocated line results differ by order: under the first order, B has the higher RAROC; under the second, A does. The raw leave-one-out contributions do not fully allocate capital at all. A reported ranking therefore needs to be interpreted alongside the allocation method and its reconciliation.

Common mistake avoided. Do not assume that leave-one-out contributions automatically form a complete allocation. Check their sum against total capital before using them to reconcile line economic profit.

Euler Allocation: A Risk-Sensitive Reconciliation

Euler allocation is available when portfolio capital is represented by a differentiable, positively homogeneous function of line risk exposures. Positive homogeneity of degree one means that scaling every exposure by the same positive factor \(c\) scales capital by \(c\): \(K(cx)=cK(x)\). Under those conditions, Euler’s theorem provides a complete allocation that reflects the model’s local sensitivity to each exposure.

Theorem: If \(K(x_1,\ldots,x_n)\) is differentiable and positively homogeneous of degree one, then its Euler allocation to line \(i\) is \(K_i=x_i\,\partial K/\partial x_i\). The allocations sum to total capital: \(\sum_i K_i=K\). This result does not directly apply at points where the capital function is not differentiable, or when the specified capital function is not positively homogeneous of degree one.

To justify the formula, define \(g(c)=K(cx_1,\ldots,cx_n)\). Homogeneity gives \(g(c)=cK(x)\), so \(g'(1)=K(x)\). By the chain rule, \(g'(1)=\sum_i x_i\,\partial K/\partial x_i\). Therefore \(\sum_i K_i=K\). This proves that the Euler contributions reconcile to total capital under the stated assumptions.

For a two-line standard-deviation capital model, let the standalone standard deviations be \(s_A\) and \(s_B\), with correlation \(\rho\). The portfolio capital is \(K=\sqrt{s_A^2+s_B^2+2\rho s_As_B}\). Differentiating this specified function gives

$$ K_A=s_A\frac{\partial K}{\partial s_A} =\frac{s_A(s_A+\rho s_B)}{K}, \qquad K_B=s_B\frac{\partial K}{\partial s_B} =\frac{s_B(s_B+\rho s_A)}{K}. $$

Adding the numerators yields \(s_A^2+s_B^2+2\rho s_As_B=K^2\); division by \(K\) gives \(K_A+K_B=K\). The derivatives and allocation formula depend on this particular model, not on an assumption that every capital model is a standard-deviation model.

Worked Example: Euler Capital and Line Performance

A hypothetical portfolio has standard-deviation exposures \(s_A=30\) and \(s_B=40\), measured in thousands of dollars, with correlation \(\rho=0.25\). Earnings are \(9\) thousand dollars for A and \(14\) thousand dollars for B. The hurdle rate is \(12\%\). Use the two-line standard-deviation model above.

Step 1: Calculate total capital. The cross term is \(2(0.25)(30)(40)=600\). Thus \(K=\sqrt{30^2+40^2+600}=\sqrt{3100}=55.6776436\) thousand dollars.

Step 2: Calculate Euler allocations. For A, the numerator is \(30(30+0.25(40))=30(40)=1200\), so \(K_A=1200/\sqrt{3100}=21.5526\) thousand dollars. For B, the numerator is \(40(40+0.25(30))=40(47.5)=1900\), so \(K_B=1900/\sqrt{3100}=34.1250\) thousand dollars. The sum is \(21.5526+34.1250=55.6776\) thousand dollars, matching \(K\) to the displayed precision.

Step 3: Calculate line results. A’s RAROC is \(9/21.5526=0.4176\), or \(41.76\%\), and its economic profit is \(9-0.12(21.5526)=6.4137\) thousand dollars, which rounds to \(6.414\) thousand dollars. B’s RAROC is \(14/34.1250=0.4103\), or \(41.03\%\), and its economic profit is \(14-0.12(34.1250)=9.9050\) thousand dollars, which rounds to \(9.905\) thousand dollars. Calculations use unrounded allocations; displayed monetary results are rounded to three decimals in thousands.

Check. Line economic profits total \(6.4137+9.9050=16.3187\) thousand dollars. Enterprise economic profit is \(23-0.12(55.6776436)=16.3187\) thousand dollars. Enterprise RAROC is \(23/55.6776436=0.4131\), or \(41.31\%\). The line RAROCs differ, but their capital-weighted average equals the enterprise RAROC.

Common mistake avoided. The Euler allocation to A is \(1200/\sqrt{3100}\), not a value obtained by dividing by a rounded or miscalculated total. Keep the same denominator \(K=\sqrt{3100}\) for both allocations, retain precision through the line calculations, and check that allocations sum to \(K\).

Interpreting the Results in a Decision

Alternative allocations answer different management questions. Proportional allocation is transparent and straightforward, but its weights need not reflect diversification. Marginal measures describe how total capital changes when a line is removed or added, but leave-one-out values may not reconcile and sequential increments may depend on order. Euler allocation is tied to a differentiable, homogeneous capital model and produces a full allocation consistent with that model’s marginal sensitivities.

None of these allocation choices changes total earnings or total capital in the examples. If allocations fully reconcile, total economic profit remains \(E-hK\), even though the distribution across lines changes. Consequently, comparing line RAROC or economic profit requires identifying the allocation rule, checking that all capital is allocated, and distinguishing the result of the rule from a change in the line’s actual earnings or risk.

  • Check whether \(\sum_i K_i=K\) before adding line economic profits.
  • For RAROC, state the numerator and allocated-capital denominator, and confirm allocated capital is positive.
  • For Euler allocation, verify differentiability and positive homogeneity before applying Euler’s theorem.
  • Do not infer that a higher line RAROC means higher total economic profit; the two measures use different comparisons.

This allocation-focused interpretation is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus: the calculation is only one part of the analysis; the reconciliation and meaning of the denominator matter as well.

Check Your Understanding

For each problem, calculate the requested line measures and check any enterprise reconciliation.

  1. A hypothetical enterprise has capital of \(80\) million dollars, earnings of \(6\) million dollars for Line A and \(10\) million dollars for Line B, and a hurdle rate of \(8\%\). Allocate capital \(40\%\) to A and \(60\%\) to B. Find each line’s RAROC and economic profit, and enterprise economic profit. (Answer: A RAROC \(18.75\%\), economic profit \(3.44\) million dollars; B RAROC \(20.83\%\), economic profit \(6.16\) million dollars; enterprise economic profit \(9.60\) million dollars.)
  2. Enterprise capital is \(100\) million dollars, with earnings of \(5\) million dollars and \(9\) million dollars. Allocations are \(25\) million dollars and \(75\) million dollars, and the hurdle rate is \(10\%\). Find both line RAROCs and economic profits, then their total. (Answer: RAROCs \(20.00\%\) and \(12.00\%\); economic profits \(2.50\) million dollars and \(1.50\) million dollars; total \(4.00\) million dollars.)
  3. Under the two-line standard-deviation model, \(s_A=18\), \(s_B=24\), and \(\rho=0.50\), with exposures in thousands of dollars. Find total capital and both Euler allocations. (Answer: \(K=\sqrt{1332}=36.4966\) thousand dollars; \(K_A=540/\sqrt{1332}=14.7952\) thousand dollars; \(K_B=792/\sqrt{1332}=21.7014\) thousand dollars.)
  4. Standalone capital for A and B is \(35\) and \(45\) million dollars, while combined capital is \(60\) million dollars. Find the raw leave-one-out contributions and the shortfall from total capital if they are used directly as allocations. (Answer: A contributes \(15\) million dollars and B contributes \(25\) million dollars; they allocate \(40\) million dollars, leaving \(20\) million dollars unallocated.)
  5. A capital function is differentiable and positively homogeneous of degree one. Explain why its Euler allocations sum to total capital, and name one condition that would prevent direct use of this result. (Answer: Euler’s theorem gives \(\sum_i x_i\,\partial K/\partial x_i=K\); direct use is not justified at a nondifferentiable point or for a capital function that is not positively homogeneous of degree one.)

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