Why Allocate Capital Relief?
A proposed reinsurance or other risk-transfer action may change the capital associated with several business lines. The portfolio calculation answers how much total capital changes; an allocation answers how that change is attributed across lines. That attribution can inform internal performance measures, pricing discussions, and governance, but it depends on the allocation method. It is not automatically a measure of which line caused the action to be valuable.
The central check is reconciliation: line-level relief allocations should add to the portfolio’s total capital relief if the method is intended to allocate the full change. Different methods can meet that check while telling different stories. The basis and assumptions therefore need to be stated alongside the numbers.
In this tutorial, exposures are nonnegative quantities on a consistent scale, and both portfolio capital values use the same capital measure and valuation basis. The action is represented by the change in retained exposures. Any premium, transaction cost, counterparty adjustment, or other effect must either be included consistently in the capital measure or analyzed separately; it should not be silently mixed into a capital-only allocation.
Proportional Allocation
A proportional allocation divides total relief according to an agreed set of baseline weights. A common basis is each line’s share of pre-action allocated capital. If the baseline allocations \(b_i\) are nonnegative and sum to pre-action capital, the weights are \(w_i=b_i/C(\mathbf{x})\), and relief is allocated as \(r_i=w_iR\). Another weighting basis may be appropriate, but it must be named and normalized so the weights sum to one.
This method is straightforward and reconciles by construction. It does not calculate each line’s incremental contribution to the change. In particular, proportional allocation can assign relief to a line whose own exposure did not change, because its share follows from the selected weights rather than a line-specific causal calculation.
Worked Example: Proportional Allocation Using Baseline Shares
A hypothetical insurer has pre-action allocated capital of $60 million for Property and $40 million for Casualty. These allocations sum to total pre-action capital of $100 million. A proposed risk-transfer action reduces portfolio capital to $88 million. Allocate the relief proportionally using the pre-action capital shares.
Method. First calculate total relief: \(R=100-88=12\) million. The weights are \(60/100=0.60\) for Property and \(40/100=0.40\) for Casualty.
Property receives \(0.60(12)=7.20\) million of attributed relief. Casualty receives \(0.40(12)=4.80\) million.
Check. The allocated amounts sum to \(7.20+4.80=12.00\) million, equal to the portfolio relief. The common mistake is to use the post-action capital of $88 million as the denominator while retaining the pre-action weights’ numerators. That would produce weights \(60/88\) and \(40/88\), which sum to \(100/88\), not one. The correction is to use a consistent weighting basis and normalize the weights.
Marginal Allocation and Path Dependence
A marginal allocation attributes a capital change using the effect of changing a particular line while holding the other exposures fixed at specified values. For a small change in line \(i\), the local capital effect is described by the partial derivative \(\partial C/\partial x_i\). For a finite action, contributions can be calculated by changing lines one at a time. The result depends on the order whenever the capital function is nonlinear or captures interactions among lines.
A more general description specifies a path \(\mathbf{x}(t)\), where \(t\) runs from 0 before the action to 1 after it. Along a differentiable path, the line-\(i\) contribution to relief is the integral of the line’s marginal capital sensitivity times its reduction in exposure:
The chain rule gives the reconciliation result: summing these contributions over lines gives \(C(\mathbf{x}(0))-C(\mathbf{x}(1))\). The path must be specified because another path can produce different line-level contributions even though the endpoints, and hence total relief, are unchanged.
Worked Example: Sequential Marginal Relief Depends on Order
For a hypothetical two-line portfolio, exposures \(x\) and \(y\) are measured in normalized units, and capital units are given by the illustrative smooth function \(C(x,y)=x+y+0.1xy\). The function is used only to demonstrate allocation mechanics; it is not asserted to be a standard regulatory formula. Exposures change from \((4,6)\) to \((2,3)\). Calculate line contributions by reducing one exposure at a time, first reducing \(x\) and then \(y\).
Starting capital. \(C(4,6)=4+6+0.1(4)(6)=10+2.4=12.4\).
Reduce \(x\) first. At \((2,6)\), capital is \(C(2,6)=2+6+0.1(2)(6)=8+1.2=9.2\). The relief attributed to \(x\) is \(12.4-9.2=3.2\).
Then reduce \(y\). At the endpoint \((2,3)\), capital is \(C(2,3)=2+3+0.1(2)(3)=5+0.6=5.6\). The relief attributed to \(y\) is \(9.2-5.6=3.6\).
Check. Total allocated relief is \(3.2+3.6=6.8\), and portfolio relief is \(12.4-5.6=6.8\). If instead \(y\) is reduced first, the intermediate capital is \(C(4,3)=4+3+1.2=8.2\). That order attributes \(12.4-8.2=4.2\) to \(y\), then \(8.2-5.6=2.6\) to \(x\). The same total reconciles, but the line split changes. The common mistake is to present sequential marginal contributions without disclosing the sequence; the correction is to state the path or use a method designed for the chosen allocation policy.
Euler Allocation
Euler allocation is a marginal method with a useful reconciliation property when the capital function is differentiable and homogeneous of degree one. Degree-one homogeneity means that scaling every exposure by the same positive factor \(a\) scales capital by that factor: \(C(a\mathbf{x})=aC(\mathbf{x})\). For example, a positively homogeneous risk measure applied to proportionally scaled losses has this property when the underlying model and capital definition are unchanged.
The reason Euler allocations sum to capital follows from differentiating homogeneity. Set \(g(a)=C(a\mathbf{x})\). Homogeneity gives \(g(a)=aC(\mathbf{x})\), so \(g'(1)=C(\mathbf{x})\). The chain rule also gives \(g'(1)=\sum_i x_i\,\partial C(\mathbf{x})/\partial x_i\). Equating the two expressions yields the Euler identity:
Applying the identity before and after the action shows why the relief allocation reconciles: \(\sum_i r_i=\sum_i c_i(\mathbf{x})-\sum_i c_i(\mathbf{x}')=C(\mathbf{x})-C(\mathbf{x}')\). Differentiability and degree-one homogeneity are substantive conditions, not optional notation. If either fails, this identity cannot be assumed; a different allocation rule or an explicitly justified extension is needed.
Worked Example: Euler Relief Under Independent Normal Risk
A hypothetical portfolio has two independent, zero-mean normal line losses with standard deviations \(x\) and \(y\), measured in millions of dollars. Capital is defined as one fixed positive multiple of aggregate standard deviation; the common multiple cancels from all calculations, so use \(C(x,y)=\sqrt{x^2+y^2}\). This capital function is differentiable for nonzero \((x,y)\) and homogeneous of degree one. Exposure changes from \((3,4)\) to \((1,2)\). Calculate Euler allocations and attributed relief. No interest or mortality assumptions apply to this one-period loss model.
Before the action. Capital is \(C(3,4)=\sqrt{3^2+4^2}=\sqrt{25}=5\). The partial derivatives are \(\partial C/\partial x=x/C\) and \(\partial C/\partial y=y/C\). Thus the Property-like first line allocation is \(3(3/5)=9/5=1.8000\), and the second line allocation is \(4(4/5)=16/5=3.2000\). They sum to \(5.0000\).
After the action. Capital is \(C(1,2)=\sqrt{1^2+2^2}=\sqrt{5}=2.2361\), rounded to four decimals. The Euler allocations are \(1(1/\sqrt{5})=1/\sqrt{5}=0.4472\) and \(2(2/\sqrt{5})=4/\sqrt{5}=1.7889\). Their unrounded sum is \(\sqrt{5}\), as required.
Attributed relief. The first line receives \(1.8000-0.4472=1.3528\) million, and the second receives \(3.2000-1.7889=1.4111\) million, with rounding applied only to the displayed final values. Total portfolio relief is \(5-\sqrt{5}=2.7639\) million. The displayed line reliefs sum to \(2.7639\) million.
Check and common mistake. The pre-action and post-action Euler allocations each sum to their corresponding capital values, and their differences sum to portfolio relief. A tempting error is to allocate post-action capital as \(x^2/C\) and \(y^2/C\). That expression is not Euler allocation: the correct terms are \(x\,\partial C/\partial x=x^2/C\) and \(y\,\partial C/\partial y=y^2/C\), which in this particular model do equal those expressions; however, one must evaluate them using \(C=\sqrt{x^2+y^2}\). At \((1,2)\), they are \(1/\sqrt{5}\) and \(4/\sqrt{5}\), not \(0.2\) and \(0.8\). The latter divide squared exposures by \(x^2+y^2\), producing shares rather than capital allocations.
Choosing and Reviewing an Allocation
Each method answers a different attribution question. Proportional allocation applies a stated baseline split to total relief. Sequential marginal allocation measures changes along a stated sequence or path. Euler allocation attributes the capital level using marginal sensitivities and, under its conditions, reconciles without a sequence for a single portfolio state. For a before-and-after Euler relief allocation, subtract the line’s post-action Euler amount from its pre-action amount.
| Method | Basis | Key condition or limitation |
|---|---|---|
| Proportional | Normalized shares from a selected baseline | Reconciles when weights sum to one; does not measure marginal causation |
| Marginal, sequential | Changes in capital as lines are changed in sequence | Can be order-dependent when capital is nonlinear |
| Euler | Exposure multiplied by the capital function’s partial derivative | Requires differentiability and degree-one homogeneity for the stated reconciliation result |
A review should check that the total capital change is calculated on a consistent basis, that the allocation method is named, and that the line allocations reconcile at the precision used. If the action changes the capital model, the risk measure, or the treatment of costs, a simple difference of allocations may not represent a like-for-like comparison. Negative line-level relief is possible under some functions or allocation rules: it indicates an attributed increase, not necessarily a calculation error. It should be investigated and explained rather than silently reset to zero.
These distinctions matter when communicating a risk-transfer decision. A proportional split may be useful as a stable policy convention; a marginal path can describe a particular implementation sequence; and Euler attribution can provide a sensitivity-based allocation when its assumptions fit. None should be labeled “the” contribution without describing its basis. That discipline keeps line-level reporting connected to, but distinct from, the portfolio-level capital effect aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
Check Your Understanding
For each problem, identify the allocation basis and verify the stated capital arithmetic.
- A portfolio has baseline allocated capital of $45 million and $75 million by line. A proposed action reduces total capital from $120 million to $108 million. Allocate the relief proportionally using those baseline amounts. (Answer: $4.50 million and $7.50 million.)
- Use the illustrative capital function \(C(x,y)=x+y+0.05xy\). Exposures change from \((2,4)\) to \((1,2)\), first reducing \(x\), then \(y\). Find the two sequential marginal relief amounts and check the total. (Answer: \(x\) receives 1.40 units, \(y\) receives 2.10 units; total relief is 3.50 units.)
- For \(C(x,y)=\sqrt{x^2+y^2}\), exposures change from \((6,8)\) to \((3,4)\). Find the Euler allocations before and after, then the relief by line. (Answer: before, 3.60 and 6.40; after, 1.80 and 3.20; relief, 1.80 and 3.20, totaling 5.00.)
- Using \(C(x,y)=x+y+0.1xy\), change exposures from \((3,5)\) to \((1,2)\). Calculate sequential marginal relief when \(x\) is reduced first, and then when \(y\) is reduced first. (Answer: total relief is 6.50 units in either order; reducing \(x\) first attributes 2.50 to \(x\) and 4.00 to \(y\), while reducing \(y\) first attributes 4.20 to \(y\) and 2.30 to \(x\).)
- State why Euler allocation can be used to reconcile capital by line, and name two conditions required for the identity used here. (Answer: the Euler identity makes the line allocations sum to total capital; the capital function must be differentiable and homogeneous of degree one.)