Why Euler Allocation Works
A capital allocation should assign the aggregate capital requirement to components in a way that reconciles to the total. Marginal analysis provides a sensitivity for each component: how much aggregate capital changes locally when that component’s exposure changes. Euler allocation turns each sensitivity into a capital contribution by weighting it by the component’s current exposure.
The key condition is that the capital function is homogeneous of degree one: scaling every component exposure by the same positive factor scales aggregate capital by that factor. When the function is also differentiable, Euler’s theorem proves that the exposure-weighted marginal contributions add exactly to aggregate capital. This framework is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
The partial derivative \(\partial C/\partial x_i\) is the marginal capital sensitivity for component \(i\), holding the other exposures fixed. It is a rate of change, not a capital amount. Multiplying it by the current exposure \(x_i\) gives a contribution in the same units as \(C\). The allocation uses the marginal sensitivity at the portfolio being evaluated; it is not, in general, a standalone capital calculation.
Euler’s Theorem and Full Allocation
Assume that the exposure vector lies in a domain where \(C\) is differentiable and where scaling the vector by positive factors is permitted. Homogeneity of degree one gives a relationship between capital at the current portfolio and capital along the ray formed by scaling all its exposures. Differentiate that relationship with respect to the scaling factor.
Define \(g(t)=C(tx_1,\ldots,tx_n)\) for \(t>0\). Degree-one homogeneity gives \(g(t)=tC(x_1,\ldots,x_n)\).
The right-hand side has derivative \(C(x_1,\ldots,x_n)\) with respect to \(t\).
At \(t=1\), differentiability gives \(g'(1)=\sum_{i=1}^{n}x_i\frac{\partial C}{\partial x_i}(x_1,\ldots,x_n)\).
Both expressions are \(g'(1)\), so the exposure-weighted contributions sum to aggregate capital.
The proof explains the allocation rule: the derivative measures the capital sensitivity to an exposure change, while the exposure-weighted derivative measures that sensitivity at the component’s current scale. Summing those current-scale contributions recovers the total because scaling the whole portfolio scales its capital in exactly the same proportion.
Worked Allocations
The examples use hypothetical capital functions and exposures. They illustrate Euler’s theorem directly; they are not claims about any particular insurer or capital model.
Worked Example: Two Components With Unequal Marginal Contributions
A hypothetical model reports capital in dollars and uses exposures \(x\) and \(y\), measured in exposure units. Its capital function is $$ C(x,y)=\frac{x^2+xy+y^2}{x+y},\qquad x+y>0. $$ The function is differentiable wherever \(x+y>0\). Its numerator has degree two and its denominator degree one, so \(C(tx,ty)=tC(x,y)\) for \(t>0\). Allocate capital at \(x=3\) and \(y=1\).
Step 1: Calculate aggregate capital. The numerator is \(3^2+(3)(1)+1^2=9+3+1=13\), and the denominator is \(3+1=4\). Thus \(C(3,1)=13/4=\$3.25\).
Step 2: Calculate the marginal sensitivities. Using the quotient rule, $$ \frac{\partial C}{\partial x} =\frac{(2x+y)(x+y)-(x^2+xy+y^2)}{(x+y)^2}, \qquad \frac{\partial C}{\partial y} =\frac{(x+2y)(x+y)-(x^2+xy+y^2)}{(x+y)^2}. $$ At \((3,1)\), these are \((7\cdot4-13)/16=15/16\) and \((5\cdot4-13)/16=7/16\), respectively.
Step 3: Weight each sensitivity by its exposure. The Euler contributions are \(3(15/16)=45/16=\$2.8125\) for \(x\) and \(1(7/16)=7/16=\$0.4375\) for \(y\). Rounded to the nearest cent, they are \(\$2.81\) and \(\$0.44\).
Check and method. Before rounding, \(45/16+7/16=52/16=13/4=C(3,1)\). This is an Euler allocation because the function is differentiable and degree one homogeneous. A common mistake is to add the marginal sensitivities, \(15/16+7/16=22/16\), and call that capital. Those are rates, not contributions; multiplying each by its exposure gives the amounts that reconcile.
Worked Example: Allocating Across Three Components
A hypothetical model has three positive exposures \(x,y,z\), and its capital function, in dollars, is $$ C(x,y,z)=\frac{x^2+y^2+z^2}{x+y+z},\qquad x+y+z>0. $$ It is differentiable on this domain and homogeneous of degree one: scaling all three exposures multiplies the numerator by \(t^2\) and the denominator by \(t\). Find the Euler allocation at \((2,1,1)\).
Step 1: Find the total. The numerator is \(2^2+1^2+1^2=6\), and the denominator is \(2+1+1=4\). Therefore \(C(2,1,1)=6/4=\$1.50\).
Step 2: Differentiate with respect to each exposure. The quotient rule gives $$ \frac{\partial C}{\partial x} =\frac{2x(x+y+z)-(x^2+y^2+z^2)}{(x+y+z)^2}, $$ with analogous expressions replacing \(x\) by \(y\) or \(z\) in the first term of the numerator. At \((2,1,1)\), the three sensitivities are \((2\cdot2\cdot4-6)/16=10/16=5/8\), \((2\cdot1\cdot4-6)/16=2/16=1/8\), and \(1/8\).
Step 3: Calculate contributions. Weighting by exposures gives \(2(5/8)=5/4=\$1.25\) for \(x\), \(1(1/8)=\$0.125\) for \(y\), and \(1(1/8)=\$0.125\) for \(z\). Expressed to the nearest cent, the amounts are \(\$1.25\), \(\$0.13\), and \(\$0.13\).
Check and method. The exact amounts sum to \(5/4+1/8+1/8=6/4=\$1.50\). The rounded amounts sum to \(\$1.51\) because each of the two smaller contributions was rounded up; full allocation holds before rounding. The method is to differentiate first and then weight each marginal sensitivity by its own exposure.
Worked Example: Scaling a Portfolio and Its Allocation
A hypothetical two-component model reports capital in millions of dollars and uses the differentiable, degree-one homogeneous function $$ C(x,y)=\frac{(x+2y)^2}{x+y},\qquad x+y>0. $$ Calculate the allocation at \((1,1)\), then determine the allocation if both exposures are tripled. The scaling condition matters: this comparison scales both components together, not just one.
Step 1: Allocate at \((1,1)\). Aggregate capital is \((1+2)^2/(1+1)=9/2=\$4.50\) million. Differentiating gives $$ \frac{\partial C}{\partial x} =\frac{x(x+2y)}{(x+y)^2}, \qquad \frac{\partial C}{\partial y} =\frac{(x+2y)(3x+2y)}{(x+y)^2}. $$ At \((1,1)\), the sensitivities are \(3/4\) and \(15/4\). The exposure-weighted contributions are \(1(3/4)=\$0.75\) million and \(1(15/4)=\$3.75\) million.
Step 2: Scale both exposures by three. The new exposures are \((3,3)\). Homogeneity gives \(C(3,3)=3C(1,1)=\$13.50\) million. The sensitivities remain \(3/4\) and \(15/4\), since the displayed derivative formulas are unchanged when both exposures are scaled by the same factor. The new contributions are \(3(3/4)=\$2.25\) million and \(3(15/4)=\$11.25\) million.
Check and method. At the original portfolio, \(0.75+3.75=4.50\) million; at the scaled portfolio, \(2.25+11.25=13.50\) million. Each component’s allocation triples along with aggregate capital. A common mistake is to triple only the total and leave the component allocations unchanged; that would no longer represent the Euler contributions at the scaled exposures.
Conditions, Interpretation, and Common Traps
The theorem is an exact reconciliation result for a differentiable, degree-one homogeneous capital function at the portfolio evaluated. It does not establish that the model is well specified, that the exposures are the right allocation basis, or that a component allocation must be positive. Those are separate modeling and interpretation questions. In particular, the derivative of capital with respect to an exposure can be negative, so a negative Euler contribution is possible even when all exposures are positive.
Differentiability is important because the formula uses a well-defined partial derivative at the portfolio. At points where the function is not differentiable, this derivative-based rule cannot be applied as stated. Likewise, if scaling all component exposures by \(t\) does not scale capital by \(t\), the degree-one condition fails and the proof does not establish full allocation. A numerical coincidence between an allocation sum and total capital at one portfolio is not a substitute for checking the assumptions.
| Item | Meaning | Role in Euler allocation |
|---|---|---|
| \(C(x_1,\ldots,x_n)\) | Aggregate capital at the current exposures | The total to be allocated |
| \(\partial C/\partial x_i\) | Marginal change in capital per unit change in exposure \(i\) | A sensitivity, not by itself a capital amount |
| \(x_i(\partial C/\partial x_i)\) | Exposure-weighted sensitivity for component \(i\) | The Euler contribution |
| Degree-one homogeneity and differentiability | Conditions on the capital function at the portfolio | Support the full-allocation identity |
The most important exam trap is to confuse marginal sensitivities with allocated capital. A sensitivity can be compared across components as a local rate, but summing rates generally does not produce the portfolio capital. The Euler contribution includes the component’s exposure, and it is those exposure-weighted values that sum to the total under the theorem’s conditions. Keep exact values through the reconciliation check, then round the reported amounts; rounding individual components can create a small displayed difference.
Check Your Understanding
For each problem, apply the stated assumptions and check the allocation against aggregate capital.
- A hypothetical differentiable capital function is \(C(x,y)=(x^2+xy+y^2)/(x+y)\), with \(x+y>0\), and capital is in dollars. At \((4,2)\), find the total and both Euler contributions. (Answer: total \(\$14/3\); contributions \(32/9\) and \(10/9\) dollars, which sum to \(\$14/3\).)
- A hypothetical differentiable, degree-one homogeneous function is \(C(x,y,z)=(x^2+y^2+z^2)/(x+y+z)\), with positive exposures and capital in dollars. At \((2,1,1)\), find the three marginal sensitivities and contributions. (Answer: sensitivities \(5/8,1/8,1/8\); contributions \(\$1.25,\$0.125,\$0.125\), totaling \(\$1.50\).)
- For \(C(x,y)=(x+2y)^2/(x+y)\), where \(x,y\) are positive exposure units and capital is in millions of dollars, calculate the total and Euler contributions at \((1,3)\). (Answer: total \(\$12.25\) million; contributions \(\$0.4375\) million and \(\$11.8125\) million.)
- Use the same function \(C(x,y)=(x+2y)^2/(x+y)\), in millions of dollars. If exposures change from \((1,1)\) to \((3,3)\), what are the new total and Euler contributions? (Answer: total \(\$13.50\) million; contributions \(\$2.25\) million and \(\$11.25\) million.)
- A differentiable, degree-one homogeneous capital function has exposures \((2,5,1)\). At that portfolio its marginal sensitivities are \(0.4,0.1,2\), respectively, and the capital amounts are in millions of dollars. Use Euler allocation to find each contribution and aggregate capital. (Answer: contributions \(\$0.8\), \(\$0.5\), and \(\$2.0\) million; aggregate capital is \(\$3.3\) million.)