Integrated Capital Allocation Workshop

Why Compare Allocation Methods?

A portfolio’s total economic capital is one amount, but managers often need to attribute it to business units or risk-bearing components. The allocation method matters: it determines how much of the portfolio’s capital each unit is said to use and how the portfolio’s diversification benefit is attributed. Several methods can be internally consistent while producing different unit-level results, because they answer different allocation questions.

This workshop compares proportional, marginal, Euler, and Shapley allocations. The examples use hypothetical capital models and values. Capital amounts are stated in millions of dollars unless noted otherwise. The methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $N$ be the set of portfolio units, and let $C(S)$ be the capital measure for the portfolio containing units in subset $S\subseteq N$, with $C(\varnothing)=0$. The total portfolio capital is $C(N)$. If $C(\{i\})$ is unit $i$'s standalone capital, the diversification benefit is $$ D=\sum_{i\in N}C(\{i\})-C(N). $$ When $D>0$, the combined portfolio requires less capital than the sum of standalone capitals under the specified measure.

Proportional Allocation and Diversification Attribution

A proportional allocation distributes total capital according to each unit’s share of the sum of standalone capitals. It is straightforward and uses only standalone values and portfolio total capital; it does not require a differentiable capital model or capital values for every possible subset of units.

A_i^{\mathrm{prop}}=C(N)\frac{C(\{i\})}{\sum_{j\in N}C(\{j\})}.

The allocations sum to total capital whenever the denominator is positive. The method also gives a simple way to attribute diversification: assign each unit the same proportion of total diversification benefit as its share of standalone capital. This is an attribution rule, not a claim that the unit caused precisely that share of the benefit.

Worked Example: Proportional Allocation and Diversification

A hypothetical insurer has three units with standalone capital of $120 million, $80 million, and $50 million. The capital measure for the combined portfolio is $200 million. Calculate proportional allocations and each unit’s attributed diversification benefit.

Step 1: Sum the standalone capital. $$ 120+80+50=250. $$

Step 2: Calculate total diversification benefit. $$ D=250-200=50. $$

Step 3: Allocate portfolio capital proportionally. The units’ standalone shares are $120/250=0.48$, $80/250=0.32$, and $50/250=0.20$. Thus the allocations are $200(0.48)=96$, $200(0.32)=64$, and $200(0.20)=40$ million dollars.

Step 4: Attribute diversification. Subtract each allocation from its standalone capital: $120-96=24$, $80-64=16$, and $50-40=10$ million dollars.

Check. The capital allocations sum to $96+64+40=200$. The attributed diversification amounts sum to $24+16+10=50$, equal to the total benefit. The method avoids the mistake of allocating the standalone total of $250 million instead of the portfolio total of $200 million.

Marginal Sensitivities and Euler Contributions

For a differentiable capital model, let $w_i$ denote the exposure or scale of unit $i$, and write total capital as $C(w_1,\ldots,w_n)$. The marginal capital sensitivity for unit $i$ is the partial derivative $\partial C/\partial w_i$: the local change in total capital per unit increase in that exposure, holding other exposures fixed. It is a sensitivity, not generally an allocation of the current capital.

The Euler contribution multiplies that sensitivity by the unit’s current exposure. The distinction is important when exposures are not all one, or when comparing sensitivity units with dollar capital amounts.

M_i=\frac{\partial C}{\partial w_i}, \qquad A_i^{\mathrm{Euler}}=w_i\frac{\partial C}{\partial w_i}.

Euler allocations have a useful reconciliation property under specific conditions. If $C$ is differentiable and positively homogeneous of degree one, meaning $C(tw)=tC(w)$ for every $t>0$, Euler’s theorem gives $$ \sum_i w_i\frac{\partial C}{\partial w_i}=C(w). $$ Thus the Euler contributions fully allocate total capital. Differentiability and degree-one homogeneity are essential to this result; a capital measure that is not differentiable at the current portfolio, or is not degree-one homogeneous, need not have this property in this form.

Justification: Define $g(t)=C(tw)$. By degree-one homogeneity, $g(t)=tC(w)$, so $g'(1)=C(w)$. By the chain rule, $g'(1)=\sum_i w_i(\partial C/\partial w_i)$. Equating the two expressions proves the Euler allocation identity.

Worked Example: Marginal Sensitivities Versus Euler Contributions

A hypothetical portfolio has two independent, zero-mean loss components with standard deviations 3 and 4. Their exposures are $w_1=2$ and $w_2=1$. Define capital as the standard deviation of the aggregate loss. Under independence, calculate total capital, the marginal sensitivities, and the Euler contributions.

Method and assumptions. The components are independent and have finite variances. For this model, $C(w_1,w_2)=\sqrt{9w_1^2+16w_2^2}$. No normality assumption is needed when capital is defined as standard deviation.

Step 1: Calculate portfolio capital. $$ C(2,1)=\sqrt{9(2^2)+16(1^2)} =\sqrt{36+16} =\sqrt{52} \approx 7.2111. $$

Step 2: Differentiate the capital function. The marginal sensitivities are $$ \frac{\partial C}{\partial w_1}=\frac{9w_1}{C}, \qquad \frac{\partial C}{\partial w_2}=\frac{16w_2}{C}. $$ At $(w_1,w_2)=(2,1)$, these equal $18/\sqrt{52}\approx2.4962$ and $16/\sqrt{52}\approx2.2188$, respectively.

Step 3: Calculate the Euler contributions. $$ A_1^{\mathrm{Euler}}=2\left(\frac{18}{\sqrt{52}}\right) =\frac{36}{\sqrt{52}}\approx4.9923, \qquad A_2^{\mathrm{Euler}}=1\left(\frac{16}{\sqrt{52}}\right) =\frac{16}{\sqrt{52}}\approx2.2188. $$

Check. The contributions sum to $52/\sqrt{52}=\sqrt{52}\approx7.2111$, as required. The common mistake is to report the two marginal sensitivities as the allocations: they sum to about 4.7150, not to portfolio capital. Multiplying each sensitivity by its exposure produces the Euler contributions.

Shapley Allocation: Averaging Across Entry Orders

The Shapley allocation uses the capital of every possible subset of units. For a given unit, consider every possible ordering of the units and ask how much total capital increases when that unit joins the units that precede it. The Shapley contribution is the average of these incremental contributions over all orderings. This makes the method sensitive to the portfolio’s coalition values, rather than only its standalone values or its local behavior at the current portfolio.

For a portfolio with $n$ units, the Shapley allocation to unit $i$ is $$ \phi_i=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(n-|S|-1)!}{n!} \left[C(S\cup\{i\})-C(S)\right]. $$ The bracketed term is the capital added by unit $i$ when it joins subset $S$. The coefficient is the fraction of all unit orderings in which exactly the members of $S$ precede $i$.

For three units, predecessor sets of sizes zero, one, and two have weights $1/3$, $1/6$, and $1/3$, respectively. Each particular predecessor set of size one has weight $1/6$. The size-one weight applies separately to each of the two possible one-unit subsets.

Worked Example: Shapley Allocation for Three Units

A hypothetical portfolio has units $A$, $B$, and $C$. Its capital values are $C(A)=3$, $C(B)=4$, $C(C)=4$, $C(AB)=5$, $C(AC)=5$, $C(BC)=\sqrt{32}$, and $C(ABC)=\sqrt{41}$, all in millions of dollars. Calculate the Shapley allocations.

Step 1: Allocate to $A$. Its marginal increments are $3$ when it enters first, $5-4=1$ after $B$, $5-4=1$ after $C$, and $\sqrt{41}-\sqrt{32}$ after $B$ and $C$. Therefore, $$ \phi_A=\frac{1}{3}(3)+\frac{1}{6}(1+1) +\frac{1}{3}(\sqrt{41}-\sqrt{32}) \approx1.5821. $$

Step 2: Allocate to $B$. Its increments are $4$ when it enters first, $5-3=2$ after $A$, $\sqrt{32}-4$ after $C$, and $\sqrt{41}-5$ after $A$ and $C$: $$ \phi_B=\frac{1}{3}(4)+\frac{1}{6}\left(2+\sqrt{32}-4\right) +\frac{1}{3}(\sqrt{41}-5) \approx2.4105. $$

Step 3: Allocate to $C$. By the same calculation, with its own predecessor increments, $$ \phi_C=\frac{1}{3}(4)+\frac{1}{6}\left(5-3+\sqrt{32}-4\right) +\frac{1}{3}(\sqrt{41}-5) \approx2.4105. $$

Check. The allocations sum to approximately $1.5821+2.4105+2.4105=6.4031$, which is $\sqrt{41}$ to four decimal places. Using $1/6$ for the size-two predecessor set, or using $1/3$ for each individual size-one set, would assign incorrect weights and generally break the reconciliation.

Integrated Comparison of the Four Methods

The following example applies all four methods to the same hypothetical portfolio. For independent losses, capital measured by standard deviation gives a transparent coalition model; the model is used here to compare allocation rules, not to imply that every economic capital measure is a standard deviation.

Worked Example: Comparing Proportional, Marginal, Euler, and Shapley Results

Three independent, zero-mean hypothetical loss components have standard deviations 3, 4, and 4 million dollars. Each exposure is one. Capital is the standard deviation of aggregate loss. Calculate all four allocations and the diversification attribution.

Step 1: Establish the capital values. The standalone capitals are $3$, $4$, and $4$, for a standalone total of $11$. Pair and portfolio capitals are $C(AB)=\sqrt{3^2+4^2}=5$, $C(AC)=5$, $C(BC)=\sqrt{4^2+4^2}=\sqrt{32}$, and $C(ABC)=\sqrt{3^2+4^2+4^2}=\sqrt{41}\approx6.4031$. Thus total diversification benefit is $11-\sqrt{41}\approx4.5969$ million dollars.

Step 2: Calculate proportional allocations. The standalone shares are $3/11$, $4/11$, and $4/11$. The allocations are $$ \left(\sqrt{41}\frac{3}{11},\sqrt{41}\frac{4}{11},\sqrt{41}\frac{4}{11}\right) \approx(1.7463,\ 2.3284,\ 2.3284). $$

Step 3: Calculate marginal and Euler contributions. At unit exposures, the marginal sensitivities are $9/\sqrt{41}$, $16/\sqrt{41}$, and $16/\sqrt{41}$, or approximately $1.4056$, $2.499$, and $2.499$. Since each exposure is one, the Euler contributions have the same numerical values: $$ \left(\frac{9}{\sqrt{41}},\frac{16}{\sqrt{41}},\frac{16}{\sqrt{41}}\right) \approx(1.4056,\ 2.499,\ 2.499). $$ Their sum is $(9+16+16)/\sqrt{41}=\sqrt{41}$, so they fully allocate the portfolio capital.

Step 4: Calculate Shapley contributions. For unit $A$, $$ \phi_A=\frac{3}{3}+\frac{(5-4)+(5-4)}{6} +\frac{\sqrt{41}-\sqrt{32}}{3} \approx1.5821. $$ For unit $B$, $$ \phi_B=\frac{4}{3}+\frac{(5-3)+(\sqrt{32}-4)}{6} +\frac{\sqrt{41}-5}{3} \approx2.4105. $$ By symmetry, $\phi_C=\phi_B\approx2.4105$.

Step 5: Attribute diversification under Shapley. Subtract each Shapley allocation from its standalone capital. The attributed benefits are approximately $3-1.5821=1.4179$, $4-2.4105=1.5895$, and $4-2.4105=1.5895$ million dollars. They sum to $4.5969$, the total diversification benefit.

Check and interpretation. The four allocations each sum to total capital (allowing for rounding). Proportional allocation assigns capital by standalone share; Euler reflects local exposure sensitivities; Shapley averages incremental contributions across all entry orders. The methods differ because their attribution principles differ, not because one of these reconciliations is arithmetically wrong. The common mistake is to compare marginal sensitivities directly with capital allocations without checking whether they have been scaled by exposure.

How to Choose and Check a Method

An allocation is most useful when its purpose and assumptions are explicit. Proportional allocation is easy to communicate when only standalone and total capital are available. Marginal sensitivity is useful for describing the local impact of a small exposure change, but is not necessarily a full allocation. Euler allocation is appropriate when the capital model is differentiable and degree-one homogeneous at the portfolio being analyzed. Shapley allocation provides a systematic average of discrete incremental effects, but requires the capital measure for every relevant subset and can become computationally intensive as the number of units grows.

Before interpreting unit-level results, check the following:

Confirm the object being allocated. Specify the portfolio, capital measure, units, and exposure definition.
Check full allocation. Sum the contributions and compare them with $C(N)$; explain any difference rather than silently treating rounded values as exact.
Separate capital from diversification attribution. A unit’s allocated capital and its attributed diversification benefit are related, but they are not the same quantity.
Match the method to its conditions. Euler reconciliation depends on differentiability and degree-one homogeneity. Proportional allocation needs a positive standalone total. Shapley requires coalition capital values.
Interpret differences rather than averaging them away. Different methods can be reconciled and still assign different amounts because they encode different views of contribution.

Key takeaway: Proportional, Euler, and Shapley allocations can each fully allocate portfolio capital, but they do so using different information and attribution principles. A marginal sensitivity describes local change; it becomes an Euler capital contribution only after multiplication by the corresponding exposure.

Check Your Understanding

For each problem, calculate the requested amounts and check the stated reconciliation.

A hypothetical portfolio has standalone capitals of $70 million, $50 million, and $30 million, and total capital of $120 million. Find proportional allocations and the diversification benefit attributed to each unit. (Answer: allocations $56 million, $40 million, and $24 million; attributed benefits $14 million, $10 million, and $6 million; total benefit $30 million.)
Two independent, zero-mean loss components have standard deviations 6 and 8, with exposures $w_1=1$ and $w_2=2$. Capital is aggregate standard deviation. Find total capital, marginal sensitivities, and Euler contributions. (Answer: capital $\sqrt{292}\approx17.0880$; sensitivities $36/\sqrt{292}\approx2.1061$ and $128/\sqrt{292}\approx7.4906$; Euler contributions approximately $2.1061$ and $14.9812$, summing to $\sqrt{292}$ before rounding.)
Three units have standalone capitals $C(A)=40$, $C(B)=30$, $C(C)=20$; pair capitals $C(AB)=55$, $C(AC)=48$, $C(BC)=42$; and total capital $C(ABC)=70$. Calculate the Shapley allocations and total diversification benefit. (Answer: $\phi_A=31.5$, $\phi_B=23.5$, $\phi_C=15$; total diversification benefit $20$.)
A portfolio has standalone capitals of $50 million, $30 million, and $20 million, with total capital of $80 million. Calculate proportional capital allocations and the proportional diversification attribution. (Answer: allocations $40 million, $24 million, and $16 million; attributed benefits $10 million, $6 million, and $4 million; total benefit $20 million.)
For a three-unit Shapley calculation, state the weight for a predecessor set of size zero, for each particular predecessor set of size one, and for a predecessor set of size two. (Answer: $1/3$, $1/6$, and $1/3$, respectively; there are two distinct size-one predecessor sets, each weighted $1/6$.)

What You'll Learn