Numerical Estimation of Marginal and Euler Contributions

Why Estimate Capital Contributions Numerically?

A capital model may produce portfolio capital through a simulation engine rather than a convenient closed-form formula. Even when the model can be run at different exposure levels, differentiating its output analytically may be impractical. A numerical perturbation provides an estimate of how portfolio capital changes when one component's exposure changes, while holding the other components fixed.

The estimate is useful only if its design matches the question. The perturbation must change the intended exposure, the compared calculations should use the same simulated scenarios, and the result should be checked for sensitivity to the perturbation size. These checks help distinguish a genuine marginal effect from simulation noise or numerical instability.

Definition: Let \(L(\mathbf{w})=\sum_{i=1}^n w_iX_i\) be portfolio loss, where \(X_i\) is the simulated loss from component \(i\) at unit exposure and \(w_i\) is its exposure weight. For a capital function \(\rho(\mathbf{w})=\rho(L(\mathbf{w}))\), the marginal capital contribution of component \(i\) at the current weights is \(M_i=\partial \rho(\mathbf{w})/\partial w_i\), when that derivative exists. Its Euler contribution is \(C_i=w_iM_i\). A marginal contribution is per unit of weight; an Euler contribution is the marginal contribution multiplied by the component's current weight.

In a simulation, the capital function is usually evaluated on a finite set of scenarios. Write the simulated component losses as \(X_{ij}\), where \(j=1,\ldots,N\) indexes scenarios. The portfolio loss in scenario \(j\) is \(L_j(\mathbf{w})=\sum_i w_iX_{ij}\). With equal scenario weights, a simulated capital estimate can be written \(\hat{\rho}(\mathbf{w})\). Unequal scenario probabilities can also be used, provided the same weighting convention is applied consistently to every perturbed run.

Finite Differences and Scenario Pairing

For a small positive step \(h\), a forward difference estimates the marginal contribution as \([\hat{\rho}(\mathbf{w}+h\mathbf{e}_i)-\hat{\rho}(\mathbf{w})]/h\), where \(\mathbf{e}_i\) changes only component \(i\). A backward difference uses \([\hat{\rho}(\mathbf{w})-\hat{\rho}(\mathbf{w}-h\mathbf{e}_i)]/h\). If the capital function is differentiable around the current weights, the central difference is generally more accurate for a given small step:

\widehat{M}_i(h)= \frac{\hat{\rho}(\mathbf{w}+h\mathbf{e}_i)-\hat{\rho}(\mathbf{w}-h\mathbf{e}_i)}{2h}, \qquad \widehat{C}_i(h)=w_i\widehat{M}_i(h).

The plus and minus calculations should use common random numbers: the same underlying simulated scenarios and random inputs are used in both runs. Only the exposure being tested changes. This pairing makes the difference more informative because scenario-to-scenario variation is shared rather than added independently to each result. If the paired capital estimates have positive covariance, then

\operatorname{Var}(\hat{\rho}_+-\hat{\rho}_-) = \operatorname{Var}(\hat{\rho}_+)+\operatorname{Var}(\hat{\rho}_-) -2\operatorname{Cov}(\hat{\rho}_+,\hat{\rho}_-).

The covariance term reduces the variance of the difference when it is positive. Common random numbers do not remove finite-sample error, but they often make the perturbation estimate substantially less noisy than comparing unrelated simulation runs.

Worked Example: Central Differences and a Step-Size Check

A hypothetical two-component portfolio has four equally weighted simulated scenarios. Component A has losses \(X_A=(0,2,4,2)\), and component B has losses \(X_B=(2,0,2,4)\). Both exposure weights are 1. The model's capital estimate is the simulated mean loss plus one simulated standard deviation, where variance is calculated with divisor \(N=4\). Estimate A's marginal and Euler contributions using central differences with \(h=0.10\) and \(h=0.01\).

Method and baseline. The portfolio scenario losses are \(L=(2,2,6,6)\). Their mean is \(4\); their deviations from the mean are \((-2,-2,2,2)\), so the variance is \((4+4+4+4)/4=4\), and the standard deviation is \(2\). Thus baseline capital is \(\hat{\rho}=4+2=6\).

When A's weight changes from 1 to \(1+h\), the scenario losses become \((2,2+2h,6+4h,6+2h)\). The mean is \(4+2h\). The baseline variance is 4, the covariance of \(L\) with \(X_A\) is 2, and the variance of \(X_A\) is 2. Expanding the variance of \(L+hX_A\) therefore gives \(4+4h+2h^2\). The perturbed capital is

\hat{\rho}(1+h,1)=4+2h+\sqrt{4+4h+2h^2}.

Step \(h=0.10\). At \(h=0.10\), the variance is \(4+0.4+0.02=4.42\), the mean is \(4.20\), and capital is \(4.20+\sqrt{4.42}=6.30238\). At \(h=-0.10\), the variance is \(4-0.4+0.02=3.62\), the mean is \(3.80\), and capital is \(3.80+\sqrt{3.62}=5.70263\). Therefore,

\widehat{M}_A(0.10)=\frac{6.30238-5.70263}{0.20}=2.99875.

Step \(h=0.01\). At \(h=0.01\), the variance is \(4.0402\), the mean is \(4.02\), and capital is \(4.02+\sqrt{4.0402}=6.03002\). At \(h=-0.01\), the variance is \(3.9602\), the mean is \(3.98\), and capital is \(3.98+\sqrt{3.9602}=5.97003\). Thus \(\widehat{M}_A(0.01)=(6.03002-5.97003)/0.02=3.0000\), to four decimal places. Since the current weight is 1, the Euler contribution is also approximately \(3.0000\).

The two estimates are close, and the smaller-step estimate agrees with the derivative. The common mistake is to divide the central capital difference by \(h\) rather than \(2h\). The two evaluations are \(2h\) apart, so the denominator must be \(2h\).

A Smooth Capital Function: Derivative and Euler Check

A useful way to check simulation estimates is to compare them with a formula in a model where the derivative is available. Consider the capital function \(\rho(L)=E[L]+k\sqrt{\operatorname{Var}(L)}\), with a fixed constant \(k\), finite second moments, and positive standard deviation. For a portfolio \(L=\sum_iw_iX_i\), assume the simulated distribution is held fixed while weights change. Differentiating the mean and variance gives the marginal contribution

\frac{\partial \rho}{\partial w_i} =E[X_i]+k\frac{\operatorname{Cov}(L,X_i)} {\sqrt{\operatorname{Var}(L)}}.

The formula requires positive portfolio standard deviation; at zero standard deviation its displayed quotient is undefined. In practice, the corresponding simulation estimate replaces expectations, variances, and covariances by consistently calculated scenario estimates. The direct finite-difference procedure remains useful when a model's capital output is available but this analytic expression is not.

Justification: Since \(\partial L/\partial w_i=X_i\), the derivative of \(E[L]\) with respect to \(w_i\) is \(E[X_i]\). Also, \(\operatorname{Var}(L)=E[L^2]-E[L]^2\), whose derivative is \(2E[LX_i]-2E[L]E[X_i]=2\operatorname{Cov}(L,X_i)\). Applying the chain rule to the square root gives \(\partial\sqrt{\operatorname{Var}(L)}/\partial w_i=\operatorname{Cov}(L,X_i)/\sqrt{\operatorname{Var}(L)}\). Multiplying this derivative by \(k\) and adding the mean derivative gives the stated result.

When \(\rho\) is positively homogeneous of degree one, meaning \(\rho(c\mathbf{w})=c\rho(\mathbf{w})\) for \(c>0\), Euler's theorem gives \(\rho(\mathbf{w})=\sum_iw_i(\partial\rho/\partial w_i)\) at differentiable points. The mean-plus-standard-deviation function above has this property for positive scaling of all portfolio weights. Thus estimated Euler contributions should approximately sum to estimated portfolio capital when the assumptions hold. A discrepancy can signal finite-difference error, simulation noise, inconsistent scenarios, or a failure of the homogeneity or differentiability assumptions; reconciliation is a diagnostic, not a reason to force the estimates to match.

Worked Example: Paired Scenarios Versus Independent Runs

A hypothetical model reports expected loss as its capital measure. Four common scenarios produce other-portfolio losses \(B=(80,100,120,140)\) and component losses \(A=(10,20,30,40)\). At weight 1 for A, estimate its marginal contribution by increasing the weight to 1.10. Then compare with a calculation that mistakenly compares the perturbed run with an unrelated baseline run.

Paired calculation. At weight 1, total losses are \(B+A=(90,120,150,180)\), with mean \(135\). At weight 1.10, using the same scenarios, losses are \(B+1.10A=(91,122,153,184)\), with mean \(137.5\). The estimated marginal contribution is \((137.5-135)/0.10=25\). This also equals the simulated mean component loss, \((10+20+30+40)/4=25\), as expected for an expected-loss measure. At the current weight of 1, the Euler contribution is 25.

Unpaired comparison. Suppose an unrelated baseline run happens to have total losses \((100,120,170,210)\), with mean \(150\), while the perturbed run still has mean \(137.5\). Comparing those means gives \((137.5-150)/0.10=-125\), not 25. The unrelated scenarios have overwhelmed the exposure effect. The correct method pairs each perturbation with the same underlying scenarios; the common-random-number estimate is 25.

The trap is to treat two noisy simulation outputs as comparable merely because they estimate the same model. For a marginal estimate, the intended change must be isolated. Pairing does that here; independent runs can produce a result with the wrong magnitude or even the wrong sign.

Worked Example: Estimating and Reconciling Two Euler Contributions

A hypothetical portfolio uses four equally weighted simulated scenarios with \(X_A=(0,2,4,2)\) and \(X_B=(0,0,2,2)\), both at weight 1. Capital is simulated mean loss plus one simulated standard deviation, with divisor \(N=4\). Use the covariance derivative to calculate each marginal contribution, then check the Euler total.

The portfolio losses are \(L=(0,2,6,4)\), with mean \(3\) and deviations \((-3,-1,3,1)\). Their squared deviations average to \((9+1+9+1)/4=5\), so the standard deviation is \(\sqrt{5}=2.2361\). Portfolio capital is \(3+\sqrt{5}=5.2361\).

For A, its mean is 2. The covariance of \(L\) and A is \([(-3)(-2)+(-1)(0)+(3)(2)+(1)(0)]/4=3\). Therefore, A's marginal contribution is \(2+3/\sqrt{5}=3.3416\), and its Euler contribution at weight 1 is \(3.3416\).

For B, its mean is 1. Its deviations from its mean are \((-1,-1,1,1)\), so the covariance of \(L\) and B is \([3+1+3+1]/4=2\). B's marginal contribution is \(1+2/\sqrt{5}=1.8944\); its Euler contribution is also \(1.8944\). The two contributions sum to \(3.3416+1.8944=5.2360\), agreeing with capital \(5.2361\) to rounding. The small difference is rounding error.

This check illustrates the role of portfolio-wide dependence: a component's marginal capital is not generally just its own mean plus a standalone volatility amount. Its covariance with the total portfolio enters the derivative. A numerical implementation should use the covariance with the aggregate loss and should use the same scenario weights for both components.

Practical Checks and Common Traps

A reliable numerical estimate requires more than one perturbation run. Use a short sequence of step sizes, such as \(h\), \(h/2\), and \(h/4\), while keeping the underlying simulation scenarios fixed. A stable estimate across appropriately chosen steps supports the local derivative interpretation. If the step is too large, the quotient can reflect curvature over a wide range rather than the local marginal effect. If it is too small, rounding, simulation noise, or model-engine tolerances can dominate the difference.

Change one weight at a time. Perturbing several components together estimates a combined direction, not an individual marginal contribution.
Keep scenarios paired. Resampling between the baseline and perturbed runs adds avoidable variation to their difference.
Use the right denominator. A central difference spans \(2h\); a forward or backward difference spans \(h\).
Distinguish units. A marginal contribution is per unit of exposure weight. Multiply by the current weight to obtain the Euler contribution.
Check local smoothness and scale. A derivative interpretation requires a locally differentiable capital function; a step-size check cannot establish differentiability by itself.
Check the model's assumptions before reconciling. Euler contributions should sum to capital only when the relevant homogeneity and differentiability conditions hold.

Numerical estimation is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus because it connects simulated capital outputs to marginal and Euler allocation. In professional use, document the scenario pairing, perturbation size, capital calculation, and stability checks so another reviewer can reproduce the estimate.

Check Your Understanding

For each problem, use the stated simulation and capital convention. Assume equal scenario weights unless specified otherwise.

A hypothetical expected-loss model has four paired baseline losses with mean 72. Increasing one component's weight by 0.05 produces paired losses with mean 73.5. Estimate its marginal contribution and, if its current weight is 1, its Euler contribution. (Answer: marginal contribution 30; Euler contribution 30.)
For \(\rho=E[L]+\sqrt{\operatorname{Var}(L)}\), simulated losses are \(L=(1,3,5)\) and component losses are \(X=(0,1,2)\), with current weight 1. Using divisor \(N=3\), calculate the analytic marginal contribution. (Answer: \(1+(4/3)/\sqrt{8/3}\approx1.8165\); the Euler contribution is approximately 1.8165.)
A central-difference calculation gives capital 208.6 at weight \(w_i+h\) and 205.4 at \(w_i-h\), where \(h=0.20\) and \(w_i=0.75\). Calculate the marginal estimate and Euler estimate. (Answer: marginal contribution \((208.6-205.4)/0.40=8\); Euler contribution \(0.75(8)=6\).)
A positively homogeneous capital estimate is 120. Three components have weights \(0.5,1,\) and \(2\), with estimated marginal contributions \(20,30,\) and \(25\). Calculate their Euler contributions and the reconciliation difference, defined as their sum minus portfolio capital. (Answer: Euler contributions \(10,30,50\); total \(90\); reconciliation difference \(-30\), indicating the estimates or assumptions do not reconcile.)

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