Negative and Disproportionate Capital Contributions

Why a Capital Contribution Can Look Surprising

An allocated contribution is not necessarily a measure of how much risk a business line creates in isolation. It describes the line’s role within a specified portfolio and under a specified capital measure. When a line offsets other risks, its contribution can be negative. Under some risk measures and allocation methods, a line’s allocated share can also exceed its standalone capital.

These outcomes are not automatically errors. They depend on the risk measure, the dependence among losses, and the definition of contribution. A negative result may reflect genuine diversification; a contribution exceeding standalone capital may reveal a non-subadditive measure or an allocation rule that assigns interaction effects unevenly. The distinctions are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: A standalone capital for risk $i$ is the capital measure applied to that risk on its own. A portfolio contribution is an amount assigned to risk $i$ within a portfolio. A contribution is negative if it is below zero and disproportionate if, for example, it exceeds the risk’s standalone capital. These comparisons are meaningful only after specifying the capital measure and allocation method.

Three amounts should not be confused. A marginal contribution may be the change in portfolio capital after adding a risk; an Euler contribution allocates capital using a derivative; and a Shapley contribution averages marginal changes over different orders of addition. They answer related but different questions.

Negative Contributions Under Covariance Aggregation

Suppose each risk is represented by a centered loss, such as unexpected loss, with finite variance. Let $\sigma_i$ be its standalone standard deviation, and let $\rho_{ij}$ be the correlation between risks $i$ and $j$. Assume the correlation matrix is positive semidefinite, so it describes a valid joint covariance structure. Define portfolio capital as the standard deviation of the aggregate loss. At unit exposures, $S=\sum_i X_i$, this gives:

C=\sqrt{\operatorname{Var}(S)} =\sqrt{\sum_i \sigma_i^2+2\sum_{i<j}\rho_{ij}\sigma_i\sigma_j}.

For a homogeneous capital measure, the Euler contribution of risk $i$ is its exposure multiplied by the derivative of portfolio capital with respect to that exposure. At unit exposures, this simplifies to:

EC_i=\frac{\operatorname{Cov}(X_i,S)}{C} =\frac{\sigma_i^2+\sum_{j\ne i}\rho_{ij}\sigma_i\sigma_j}{C}, \qquad C>0.

The numerator is the covariance of risk $i$ with the whole portfolio. It can be negative even though $\sigma_i^2$ is positive: sufficiently negative covariance with the other risks can outweigh the component’s own variance. If $C=0$, the displayed derivative formula is undefined; the portfolio needs separate treatment at that degenerate point.

Derivation: Give risk $i$ an exposure $w_i$, so $C(\mathbf w)=\sqrt{\mathbf w^\mathsf{T}\Sigma\mathbf w}$, where $\Sigma$ is the covariance matrix. Differentiating the square root gives $\frac{\partial C}{\partial w_i}=\frac{(\Sigma\mathbf w)_i}{C(\mathbf w)}$. Multiplying by $w_i$ gives the Euler contribution $w_i\frac{\partial C}{\partial w_i}$. At unit exposures, $(\Sigma\mathbf w)_i=\operatorname{Cov}(X_i,S)$, yielding the formula above. Euler contributions sum to $C$ because this standard-deviation capital function is homogeneous of degree one.

Worked Example: A Negative Euler Contribution

A hypothetical insurer models two centered risk losses with standard deviations of $100 and $80. Their correlation is $-0.90$, and capital is the standard deviation of the aggregate loss. Calculate total capital and both Euler contributions. Round only final dollar amounts to the nearest cent.

Method and assumptions. Use covariance aggregation, assuming the stated correlation is valid and both risks have finite variances. The variance of the portfolio is:

C^2=100^2+80^2+2(-0.90)(100)(80) =10{,}000+6{,}400-14{,}400=2{,}000.

Thus $C=\sqrt{2{,}000}=\$44.72135955\ldots$. The first risk’s covariance with the portfolio is $100^2+(-0.90)(100)(80)=2{,}800$, so its contribution is $2{,}800/44.72135955\ldots=\$62.61$. The second risk’s covariance with the portfolio is $80^2+(-0.90)(100)(80)=-800$, so its contribution is $-800/44.72135955\ldots=-\$17.89$.

The unrounded contributions sum to $C$; their rounded values sum to $44.72. The negative contribution does not mean the second risk has negative standalone variability: its standalone standard deviation is $80. It means that, at this portfolio composition, its covariance with the aggregate is negative. The common mistake is to use only the second risk’s variance, $80^2$, in its Euler numerator. Including its covariance with the first risk is essential.

For two risks, the contribution of risk $i$ is negative exactly when $\sigma_i^2+\rho_{ij}\sigma_i\sigma_j<0$. If both standard deviations are positive, this is equivalent to $\rho_{ij}<-\sigma_i/\sigma_j$. Such a correlation is feasible only if the other risk has larger standard deviation than risk $i$, since a correlation cannot be less than $-1$. With several risks, the corresponding condition is $\operatorname{Cov}(X_i,S)<0$: the total covariance with the portfolio, not any one pairwise correlation by itself, determines the sign.

When a Contribution Exceeds Standalone Capital

Under covariance aggregation, an Euler contribution cannot exceed its standalone standard deviation. For $C>0$, the Cauchy–Schwarz inequality gives $|\operatorname{Cov}(X_i,S)|\leq \sigma_i C$. Dividing by $C$ shows $|EC_i|\leq\sigma_i$. Thus a covariance-based Euler contribution may be negative, but it cannot exceed $\sigma_i$. A claim of such an excess under this model should prompt a check of the formula, inputs, and meaning of “standalone capital.”

That bound is specific to this capital measure and allocation. Other measures, including Value at Risk (VaR), can fail subadditivity: for some risks, the capital of the aggregate can exceed the sum of standalone capitals. Allocation methods applied to such a measure can assign a positive share that exceeds a risk’s standalone capital.

Worked Example: VaR Allocation Exceeding Standalone Capital

Consider two hypothetical lines of business at a 95% VaR level. With probability $0.92$, both have loss zero. With probability $0.04$, only line A has a loss of $100. With probability $0.04$, only line B has a loss of $100. Find each standalone VaR, the portfolio VaR, and each line’s two-risk Shapley allocation. Use the quantile definition $\operatorname{VaR}_{0.95}(X)=\inf\{x:P(X\leq x)\geq0.95\}$.

Method. First calculate the VaR for each subset of lines, then average each line’s marginal contribution over the two possible orders of addition. For A alone, $P(X_A=0)=0.92+0.04=0.96$, so $\operatorname{VaR}_{0.95}(X_A)=0$. By the same calculation, $\operatorname{VaR}_{0.95}(X_B)=0$.

In the aggregate, the loss is $100 with probability $0.04+0.04=0.08$ and zero with probability $0.92$. Since $P(S\leq0)=0.92<0.95$ and $P(S\leq100)=1$, the portfolio VaR is $100. The two Shapley contributions are:

\phi_A=\frac{1}{2}\big[(0-0)+(100-0)\big]=\$50, \qquad \phi_B=\frac{1}{2}\big[(0-0)+(100-0)\big]=\$50.

Each line receives $50, exceeding its standalone VaR of zero. The allocated amounts reconcile to portfolio VaR: $50 + $50 = $100. The common mistake is to add the standalone VaRs and conclude the portfolio has zero capital. The probability of at least one loss is 8%, which puts the 95th percentile at $100. This example also shows why the quantile convention and confidence level must be stated.

Contribution Is Not the Same as Marginal Change

A finite-difference marginal contribution compares portfolio capital with and without a risk. It need not equal that risk’s Euler contribution. The difference is especially important when a line diversifies the existing portfolio: the total portfolio capital can fall, while the Euler rule assigns a different amount based on the local derivative at the current exposures.

Worked Example: A Negative Marginal Change and a Negative Euler Contribution

A hypothetical portfolio has two centered losses with standard deviations of $120 and $100 and correlation $-0.90$. Capital is standard deviation. Calculate the capital with both risks, the finite-difference marginal change from adding the second risk to the first, and the second risk’s Euler contribution. Round only final amounts to the nearest cent.

Method. The combined variance is $120^2+100^2+2(-0.90)(120)(100)=14{,}400+10{,}000-21{,}600=2{,}800$. Combined capital is therefore $\sqrt{2{,}800}=\$52.915026\ldots$. Capital with only the first risk is $120, so the finite-difference change is $\$52.915026\ldots-\$120=-\$67.08$.

The second risk’s Euler numerator is $100^2+(-0.90)(120)(100)=10{,}000-10{,}800=-800$. Its Euler contribution is $-800/\sqrt{2{,}800}=-\$15.12$. The first risk’s Euler contribution is $(14{,}400-10{,}800)/\sqrt{2{,}800}=\$68.03$. Before rounding, the two Euler contributions sum to the combined capital of $52.915026… .

The finite-difference change is not the Euler contribution: $-\$67.08$ and $-\$15.12$ answer different questions. The first compares two portfolios; the second allocates total capital at the current portfolio. Treating them as interchangeable is the common mistake. The model’s covariance assumptions are the same as in the prior example: finite variances, valid correlation, and standard-deviation capital.

Assessing a Surprising Allocation

When a contribution is negative or exceeds standalone capital, assess it in a fixed sequence. First identify the capital measure and the allocation rule; do not compare an Euler contribution with a Shapley contribution as if they were defined identically. Then check the portfolio dependence and exposure assumptions. Finally, test reconciliation: for a full Euler or Shapley allocation, the contributions should sum to the stated portfolio capital, subject to rounding.

State the measure.
For covariance-based standard-deviation capital, compute the aggregate covariance and use the Euler covariance formula. For VaR, use the stated quantile convention and confidence level.

Identify what is being allocated.
A finite-difference change, an Euler contribution, and a Shapley allocation are different quantities. Record the definition before interpreting the sign or size.

Check the result against the model.
Under covariance Euler allocation, a negative contribution is possible, but its magnitude cannot exceed standalone standard deviation. An amount above that bound signals a calculation or interpretation issue.

Check reconciliation and context.
Confirm that the full allocation sums to portfolio capital and explain whether diversification or a non-subadditive measure accounts for the result.

A negative allocation is not automatically a reason to remove a risk: it is a statement about its modeled contribution in the current portfolio, not a complete business decision. Likewise, an allocation above standalone capital is not automatically an error if the measure and allocation method permit it. A sound interpretation names both the model condition and the allocation rule that produce the result.

Key takeaway: Under covariance-based Euler allocation, negative contributions arise when a risk has negative covariance with the portfolio, while the Cauchy–Schwarz bound prevents a contribution from exceeding standalone standard deviation. Contributions exceeding standalone capital can occur under other measures and allocation methods, as VaR non-subadditivity demonstrates.

Check Your Understanding

For each hypothetical portfolio, state the method and keep the capital measure and allocation definition clear.

Two centered risks have standard deviations $70 and $50 and correlation $-0.60$. Using standard-deviation capital, calculate total capital and both Euler contributions. (Answer: $C=\sqrt{3{,}200}=\$56.57$; contributions are $\$49.50$ and $\$7.07$, respectively.)
Two centered risks have standard deviations $90 and $60 and correlation $-0.80$. Calculate total capital and the Euler contribution of the $60 risk. (Answer: $C=\sqrt{3{,}060}=\$55.32$; the $60 risk contributes $-\$13.02$.)
At the 95% VaR level, two lines each have a $100 loss with probability 0.03, and the losses are mutually exclusive; otherwise both losses are zero. Find the standalone VaRs, portfolio VaR, and two-risk Shapley allocations. (Answer: Both standalone VaRs are $0; portfolio VaR is $100; Shapley allocations are $50 each.)
A covariance-based Euler calculation gives $\operatorname{Cov}(X_i,S)=\$4{,}200$, $\sigma_i=\$70$, and portfolio capital $C=\$50$. Can these values be consistent with the model? Explain using the applicable bound. (Answer: No. The bound is $|\operatorname{Cov}(X_i,S)|\leq\sigma_i C=\$3{,}500$, but the stated covariance is $4,200.)
Two centered risks have standard deviations $70 and $100 and correlation $-0.80$, with standard-deviation capital. Calculate total capital and the $70 risk’s Euler contribution, and state whether it is negative. (Answer: $C=\sqrt{3{,}700}=\$60.83$; the $70 risk contributes $(4{,}900-5{,}600)/\sqrt{3{,}700}=-\$11.51$, so its contribution is negative.)

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