Why Attribute Catastrophe Capital?
A catastrophe model can estimate capital for a combined portfolio, but managers also need to understand how much each region, peril, or business unit contributes to that total. This supports performance assessment, risk limits, planning, and decisions about where to deploy capacity. The challenge is that the capital requirement for the combined portfolio generally is not the sum of capital requirements calculated for each unit alone.
Capital attribution starts by choosing a risk measure and a definition of capital. It then assigns contributions to the units in a way that is consistent with that choice. This tutorial focuses on standalone and marginal contributions, including allocations based on the portfolio’s tail losses. The methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
An allocation is not automatically a standalone capital estimate. Standalone amounts answer what each unit would require in isolation; an allocation answers how the chosen measure of the combined portfolio is assigned. Because the combined result reflects dependence among units, the two sets of numbers can differ substantially.
Tail-Based Allocation for Discrete Catastrophe Scenarios
Suppose a catastrophe model supplies a finite set of loss scenarios with probabilities. For confidence level \(\alpha\), define \(\operatorname{VaR}_{\alpha}(L)\) as the \(\alpha\)-quantile of portfolio loss. Define \(\operatorname{TVaR}_{\alpha}(L)\) as the average loss in the upper tail beyond that quantile, including only as much probability mass at the cutoff as is needed to form a tail of total probability \(1-\alpha\). This fractional treatment matters when losses are discrete or tied.
For a discrete model, a convenient allocation of TVaR assigns each unit its probability-weighted loss in that same portfolio tail, divided by the tail probability. If a scenario at the cutoff is only partly included, each unit’s loss in that scenario receives the same fractional weight. The resulting contributions add to portfolio TVaR because the portfolio loss in every scenario is the sum of the unit losses.
Worked Example: Allocating Tail Capital Across Three Regions
A hypothetical insurer models five equally likely catastrophe scenarios. Losses are in millions of dollars. Use \(\alpha=60\%\), so the upper tail has probability \(40\%\), or two of the five scenarios. Calculate portfolio TVaR, allocate it to the three regions using portfolio-tail losses, and compare with standalone TVaR.
| Scenario | Region A | Region B | Region C | Portfolio total |
|---|---|---|---|---|
| 1 | 8 | 1 | 0 | 9 |
| 2 | 1 | 5 | 1 | 7 |
| 3 | 6 | 1 | 1 | 8 |
| 4 | 2 | 3 | 1 | 6 |
| 5 | 3 | 1 | 1 | 5 |
Step 1: Identify the portfolio tail. The portfolio totals in descending order are \(9,8,7,6,5\). The worst two scenarios are 1 and 3. Their totals are distinct, so the tail has no tied cutoff and no fractional scenario weight is needed.
Step 2: Calculate aggregate TVaR. Each scenario has probability \(0.20\), and the upper tail has probability \(0.40\). Therefore:
Step 3: Allocate the same tail to each region. Region A contributes \((8+6)/2=7.00\), Region B contributes \((1+1)/2=1.00\), and Region C contributes \((0+1)/2=0.50\). The reconciliation check is \(7.00+1.00+0.50=8.50\), equal to aggregate TVaR.
Step 4: Compare with standalone TVaR. For Region A alone, the two largest losses are 8 and 6, giving standalone TVaR \(7.00\). For Region B they are 5 and 3, giving \(4.00\). For Region C they are 1 and 1, giving \(1.00\). The sum of standalone TVaRs is \(12.00\), while portfolio TVaR is \(8.50\).
The portfolio-tail allocation is not obtained by adding the standalone amounts. Regions B and C have smaller allocated contributions because their worst losses do not occur in the same portfolio scenarios as Region A’s largest losses. The method avoids the common mistake of treating the sum of standalone capital as the portfolio requirement.
Why Tail Contributions Reconcile
The reconciliation is a direct consequence of additivity of losses within each scenario. Let \(w_s\) be the probability weight assigned to scenario \(s\) in the selected tail, with the weights summing to \(1-\alpha\). They include fractional weights where needed at the cutoff. If \(X_{i,s}\) is unit \(i\)’s loss in scenario \(s\), then the tail allocation is
Summing across units and using \(L_s=\sum_i X_{i,s}\) gives:
This argument also explains why all units must use the same portfolio tail weights. Calculating a separate tail for each unit produces standalone TVaRs, not an allocation of aggregate TVaR.
Marginal Contributions and Euler Allocation
A marginal contribution measures the local change in portfolio capital as a unit’s exposure changes. Write \(L(\boldsymbol{x})=\sum_i x_iX_i\), where \(x_i\) is a scale applied to unit \(i\), and let \(R(\boldsymbol{x})=\rho(L(\boldsymbol{x}))\). When \(R\) is differentiable, the local change per unit increase in \(x_i\) is \(\partial R/\partial x_i\). At the current scale, the Euler allocation is \(C_i=x_i\,\partial R/\partial x_i\).
For a differentiable, positively homogeneous risk measure, Euler’s theorem gives \(\sum_i x_i\,\partial R/\partial x_i=R\). Positive homogeneity means that scaling every exposure by the same positive factor scales the risk measure by that factor. Thus, when the conditions hold, Euler contributions reconcile exactly to aggregate capital. They are not generally equal to standalone capital.
Worked Example: Euler Allocation for Independent Normal Losses
Assume two hypothetical regional losses are independent and normally distributed: \(X_A\sim N(2,9)\) and \(X_B\sim N(3,16)\), with losses in millions of dollars. The second parameter is the variance, so the standard deviations are 3 and 4. Use the VaR measure at \(\alpha=\Phi(2)\), where the standard normal quantile is \(z_{\alpha}=2\). Calculate aggregate VaR and its Euler allocation.
Step 1: Calculate aggregate parameters. Independence gives \(E[L]=2+3=5\). The aggregate variance is \(9+16=25\), so the aggregate standard deviation is 5. The aggregate loss is normal under the stated assumptions.
Step 2: Calculate aggregate VaR. For a normal loss with mean \(\mu\) and standard deviation \(\sigma\), \(\operatorname{VaR}_{\alpha}=\mu+z_{\alpha}\sigma\). Thus aggregate VaR is \(5+2(5)=15.00\) million dollars.
Step 3: Derive and calculate the contributions. For the scaled portfolio, independence gives variance \(9x_A^2+16x_B^2\), so
Differentiating and evaluating at \(x_A=x_B=1\) gives:
The check is \(5.60+9.40=15.00\), the aggregate VaR. In contrast, standalone VaRs are \(2+2(3)=8.00\) and \(3+2(4)=11.00\), which sum to \(19.00\). That sum is not the portfolio VaR.
Why the derivative formula works. For independent normal losses, the portfolio standard deviation at scale \(\boldsymbol{x}\) is \(\sigma(\boldsymbol{x})=\sqrt{9x_A^2+16x_B^2}\). Differentiating it gives \(\partial\sigma/\partial x_A=9x_A/\sigma\) and \(\partial\sigma/\partial x_B=16x_B/\sigma\). The VaR is the scaled mean plus \(z_{\alpha}\sigma\); differentiating therefore assigns each unit its mean contribution plus its scaled effect on portfolio standard deviation. At the current scales, those derivatives produce the Euler amounts above.
This calculation assumes normal aggregate losses, independence, and a differentiable risk measure at the current portfolio. The use of \(z_{\alpha}=2\) corresponds to \(\alpha=\Phi(2)\), approximately \(0.9772\), not a 99% confidence level. The standalone-versus-aggregate comparison illustrates how diversification can reduce the total while the allocation still sums to that total.
When a Finite Difference Is More Useful
A derivative is a local quantity. It can be difficult to interpret when a discrete scenario model has a tied tail, when a small exposure change alters the ordering of scenarios, or when the selected risk measure is not differentiable. In those cases, a finite change in portfolio capital can answer a practical planning question: how much does the chosen risk measure change when a specified exposure is added?
Worked Example: Measuring the Incremental Effect of a New Exposure
A hypothetical portfolio has four equally likely scenarios and uses \(\operatorname{TVaR}_{0.75}\), the worst 25% of outcomes. Existing portfolio losses are \(1,3,6,8\) million dollars across scenarios 1 through 4. A proposed unit would add losses of \(4,0,2,1\) million dollars, respectively. Calculate the change in portfolio TVaR when the unit is added at full scale and compare it with its standalone TVaR.
Step 1: Calculate existing portfolio capital. The worst quarter is scenario 4, so existing \(\operatorname{TVaR}_{0.75}=8.00\) million dollars.
Step 2: Add the proposed losses by scenario. The new totals are \(1+4=5\), \(3+0=3\), \(6+2=8\), and \(8+1=9\). The worst quarter is still scenario 4, so new portfolio TVaR is \(9.00\) million dollars.
Step 3: Calculate the finite increment and standalone amount. The portfolio increase is \(9.00-8.00=1.00\) million dollars. In isolation, the proposed unit’s worst-quarter loss is \(4.00\) million dollars, so its standalone TVaR is \(4.00\) million dollars.
The distinction is important: the incremental effect at this portfolio position is \(1.00\), not the standalone amount of \(4.00\). Here, scenario 4 remains the worst scenario as the new exposure is scaled from zero to one, so the local marginal effect is also \(1.00\). If the exposure were large enough to change the tail ordering, the marginal effect could change; a finite increment should not be treated as a derivative in general.
Interpretation and Common Traps
Capital attribution is useful only when its basis is explicit. State the risk measure, confidence level, gross or net loss basis, and whether the result is full risk measure or economic capital net of expected loss. For catastrophe applications, also identify which modeled portfolio and exposure set the allocation represents. Changing the peril, reinsurance treatment, or scenario set can change both aggregate capital and unit contributions.
- Adding standalone amounts and calling the sum portfolio capital. Standalone measures use separate unit tails. They need not equal the aggregate risk measure.
- Using each unit’s own tail to allocate aggregate tail capital. This does not reconcile to the aggregate TVaR. Use common portfolio-tail weights for a tail allocation.
- Ignoring probability mass at a tied cutoff. Include only the fraction of tied probability needed to reach the required tail probability, and apply that fraction consistently to every unit.
- Confusing marginal and allocated contributions. A derivative is a change per unit of exposure scale. The Euler allocation multiplies that derivative by the current scale.
- Assuming every contribution must be positive. A contribution is defined by the allocation method and the combined loss behavior. In particular, an economic-capital risk-margin contribution can be negative.
A good attribution should reconcile arithmetically and make its interpretation clear. Reconciliation is necessary, but it does not prove that the chosen allocation is appropriate for every business decision. A tail allocation describes the units’ losses in the portfolio tail; an Euler allocation describes local sensitivity under its model and smoothness assumptions. Neither should be presented as a universal measure of management responsibility.
Check Your Understanding
For each problem, identify the portfolio-level measure before comparing it with standalone amounts.
- Three equally likely scenarios have losses for units A and B of \((2,1)\), \((5,0)\), and \((1,3)\), respectively. Using \(\operatorname{TVaR}_{2/3}\), calculate the portfolio value and the two tail contributions. (Answer: Portfolio TVaR \(=5\); contributions A \(=5\), B \(=0\).)
- Independent losses have \(X_A\sim N(1,4)\) and \(X_B\sim N(2,9)\). Use \(z_{\alpha}=1\) and VaR. Give the aggregate VaR and Euler contributions. (Answer: Aggregate VaR \(=3+\sqrt{13}\approx6.6056\); \(C_A=1+4/\sqrt{13}\approx2.1094\), \(C_B=2+9/\sqrt{13}\approx4.4961\).)
- Four equally likely scenarios have existing portfolio losses \(2,4,7,9\) and proposed-unit losses \(3,1,0,2\). Under \(\operatorname{TVaR}_{0.75}\), find the full-scale increase in portfolio TVaR and the proposed unit’s standalone TVaR. (Answer: Portfolio TVaR increases from \(9\) to \(11\), an increment of \(2\); standalone TVaR is \(3\).)
- Five equally likely scenarios have unit losses A \(=(8,6,5,2,0)\) and B \(=(4,3,4,2,1)\). Allocate \(\operatorname{TVaR}_{0.70}\), carefully handling the tied portfolio cutoff. (Answer: Portfolio totals are \(12,9,9,4,1\); contributions are A \(=7.1667\), B \(=3.8333\), summing to portfolio TVaR \(=11.0000\).)