Why Capital Assessments Can Combine Different Methods
An insurer may have a risk for which its own data and modelling approach are particularly informative, while relying on prescribed calculations for other risks. A partial internal model makes this mixed approach possible: it models an approved part of the risk profile and retains standard-formula calculations for the rest. The central task is to define the boundary between those parts and combine them on a coherent basis.
A partial model is not simply a standard formula with a discretionary adjustment. Its modelled scope, the standard-formula scope that remains, and the method used to combine the resulting capital amounts must be specified. The calculations below use hypothetical capital amounts and explicit aggregation rules to illustrate that architecture. They are not representations of a particular insurer’s results or a universal regulatory formula.
The modelled part might be a particular risk or sub-risk, a business unit, or a defined combination of these. The precise boundary matters more than the label. A partial model could replace one prescribed component, or it could cover several components within a specified business scope. It could also capture a risk that is not represented in the standard formula in the same way. In each case, the assessment must say what is included in the model output and what remains prescribed.
For capital amounts to be combined meaningfully, they must refer to a consistent assessment basis. For example, they should be defined for a compatible time horizon and confidence level, use compatible valuation conventions, and be expressed in consistent units. In a Solvency II context, the SCR calibration is based on a one-year 99.5% value-at-risk measure of basic own funds. The examples here assume that all stated component amounts have already been produced on the common basis required by the hypothetical assessment.
Define the Boundary Before Calculating
The first practical step is to identify the risks represented by each capital amount. A component labelled “property risk” might include catastrophe risk as well as ordinary underwriting risk, or it might exclude catastrophe risk and leave that exposure to a separate prescribed calculation. The label alone does not resolve the scope.
A scope map should make clear which risk exposures are modelled, which are calculated by the standard formula, and how dependencies across the two parts are handled. It should also identify any risks that are outside the calculation being illustrated. This is essential both for complete coverage and for avoiding duplicate charges.
Identify the risks, sub-risks, or business activities included in the partial internal model and confirm what its output represents.
List the standard-formula components retained for risks outside the modelled scope. Check that these components cover the intended remaining risks.
Check for overlap and gaps. If the modelled output already includes an exposure, do not charge that same exposure again as a separate prescribed component unless the calculation has explicitly split it out.
Use the prescribed or approved aggregation method for the mixed components. Do not assume that capital amounts should simply be added or that one universal formula applies to every partial model.
Combining Capital Components
The aggregation method depends on the applicable framework and the approved design of the assessment. It may use prescribed dependency parameters, an approved method that accounts for dependencies, or a specified combination of methods. The following covariance-style equation is useful when the problem explicitly gives a two-component correlation parameter and directs you to use this aggregation rule.
The equation follows from the variance identity for two random quantities. If \(Z_1\) and \(Z_2\) are loss measures with standard deviations \(\sigma_1\) and \(\sigma_2\), then the variance of their sum is
When \(\operatorname{Cov}(Z_1,Z_2)=\rho\sigma_1\sigma_2\), taking the square root gives the standard deviation of the sum:
The illustrative capital equation has this form when the specified component capital amounts are used in the aggregation rule. That derivation explains the structure of the equation; it does not establish that a particular regulatory framework must use this equation for every mixed assessment. Use the rule stated for the assessment, rather than inferring one from the names of the components.
Worked Example: Replacing One Prescribed Component
A hypothetical insurer’s partial internal model replaces the standard-formula property underwriting component. On a common assessment basis, the model produces property underwriting capital of 180 thousand dollars. The other risks are represented by a prescribed standard-formula block of 160 thousand dollars. For this illustration, the approved combination rule is the two-component equation above with \(\rho=0.25\). Calculate the aggregate capital in thousands of dollars, to two decimal places.
Method. Treat 180 as the modelled block and 160 as the remaining prescribed block. Apply the given aggregation rule once to these two amounts.
Step 1: Square each component. \(180^2=32{,}400\) and \(160^2=25{,}600\).
Step 2: Calculate the cross term. \(2(0.25)(180)(160)=0.5(28{,}800)=14{,}400\).
Step 3: Add the terms and take the square root. \(32{,}400+25{,}600+14{,}400=72{,}400\), so \(C_{\mathrm{agg}}=\sqrt{72{,}400}=269.072...\) thousand dollars.
Rounded only at the final step, the aggregate is 269.07 thousand dollars. The result is between the larger component, 180, and their sum, 340, as expected for this positive but sub-unit correlation. The common mistake this method avoids is adding 180 and 160 as though the stated aggregation rule did not apply.
Worked Example: Combining a Modelled Business Unit With Prescribed Risks
Worked Example: Combining a Modelled Business Unit With Prescribed Risks
A hypothetical insurer models the underwriting risk for its marine business as a single component of 90 thousand dollars. Standard-formula calculations for the remaining business produce two components: 120 thousand dollars for other underwriting risk and 70 thousand dollars for catastrophe risk. The assessment specifies a three-component covariance-style aggregation rule. The correlations are 0.20 between marine and other underwriting risk, 0.10 between marine and catastrophe risk, and 0.25 between other underwriting and catastrophe risk. Calculate aggregate capital in thousands of dollars, to two decimal places.
Method. Keep the three non-overlapping components distinct and include all three stated pairwise covariance terms. The assumed rule is the square root of the sum of squared components plus twice the sum of correlation-weighted component products.
Step 1: Square the three component amounts. \(90^2=8{,}100\), \(120^2=14{,}400\), and \(70^2=4{,}900\). Their sum is \(8{,}100+14{,}400+4{,}900=27{,}400\).
Step 2: Calculate each pairwise term. For marine and other underwriting risk, \(2(0.20)(90)(120)=4{,}320\). For marine and catastrophe risk, \(2(0.10)(90)(70)=1{,}260\). For other underwriting and catastrophe risk, \(2(0.25)(120)(70)=4{,}200\).
Step 3: Aggregate. The full quantity under the square root is \(27{,}400+4{,}320+1{,}260+4{,}200=37{,}180\). Therefore, \(C_{\mathrm{agg}}=\sqrt{37{,}180}=192.820...\) thousand dollars.
The answer, rounded only at the final step, is 192.82 thousand dollars. The calculation includes the modelled marine component and both prescribed components exactly once. A common mistake is to include the marine business again in the prescribed underwriting component after it has already been replaced by the modelled amount.
Scope Overlap and Reconciliation
A model output can cover more than its short label suggests. For example, a partial model for a property portfolio may include catastrophe losses within its property result. If the same portfolio’s catastrophe exposure is then added again as a separate standard-formula component, the combined calculation counts the overlapping exposure twice. Conversely, if the partial model excludes catastrophe risk and the prescribed calculation omits it, the assessment has a gap.
The remedy is to reconcile the components before aggregating. Either define the model output to exclude a risk that will be calculated separately, or retain that risk within the modelled output and remove the overlapping prescribed charge from the combination. The approach must follow the defined scope and the applicable calculation requirements; it cannot be decided just by choosing whichever component produces a preferred total.
Worked Example: Correcting an Overlapping Component
A hypothetical partial internal model produces property capital of 140 thousand dollars, and its documented scope includes catastrophe exposure for that same property portfolio. A separate standard-formula catastrophe result of 60 thousand dollars also covers that exposure. A standard-formula market component is 100 thousand dollars and does not overlap the modelled property scope. For this illustration, combine the property and market amounts using the stated two-component rule with \(\rho=0.25\). Calculate the corrected aggregate in thousands of dollars, to two decimal places.
Method. First reconcile the scope, then aggregate. Because the catastrophe exposure is already included in the 140 modelled amount, the separate 60 must not be included as a second charge for the same exposure in this illustration.
Step 1: Identify the non-overlapping components. Retain modelled property capital \(C_1=140\) and prescribed market capital \(C_2=100\). Exclude the duplicate 60 from this illustrative combination.
Step 2: Calculate the squared terms. \(140^2=19{,}600\) and \(100^2=10{,}000\).
Step 3: Calculate the cross term. \(2(0.25)(140)(100)=7{,}000\).
Step 4: Aggregate. \(19{,}600+10{,}000+7{,}000=36{,}600\), and \(\sqrt{36{,}600}=191.311...\).
The corrected illustrative result is 191.31 thousand dollars, rounded to two decimal places. Simply adding \(140+60+100=300\) would both double count the overlapping catastrophe exposure and ignore the explicitly stated aggregation rule. In an actual assessment, the required treatment depends on the approved component definitions and applicable rules.
What to Check in a Mixed Capital Assessment
A partial internal model requires more than substituting one number for another. A sound calculation makes the model boundary, prescribed boundary, and combination method visible. Before accepting an aggregate, check the following:
- Scope: Does each component state which exposures it covers? Are modelled business units or risks clearly distinguished from those calculated by the standard formula?
- Completeness: Is every intended risk represented somewhere in the assessment, with no unaddressed exposure at the boundary?
- Overlap: Does a model output already include a risk that appears again as a prescribed component?
- Basis: Are the amounts expressed in the same units and calculated on compatible valuation, time-horizon, and calibration bases?
- Aggregation: Is the method and its dependency input explicitly specified? Is it appropriate to the mixed components under the applicable framework?
A component-level calculation is only as meaningful as its definitions. A precisely evaluated formula can still produce an inappropriate result if it combines amounts with incompatible scopes or bases. For exam-style calculations, state the components and their scopes, write the given aggregation rule, calculate each term, and then interpret the result in context.
Check Your Understanding
For each calculation, use the aggregation rule and scope information stated in the question. Amounts are in thousands of dollars; round numerical results to two decimal places.
- A modelled block is 80 and a prescribed block is 60. The stated two-component rule uses \(\rho=0.50\). What is the aggregate? (Answer: 121.66 thousand dollars.)
- A partial model covers one business unit and produces 70. The non-overlapping prescribed block is 90. The stated rule uses \(\rho=0.20\). Calculate the aggregate. (Answer: 124.58 thousand dollars.)
- Three non-overlapping components are 50, 60, and 80. Their respective correlations are \(\rho_{12}=0.10\), \(\rho_{13}=0.25\), and \(\rho_{23}=0\). Apply the three-component rule from the tutorial. (Answer: 122.88 thousand dollars.)
- A modelled property result includes catastrophe exposure for a portfolio. A prescribed catastrophe amount covers that same exposure, while a prescribed market amount covers a distinct risk. Before aggregation, what scope issue must be resolved? (Answer: The catastrophe exposure overlaps; reconcile the component definitions so it is not omitted or counted twice.)
- A modelled component is 110 and a non-overlapping prescribed component is 150. The stated two-component rule uses \(\rho=0.30\). Calculate the aggregate. (Answer: 210.95 thousand dollars.)