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In the previous lesson, we looked at the types of risk covered by the Solvency Capital Requirement. The next step is to understand how these risks are combined to produce a single SCR figure. As with the Risk Margin, the aim here is not to master every technical detail, but to understand the structure and logic of the calculation.
Under Solvency II, insurers can calculate SCR using one of two approaches. The first is the Standard Formula, which is prescribed by regulation and designed to be applicable to a wide range of insurers. The second is an Internal Model, which is developed by the insurer to reflect its specific risk profile and must be approved by the regulator. While Internal Models can be more tailored, most insurers use the Standard Formula, and this is the approach we focus on in this course.
The Standard Formula follows a modular structure. Different types of risk are first assessed separately and then combined, allowing for diversification effects between them. At a high level, the calculation proceeds in three stages.
First, capital requirements are calculated for individual risk modules, such as market risk, life underwriting risk, non-life underwriting risk, health underwriting risk and counterparty default risk. Each module represents the impact of a severe but plausible one-year shock.
Second, these module-level capital requirements are aggregated to form the Basic Solvency Capital Requirement (BSCR). This aggregation allows for diversification by applying correlations between risk modules.
Finally, adjustments are made for operational risk and for the loss-absorbing capacity of technical provisions and deferred taxes. In this lesson, we focus on the core aggregation logic within the BSCR.
In simplified form, the aggregation of risk modules in the Standard Formula can be written as:
\( \mathrm{BSCR} = \sqrt{\sum_{i} \sum_{j} \mathrm{SCR}_{i} \cdot \rho_{ij} \cdot \mathrm{SCR}_{j}} \)where:
This formula reflects the fact that not all risks materialise simultaneously or move in the same direction.
Consider a simplified insurer with the following Standard Formula capital requirements:
Assume the following simplified correlation matrix:
Using the aggregation formula, the BSCR is:
\( \mathrm{BSCR} = \sqrt{120^2 + 80^2 + 40^2 + 2 \cdot 0.25 \cdot (120 \cdot 80 + 120 \cdot 40 + 80 \cdot 40)} \)Cross terms:
Sum of cross terms: \( 9600 + 4800 + 3200 = 17600 \)
Apply correlation and factor 2: \( 2 \cdot 0.25 \cdot 17600 = 8800 \)
The Basic Solvency Capital Requirement is therefore approximately 177.
In a full Standard Formula calculation, the final SCR would then be obtained by:
Under an Internal Model, the same economic idea applies, but risks are usually modelled jointly rather than aggregated using a prescribed correlation matrix. Internal Models can better reflect company-specific risk profiles, but they require significant modelling effort and regulatory approval.
Let's practise calculating SCR!