Aggregation of capital requirements
Operation that combines capital needs for several risks into a single figure and returns less than their sum as soon as the risks are not perfectly dependent.
Annex IV of Directive 2009/138/EC of November 25, 2009, Solvency II, aggregates the modules of the basic solvency capital requirement through a correlation matrix. A company exposed to underwriting, market and counterparty risk will not suffer its worst years on the same day, and capital reflects it. Three techniques coexist, in increasing order of faithfulness and cost. The matrix formula, the one in the standard formula, takes the square root of a quadratic form over module capital needs. It is simple and auditable, but it captures only linear dependence and implicitly assumes elliptical distributions. Copula aggregation simulates marginals separately then couples them, capturing tail dependence. Integrated simulation generates scenarios for every source inside one model. One rule holds for all three: aggregation only makes sense with a subadditive risk measure, which TVaR is and VaR not always. Otherwise you can get aggregate capital above the sum of the parts, an absurd result that indicts the measure rather than the portfolio.
Internal model of a composite insurer as of December 31, 2025. Standalone needs: non-life underwriting 420M EUR, market 310M EUR, counterparty 70M EUR, operational 95M EUR, sum 895M EUR. Matrix aggregation with correlations from 0.25 to 0.50: 712M EUR, a 183M EUR diversification benefit. The same aggregation under a Student t copula with four degrees of freedom returns 760M EUR, and the 48M EUR gap is the price of the tail dependence the matrix ignores.
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