The operation combining capital needs for several risks into a single figure, returning less than their sum as soon as the risks are not perfectly dependent.
A company exposed to underwriting, market and operational risk will not suffer its three worst years on the same day. Aggregation turns that obvious point into capital, and three techniques coexist, in increasing order of faithfulness and cost. The correlation matrix formula, the one in the Solvency II standard formula, takes the square root of a quadratic form over module capital needs: simple, 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, finally, generates scenarios for every source inside one model. One rule survives all three: aggregation only makes sense if the risk measure used is subadditive, which TVaR always is and VaR is not, failing which 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.
agrégation du capital, capital requirement aggregation, sommation des modules de risque, agrégation par matrice de corrélation